{"id":7091,"date":"2022-03-01T08:33:12","date_gmt":"2022-03-01T08:33:12","guid":{"rendered":"https:\/\/10minuteschool.com\/content\/?p=7091"},"modified":"2023-07-03T15:27:31","modified_gmt":"2023-07-03T09:27:31","slug":"buet-admission-question-bank-2017-2018","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/buet-admission-question-bank-2017-2018\/","title":{"rendered":"\u09ac\u09c1\u09df\u09c7\u099f \u09ad\u09b0\u09cd\u09a4\u09bf \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be: \u09b8\u09ae\u09be\u09a7\u09be\u09a8\u09b8\u09b9 \u09aa\u09cd\u09b0\u09b6\u09cd\u09a8\u09ac\u09cd\u09af\u09be\u0982\u0995- (\u09e8\u09e6\u09e7\u09ed-\u09e7\u09ee) (BUET Admission Test Question Bank 2017-18)"},"content":{"rendered":"

\u09b2\u09bf\u0996\u09bf\u09a4 \u0985\u0982\u09b6 (Written Part)<\/strong><\/h2>\n

\u0989\u099a\u09cd\u099a\u09a4\u09b0 \u0997\u09a3\u09bf\u09a4 (Higher Mathematics)<\/strong><\/h3>\n

1. \u09af\u09a6\u09bf \\mathrm{f} \\left(\\frac{1-\\mathrm{x}}{1+\\mathrm{x}} \\right)=\\mathrm{x}+2 <\/span> \u09b9\u09af\u09bc, \u09a4\u09be\u09b9\u09b2\u09c7 \u09a1\u09cb\u09ae\u09c7\u09a8 \u098f\u09ac\u0982 \u09b0\u09c7\u099e\u09cd\u099c\u09b8\u09b9 \\mathrm{f}^{-1}(\\mathrm{x}) <\/span> \u09ac\u09c7\u09b0 \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u09a7\u09b0\u09bf, \\mathrm{ \\frac{1-x}{1+x}=y \\Rightarrow 1-x=y+x y \\Rightarrow x y+x=1-y \\Rightarrow x=\\frac{1-y}{1+y}} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \\mathrm{f(y)=\\frac{1-y}{1+y}+2=\\frac{1-y+2+2 y}{1+y}=\\frac{3+y}{1+y}} <\/span>
\n\u098f\u0996\u09a8, \\mathrm{f(x)=\\frac{3+x}{1+x}} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \\mathrm{f(x)}<\/span> \u098f\u09b0 \u09a1\u09cb\u09ae\u09c7\u09a8 \\mathrm{=f^{-1}(x)} <\/span> \u098f\u09b0 \u09b0\u09c7\u099e\u09cd\u099c = R-{-1} (Ans)<\/p>\n

2. \u09af\u09a6\u09bf \\left[\\begin{array}{l}4 \\\\ 1 \\\\ 3\\end{array}\\right] \\times A=\\left[\\begin{array}{lll}-4 & 8 & 4 \\\\ -1 & 2 & 1 \\\\ -3 & 6 & 3\\end{array}\\right] <\/span>, \u09b9\u09af\u09bc, \u09a4\u09be\u09b9\u09b2\u09c7 A \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u09a7\u09b0\u09bf, B=\\left[\\begin{array}{l}4 \\\\ 1 \\\\ 3\\end{array}\\right] ; \\quad C=\\left[\\begin{array}{lll}-4 & 8 & 4 \\\\ -1 & 2 & 1 \\\\ -3 & 6 & 3\\end{array}\\right] \u09b8\u09c1\u09a4\u09b0\u09be\u0982 B \\times A=C <\/span>
\n\u098f\u0996\u09a8 B \u098f\u09b0 \u0995\u09cd\u09b0\u09ae =3 \\times 1 <\/span>; C \u098f\u09b0 \u0995\u09cd\u09b0\u09ae =3 \\times 3 <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982 A \u098f\u09b0 \u0995\u09cd\u09b0\u09ae =1 \\times 3 \\quad\\left[\\because \\mathrm{B}_{(3 \\times 1)} \\times \\mathrm{A}_{(1 \\times 3)}=\\mathrm{C}_{(3 \\times 3)}\\right] <\/span>
\n\u09a7\u09b0\u09bf, A =\\left[\\begin{array}{lll}x & y & z\\end{array}\\right] <\/span>
\n \\left[\\begin{array}{l}4 \\\\ 1 \\\\ 3\\end{array}\\right] \\times\\left[\\begin{array}{lll}x & y & z\\end{array}\\right]=\\left[\\begin{array}{lll}-4 & 8 & 4 \\\\ -1 & 2 & 1 \\\\ -3 & 6 & 3\\end{array}\\right] \\ldots \\ldots \\ldots<\/span> (i)
\n(i) \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a5\u09c7\u0995\u09c7 \u09aa\u09be\u0987, 4 \\mathrm{x}=-4 \\Rightarrow \\mathrm{x}=-1 ; 4 \\mathrm{y}=8 \\Rightarrow \\mathrm{y}=2 ; 4 \\mathrm{z}=4 \\Rightarrow \\mathrm{z}=1 <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09df A =\\left[\\begin{array}{lll}x & y & z\\end{array}\\right]=\\left[\\begin{array}{lll}-1 & 2 & 1\\end{array}\\right] <\/span><\/p>\n

3. \u09af\u09a6\u09bf \\cos \\alpha+\\sec \\alpha=\\frac{5}{2} <\/span> \u09b9\u09af\u09bc, \\cos ^{\\mathrm{n}} \\alpha+\\sec ^{\\mathrm{n}} \\alpha <\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u09a6\u09c7\u0993\u09df\u09be \u0986\u099b\u09c7,
\n \\cos \\alpha+\\sec \\alpha=\\frac{5}{2} <\/span>
\n \\Rightarrow \\cos \\alpha+\\frac{1}{\\cos \\alpha}=\\frac{5}{2} <\/span>
\n \\Rightarrow \\frac{\\cos ^{2} \\alpha+1}{\\cos \\alpha}=\\frac{5}{2} <\/span>
\n \\Rightarrow 2 \\cos ^{2} \\alpha+2=5 \\cos \\alpha <\/span>
\n \\Rightarrow 2 \\cos ^{2} \\alpha-5 \\cos \\alpha+2=0 <\/span>
\n \\Rightarrow 2 \\cos ^{2} \\alpha-4 \\cos \\alpha-\\cos \\alpha+2=0 <\/span>
\n \\Rightarrow 2 \\cos \\alpha(\\cos \\alpha-2)-1(\\cos \\alpha-2)=0 <\/span>
\n \\Rightarrow(\\cos \\alpha-2)(2 \\cos \\alpha-1)=0 <\/span>
\n \\Rightarrow \\cos \\alpha-2=0 \\Rightarrow \\cos \\alpha=2 ; <\/span> \u09af\u09c7\u099f\u09bf \u0985\u09ac\u09be\u09b8\u09cd\u09a4\u09ac\u0964
\n\u0985\u09a5\u09ac\u09be, 2 \\cos \\alpha-1=0 \\Rightarrow \\cos \\alpha=\\frac{1}{2} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09df \u09ae\u09be\u09a8 =\\cos ^{n} \\alpha+\\sec ^{n} \\alpha <\/span>
\n =\\cos ^{n} \\alpha+\\frac{1}{\\cos ^{n} \\alpha} <\/span>
\n =\\left(\\frac{1}{2}\\right)^{n}+\\left(\\frac{1}{\\frac{1}{2}}\\right)^{n} <\/span>
\n =2^{n}+\\frac{1}{2^{n}} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09df \u09ae\u09be\u09a8 =2^{n}+\\frac{1}{2^{n}} <\/span><\/p>\n

4. \\mathrm{\\left(x^{2}-2+\\frac{1}{x^{2}}\\right)^{6}} <\/span> \u098f\u09b0 \u09ac\u09bf\u09b8\u09cd\u09c3\u09a4\u09bf\u09a4\u09c7 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u09aa\u09a6\u09c7\u09b0 \u09ae\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \\mathrm{\\left(x^{2}-2+\\frac{1}{x^{2}}\\right)^{6}=\\left(x^{2}-2 \\cdot x \\cdot \\frac{1}{x}+\\frac{1}{x^{2}}\\right)^{6}} <\/span>
\n \\mathrm{=\\left\\{\\left(x-\\frac{1}{x}\\right)^{2}\\right\\}^{6}=\\left(x-\\frac{1}{x}\\right)^{12}} <\/span>
\n\u09a7\u09b0\u09bf, \u09ac\u09bf\u09b8\u09cd\u09c3\u09a4\u09bf\u09b0 (r+1) \u09a4\u09ae \u09aa\u09a6 \u09a7\u09cd\u09b0\u09c1\u09ac\u09aa\u09a6\u0964
\n\u098f\u0996\u09a8 (r+1) \u09a4\u09ae \u09aa\u09a6 =\\mathrm{T}_{\\mathrm{r}+1}={ }^{12} \\mathrm{C}_{\\mathrm{r}} \\cdot \\mathrm{x}^{12-\\mathrm{r}} \\cdot(-1)^{\\mathrm{r}} \\cdot \\frac{1}{\\mathrm{x}^{\\mathrm{r}}}={ }^{12} \\mathrm{C}_{\\mathrm{r}} \\cdot(-1)^{\\mathrm{r}} \\cdot \\mathrm{x}^{12-2 \\mathrm{r}} <\/span>
\n\u0986\u09ac\u09be\u09b0 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u09aa\u09a6\u09c7\u09b0 \u099c\u09a8\u09cd\u09af 12-2 \\mathrm{r}=0 \\Rightarrow 2 \\mathrm{r}=12 \\Rightarrow \\mathrm{r}=6 <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u09aa\u09a6 =\\mathrm{T}_{6+1}={ }^{12} \\mathrm{C}_{6} \\cdot(-1)^{6} \\cdot \\mathrm{x}^{0}={ }^{12} \\mathrm{C}_{6}=924 <\/span> (Ans.)<\/p>\n

5. \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 3 \\mathrm{\\hat{i}+\\hat{j}+\\hat{k}} <\/span> \u098f\u09ac\u0982 \\mathrm{2 \\hat{i}+2 \\hat{j}-\\hat{k}} <\/span> \u098f\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u09af\u09c7\u09b9\u09c7\u09a4\u09c1 3 \\mathrm{\\hat{i}+\\hat{j}+\\hat{k}} <\/span> \u098f\u09ac\u0982 \\mathrm{2 \\hat{i}+2 \\hat{j}-\\hat{k}} <\/span> \u098f\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09ae\u09c2\u09b2 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7, \u09a4\u09be\u0987 \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u09af\u09c7\u0995\u09cb\u09a8 \u09ad\u09c7\u0995\u09cd\u099f\u09b0 3 \\mathrm{\\hat{i}+\\hat{j}+\\hat{k}} <\/span> \u098f\u09ac\u0982 \\mathrm{2 \\hat{i}+2 \\hat{j}-\\hat{k}} <\/span> \u098f\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u0997\u09be\u09ae\u09c0 \u09b9\u09ac\u09c7\u0964
\n\u09a4\u09be\u0987 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3, \\overrightarrow{\\mathrm{r}}=\\langle 0,0,0\\rangle+\\mathrm{t}\\langle\\mathrm{a}, \\mathrm{b}, \\mathrm{c}\\rangle <\/span>
\n \\therefore \\mathrm{x}=\\mathrm{at}, \\mathrm{y}=\\mathrm{bt}, \\mathrm{z}=\\mathrm{ct} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09df \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \\frac{\\mathrm{x}}{\\mathrm{a}}=\\frac{\\mathrm{y}}{\\mathrm{b}}=\\frac{\\mathrm{z}}{\\mathrm{c}} <\/span> \u09af\u09c7\u0996\u09be\u09a8\u09c7 \\mathrm{a} \\neq 0, \\mathrm{~b} \\neq 0, \\mathrm{c} \\neq 0 <\/span><\/p>\n

6. \u09af\u09a6\u09bf \\mathrm{r e^{i \\theta}=\\frac{3+2 i}{2+3 i}+\\frac{1+5 i}{1-2 i}} <\/span>, \u09a4\u09ac\u09c7 r,~\\theta <\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u098f\u0996\u09be\u09a8\u09c7,
\n\n\n\\begin{array}{l}\n\n\\mathrm{r} \\cdot \\mathrm{e}^{\\mathrm{i} \\theta}=\\frac{3+2 \\mathrm{i}}{2+3 \\mathrm{i}}+\\frac{1+5 \\mathrm{i}}{1-2 \\mathrm{i}} \\\\\n\n\\Rightarrow \\mathrm{r} \\cdot \\mathrm{e}^{\\mathrm{i} \\theta}=\\frac{(3+2 \\mathrm{i})(2-3 \\mathrm{i})}{(2+3 \\mathrm{i})(2-3 \\mathrm{i})}+\\frac{(1+5 \\mathrm{i})(1+2 \\mathrm{i})}{(1-2 \\mathrm{i})(1+2 \\mathrm{i})} \\Rightarrow \\mathrm{r} \\cdot \\mathrm{e}^{\\mathrm{i} \\theta}=\\frac{6-9 \\mathrm{i}+4 \\mathrm{i}-6 \\mathrm{i}^{2}}{4-9 \\mathrm{i}^{2}}+\\frac{1+2 \\mathrm{i}+5 \\mathrm{i}+10 \\mathrm{i}^{2}}{1-4 \\mathrm{i}^{2}} \\\\\n\n\\Rightarrow \\mathrm{r} \\cdot \\mathrm{e}^{\\mathrm{i} \\theta}=\\frac{(6+6)+\\mathrm{i}(4-9)}{4+9}+\\frac{(1-10)+\\mathrm{i}(2+5)}{1+4}\\left[\\because \\mathrm{i}^{2}=-1\\right] \\\\\n\n\\Rightarrow \\mathrm{r} \\cdot \\mathrm{e}^{\\mathrm{i} \\theta}=\\frac{12-5 \\mathrm{i}}{13}+\\frac{-9+7 \\mathrm{i}}{5} \\Rightarrow \\mathrm{r} \\cdot \\mathrm{e}^{\\mathrm{i} \\theta}=\\frac{12}{13}-\\frac{5}{13} \\mathrm{i}-\\frac{9}{5}+\\frac{7}{5} \\mathrm{i} \\Rightarrow \\mathrm{r} \\cdot \\mathrm{e}^{\\mathrm{i} \\theta}=\\frac{-57}{65}+\\frac{66}{65} \\mathrm{i} \\\\\n\n\\therefore \\mathrm{r}=\\sqrt{\\left(\\frac{-57}{65}\\right)^{2}+\\left(\\frac{66}{65}\\right)^{2}}=\\frac{\\sqrt{32+19+4356}}{65}=\\frac{\\sqrt{7605}}{65}=\\frac{39 \\sqrt{5}}{65}=\\frac{3}{\\sqrt{5}} \\text { (Ans.) } \\\\\n\n\\theta=\\pi-\\tan ^{-1}\\left(\\frac{\\frac{66}{65}}{\\frac{57}{65}}\\right)=\\pi-\\tan ^{-1}\\left(\\frac{66}{57}\\right)(\\text { Ans. })\n\n\\end{array}\n\n<\/span><\/p>\n

7. \u098f\u09b0\u09c2\u09aa \u09a6\u09c1\u0987\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0 \u09af\u09be\u09a6\u09c7\u09b0 \u09aa\u09cd\u09b0\u09a4\u09cd\u09af\u09c7\u0995\u099f\u09bf\u09b0 \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \\mathrm{(3,4)}<\/span> \u098f\u09ac\u0982 \u09af\u09be\u09b0\u09be \\mathrm{x}^{2}+\\mathrm{y}^{2}=9 <\/span> \u09ac\u09c3\u09a4\u09cd\u09a4\u0995\u09c7 \u09b8\u09cd\u09aa\u09b0\u09cd\u09b6 \u0995\u09b0\u09c7\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \\mathrm{x}^{2}+\\mathrm{y}^{2}=9 <\/span> \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \\mathrm{C}_{1}(0,0) <\/span> \u098f\u09ac\u0982 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7 \\mathrm{r}_{1}=3 <\/span>
\n\u098f\u0996\u09a8 \u09a7\u09b0\u09bf, \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09af\u09bc \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7 \\mathrm{=r_{2}} <\/span> \u098f\u09ac\u0982 \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \\mathrm{C_{2}(3,4)} <\/span>
\n \\mathrm{C_{1} C_{2}=r_{2} \\pm r_{1} \\Rightarrow \\sqrt{(3-0)^{2}+(4-0)^{2}}=r_{2} \\pm 3} <\/span>
\n \\mathrm{\\Rightarrow 5=r_{2} \\pm 3 \\Rightarrow r_{2}=5 \\pm 3} <\/span>
\n \\mathrm{\\Rightarrow r_{2}=8}~~ [\\text{for} ~(+) ] ~~or, ~~~~\\mathrm{r_{2}=2}~~ [\\text{for} ~(-)] <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09af\u09bc \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0983 \\mathrm{(x-3)^{2}+(y-4)^{2}=2^{2}=4} <\/span> \u098f\u09ac\u0982 \\mathrm{(x-3)^{2}+(y-4)^{2}=8^{2}=64} <\/span> (Ans.)<\/p>\n

8. \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u0995\u09b0\u0983 \\mathrm{\\cos ^{-1} x-\\sin ^{-1} x=\\sin ^{-1}(1-x)} <\/span>
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983\"\"<\/strong><\/p>\n

\u09a6\u09c7\u0993\u09af\u09bc\u09be \u0986\u099b\u09c7,
\n\\mathrm{\n\n\\begin{array}{l}\n\n\\cos ^{-1}(x)-\\sin ^{-1} x=\\sin ^{-1}(1-x) \\\\\n\n\\Rightarrow \\sin ^{-1}\\left(\\sqrt{1-x^{2}}\\right)-\\sin ^{-1}(x)=\\sin ^{-1}(1-x) \\\\\n\n\\Rightarrow \\sin ^{-1}\\left\\{\\sqrt{1-x^{2}} \\cdot \\sqrt{1-x^{2}}-x \\cdot \\sqrt{1-\\left(\\sqrt{1-x^{2}}\\right)^{2}}\\right\\}=\\sin ^{-1}(1-x) \\\\\n\n\\Rightarrow \\sin ^{-1}\\left\\{1-x^{2}-x \\cdot \\sqrt{1-1+x^{2}}\\right\\}=\\sin ^{-1}(1-x) \\\\\n\n\\Rightarrow \\sin ^{-1}\\left\\{1-x^{2}-x \\cdot x\\right\\}=\\sin ^{-1}(1-x) \\\\\n\n\\Rightarrow 1-x^{2}-x^{2}=1-x \\\\\n\n\\Rightarrow 2 x^{2}-x=0 \\\\\n\n\\Rightarrow x(2 x-1)=0 \\\\\n\n\\end{array}} <\/span>
\n\u09a4\u09be\u09b9\u09b2\u09c7, x=0 <\/span> \u0985\u09a5\u09ac\u09be x=\\frac{1}{2} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09af\u09bc \u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983 x=0, \\frac{1}{2} <\/span> (Ans.)<\/p>\n

9. \\mathrm{\\lim _{x \\rightarrow 0} \\frac{e^{x^{2}}-\\cos x}{x^{2}}} <\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong>
\n\\mathrm{\n\n\\begin{array}{l}\n\n\\lim _{x \\rightarrow 0} \\frac{{e^{x^{2}}-\\cos x}}{x^{2}}\\\\\n\n=\\lim _{x \\rightarrow 0} \\frac{e^{x^{2}}\\cdot 2 x+\\sin x}{2 x} ~~~~~[\\text{L' Hospital Rule}] \\\\\n\n=\\lim _{x \\rightarrow 0}\\left\\{e^{x^{2}} \\cdot \\frac{2 x}{2 x}+\\frac{\\sin x}{2 x}\\right\\}\\\\\n\n=\\lim _{x \\rightarrow 0} e^{x^{2}}+\\frac{1}{2} \\lim _{x \\rightarrow 0} \\frac{\\sin x}{x}\\\\\n\n=e^{0}+\\frac{1}{2} \\cdot 1\\\\\n\n=1+\\frac{1}{2}=\\frac{3}{2} \\text { (Ans.) }\n\n\\end{array}}\n\n<\/span><\/p>\n

10. \u098f\u0995\u099f\u09bf \u0989\u09aa\u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0 \u09af\u09be\u09b0 \u098f\u0995\u099f\u09bf \u0989\u09aa\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09c7\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0982\u0995 \\mathrm{(1,-1)}<\/span>, \u0985\u09a8\u09c1\u09b0\u09c2\u09aa \u09a6\u09bf\u0995\u09be\u0995\u09cd\u09b7 \\mathrm{x-y-4=0}<\/span> \u098f\u09ac\u0982 \u09af\u09be \\mathrm{(1,1)}<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09bf\u09af\u09bc\u09c7 \u0985\u09a4\u09bf\u0995\u09cd\u09b0\u09ae \u0995\u09b0\u09c7\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983 \"\"<\/strong><\/p>\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u0989\u09aa\u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u0989\u09ce\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09bf\u0995\u09a4\u09be = e
\n\u09a6\u09c7\u09af\u09bc\u09be \u0986\u099b\u09c7, S \\equiv(1,-1) <\/span>; \u09a6\u09bf\u0995\u09be\u0995\u09cd\u09b7 \u09ac\u09be \u09a8\u09bf\u09af\u09bc\u09be\u09ae\u0995 \u09b0\u09c7\u0996\u09be\u0983 \\mathrm{x-y-4=0} <\/span>
\n\u09a7\u09b0\u09bf, \u0989\u09aa\u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u0989\u09aa\u09b0 \\mathrm{P}(\\mathrm{x}, \\mathrm{y}) <\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1
\n \\therefore \\mathrm{SP}=\\mathrm{e} \\cdot \\mathrm{pm} <\/span>
\n \\Rightarrow \\sqrt{(\\mathrm{x}-1)^{2}+(\\mathrm{y}+1)^{2}}=\\mathrm{e} \\cdot\\left|\\frac{\\mathrm{x}-\\mathrm{y}-4}{\\sqrt{1^{2}+(-1)^{2}}}\\right| <\/span>
\n \\Rightarrow(\\mathrm{x}-1)^{2}+(\\mathrm{y}+1)^{2}=\\mathrm{e}^{2} \\cdot \\frac{(\\mathrm{x}-\\mathrm{y}-4)^{2}}{2} \\ldots \\ldots \\ldots <\/span> (i)
\n(i) \u09a8\u0982 \u0989\u09aa\u09ac\u09c3\u09a4\u09cd\u09a4\u099f\u09bf (1,1) <\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0
\n \\therefore(1-1)^{2}+(1+1)^{2}=\\mathrm{e}^{2} \\cdot \\frac{(1-1-4)^{2}}{2} <\/span>
\n \\Rightarrow 0+4=\\mathrm{e}^{2} \\cdot \\frac{16}{2} <\/span>
\n \\Rightarrow \\mathrm{e}^{2}=\\frac{1}{2} <\/span>
\n \\Rightarrow \\mathrm{e}=\\frac{1}{\\sqrt{2}} <\/span>
\n(i) \u09a8\u0982 \u09a5\u09c7\u0995\u09c7 \u09aa\u09be\u0987, \\mathrm{(x-1)^{2}+(y+1)^{2}=\\frac{1}{2} \\cdot \\frac{(x-y-4)^{2}}{2}} <\/span>
\n\\mathrm{ \\Rightarrow 4 x^{2}+4 y^{2}-8 x+8 y+8=x^{2}+y^{2}+16-2 x y-8 x+8 y} <\/span>
\n \\mathrm{\\Rightarrow 3 x^{2}+3 y^{2}+2 x y-8=0} <\/span> (Ans.)<\/p>\n

11. \u09af\u09a6\u09bf \\mathrm{\\underline{a}+\\underline{b}+\\underline{c}=\\underline{0}} <\/span> \u098f\u09ac\u0982 \\mathrm{|\\underline{a}|=3,|\\underline{b}|=5,|\\underline{c}|=7} <\/span> \u09b9\u09af\u09bc, \u09a4\u09be\u09b9\u09b2\u09c7 \\mathrm{\\underline{a}} <\/span> \u098f\u09ac\u0982 \\mathrm{\\underline{b}} <\/span> \u098f\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u0995\u09cb\u09a3 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u09a6\u09c7\u0993\u09af\u09bc\u09be \u0986\u099b\u09c7,
\n \\mathrm{\\underline{a}+\\underline{b}+\\underline{c}=0} <\/span>
\n \\mathrm{\\Rightarrow \\underline{a}+\\underline{b}=-\\underline{c}} <\/span>
\n\u09a7\u09b0\u09bf, \\underline{\\mathrm{a}} <\/span> \u0993 \\underline{\\mathrm{b}} <\/span> \u098f\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u0995\u09cb\u09a3 =\\theta <\/span>
\n\u09a4\u09be\u09b9\u09b2\u09c7,
\n (|\\mathrm{a}|)^{2}+(|\\mathrm{b}|)^{2}+2 \\cdot|\\mathrm{a}| \\cdot|\\mathrm{b}| \\cdot \\cos \\theta=(|\\mathrm{c}|)^{2} <\/span>
\n \\mathrm{\\Rightarrow 3^{2}+5^{2}+2 \\times 3 \\times 5 \\cos \\theta=7^{2}} <\/span>
\n \\mathrm{\\Rightarrow \\cos \\theta=\\frac{7^{2}-3^{2}-5^{2}}{2 \\times 3 \\times 5}=\\frac{1}{2}} <\/span>
\n \\therefore \\mathrm{ \\theta=\\cos ^{-1}\\left(\\frac{1}{2}\\right)=\\frac{\\pi}{3}} <\/span> (Ans.)<\/p>\n

12. \\mathrm{O} <\/span> \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \u098f\u0995\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7 10 \\mathrm{~cm} <\/span> \u098f\u09ac\u0982 \\mathrm{AB} <\/span> \u099a\u09be\u09aa\u09c7\u09b0 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af 14 \\mathrm{~cm} <\/span>\u0964 \u0995\u09cb\u09a3 \\angle \\mathrm{AOB} <\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \u09ac\u09c7\u09b0 \u0995\u09b0 \u098f\u09ac\u0982 \u099a\u09be\u09aa \\mathrm{AB} <\/span> \u0993 \u099c\u09cd\u09af\u09be \\mathrm{AB} <\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0986\u09ac\u09a6\u09cd\u09a7 \u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong><\/p>\n

\"\"
\n\u09a6\u09c7\u09af\u09bc\u09be \u0986\u099b\u09c7, \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7, \\mathrm{r}=10 \\mathrm{~cm}, ~~\\mathrm{AB} <\/span> \u099a\u09be\u09aa\u09c7\u09b0 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af \\mathrm{s}=14 \\mathrm{~cm} <\/span>
\n\u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf,
\n \\mathrm{s}=\\mathrm{r} \\theta <\/span>
\n \\Rightarrow \\theta=\\frac{\\mathrm{s}}{\\mathrm{r}}=\\frac{14}{10}=1.4^{\\mathrm{c}} <\/span> (Ans.)
\n\u0995\u09cd\u09b7\u09c1\u09a6\u09cd\u09b0\u09a4\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 = \u09ac\u09c3\u09a4\u09cd\u09a4\u0995\u09b2\u09be AOB \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 - <\/span> \\triangle \\mathrm{AOB} <\/span> \u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 =\\frac{1}{2} \\times \\mathrm{r}^{2} \\theta-\\frac{1}{2} \\times \\mathrm{r}^{2} \\sin \\theta <\/span>
\n =\\frac{1}{2} \\times r^{2}(\\theta-\\sin \\theta) <\/span>
\n =\\frac{1}{2} r^{2}\\left\\{1.4-\\sin \\left(1.4^{\\mathrm{c}}\\right)\\right\\} <\/span>
\n =\\frac{1}{2} \\times 10 \\times 10\\left\\{1.4-\\sin \\left(1.4^{\\mathrm{c}}\\right)\\right\\} <\/span>
\n =50 \\cdot\\{1.4-0.985\\} <\/span>
\n =50 \\times 0.415=20.73 <\/span> \u09ac\u09b0\u09cd\u0997 \u09b8\u09c7.\u09ae\u09bf. (Ans.)<\/p>\n

13. \\mathrm{y}=\\left(\\mathrm{x}+\\sqrt{1+\\mathrm{x}^{2}}\\right)^{\\mathrm{m}} <\/span> \u09b9\u09b2\u09c7 \u09aa\u09cd\u09b0\u09ae\u09be\u09a3 \u0995\u09b0 \u09af\u09c7, \\left(1+\\mathrm{x}^{2}\\right) \\frac{\\mathrm{d}^{2} \\mathrm{y}}{\\mathrm{dx}^{2}}+\\mathrm{x} \\frac{\\mathrm{dy}}{\\mathrm{dx}}-\\mathrm{m}^{2} \\mathrm{y}=0 <\/span> \u0964 \u0985\u09a4\u0983\u09aa\u09b0 \\mathrm{x}=0 <\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09cd\u09a4\u09c7 \\frac{\\mathrm{d}^{3} \\mathrm{y}}{\\mathrm{dx}^{3}} <\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \u09ac\u09c7\u09b0 \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong>
\n\u09a6\u09c7\u0993\u09af\u09bc\u09be \u0986\u099b\u09c7, \\mathrm{y}=\\left(\\mathrm{x}+\\sqrt{1+\\mathrm{x}^{2}}\\right)^{\\mathrm{m}} <\/span>
\n \\frac{\\mathrm{dy}}{\\mathrm{dx}}=\\frac{\\mathrm{m} \\cdot\\left(\\mathrm{x}+\\sqrt{1+\\mathrm{x}^{2}}\\right)^{\\mathrm{m}}}{\\left(\\mathrm{x}+\\sqrt{1+\\mathrm{x}^{2}}\\right)}\\left\\{1+\\frac{2 \\mathrm{x}}{2 \\sqrt{1+\\mathrm{x}^{2}}}\\right\\} <\/span>
\n \\mathrm{\\frac{d y}{d x}=m \\cdot \\frac{\\left(x+\\sqrt{1+x^{2}}\\right)^{m}}{\\left(x+\\sqrt{1+x^{2}}\\right)}\\left\\{\\frac{x+\\sqrt{1+x^{2}}}{\\sqrt{1+x^{2}}}\\right\\}} <\/span>
\n \\mathrm{\\Rightarrow \\sqrt{1+x^{2}} \\cdot \\frac{d y}{d x}=m \\cdot\\left(x+\\sqrt{1+x^{2}}\\right)^{m} \\ldots \\ldots \\ldots} <\/span> (i)
\n \\mathrm{\\Rightarrow\\left(1+\\mathrm{x}^{2}\\right) \\cdot\\left(\\frac{\\mathrm{dy}}{\\mathrm{dx}}\\right)^{2}=\\mathrm{m}^{2} \\mathrm{y}^{2}} <\/span> [\u09ac\u09b0\u09cd\u0997 \u0995\u09b0\u09c7]
\n \\mathrm{\\Rightarrow\\left(1+x^{2}\\right) \\cdot 2 \\frac{d y}{d x} \\cdot \\frac{d^{2} y}{d x^{2}}+\\left(\\frac{d y}{d x}\\right)^{2} \\cdot 2 x=m^{2} \\cdot 2 y \\cdot \\frac{d y}{d x}} <\/span> [x \u098f\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 \u0985\u09a8\u09cd\u09a4\u09b0\u09c0\u0995\u09b0\u09a3 \u0995\u09b0\u09c7]
\n \\mathrm{\\Rightarrow\\left(1+x^{2}\\right) \\cdot \\frac{d^{2} y}{d x^{2}}+\\left(\\frac{d y}{d x}\\right) \\cdot x-m^{2} \\cdot y=0 \\ldots \\ldots \\ldots (ii) [2 \\frac{d y}{d x}} <\/span> \u09a6\u09bf\u09df\u09c7 \u09ad\u09be\u0997 \u0995\u09b0\u09c7]
\n(ii) \u09a8\u0982 \u0995\u09c7 \\mathrm{x} <\/span> \u098f\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 \u0985\u09a8\u09cd\u09a4\u09b0\u09c0\u0995\u09b0\u09a3 \u0995\u09b0\u09c7 \u09aa\u09be\u0987,
\n\\mathrm{ \\left(1+x^{2}\\right) \\cdot \\frac{d^{3} y}{d x^{3}}+\\frac{d^{2} y}{d x^{2}} \\cdot 2 x+x \\cdot \\frac{d^{2} y}{d x^{2}}+\\frac{d y}{d x}-m^{2} \\frac{d y}{d x}=0 \\ldots \\ldots \\ldots} <\/span> (iii)
\n \\mathrm{x=0} <\/span> \u09b9\u09b2\u09c7, \\mathrm{y=\\left(0+\\sqrt{1+0^{2}}\\right)^{m}=1]} <\/span>
\n(i) \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7 \\mathrm{x}=0 <\/span> \u09ac\u09b8\u09bf\u09df\u09c7 \u09aa\u09be\u0987, \\mathrm{1 \\cdot \\frac{\\mathrm{dy}}{\\mathrm{dx}}=\\mathrm{m} \\Rightarrow \\frac{\\mathrm{dy}}{\\mathrm{dx}}=\\mathrm{m}} <\/span>
\n(ii) \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7 \\mathrm{x=0, \\frac{d y}{d x}=m, y=1} <\/span> \u09ac\u09b8\u09bf\u09df\u09c7 \u09aa\u09be\u0987,
\n \\mathrm{(1+0) \\cdot \\frac{d^{2} y}{d x^{2}}+0 \\cdot m-m^{2} \\times 1=0 \\Rightarrow \\frac{d^{2} y}{d x^{2}}=m^{2}} <\/span>
\n(iii) \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7 \u09ae\u09be\u09a8 \u09ac\u09b8\u09bf\u09af\u09bc\u09c7 \u09aa\u09be\u0987,
\n \\mathrm{(1+0) \\cdot \\frac{\\mathrm{d}^{3} \\mathrm{y}}{\\mathrm{dx}^{3}}+\\mathrm{m}^{2} \\times 2 \\times 0+0 \\times \\mathrm{m}^{2}+\\mathrm{m}-\\mathrm{m}^{2} \\times \\mathrm{m}=0 \\Rightarrow \\frac{\\mathrm{d}^{3} \\mathrm{y}}{\\mathrm{dx}^{3}}=\\mathrm{m}^{3}-\\mathrm{m}~~~~\\text{(Ans.)}} <\/span><\/p>\n

14. \u0985\u09b8\u09c0\u09ae \u09a7\u09be\u09b0\u09be\u099f\u09bf\u09b0 \u09af\u09cb\u0997\u09ab\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u0983 1+\\frac{3}{4}+\\frac{3.5}{4.8}+\\frac{3.5 .7}{4.8 .12}+\\cdots \\infty <\/span> \u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983\u00a0<\/strong>\u09a7\u09b0\u09bf, S=1+\\frac{3}{4}+\\frac{3 \\cdot 5}{4 \\cdot 8}+\\frac{3 \\cdot 5 \\cdot 7}{4 \\cdot 8 \\cdot 12}+\\cdots \\infty \\ldots <\/span> (i) \\\\<\/span>\u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf, (1+\\mathrm{x})^{\\mathrm{n}}=1+\\mathrm{nx}+\\frac{\\mathrm{n}(\\mathrm{n}-1)}{2 !} \\cdot \\mathrm{x}^{2}+\\frac{\\mathrm{n}(\\mathrm{n}-1)(\\mathrm{n}-2)}{3 !} \\mathrm{x}^{3}+\\cdots \\infty \\ldots <\/span> (ii)\\\\<\/span>(i) \u09a8\u0982 \u0993 (ii) \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u09a4\u09c1\u09b2\u09a8\u09be \u0995\u09b0\u09c7 \u09aa\u09be\u0987,
\n \\mathrm{nx}=\\frac{3}{4} \\ldots \\ldots \\ldots <\/span> (iii) ;
\n \\frac{\\mathrm{n}(\\mathrm{n}-1)}{2 \\cdot 1} \\cdot \\mathrm{x}^{2}=\\frac{3 \\cdot 5}{4 \\cdot 8} \\ldots \\ldots \\ldots <\/span> (iv)<\/p>\n

(iv) \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0995\u09c7 (iii) \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09ac\u09b0\u09cd\u0997 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ad\u09be\u0997 \u0995\u09b0\u09c7 \u09aa\u09be\u0987,
\n \\frac{\\mathrm{n}(\\mathrm{n}-1) \\mathrm{x}^{2}}{2 \\times \\mathrm{n}^{2} \\mathrm{X}^{2}}=\\frac{3 \\cdot 5 \\cdot 4 \\cdot 4}{4 \\cdot 8 \\cdot 3 \\cdot 3} \\Rightarrow \\frac{\\mathrm{n}^{2}-\\mathrm{n}}{\\mathrm{n}^{2}}=\\frac{5}{3} \\Rightarrow 1-\\frac{1}{\\mathrm{n}}=\\frac{5}{3} \\Rightarrow \\frac{1}{\\mathrm{n}}=\\frac{-2}{3} \\Rightarrow \\mathrm{n}=\\frac{-3}{2} \\ldots \\ldots <\/span> (v)<\/p>\n

(iii) \\Rightarrow \\mathrm{nx}=\\frac{3}{4} \\Rightarrow \\frac{-3}{2} \\times \\mathrm{x}=\\frac{3}{4} \\Rightarrow \\mathrm{x}=\\frac{-1}{2} \\ldots \\ldots <\/span> (vi)<\/p>\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09df \u09af\u09cb\u0997\u09ab\u09b2,
\n S=(1+x)^{n}=\\left(1-\\frac{1}{2}\\right)^{\\frac{-3}{2}}=\\left(\\frac{1}{2}\\right)^{\\frac{-3}{2}}=2^{\\frac{3}{2}}=(\\sqrt{2})^{3}=2 \\sqrt{2} <\/span> (Ans.)<\/p>\n

15. \\mathrm{A}(5,3), \\mathrm{B}(-2,0) <\/span> \u098f\u09ac\u0982 \\mathrm{C}(1,1) <\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a4\u09bf\u09a8\u099f\u09bf \u098f\u0995\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u0989\u09aa\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09b9\u09b2\u09c7 \u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \u0993 \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c \\mathrm{ABC} <\/span> \u098f\u09b0 \u09ad\u09b0\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong>
\n\u09ac\u09c3\u09a4\u09cd\u09a4\u09c7\u09b0 \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \\mathrm{O}\\left(\\mathrm{x}_{1}, \\mathrm{y}_{1}\\right) <\/span> \u09b9\u09b2\u09c7,
\n \\mathrm{OA}=\\mathrm{OC} \\Rightarrow \\sqrt{\\left(\\mathrm{x}_{1}-5\\right)^{2}+\\left(\\mathrm{y}_{1}-3\\right)^{2}}=\\sqrt{\\left(\\mathrm{x}_{1}-1\\right)^{2}+\\left(\\mathrm{y}_{1}-1\\right)^{2}} <\/span>
\n \\Rightarrow \\mathrm{x}_{1}^{2}+\\mathrm{y}_{1}^{2}-10 \\mathrm{x}_{1}-6 \\mathrm{y}_{1}+34=\\mathrm{x}_{1}^{2}+\\mathrm{y}_{1}^{2}-2 \\mathrm{x}_{1}-2 \\mathrm{y}_{1}+2 <\/span>
\n \\Rightarrow-8 \\mathrm{x}_{1}-4 \\mathrm{y}_{1}+32=0 \\Rightarrow 2 \\mathrm{x}_{1}+\\mathrm{y}_{1}-8=0 \\ldots \\ldots \\ldots <\/span> (i)
\n\u098f\u09ac\u0982 \\mathrm{OA}=\\mathrm{OB} \\Rightarrow \\sqrt{\\left(\\mathrm{x}_{1}-5\\right)^{2}+\\left(\\mathrm{y}_{1}-3\\right)^{2}}=\\sqrt{\\left(\\mathrm{x}_{1}+2\\right)^{2}+\\left(\\mathrm{y}_{1}-0\\right)^{2}} <\/span>
\n \\Rightarrow \\mathrm{x}_{1}^{2}+\\mathrm{y}_{1}^{2}-10 \\mathrm{x}_{1}-6 \\mathrm{y}_{1}+34=\\mathrm{x}_{1}^{2}+\\mathrm{y}_{1}^{2}+4 \\mathrm{x}_{1}+4 <\/span>
\n \\Rightarrow-14 x_{1}-6 y_{1}+30=0 \\Rightarrow 7 x_{1}+3 y_{1}-15=0 \\ldots \\ldots <\/span> (ii)
\n(i) \u09a8\u0982 \u0993 (ii) \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 \u0995\u09b0\u09c7 \u09aa\u09be\u0987, \\mathrm{x}_{1}=-9 ; \\mathrm{y}_{1}=26 <\/span>
\n\u098f\u0996\u09a8 \\triangle \\mathrm{ABC} <\/span> \u098f\u09b0 \u09ad\u09b0\u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 0\\left(\\mathrm{x}_{2}, \\mathrm{y}_{2}\\right) <\/span> \u09b9\u09b2\u09c7 \\mathrm{x}_{2}=\\frac{5+1-2}{3} ; \\mathrm{y}_{2}=\\frac{3+1+0}{3} \\therefore \\mathrm{D} \\equiv\\left(\\frac{4}{3}, \\frac{4}{3}\\right) <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac,
\n \\mathrm{OD}=\\sqrt{\\left(\\mathrm{x}_{1}-\\mathrm{x}_{2}\\right)^{2}+\\left(\\mathrm{y}_{1}-\\mathrm{y}_{2}\\right)^{2}}=\\sqrt{\\left(-9-\\frac{4}{3}\\right)^{2}+\\left(26-\\frac{4}{3}\\right)^{2}} <\/span>
\n =\\sqrt{\\frac{961}{9}+\\frac{5476}{9}}=\\frac{\\sqrt{6437}}{3}=26.74 <\/span> (\u09aa\u09cd\u09b0\u09be\u09df) (Ans.)<\/p>\n

16. \"\"<\/p>\n

\\mathrm{x=\\frac{1}{y^{2}}, x=y} <\/span> \u098f\u09ac\u0982 \\mathrm{y=2} <\/span> \u09b0\u09c7\u0996\u09be\u0997\u09c1\u09b2\u09bf\u09b0 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0986\u09ac\u09a6\u09cd\u09a7 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u0964 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u099f\u09bf\u09b0 \u099a\u09bf\u09a4\u09cd\u09b0 \u0985\u0982\u0995\u09a8 \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \\mathrm{x=\\frac{1}{y^{2}}} <\/span> \u0993 \\mathrm{x=y} <\/span> \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 \u099b\u09c7\u09a6 \u0995\u09b0\u09c7, \u09a4\u09be\u09a6\u09c7\u09b0 \u099b\u09c7\u09a6\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7,
\n\\mathrm{ \\frac{1}{y^{2}}=y \\Rightarrow y^{3}=1 \\Rightarrow y=1 \\therefore x=\\frac{1}{y^{2}}=1} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u0986\u09ac\u09a6\u09cd\u09a7 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2
\n \\mathrm{=\\int^{2}\\left(y-\\frac{1}{y^{2}}\\right) d y=\\left[\\frac{y^{2}}{2}+\\frac{1}{y}\\right]_{1}^{2}=\\left(2+\\frac{1}{2}-\\frac{1}{2}-1\\right)} <\/span> \u09ac\u09b0\u09cd\u0997 \u098f\u0995\u0995 =1 <\/span> \u09ac\u09b0\u09cd\u0997 \u098f\u0995\u0995<\/p>\n

17. EXAMINATION \u09b6\u09ac\u09cd\u09a6\u099f\u09bf\u09b0 \u09ac\u09cd\u09af\u099e\u09cd\u099c\u09a8\u09ac\u09b0\u09cd\u09a3\u0997\u09c1\u09b2\u09bf\u0995\u09c7 \u098f\u0995\u09a4\u09cd\u09b0\u09c7 \u09a8\u09be \u09b0\u09c7\u0996\u09c7 \u0995\u09a4 \u09b0\u0995\u09ae\u09c7 \u09b8\u09be\u099c\u09be\u09a8\u09cb \u09af\u09be\u09df?
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \\underset{\\mathrm{~~~~A~I}}{\\mathrm{EAI}}\\underset{\\mathrm{~~~~~~~~~~N}}{\\mathrm{OXMNT}} <\/span>
\n\u09b6\u09ac\u09cd\u09a6\u099f\u09bf\u09a4\u09c7 \u09ae\u09cb\u099f \u09ac\u09b0\u09cd\u09a3 = 11 \u099f\u09bf
\n\u098f\u0995\u0987 \u09b0\u0995\u09ae \u09ac\u09b0\u09cd\u09a3 A=2 <\/span> \u099f\u09bf, I=2 <\/span> \u099f\u09bf, N=2 <\/span> \u099f\u09bf
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09b6\u09ac\u09cd\u09a6\u099f\u09bf\u09b0 \u09ac\u09bf\u09a8\u09cd\u09af\u09be\u09b8 \u09b8\u0982\u0996\u09cd\u09af\u09be =\\frac{11 !}{2 ! \\times 2 ! \\times 2 !} <\/span>
\n\u09b6\u09ac\u09cd\u09a6\u099f\u09bf\u09a4\u09c7 5 \u099f\u09bf \u09ac\u09cd\u09af\u09be\u099e\u09cd\u099c\u09a8\u09ac\u09b0\u09cd\u09a3 \u0986\u099b\u09c7 \u09af\u09be\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 2 \u099f\u09bf \u098f\u0995\u0987\u09b0\u0995\u09ae\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09ac\u09cd\u09af\u09be\u099e\u09cd\u099c\u09a8\u09ac\u09b0\u09cd\u09a3\u0995\u09c7 \u098f\u0995\u09a4\u09cd\u09b0\u09c7 \u09b0\u09c7\u0996\u09c7 \u09ac\u09bf\u09a8\u09cd\u09af\u09be\u09b8 \u09b8\u0982\u0996\u09cd\u09af\u09be =\\frac{7 !}{2 ! \\times 2 !} \\times \\frac{5 !}{2 !} <\/span> \u0964
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u09ac\u09cd\u09af\u09be\u099e\u09cd\u099c\u09a8\u09ac\u09b0\u09cd\u09a3\u0997\u09c1\u09b2\u09bf\u0995\u09c7 \u098f\u0995\u09a4\u09cd\u09b0\u09c7 \u09a8\u09be \u09b0\u09c7\u0996\u09c7 \u09ac\u09bf\u09a8\u09cd\u09af\u09be\u09b8 \u09b8\u0982\u0996\u09cd\u09af\u09be
\n =\\frac{11 !}{2 ! \\times 2 ! \\times 2 !}-\\frac{7 !}{2 ! \\times 2 !} \\times \\frac{5 !}{2 !}=4914000 <\/span> (Ans.)<\/p>\n

18. \u0997\u09a3\u09bf\u09a4 \u0993 \u09aa\u09b0\u09bf\u09b8\u0982\u0996\u09cd\u09af\u09be\u09a8 \u09ac\u09bf\u09b7\u09af\u09bc\u09c7 200 \u099c\u09a8 \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be\u09b0\u09cd\u09a5\u09c0\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 20 \u099c\u09a8 \u09aa\u09b0\u09bf\u09b8\u0982\u0996\u09cd\u09af\u09be\u09a8\u09c7 \u098f\u09ac\u0982 40 \u099c\u09a8 \u0997\u09a3\u09bf\u09a4\u09c7 \u09ab\u09c7\u09b2 \u0995\u09b0\u09c7 \u098f\u09ac\u0982 \u0989\u09ad\u09af\u09bc \u09ac\u09bf\u09b7\u09af\u09bc\u09c7 10 \u099c\u09a8 \u09ab\u09c7\u09b2 \u0995\u09b0\u09c7\u0964 \u09a8\u09bf\u09b0\u09aa\u09c7\u0995\u09cd\u09b7\u09ad\u09be\u09ac\u09c7 \u098f\u0995\u099c\u09a8 \u099b\u09be\u09a4\u09cd\u09b0\u0995\u09c7 \u09ac\u09be\u099b\u09be\u0987 \u0995\u09b0\u09b2\u09c7 \u09a4\u09be\u09b0 \u09aa\u09b0\u09bf\u09b8\u0982\u0996\u09cd\u09af\u09be\u09a8\u09c7 \u09aa\u09be\u09b8 \u0993 \u0997\u09a3\u09bf\u09a4\u09c7 \u09ab\u09c7\u09b2 \u09b9\u0993\u09af\u09bc\u09be\u09b0 \u09b8\u09ae\u09cd\u09ad\u09be\u09ac\u09a8\u09be \u09ac\u09c7\u09b0 \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u09ae\u09cb\u099f \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be\u09b0\u09cd\u09a5\u09c0 =\\mathrm{n}(\\mathrm{u})=200 <\/span>
\n\u09aa\u09b0\u09bf\u09b8\u0982\u0996\u09cd\u09af\u09be\u09a8\u09c7 \u09ab\u09c7\u09b2 \u0995\u09b0\u09c7 =\\mathrm{n}(\\mathrm{s})=20 <\/span>
\n\u0997\u09a3\u09bf\u09a4\u09c7 \u09ab\u09c7\u09b2 \u0995\u09b0\u09c7 =\\mathrm{n}(\\mathrm{m})=40 <\/span>
\n\u0997\u09a3\u09bf\u09a4 \u0993 \u09aa\u09b0\u09bf\u09b8\u0996\u09cd\u09af\u09be\u09a8 \u0989\u09ad\u09af\u09bc \u09ac\u09bf\u09b7\u09af\u09bc\u09c7 \u09ab\u09c7\u09b2 \u0995\u09b0\u09c7 \\mathrm{n}(\\mathrm{s} \\cap \\mathrm{m})=10 <\/span>
\n\u09aa\u09b0\u09bf\u09b8\u0982\u0996\u09cd\u09af\u09be\u09a8\u09c7 \u09aa\u09be\u09b8 \u0993 \u0997\u09a3\u09bf\u09a4\u09c7 \u09ab\u09c7\u09b2 \u0995\u09b0\u09be \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be\u09b0\u09cd\u09a5\u09c0 \u09b8\u0982\u0996\u09cd\u09af\u09be=
\n \\mathrm{n}\\left(\\mathrm{s}^{\\prime} \\cap \\mathrm{m}\\right)=\\mathrm{n}(\\mathrm{m})-\\mathrm{n}(\\mathrm{m} \\cap \\mathrm{s})=40-10=30 <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09aa\u09b0\u09bf\u09b8\u0982\u0996\u09cd\u09af\u09be\u09a8\u09c7 \u09aa\u09be\u09b8 \u0993 \u0997\u09a3\u09bf\u09a4\u09c7 \u09ab\u09c7\u09b2 \u0995\u09b0\u09be\u09b0 \u09b8\u09ae\u09cd\u09ad\u09be\u09ac\u09a8\u09be =\\mathrm{p}\\left(\\mathrm{s}^{\\prime} \\cap \\mathrm{m}\\right)=\\frac{\\mathrm{n}\\left(\\mathrm{s}^{\\prime} \\cap \\mathrm{m}\\right)}{\\mathrm{n}(\\mathrm{u})}=\\frac{30}{200}=\\frac{3}{20} <\/span> (Ans.)<\/p>\n

19. 20 \\mathrm{~cm} <\/span> \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af\u09c7\u09b0 \u09b9\u09be\u09b2\u0995\u09be \\mathrm{AB} <\/span> \u09a6\u09a3\u09cd\u09a1\u099f\u09bf 10 \\mathrm{~cm} <\/span> \u09ac\u09cd\u09af\u09ac\u09a7\u09be\u09a8\u09c7 \u09a6\u09c1\u0987\u099f\u09bf \u09aa\u09c7\u09b0\u09c7\u0995\u09c7\u09b0 \u0989\u09aa\u09b0 \u0986\u09a8\u09c1\u09ad\u09c2\u09ae\u09bf\u0995 \u09ad\u09be\u09ac\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4\u0964 \\mathrm{A} <\/span> \u0993 \\mathrm{B} <\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 2 \\mathrm{~W} <\/span> \u0993 3 \\mathrm{~W} <\/span> \u0993\u099c\u09a8 \u099d\u09c1\u09b2\u09be\u09a8\u09cb \u09b9\u09b2\u09c7, \u09aa\u09c7\u09b0\u09c7\u0995 \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u0995\u09cb\u09a8 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u098f\u09a6\u09c7\u09b0 \u0989\u09aa\u09b0 \u099a\u09be\u09aa \u09b8\u09ae\u09be\u09a8 \u09b9\u09ac\u09c7?
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983 \"\"<\/strong><\/p>\n

\u09a6\u09a8\u09cd\u09a1\u09c7\u09b0 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af \\mathrm{AB}=20 \\mathrm{~cm} <\/span>
\n\u09a7\u09b0\u09bf, \u09aa\u09c7\u09b0\u09c7\u0995 \u09a6\u09c1\u099f\u09bf\u09b0 \u0989\u09aa\u09b0 \u09b8\u09ae\u09be\u09a8 \u099a\u09be\u09aa = p
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \\mathrm{p}+\\mathrm{p}=2 \\mathrm{w}+3 \\mathrm{w} \\Rightarrow 2 \\mathrm{p}=5 \\mathrm{w} \\Rightarrow \\mathrm{p}=\\frac{5 \\mathrm{w}}{2} \\ldots \\ldots \\ldots <\/span> (i)<\/p>\n

\u09a7\u09b0\u09bf A \u09aa\u09cd\u09b0\u09be\u09a8\u09cd\u09a4 \u09a5\u09c7\u0995\u09c7 \u0996\u09c1\u099f\u09bf\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac x, x + 10
\n \\mathrm{A} <\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09ac\u09b2\u09c7\u09b0 \u09ad\u09cd\u09b0\u09be\u09ae\u0995\u09c7\u09b0 \u09b8\u09c2\u09a4\u09cd\u09b0\u09be\u09a8\u09c1\u09af\u09be\u09af\u09bc\u09c0, 2 \\mathrm{w} \\times 0+3 \\mathrm{w} \\times \\mathrm{AB}=\\mathrm{p} \\times \\mathrm{AC}+\\mathrm{p} \\times \\mathrm{AD} <\/span>
\n \\Rightarrow 0+3 \\mathrm{w} \\times 20=\\frac{5 \\mathrm{w}}{2} \\times \\mathrm{x}+\\frac{5 \\mathrm{w}}{2} \\times(\\mathrm{x}+10) <\/span>
\n \\mathrm{\\Rightarrow 3 w \\times 20=\\frac{5 w}{2} \\times(2 x+10) \\Rightarrow 2 x+10=24 \\Rightarrow 2 x=14 \\Rightarrow x=7} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09aa\u09c7\u09b0\u09c7\u0995 \u09a6\u09c1\u099f\u09bf A \u09aa\u09cd\u09b0\u09be\u09a8\u09cd\u09a4 \u09a5\u09c7\u0995\u09c7 7 \\mathrm{~cm} <\/span> \u0993 17 \\mathrm{~cm} <\/span> \u09a6\u09c2\u09b0\u09c7 \u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/p>\n

20. \u09b8\u09cd\u09a5\u09bf\u09b0\u09be\u09ac\u09b8\u09cd\u09a5\u09be \u09a5\u09c7\u0995\u09c7 \u098f\u0995\u099f\u09bf \u09ac\u09be\u09b8\u0995\u09c7 3 \\mathrm{~ms}^{-2}<\/span> \u09b8\u09ae\u09a4\u09cd\u09ac\u09b0\u09a3\u09c7 \u099a\u09b2\u09a4\u09c7 \u09a6\u09c7\u0996\u09c7 \u09ac\u09be\u09b8\u099f\u09bf\u0995\u09c7 \u09a7\u09b0\u09be\u09b0 \u0989\u09a6\u09cd\u09a6\u09c7\u09b6\u09cd\u09af\u09c7 \u098f\u0995\u099c\u09a8 \u09b2\u09cb\u0995 \u09ac\u09be\u09b8\u09c7\u09b0 \u09aa\u09c7\u099b\u09a8\u09c7 \u0995\u09bf\u099b\u09c1\u09a6\u09c2\u09b0 \u09a5\u09c7\u0995\u09c7 12 \\mathrm{~m} \/ \\mathrm{sec}<\/span> \u09b8\u09ae\u09ac\u09c7\u0997\u09c7 \u09a6\u09cc\u09a1\u09bc\u09be\u09a4\u09c7 \u0986\u09b0\u09ae\u09cd\u09ad \u0995\u09b0\u09c7\u0964 \u09ac\u09be\u09b8 \u09a5\u09c7\u0995\u09c7 \u09b2\u09cb\u0995\u099f\u09bf \u09b8\u09b0\u09cd\u09ac\u09cb\u099a\u09cd\u099a \u0995\u09a4 \u09a6\u09c2\u09b0\u09c7 \u09a5\u09be\u0995\u09b2\u09c7 \u09ac\u09be\u09b8\u099f\u09bf\u0995\u09c7 \u09a7\u09b0\u09a4\u09c7 \u09aa\u09be\u09b0\u09ac\u09c7? \\\\<\/span><\/p>\n

\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983 \"\"<\/strong><\/p>\n

\u09a7\u09b0\u09bf, B \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 \u09ac\u09be\u09b8\u099f\u09bf 3 \\mathrm{~ms}^{-2} <\/span> \u09a4\u09cd\u09ac\u09b0\u09a3\u09c7 \\mathrm{t} <\/span> \u09b8\u09ae\u09af\u09bc\u09c7 \\mathrm{C} <\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09aa\u09cc\u099b\u09be\u09af\u09bc\u0964 \u09b2\u09cb\u0995\u099f\u09bf A \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 12 \\mathrm{~ms}^{-1} <\/span> \u09ac\u09c7\u0997\u09c7 \u09a6\u09cc\u09a1\u09bc\u09bf\u09af\u09bc\u09c7 \\mathrm{t} <\/span> \u09b8\u09ae\u09af\u09bc\u09c7 \\mathrm{C} <\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09aa\u09cc\u099b\u09be\u09af\u09bc\u0964
\nA \u0993 B \u098f\u09b0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac x \u09b9\u09b2\u09c7, \\mathrm{12 t-\\frac{1}{2} 3 t^{2}=x \\Rightarrow-3 t^{2}-24 t=2 x \\Rightarrow 3 t^{2}+24 t+2 x=0 \\ldots \\ldots \\ldots} <\/span> (i) \u098f\u0996\u09a8 \u09b2\u09cb\u0995\u099f\u09bf \u09ac\u09be\u09b8\u0995\u09c7 \u09a7\u09b0\u09a4\u09c7 \u09b9\u09b2\u09c7 t <\/span> \u098f\u09b0 \u09ae\u09be\u09a8 \u09ac\u09be\u09b8\u09cd\u09a4\u09ac \u09b9\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982 (i) \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u09c7\u09b0 \u09a8\u09bf\u09b6\u09cd\u099a\u09be\u09af\u09bc\u0995 \\geq 0 <\/span>
\n \\Rightarrow(24)^{2}-4 \\times 3 \\times 2 \\mathrm{x} \\geq 0 \\Rightarrow 24 \\times 24 \\geq 4 \\times 3 \\times 2 \\mathrm{x} \\Rightarrow \\mathrm{x} \\leq 24 <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09b2\u09cb\u0995\u099f\u09bf \u09b8\u09b0\u09cd\u09ac\u09cb\u099a\u09cd\u099a 24 \u09ae\u09bf. \u09a6\u09c2\u09b0 \u09a5\u09c7\u0995\u09c7 \u09ac\u09be\u09b8\u0995\u09c7 \u09a7\u09b0\u09a4\u09c7 \u09aa\u09be\u09b0\u09ac\u09c7\u0964<\/p>\n

\u09aa\u09a6\u09be\u09b0\u09cd\u09a5\u09ac\u09bf\u099c\u09cd\u099e\u09be\u09a8 (Physics)<\/strong><\/h3>\n

21. \u098f\u0995\u099f\u09bf \u0995\u09cd\u09b0\u09bf\u0995\u09c7\u099f \u09ac\u09b2\u09c7\u09b0 \u0993\u099c\u09a8 0.65 \\mathrm{~kg} <\/span> \u0964 \u098f\u0995\u099c\u09a8 \u09ab\u09bf\u09b2\u09cd\u09a1\u09be\u09b0 \u09ac\u09b2\u099f\u09bf\u0995\u09c7 \u09b8\u09cd\u09ac\u09b2\u09cd\u09aa\u09a4\u09ae \u09b8\u09ae\u09af\u09bc\u09c7 100 \\mathrm{~m} <\/span> \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac\u09c7 \u09a5\u09be\u0995\u09be \u0989\u0987\u0995\u09c7\u099f \u09b0\u0995\u09cd\u09b7\u0995\u09c7\u09b0 \u0995\u09be\u099b\u09c7 \u09aa\u09cc\u0981\u099b\u09be\u09a4\u09c7 \u099a\u09be\u0987\u09b2\u09c7, \u09a8\u09cd\u09af\u09c2\u09a8\u09a4\u09ae \u0995\u09a4 \\mathrm{km\/h} <\/span> \u0997\u09a4\u09bf\u09a4\u09c7 \u09ac\u09b2\u099f\u09bf \u099b\u09c1\u0981\u09dc\u09a4\u09c7 \u09b9\u09ac\u09c7? \\\\<\/span>\u098f\u0987 \u0997\u09a4\u09bf\u09a4\u09c7 \u099b\u09c1\u0981\u09dc\u09b2\u09c7 \u0995\u09a4\u0995\u09cd\u09b7\u09a3 \u09aa\u09b0 \u09a4\u09be \u0989\u0987\u0995\u09c7\u099f\u09b0\u0995\u09cd\u09b7\u0995\u09c7\u09b0 \u0995\u09be\u099b\u09c7 \u0997\u09bf\u09df\u09c7 \u09aa\u09cc\u099b\u09be\u09ac\u09c7?
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983 \"\"<\/strong><\/p>\n

\u09a6\u09c7\u0993\u09af\u09bc\u09be \u0986\u099b\u09c7, \\mathrm{R = 100 ~m} <\/span>
\n\u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf, \\mathrm{R}=\\frac{\\mathrm{v}_{0}^{2} \\sin 2 \\theta_{0}}{\\mathrm{~g}} \\Rightarrow \\mathrm{v}_{0}^{2}=\\mathrm{Rg} \\operatorname{cosec} 2 \\theta_{0} \\Rightarrow \\mathrm{v}_{0}=\\sqrt{\\mathrm{Rg} \\operatorname{cosec} 2 \\theta_{0}} \\ldots \\ldots \\ldots <\/span> (i)
\n\u098f\u0996\u0928 (i) \u09a8\u0982 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a5\u09c7\u0995\u09c7,
\n \\left(\\mathrm{v}_{0}\\right)_{\\min }=\\sqrt{\\mathrm{Rg}} <\/span> \u09af\u0996\u09a8 \\operatorname{cosec} 2 \\theta_{0}=1 \\Rightarrow \\sin 2 \\theta_{0}=1 \\Rightarrow 2 \\theta_{0}=\\frac{\\pi}{2} \\Rightarrow \\theta_{0}=\\frac{\\pi}{4} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09a8\u09cd\u09af\u09c2\u09a8\u09a4\u09ae \u09ac\u09c7\u0997 \\left(\\mathrm{v}_{0}\\right)_{\\min }=\\sqrt{100 \\times 9.8}=14 \\sqrt{5} \\mathrm{~ms}^{-1}=14 \\sqrt{5} \\times \\frac{3600}{100} \\mathrm{kmh}^{-1} <\/span>
\n =112.7 \\mathrm{kmh}^{-1} <\/span> (Ans.)
\n\u09aa\u09cd\u09b0\u09df\u09cb\u099c\u09a8\u09c0\u09df \u09b8\u09ae\u09af\u09bc, \\mathrm{T}=\\frac{2 \\mathrm{v}_{0} \\sin \\theta_{0}}{\\mathrm{~g}}=\\frac{2 \\times 14 \\sqrt{5} \\times \\sin \\left(\\frac{\\pi}{4}\\right)}{9.8}=4.52 \\mathrm{~s} <\/span> (Ans.)<\/p>\n

22. \u09e7\u099f\u09bf \u0995\u09cd\u09b0\u09c7\u09a8 \u09aa\u09cd\u09b0\u09a4\u09bf\u099f\u09bf \\mathrm{50~kg}<\/span> \u0993\u099c\u09a8\u09c7\u09b0 \\mathrm{12}<\/span> \u099f\u09bf \u09b8\u09bf\u09ae\u09c7\u09a8\u09cd\u099f\u09c7\u09b0 \u09ac\u09cd\u09af\u09be\u0997 \u09b8\u09ae\u09a6\u09cd\u09b0\u09c1\u09a4\u09bf\u09a4\u09c7 \\mathrm{160 ~m}<\/span> \u0989\u0981\u099a\u09c1 \u098f\u0995\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09ae\u09be\u09a8\u09be\u09a7\u09c0\u09a8 \u09ad\u09ac\u09a8\u09c7\u09b0 \u099b\u09be\u09a6\u09c7 \u0993\u09a0\u09be\u09a4\u09c7 \\mathrm{1 ~min~ 10 ~sec}<\/span> \u09b8\u09ae\u09df \u09a8\u09c7\u09df\u0964 \u0995\u09cd\u09b0\u09c7\u09a8\u099f\u09bf\u09b0 \u0995\u09cd\u09b7\u09ae\u09a4\u09be \u0985\u09b6\u09cd\u09ac\u09b6\u0995\u09cd\u09a4\u09bf\u09a4\u09c7 \u09ac\u09c7\u09b0 \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf, \u0995\u09cd\u09b7\u09ae\u09a4\u09be \\mathrm{P=FV} <\/span>
\n\u098f\u0996\u09be\u09a8\u09c7, \\mathrm{F}=\\mathrm{Nmg}=12 \\times 50 \\times 9.8 \\mathrm{~N}=5880 \\mathrm{~N} <\/span>
\n \\mathrm{v}=\\frac{\\mathrm{s}}{\\mathrm{t}}=\\frac{160 \\mathrm{~m}}{70 \\mathrm{~s}}=\\frac{16}{7} \\mathrm{~ms}^{-1} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u09a8\u09bf\u09b0\u09cd\u09a3\u09c7\u09df \u0995\u09cd\u09b7\u09ae\u09a4\u09be, \\mathrm{P}=\\mathrm{Fv}=5880 \\times \\frac{16}{7} \\mathrm{~W} <\/span>
\n =13440 \\mathrm{~W}=\\frac{13440}{746} \\mathrm{HP}=18.02 \\mathrm{HP} <\/span><\/p>\n

23. 80 \\mathrm{~kg} <\/span> \u0993\u099c\u09a8\u09c7\u09b0 \u098f\u0995\u099f\u09bf \u0995\u09c3\u09a4\u09cd\u09b0\u09bf\u09ae \u0989\u09aa\u0997\u09cd\u09b0\u09b9 \u09ad\u09c2\u09aa\u09c3\u09b7\u09cd\u09a0 \u09a5\u09c7\u0995\u09c7 \u0995\u09a4 \u0989\u099a\u09cd\u099a\u09a4\u09be\u09af\u09bc \u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u0995\u09b0\u09b2\u09c7 \u09a4\u09be \u09aa\u09cd\u09b0\u09a4\u09bf 24 \u0998\u09a8\u09cd\u099f\u09be\u09af\u09bc 2 \u09ac\u09be\u09b0 \u098f\u0995\u0987 \u09b8\u09cd\u09a5\u09be\u09a8 \u09aa\u09b0\u09cd\u09af\u09ac\u09c7\u0995\u09cd\u09b7\u09a3 \u0995\u09b0\u09a4\u09c7 \u09aa\u09be\u09b0\u09ac\u09c7? [\u09aa\u09c3\u09a5\u09bf\u09ac\u09c0\u09b0 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7 6400 \\mathrm{~km} <\/span> \u0993 \u09a4\u09be\u09b0 \u09ad\u09b0 6 \\times 10^{21}\\mathrm{~Ton} <\/span> ]
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u09a6\u09c7\u0993\u09df\u09be \u0986\u099b\u09c7, \u09aa\u09c3\u09a5\u09bf\u09ac\u09c0\u09b0 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7, \\mathrm{R}=6400 \\mathrm{~km}=6.4 \\times 10^{6} \\mathrm{~m} <\/span>
\n\u09aa\u09c3\u09a5\u09bf\u09ac\u09c0\u09b0 \u09ad\u09b0 \\mathrm{M=6 \\times 10^{21}~ Ton =6 \\times 10^{21} \\times 10^{3} ~kg=6 \\times 10^{24} ~kg}<\/span>
\n\u0995\u09c3\u09a4\u09cd\u09b0\u09bf\u09ae \u0989\u09aa\u0997\u09cd\u09b0\u09b9\u09c7\u09b0 \u09aa\u09b0\u09cd\u09af\u09be\u09df\u0995\u09be\u09b2, \\mathrm{T=\\frac{24}{2} ~hours =12 ~hours =12 \\times 3600 ~s=43200 ~s}<\/span>
\n\u09a7\u09b0\u09bf, \u0995\u09c3\u09a4\u09cd\u09b0\u09bf\u09ae \u0989\u09aa\u0997\u09cd\u09b0\u09b9\u09c7\u09b0 \u0989\u099a\u09cd\u099a\u09a4\u09be =\\mathrm{h}<\/span>
\n\u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf,
\n \\mathrm{T}^{2}=\\frac{4 \\pi^{2}}{\\mathrm{GM}}(\\mathrm{R}+\\mathrm{h})^{3} \\Rightarrow \\frac{\\mathrm{T}^{2} \\mathrm{GM}}{4 \\pi^{2}}=(\\mathrm{R}+\\mathrm{h})^{3} \\Rightarrow \\mathrm{R}+\\mathrm{h}=\\left(\\frac{\\mathrm{T}^{2} \\mathrm{GM}}{4 \\pi^{2}}\\right)^{\\frac{1}{3}} <\/span>
\n \\Rightarrow \\mathrm{h}=\\left(\\frac{\\mathrm{T}^{2} \\mathrm{GM}}{4 \\pi^{2}}\\right)^{\\frac{1}{3}}-\\mathrm{R}=\\left\\{\\frac{(43200)^{2} \\times 6.673 \\times 10^{-11} \\times 6 \\times 10^{24}}{4 \\times \\pi^{2}}\\right\\}^{\\frac{1}{3}}-6.4 \\times 10^{6} \\mathrm{~m}=20.24976 \\times 10^{6} \\mathrm{~m} <\/span> (Ans.)<\/p>\n

24. \u098f\u0995\u099f\u09bf \u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1 \u09a6\u09cb\u09b2\u0995 \u0998\u09a1\u09bc\u09bf \u09aa\u09be\u09b9\u09be\u09a1\u09bc\u09c7\u09b0 \u09aa\u09be\u09a6\u09a6\u09c7\u09b6\u09c7 \u09a0\u09bf\u0995 \u09b8\u09ae\u09af\u09bc \u09a6\u09c7\u09af\u09bc \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 \u09aa\u09be\u09b9\u09be\u09a1\u09bc\u09c7\u09b0 \u099a\u09c2\u09a1\u09bc\u09be\u09af\u09bc \u0989\u09a0\u09be\u09b2\u09c7 2 \u0998\u09a8\u09cd\u099f\u09be\u09af\u09bc 8 \u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1 \u09b8\u09ae\u09af\u09bc\u09c7\u09b0 \u09aa\u09be\u09b0\u09cd\u09a5\u0995\u09cd\u09af \u09a6\u09c7\u0996\u09be\u09af\u09bc\u0964 \u09aa\u09c3\u09a5\u09bf\u09ac\u09c0\u09b0 \u09ac\u09cd\u09af\u09be\u09b8 12800 \\mathrm{~km} <\/span> \u09b9\u09b2\u09c7-
\n(i) \u09aa\u09be\u09b9\u09be\u09a1\u09bc\u09c7\u09b0 \u0989\u099a\u09cd\u099a\u09a4\u09be \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964
\n(ii) \u09aa\u09be\u09b9\u09be\u09a1\u09bc\u09c7\u09b0 \u099a\u09c2\u09a1\u09bc\u09be\u09af\u09bc \u09b8\u09a0\u09bf\u0995\u09ad\u09be\u09ac\u09c7 \u0995\u09be\u099c \u0995\u09b0\u09a4\u09c7 \u09b9\u09b2\u09c7 \u09a6\u09cb\u09b2\u0995\u09c7\u09b0 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af \u0995\u09a4 \\% <\/span> \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7?
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong>
\n(i) \u09aa\u09cd\u09b0\u09b6\u09cd\u09a8\u09ae\u09a4\u09c7, \u09aa\u09be\u09b9\u09be\u09a1\u09bc\u09c7\u09b0 \u0989\u09aa\u09b0 \u0998\u09a1\u09bc\u09bf 2 \u0998\u09a8\u09cd\u099f\u09be\u09af\u09bc 8s \u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1 \u09a7\u09c0\u09b0\u09c7 \u099a\u09b2\u09c7\u0964
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982, 24 \u0998\u09a8\u09cd\u099f\u09be\u09af\u09bc \u09a7\u09c0\u09b0\u09c7 \u099a\u09b2\u09c7 \\frac{8 \\times 24}{2}=96 <\/span> \u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1\u0964\\\\<\/span>\u098f\u0996\u09a8,
\n\u09aa\u09be\u09b9\u09be\u09dc\u09c7\u09b0 \u09aa\u09be\u09a6\u09a6\u09c7\u09b6\u09c7 \u09a6\u09cb\u09b2\u09a8\u0995\u09be\u09b2 \\mathrm{T} <\/span> \u098f\u09ac\u0982 \u099a\u09c2\u09a1\u09bc\u09be\u09af\u09bc \u09a6\u09cb\u09b2\u09a8\u0995\u09be\u09b2 \\mathrm{T}^{\\prime} <\/span> \u09b9\u09b2\u09c7, \\frac{\\mathrm{T}^{\\prime}}{\\mathrm{T}}=\\frac{86400}{86400-96}=\\frac{900}{899} <\/span> \\\\<\/span>\u09aa\u09be\u09b9\u09be\u09dc\u09c7\u09b0 \u0989\u099a\u09cd\u099a\u09a4\u09be \\mathrm{h} <\/span>, \u09aa\u09be\u09a6\u09a6\u09c7\u09b6\u09c7 \u0985\u09ad\u09bf\u0995\u09b0\u09cd\u09b7\u099c \u09a4\u09cd\u09ac\u09b0\u09a3 \\mathrm{g} <\/span> \u098f\u09ac\u0982 \u099a\u09c2\u09dc\u09be\u09df \u0985\u09ad\u09bf\u0995\u09b0\u09cd\u09b7\u099c \u09a4\u09cd\u09ac\u09b0\u09a3 \\mathrm{g}^{\\prime} <\/span> \u09b9\u09b2\u09c7,
\n \\frac{\\mathrm{g}^{\\prime}}{\\mathrm{g}}=\\mathrm{(\\frac{R}{R+h})^{2}} <\/span> \\\\<\/span>\u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf, \\\\<\/span> \\frac{\\mathrm{T}^{\\prime}}{\\mathrm{T}}=\\sqrt{\\frac{\\mathrm{g}}{\\mathrm{g}^{\\prime}}}=\\sqrt{\\left(\\frac{\\mathrm{R}+\\mathrm{h}}{\\mathrm{R}}\\right)^{2}}=1+\\frac{\\mathrm{h}}{\\mathrm{R}} \\Rightarrow \\frac{\\mathrm{h}}{\\mathrm{R}}=\\frac{\\mathrm{T}^{\\prime}}{\\mathrm{T}}-1 <\/span>
\n \\Rightarrow \\mathrm{h}=\\mathrm{R} \\times\\left(\\frac{\\mathrm{T}^{\\prime}}{\\mathrm{T}}-1\\right)=\\frac{12800}{2} \\times 10^{3} \\times\\left(\\frac{900}{899}-1\\right)=7119.02 \\mathrm{~m} <\/span> (Ans.)<\/p>\n

(ii) \u09aa\u09be\u09a6\u09a6\u09c7\u09b6\u09c7 \u09a6\u09cb\u09b2\u0995\u09c7\u09b0 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af l <\/span>
\n\u09aa\u09be\u09b9\u09be\u09a1\u09bc\u09c7\u09b0 \u099a\u09c2\u09a1\u09bc\u09be\u09af\u09bc \u09a6\u09cb\u09b2\u0995\u09c7\u09b0 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af l^{\\prime} <\/span> \u0995\u09b0\u09b2\u09c7 T \u09a6\u09cb\u09b2\u09a8\u0995\u09be\u09b2 \u0985\u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09bf\u09a4 \u09a5\u09be\u0995\u09b0\u09c7\u0964
\n\u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf, \\mathrm{T}=2 \\pi \\sqrt{\\frac{\\mathrm{L}}{\\mathrm{g}}} \\quad <\/span> \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \\mathrm{L} \\propto \\mathrm{g} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \\frac{\\mathrm{l}^{\\prime}}{\\mathrm{l}}=\\frac{\\mathrm{g}^{\\prime}}{\\mathrm{g}} \\Rightarrow \\frac{1-\\mathrm{l}^{\\prime}}{\\mathrm{l}}=1-\\frac{\\mathrm{g}^{\\prime}}{\\mathrm{g}}=1-\\frac{(899)^{2}}{(900)} ~~~~~~~~~~[\\frac{\\mathrm{g}^{\\prime}}{\\mathrm{g}}=\\left(\\frac{\\mathrm{T}}{\\mathrm{T}^{\\prime}}\\right)^{2}=\\left(\\frac{899}{900}\\right)^{2}] <\/span>
\n=1-0.9977790123=2.220987 \\times10^{-3}=2.220987 \\times 10^{-3} \\times100 \\%=0.22209877 \\% <\/span> (Ans.)<\/p>\n

25. 2 \\mathrm{~mm} <\/span> \u09ac\u09cd\u09af\u09be\u09b8\u09c7\u09b0 \u098f\u0995\u099f\u09bf \u0987\u09b8\u09cd\u09aa\u09be\u09a4\u09c7\u09b0 \u09a4\u09be\u09b0\u09c7\u09b0 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af 15 \\% <\/span> \u09ac\u09c3\u09a6\u09cd\u09a7\u09bf \u0995\u09b0\u09a4\u09c7 \u0995\u09a4 \\mathrm{kN} <\/span> \u09ac\u09b2 \u09aa\u09cd\u09b0\u09af\u09bc\u09cb\u0997 \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7? \u098f\u09b0 \u09ab\u09b2\u09c7 \u09a4\u09be\u09b0\u09c7\u09b0 \u09ac\u09cd\u09af\u09be\u09b8\u09c7\u09b0 \u0995\u09a4 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 \u09b9\u09ac\u09c7? [\u0987\u09b8\u09cd\u09aa\u09be\u09a4\u09c7\u09b0 Young’s Modulus 2 \\times 10^{11} \\mathrm{~N} \/ \\mathrm{m}^{2} <\/span> \u098f\u09ac\u0982 Poisson’s ratio is 0.25 <\/span> ]
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983 <\/strong>\u09a6\u09c7\u0993\u09df\u09be \u0986\u099b\u09c7,
\n\n\n\\begin{array}{l}\n\n\\mathrm{Y=2\\times 10^{11}~Nm^{-2}},\\sigma=0.25, \\frac{l}{L}=15 \\%=0.15, d=2 \\times 10^{-3}, \\mathrm{~m}=2 \\mathrm{~mm}, \\mathrm{r}=\\frac{\\mathrm{d}}{2}=1 \\times 10^{-3} \\mathrm{~m}\n\n\\end{array}\n\n<\/span><\/p>\n

\u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf, F = \\frac{\\text { YAI }}{\\text { L }} <\/span>
\n =2 \\times 10^{11} \\times \\pi \\times\\left(1 \\times 10^{-3}\\right)^{2} \\times0.15=94247.78 \\mathrm{~N} <\/span> (Ans.)<\/p>\n

\u0986\u09ac\u09be\u09b0, \u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf, \\sigma=\\frac{\\Delta \\mathrm{d} \\times \\mathrm{L}}{\\mathrm{d \\times l}} <\/span>
\n \\Rightarrow \\Delta \\mathrm{d}=\\sigma \\times \\mathrm{d} \\times \\frac{l}{\\mathrm{~L}}=0.25 \\times 2 \\times 10^{-3} \\times 0.15 <\/span>
\n \\Rightarrow \\Delta \\mathrm{d}=7.5 \\times 10^{-5} \\mathrm{~m} <\/span>
\n \\therefore<\/span> \u09a4\u09be\u09b0\u09c7\u09b0 \u09ac\u09cd\u09af\u09be\u09b8 \u0995\u09ae\u09c7 \u09af\u09be\u09ac\u09c7, \\Delta \\mathrm{d}=7.5 \\times 10^{-5} \\mathrm{~m} <\/span> (Ans.)<\/p>\n

26. \u09aa\u09be\u09a8\u09bf\u09b0 \u0997\u09ad\u09c0\u09b0\u09a4\u09be \u09ae\u09be\u09aa\u09be\u09b0 \u099c\u09a8\u09cd\u09af, \u098f\u0995\u099f\u09bf \u099c\u09b2\u09be\u09b6\u09af\u09bc\u09c7\u09b0 \u09aa\u09be\u09a8\u09bf\u09b0 \u09aa\u09c3\u09b7\u09cd\u09a0 \u09a5\u09c7\u0995\u09c7 0.005 \\mathrm{~m} <\/span> \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7\u09c7\u09b0 \u098f\u09ac\u0982 2.5 \\times 10^{3} \\mathrm{~kg} \/ \\mathrm{m}^{3} <\/span> \u0998\u09a8\u09a4\u09cd\u09ac\u09c7\u09b0 \u098f\u0995\u099f\u09bf \u09ac\u09b2 \u099b\u09c7\u09a1\u09bc\u09c7 \u09a6\u09c7\u09af\u09bc\u09be \u09b9\u09b2\u0964 [\/katex] 10 \\mathrm{~s} [\/katex] \u09aa\u09b0 \u09ac\u09b2\u099f\u09bf \u099c\u09b2\u09be\u09b6\u09af\u09bc\u09c7\u09b0 \u09a4\u09b2\u09be\u09af\u09bc \u09aa\u09a1\u09bc\u09b2\u0964 \u09af\u09a6\u09bf 9 \\mathrm{~s} <\/span> \u098f \u09ac\u09b2\u099f\u09bf \u09aa\u09cd\u09b0\u09be\u09a8\u09cd\u09a4\u09bf\u0995 \u09ac\u09c7\u0997 \u0985\u09b0\u09cd\u099c\u09a8 \u0995\u09b0\u09c7 \u09a5\u09be\u0995\u09c7, \u09a4\u09be\u09b9\u09b2\u09c7 \u0997\u09ad\u09c0\u09b0\u09a4\u09be \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964 [\u09aa\u09be\u09a8\u09bf\u09b0 \u09b8\u09be\u09a8\u09cd\u09a6\u09cd\u09b0\u09a4\u09be \\eta=1.6 \\times 10^{-3} \\mathrm{Nsm}^{-2} <\/span> \u098f\u09ac\u0982 \u0998\u09a8\u09a4\u09cd\u09ac 1000 \\mathrm{kgm}^{-3} <\/span> ]
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983 \"\"<\/strong><\/p>\n

\u09a6\u09c7\u0993\u09df\u09be \u0986\u099b\u09c7, \\\\\n\n\\begin{array}{l}\n\n\\mathrm{r}=0.005 \\mathrm{~m}, \\rho_{\\mathrm{s}}=2.5 \\times 10^{3} \\mathrm{~kg~m}^{-3}, \\mathrm{t}=10 \\mathrm{~s}, \\rho_{\\mathrm{f}}=1000 \\mathrm{kgm}^{-3}, \\eta=1.6 \\times 10^{-3} \\mathrm{Nsm}^{-2}, V=\\frac{4}{3} \\pi r^{3}\n\n\\end{array}\n\n<\/span>\\\\<\/span>\u09ac\u09b2\u09c7\u09b0 \u0989\u09aa\u09b0 \u09aa\u09cd\u09b0\u09af\u09c1\u0995\u09cd\u09a4 \u09a8\u09c0\u099f \u09ac\u09b2, \\mathrm{F}=\\mathrm{V}\\left(\\rho_{\\mathrm{s}}-\\rho_{\\mathrm{f}}\\right)-6 \\pi \\eta \\mathrm{rv} <\/span>
\n \\therefore \\mathrm{F}=\\mathrm{m} \\frac{\\mathrm{dv}}{\\mathrm{dt}} \\Rightarrow \\mathrm{V}\\left(\\rho_{\\mathrm{s}}-\\rho_{\\mathrm{f}}\\right) \\mathrm{g-6} \\pi \\eta \\mathrm{rv}=\\mathrm{m} \\frac{\\mathrm{dv}}{\\mathrm{dt}} <\/span>
\n \\Rightarrow \\frac{4}{3} \\pi \\mathrm{r^{3}}\\left(\\rho_{s}-\\rho_{f}\\right) \\mathrm{g-6} \\pi \\eta \\mathrm{r v}=\\frac{4}{3} \\pi \\mathrm{r^{3}} \\rho_{s}\\mathrm{ \\frac{d v}{d t}} \\Rightarrow \\mathrm{v}=\\mathrm{\\frac{2 r^{2}\\left(\\rho_{s}-\\rho_{f}\\right) g}{9 \\eta}-\\frac{2 r^{2} \\rho_{s}}{9 \\eta} \\frac{d v}{d t}} <\/span>
\n \\Rightarrow \\mathrm{v}=\\mathrm{v}_{\\mathrm{T}}-\\frac{2 \\mathrm{r}^{2} \\rho_{\\mathrm{s}}}{9 \\eta} \\frac{\\mathrm{dv}}{\\mathrm{dt}} \\ldots \\ldots \\ldots <\/span> (i) [\u09af\u09c7\u0996\u09be\u09a8\u09c7, \\mathrm{v}_{\\mathrm{T}}=\\frac{2 \\mathrm{r}^{2}(\\rho_{\\mathrm{s}}-\\rho_{\\mathrm{f}}) \\mathrm{g}}{9 \\eta}] <\/span>
\n\u098f\u0996\u09a8, \\mathrm{ds}=\\mathrm{vdt} \\Rightarrow \\int_{0}^{\\mathrm{s}} \\mathrm{ds}=\\int_{0}^{10} \\mathrm{vdt} \\Rightarrow[\\mathrm{s}]_{0}^{\\mathrm{s}}=\\int_{0}^{10}\\left(\\mathrm{v}_{\\mathrm{T}}-\\frac{2 \\mathrm{r}^{2} \\rho_{\\mathrm{s}}}{9 \\eta} \\frac{\\mathrm{dv}}{\\mathrm{dt}}\\right) \\mathrm{dt} <\/span>
\n \\Rightarrow(\\mathrm{s}-0)=\\int_{0}^{10} \\mathrm{v}_{\\mathrm{T}} \\mathrm{dt}-\\frac{2 \\mathrm{r}^{2} \\rho_{\\mathrm{s}}}{9 \\eta} \\int_{0}^{\\mathrm{v}_{\\mathrm{T}}} \\mathrm{dv} \\quad\\left[\\mathrm{t}=0\\right. <\/span> \u09b8\u09ae\u09df\u09c7 \\mathrm{v}=0 ; \\mathrm{t}=10 <\/span> \u09b8\u09ae\u09af\u09bc\u09c7 \\left.\\mathrm{v}=\\mathrm{v}_{\\mathrm{T}}\\right] <\/span>
\n \\Rightarrow \\mathrm{s}=\\mathrm{v}_{\\mathrm{T}}[\\mathrm{t}]_{0}^{10}-\\frac{2 \\mathrm{r}^{2} \\rho_{\\mathrm{s}}}{9 \\eta} \\cdot[\\mathrm{v}]_{0}^{\\mathrm{v}_{\\mathrm{T}}} \\Rightarrow \\mathrm{s}=\\mathrm{v}_{\\mathrm{T}} \\times 10-\\frac{2 \\mathrm{r}^{2} \\rho_{\\mathrm{s}}}{9 \\eta} \\cdot \\mathrm{v}_{\\mathrm{T}} <\/span>
\n =\\mathrm{v}_{\\mathrm{T}} \\cdot\\left(10-\\frac{2 \\mathrm{r}^{2} \\rho_{\\mathrm{s}}}{9 \\eta}\\right)=\\frac{2 \\times \\mathrm{r}^{2}\\left(\\rho_{\\mathrm{s}}-\\rho_{\\mathrm{f}}\\right) \\mathrm{g}}{9 \\eta}\\left(10-\\frac{2 \\mathrm{r}^{2} \\rho_{\\mathrm{s}}}{9 \\eta}\\right) <\/span>
\n =\\frac{2 \\times(0.005)^{2} \\times(2500-1000) \\times 9.8}{9 \\times 1.6 \\times 10^{-3}}\\left\\{10-\\frac{2 \\times(0.005)^{2} \\times 2500}{9 \\times 1.6 \\times 10^{-3}}\\right\\} <\/span>
\n =51.04 \\times(10-8.68)=67.344 \\mathrm{~m} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u09b9\u09cd\u09b0\u09a6\u09c7\u09b0 \u0997\u09ad\u09c0\u09b0\u09a4\u09be =67.344 \\mathrm{~m} <\/span><\/p>\n

27. \u098f\u0995\u099f\u09bf 8 \\mathrm{~kg} <\/span> \u09ad\u09b0\u09c7\u09b0 \u099a\u09be\u0995\u09be\u09b0 \u099a\u0995\u09cd\u09b0\u0997\u09a4\u09bf\u09b0 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7 25 \\mathrm{~cm} <\/span> \u09b9\u09b2\u09c7 \u098f\u09b0 \u099c\u09a1\u09bc\u09a4\u09be\u09b0 \u09ad\u09cd\u09b0\u09be\u09ae\u0995 \u0995\u09a4 \u09b9\u09ac\u09c7? \u099a\u09be\u0995\u09be\u099f\u09bf\u09a4\u09c7 3 ~\\mathrm{rad} \/ \\mathrm{s}^{2} <\/span> \u0995\u09cc\u09a3\u09bf\u0995 \u09a4\u09cd\u09ac\u09b0\u09a3 \u09b8\u09c3\u09b7\u09cd\u099f\u09bf \u0995\u09b0\u09a4\u09c7 \u0995\u09a4 \u09ae\u09be\u09a8\u09c7\u09b0 \u099f\u09b0\u09cd\u0995 \u09aa\u09cd\u09b0\u09af\u09bc\u09cb\u0997 \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7?
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u09a6\u09c7\u0993\u09af\u09bc\u09be \u0986\u099b\u09c7, \u099a\u09be\u0995\u09be\u09b0 \u09ad\u09b0 \\mathrm{m}=8 \\mathrm{~kg} <\/span>
\n\u099a\u0995\u09cd\u09b0\u0997\u09a4\u09bf\u09b0 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7 \\mathrm{k}=25 \\mathrm{~cm}=0.25 \\mathrm{~m} <\/span>
\n\u099c\u09a1\u09bc\u09a4\u09be\u09b0 \u09ad\u09cd\u09b0\u09be\u09ae\u0995 \\mathrm{I}=\\mathrm{mk}^{2}=8 \\times(0.25)^{2}=0.5 \\mathrm{~kgm}^{2} <\/span>
\n\u0995\u09cc\u09a3\u09bf\u0995 \u09a4\u09cd\u09ac\u09b0\u09a3 \\alpha=3 ~\\mathrm{rads}^{-2} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u09aa\u09cd\u09b0\u09af\u09c1\u0995\u09cd\u09a4 \u099f\u09b0\u09cd\u0995 \\tau=\\mathrm{I} ~\\alpha=0.5 \\times 3=1.5 ~\\mathrm{Nm} <\/span> (Ans.)<\/p>\n

28. \u0995\u09a4 \u09a1\u09bf\u0997\u09cd\u09b0\u09c0 \u09b8\u09c7\u09b2\u09b8\u09bf\u09af\u09bc\u09be\u09b8 \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09af\u09bc \u0985\u0995\u09cd\u09b8\u09bf\u099c\u09c7\u09a8 \u0985\u09a3\u09c1\u09b0 \u09ae\u09c2\u09b2 \u0997\u09a1\u09bc \u09ac\u09b0\u09cd\u0997\u09ac\u09c7\u0997 -100^{\\circ} \\mathrm{C} <\/span> \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09b0 \u09b9\u09be\u0987\u09a1\u09cd\u09b0\u09cb\u099c\u09c7\u09a8 \u0985\u09a3\u09c1\u09b0 \u09ae\u09c2\u09b2 \u0997\u09a1\u09bc \u09b8\u09ae\u09be\u09a8 \u09b9\u09ac\u09c7?
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf, \\mathrm{C}_{\\mathrm{rms}}=\\sqrt{\\frac{3 \\mathrm{RT}}{\\mathrm{M}}} \\Rightarrow \\mathrm{M}=\\frac{3 \\mathrm{R}}{\\mathrm{C}_{\\mathrm{rms}}^{2}} \\mathrm{~T} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \\mathrm{M} \\propto \\mathrm{T} <\/span>
\n\u098f\u0996\u09a8, \\mathrm{M}_{\\mathrm{O}_{2}}=32 \\mathrm{~g}, \\mathrm{~T}_{\\mathrm{O}_{2}}=?, \\mathrm{M}_{\\mathrm{H}_{2}}=2 \\mathrm{~g} <\/span>
\n \\mathrm{T}_{\\mathrm{H}_{2}}=-100^{\\circ} \\mathrm{C}=-100+273=173 \\mathrm{~K} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \\frac{\\mathrm{M}_{\\mathrm{O}_{2}}}{\\mathrm{~T}_{\\mathrm{O}_{2}}}=\\frac{\\mathrm{M}_{\\mathrm{H}_{2}}}{\\mathrm{~T}_{\\mathrm{H}_{2}}} \\Rightarrow \\mathrm{T}_{\\mathrm{O}_{2}}=\\frac{\\mathrm{M}_{\\mathrm{O}_{2}}}{\\mathrm{M}_{\\mathrm{H}_{2}}} \\times \\mathrm{T}_{\\mathrm{H}_{2}} \\Rightarrow \\mathrm{T}_{\\mathrm{O}_{2}}=\\frac{32}{2} \\times 173 \\Rightarrow \\mathrm{T}_{\\mathrm{O}_{2}}=2768 \\mathrm{~K} <\/span> (Ans.)<\/p>\n

29. \u0995\u09cb\u09a8 \u09b8\u09cd\u09a5\u09be\u09a8\u09c7\u09b0 \u09ac\u09be\u09df\u09c1\u09b0 \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be 26^{\\circ} \\mathrm{C} <\/span> \u098f\u09ac\u0982 \u0986\u09aa\u09c7\u0995\u09cd\u09b7\u09bf\u0995 \u0986\u09b0\u09cd\u09a6\u09cd\u09b0\u09a4\u09be 70 \\% <\/span> \u0964 \u09af\u09a6\u09bf \u09b8\u09c7 \u09b8\u09cd\u09a5\u09be\u09a8\u09c7\u09b0 \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be \u0995\u09ae\u09c7 18^{\\circ} \\mathrm{C} <\/span> \u09b9\u09af\u09bc, \u09a4\u09ac\u09c7 \u09ac\u09be\u09df\u09c1\u09b8\u09cd\u09a5\u09bf\u09a4 \u099c\u09b2\u09c0\u09df\u09ac\u09be\u09b7\u09cd\u09aa\u09c7\u09b0 \u0995\u09a4 \u09b6\u09a4\u09be\u0982\u09b6 \u0998\u09a8\u09c0\u09ad\u09c2\u09a4 \u09b9\u09df\u09c7 \u09a4\u09b0\u09b2 \u09aa\u09be\u09a8\u09bf \u09b9\u09ac\u09c7? [ 26^{\\circ} \\mathrm{C} <\/span> \u098f\u09ac\u0982 18^{\\circ} \\mathrm{C} <\/span> \u098f \u09b8\u09ae\u09cd\u09aa\u09c3\u0995\u09cd\u09a4 \u099c\u09b2\u09c0\u09df\u09ac\u09be\u09b7\u09cd\u09aa\u09c7\u09b0 \u099a\u09be\u09aa \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 25.21~\\mathrm{mm} <\/span> \u098f\u09ac\u0982 15.48 \\mathrm{~mm} <\/span> \u09aa\u09be\u09b0\u09a6\u09b8\u09cd\u09a4\u09ae\u09cd\u09ad\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8]
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf \u0986\u09aa\u09c7\u0995\u09cd\u09b7\u09bf\u0995 \u0985\u09b0\u09cd\u09a6\u09cd\u09b0\u09a4\u09be R =\\frac{f}{F} \\times 100 \\% <\/span><\/p>\n

\u09a6\u09c7\u0993\u09af\u09bc\u09be \u0986\u099b\u09c7, 26^{\\circ} \\mathrm{C} <\/span> \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09af\u09bc \u09b8\u09ae\u09cd\u09aa\u09c3\u0995\u09cd\u09a4 \u099c\u09b2\u09c0\u09df\u09ac\u09be\u09b7\u09cd\u09aa\u09c7\u09b0 \u099a\u09be\u09aa =25.21 \\mathrm{~mm} <\/span>
\n 26^{\\circ} \\mathrm{C} <\/span> \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09af\u09bc \u09ac\u09b0\u09cd\u09a4\u09ae\u09be\u09a8 \u099c\u09b2\u09c0\u09df\u09ac\u09be\u09b7\u09cd\u09aa\u09c7\u09b0 \u099a\u09be\u09aa =\\mathrm{f}_{\\mathrm{i}} <\/span><\/p>\n

\u098f\u0996\u09a8, \\mathrm{R}=\\frac{\\mathrm{f}_{\\mathrm{i}}}{\\mathrm{F}} \\times 100 \\% \\Rightarrow \\mathrm{f}_{\\mathrm{i}}=\\frac{70}{100} \\times 25.21=17.647 \\mathrm{~mm} <\/span><\/p>\n

\u098f\u0996\u09a8 18^{\\circ} \\mathrm{C} <\/span> \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09af\u09bc \u09b8\u09ae\u09cd\u09aa\u09c3\u0995\u09cd\u09a4 \u099c\u09b2\u09c0\u09df\u09ac\u09be\u09b7\u09cd\u09aa\u09c7\u09b0 \u099a\u09be\u09aa \\mathrm{f}_{\\mathrm{f}}=15.48 \\mathrm{~mm} <\/span>
\n\u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09ac\u09b0\u09cd\u09a4\u09ae\u09be\u09a8 \u09ac\u09be\u09b7\u09cd\u09aa\u099a\u09be\u09aa > \u09b8\u09ae\u09cd\u09aa\u09c3\u0995\u09cd\u09a4 \u09ac\u09be\u09b7\u09cd\u09aa\u099a\u09be\u09aa\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u0995\u09bf\u099b\u09c1 \u09ac\u09be\u09b7\u09cd\u09aa \u0998\u09a8\u09c0\u09a4\u09c2\u09a4 \u09b9\u09ac\u09c7\u0964
\n\u098f\u0996\u09a8 \u099c\u09b2\u09c0\u09df\u09ac\u09be\u09b7\u09cd\u09aa\u09c7\u09b0 \u09ad\u09b0 \u09ac\u09be\u09b7\u09cd\u09aa\u099a\u09be\u09aa\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8\u09c1\u09aa\u09be\u09a4\u09bf\u0995\u0964<\/p>\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u0998\u09a8\u09c0\u09ad\u09c2\u09a4 \u09b9\u09ac\u09c7 =\\frac{\\mathrm{f}_{\\mathrm{i}}-\\mathrm{f}_{\\mathrm{f}}}{\\mathrm{f}_{\\mathrm{i}}}=\\frac{17.647-15.48}{17.647} \\times 100 \\%=12.28 \\% <\/span> (Ans.)<\/p>\n

30. \u09b8\u09c2\u09b0\u09cd\u09af\u09c7\u09b0 \u0986\u09b2\u09cb\u09a4\u09c7 \u098f\u0995\u099f\u09bf \u0989\u09a4\u09cd\u09a4\u09b2 \u09b2\u09c7\u09a8\u09cd\u09b8 \u09b0\u09c7\u0996\u09c7 \u09b2\u09c7\u09a8\u09cd\u09b8 \u09a5\u09c7\u0995\u09c7 30 \\mathrm{~cm} <\/span> \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac\u09c7 \u098f\u0995\u099f\u09bf \u09aa\u09b0\u09cd\u09a6\u09be\u09df \u09b8\u09ac\u099a\u09c7\u09df\u09c7 \u09b8\u09cd\u09aa\u09b7\u09cd\u099f \u0993 \u0989\u099c\u09cd\u099c\u09cd\u09ac\u09b2 \u0986\u09b2\u09cb\u09b0 \u09b8\u09cd\u09aa\u099f \u09aa\u09be\u0993\u09df\u09be \u0997\u09c7\u09b2\u0964 \u09b2\u09c7\u09a8\u09cd\u09b8\u099f\u09bf\u09b0 \u09aa\u09cd\u09b0\u09a4\u09cd\u09af\u09c7\u0995 \u09aa\u09c3\u09b7\u09cd\u09a0\u09c7\u09b0 \u09ac\u0995\u09cd\u09b0\u09a4\u09be\u09b0 \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7 30 \\mathrm{~cm} <\/span> \u09b9\u09b2\u09c7 \u09aa\u09be\u09a8\u09bf\u09a4\u09c7 \u09a4\u09be\u09b0 \u0995\u09cd\u09b7\u09ae\u09a4\u09be \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u0964 [\u09aa\u09be\u09a8\u09bf\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u09b8\u09b0\u09be\u0982\u0995 4\/3]
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong>
\n\u09b8\u09c2\u09b0\u09cd\u09af \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u0985\u09b8\u09c0\u09ae \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac \u09a5\u09c7\u0995\u09c7 \u0986\u09b2\u09cb\u0995 \u09b0\u09b6\u09cd\u09ae\u09bf \u09b2\u09c7\u09a8\u09cd\u09b8\u09c7 \u0986\u09aa\u09a4\u09bf\u09a4 \u09b9\u09b2\u09c7 \u09b2\u09c7\u09a8\u09cd\u09b8\u09c7\u09b0 \u09ab\u09cb\u0995\u09be\u09b8 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac\u09c7 \u09aa\u09cd\u09b0\u09a4\u09bf\u09ac\u09bf\u09ae\u09cd\u09ac \u0997\u09a0\u09bf\u09a4 \u09b9\u09df\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u09ac\u09be\u09af\u09bc\u09c1\u09a4\u09c7 \u09b2\u09c7\u09a8\u09cd\u09b8\u09c7\u09b0 \u09ab\u09cb\u0995\u09be\u09b8 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac, \\mathrm{f}_{\\mathrm{a}}=0.3 \\mathrm{~m} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \\mathrm{p}_{\\mathrm{a}}=\\frac{1}{\\mathrm{f}_{\\mathrm{a}}}=\\frac{10}{3} \\mathrm{D} <\/span>
\n\u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf,
\n \\frac{1}{\\mathrm{f}_{\\mathrm{a}}}=\\left(\\frac{\\mu_{\\mathrm{g}}}{\\mu_{\\mathrm{a}}}-1\\right)\\left(\\frac{1}{r_{1}}-\\frac{1}{r_{2}}\\right) \\Rightarrow \\frac{1}{0.3}=\\left(\\frac{\\mu_{\\mathrm{g}}}{1}-1\\right)\\left(\\frac{1}{0.3}-\\frac{1}{-0.3}\\right) <\/span>
\n\n\n\\begin{array}{l}\n\n\\Rightarrow \\frac{1}{0.3}=\\left(\\mu_{g}-1\\right) \\frac{2}{0.3} \\Rightarrow \\mu_{g}=\\frac{1}{2}+1=\\frac{3}{2} \\\\\n\n\\text { \u098f\u0996\u09a8 } p_{w}=\\frac{1}{f_{w}}=\\left(\\frac{\\mu_{g}}{\\mu_{w}}-1\\right)\\left(\\frac{1}{r_{1}}-\\frac{1}{r_{2}}\\right)=\\left(\\frac{3 \\times 3}{2 \\times 4}-1\\right)\\left(\\frac{1}{0.3}-\\frac{1}{-0.3}\\right) \\\\\n\np_{w}=\\frac{1}{8} \\times \\frac{2}{0.3}=\\frac{5}{6} \\Rightarrow p_{w}=\\frac{5}{6} D \\text { (Ans.) }\n\n\\end{array}\n\n<\/span><\/p>\n

31. \u0987\u09af\u09bc\u0982 \u098f\u09b0 \u09a6\u09cd\u09ac\u09bf-\u099a\u09bf\u09a1\u09bc \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be\u09af\u09bc \u099a\u09bf\u09a1\u09bc \u09a6\u09c1\u099f\u09bf\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac 0.18 \\mathrm{~mm} <\/span>\u0964 \u099a\u09bf\u09a1\u09bc\u0997\u09c1\u09b2\u09cb \u09a5\u09c7\u0995\u09c7 90 \\mathrm{~cm} <\/span> \u09a6\u09c2\u09b0\u09c7 \u09aa\u09b0\u09cd\u09a6\u09be\u09af\u09bc \u0995\u09cb\u09a8\u09cb \u098f\u0995\u099f\u09bf \u098f\u0995\u09ac\u09b0\u09cd\u09a3\u09c0 \u0986\u09b2\u09cb\u09b0 \u09b8\u09be\u09b9\u09be\u09af\u09cd\u09af\u09c7 \u09a1\u09cb\u09b0\u09be \u09b8\u09c3\u09b7\u09cd\u099f\u09bf \u0995\u09b0\u09be \u09b9\u09b2\u09c7, \u09af\u09a6\u09bf \\mathrm{3^{rd}} <\/span> \u0989\u099c\u09cd\u099c\u09cd\u09ac\u09b2 \u09a1\u09cb\u09b0\u09be\u099f\u09bf \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09c0\u09df \u0989\u099c\u09cd\u099c\u09cd\u09ac\u09b2 \u09a1\u09cb\u09b0\u09be \u09a5\u09c7\u0995\u09c7 \\mathrm{8.1~mm} <\/span> \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09bf\u09a4 \u09b9\u09df, \u09a4\u09be\u09b9\u09b2\u09c7 \u0986\u09b2\u09cb\u09b0 \u09a4\u09b0\u0999\u09cd\u0997\u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af \u09ac\u09c7\u09b0 \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf,
\n\n\n\\begin{array}{l|l}\\Delta \\mathrm{x}=\\frac{\\mathrm{nD} \\lambda}{\\mathrm{d}} & \\mathrm{d}=0.18 \\mathrm{~mm} =1.8 \\times 10^{-4} \\mathrm{~m}\\\\\n\n\\lambda=\\frac{\\Delta \\mathrm{xd}}{\\mathrm{nD}}=\\frac{8.1 \\times 10^{-3} \\times 1.8 \\times 10^{-4}}{3 \\times 0.9} & \\begin{array}{l}\\text{n=3}\\\\\n\n\\mathrm{D}=90 \\mathrm{~cm}=0.9 \\mathrm{~m}\n\n\\end{array} \\\\\n\n=5.4 \\times 10^{-7} \\mathrm{~m}=540 \\mathrm{~nm} \\text { (Ans.) } & \\Delta \\mathrm{x}=8.1 \\mathrm{~mm}=8.1 \\times 10^{-3} \\mathrm{~m}, ~~\\lambda=?\n\n\\end{array}\n\n<\/span><\/p>\n

32. \u098f\u0995\u099f\u09bf \u09b8\u09bf\u09b2\u09bf\u09a8\u09cd\u09a1\u09be\u09b0\u09c7 300 \\mathrm{~K} <\/span> \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09af\u09bc \u0993 10^{6} \\mathrm{~Pa} <\/span> \u099a\u09be\u09aa\u09c7 0.001 \\mathrm{~m}^{3} <\/span> \u0997\u09cd\u09af\u09be\u09b8 \u0986\u099b\u09c7\u0964 \u0997\u09cd\u09af\u09be\u09b8\u099f\u09bf\u0995\u09c7 \u09aa\u09cd\u09b0\u09a5\u09ae\u09c7 \u09b8\u09ae\u09cb\u09b7\u09cd\u09a3 \u09aa\u09cd\u09b0\u09b8\u09be\u09b0\u09a3 \u0995\u09b0\u09be \u09b9\u09b2 \u098f\u09ac\u0982 \u09aa\u09b0\u09c7 \u09b0\u09c1\u09a6\u09cd\u09a7\u09a4\u09be\u09aa\u09c0\u09af\u09bc \u09aa\u09cd\u09b0\u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09af\u09bc \u0986\u09ac\u09be\u09b0\u0993 \u09aa\u09cd\u09b0\u09b8\u09be\u09b0\u09a3 \u0995\u09b0\u09be \u09b9\u09b2, \u09aa\u09cd\u09b0\u09a4\u09bf \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7\u0987 \u09aa\u09cd\u09b0\u09b8\u09be\u09b0\u09a3\u09c7\u09b0 \u0985\u09a8\u09c1\u09aa\u09be\u09a4 1: 2 <\/span> \u0964 \u09aa\u09cd\u09b0\u09b8\u09be\u09b0\u09a3\u09c7 \u09ae\u09cb\u099f \u0995\u09be\u099c\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ae\u09be\u09a3 \u09ac\u09c7\u09b0 \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u09b8\u09ae\u09cb\u09b7\u09cd\u09a3 \u09aa\u09cd\u09b0\u09b8\u09be\u09b0\u09a3\u09c7 \u0995\u09c3\u09a4\u0995\u09be\u099c \\mathrm{W}_{1}=\\mathrm{nRT} \\ln \\left(\\frac{\\mathrm{V}_{2}}{\\mathrm{~V}_{1}}\\right) <\/span>
\n\u098f\u0996\u09a8, \\mathrm{PV}=\\mathrm{nRT} \\Rightarrow \\mathrm{n}=\\frac{\\mathrm{PV}}{\\mathrm{RT}}=\\frac{10^{6} \\times 0.001}{8.314 \\times 300}=0.4 \\mathrm{~mole} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \\mathrm{W}_{1}=\\mathrm{nRT} \\ln \\left(\\frac{\\mathrm{V}_{2}}{\\mathrm{~V}_{1}}\\right)=0.4 \\times 8.314 \\times 300 \\ln \\left(\\frac{2}{1}\\right)=691.54 \\mathrm{~J} <\/span>
\n\u098f\u0996\u09a8 \u09b0\u09c1\u09a6\u09cd\u09a7\u09a4\u09be\u09aa\u09c0\u09af\u09bc \u09aa\u09cd\u09b0\u09a4\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09af\u09bc,
\n \\mathrm{T}_{1} \\mathrm{~V}_{1}^{\\gamma-1}=\\mathrm{T}_{2} \\mathrm{~V}_{2}^{\\gamma-1} \\Rightarrow \\mathrm{T}_{2}=\\mathrm{T}_{1}\\left(\\frac{\\mathrm{V}_{1}}{\\mathrm{~V}_{2}}\\right)^{\\gamma-1}=300 \\times\\left(\\frac{1}{2}\\right)^{1.4-1}=227.357 \\mathrm{~K} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09b0\u09c1\u09a6\u09cd\u09a7\u09a4\u09be\u09aa\u09c0\u09af\u09bc \u09aa\u09cd\u09b0\u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09af\u09bc \u0995\u09c3\u09a4 \u0995\u09be\u099c,
\n \\mathrm{W}_{2}=\\frac{\\mathrm{nR}}{\\gamma-1}\\left(\\mathrm{~T}_{1}-\\mathrm{T}_{2}\\right)=\\frac{0.4 \\times 8.314}{1.4-1} \\times(300-227.357)=603.95 \\mathrm{~J} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09ae\u09cb\u099f \u0995\u09be\u099c, \\mathrm{W}=\\mathrm{W}_{1}+\\mathrm{W}_{2}=1295.49 \\mathrm{~J} <\/span> (Ans.)<\/p>\n

33. \u098f\u09a8\u099f\u09cd\u09b0\u09aa\u09bf \u09ac\u09b2\u09a4\u09c7 \u0995\u09bf \u09ac\u09c1\u099d? 100^{\\circ} \\mathrm{C} <\/span> \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09b0 4 \\mathrm{~kg} <\/span> \u09aa\u09be\u09a8\u09bf\u0995\u09c7 100^{\\circ} \\mathrm{C} <\/span> \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09b0 \u09ac\u09be\u09b7\u09cd\u09aa\u09c7 \u09aa\u09b0\u09bf\u09a3\u09a4 \u0995\u09b0\u09be \u09b9\u09b2\u0964 \u098f\u09a8\u099f\u09cd\u09b0\u09aa\u09bf\u09b0 \u09ac\u09c3\u09a6\u09cd\u09a7\u09bf \u09ac\u09c7\u09b0 \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u098f\u09a8\u099f\u09cd\u09b0\u09aa\u09bf \u09b9\u099a\u09cd\u099b\u09c7 \u098f\u09ae\u09a8 \u098f\u0995\u099f\u09bf \u0986\u09ac\u09a6\u09cd\u09a7 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8\u09b6\u09c0\u09b2 \u09b8\u09bf\u09b8\u09cd\u099f\u09c7\u09ae\u09c7\u09b0 \u09ac\u09bf\u09b6\u09c3\u0999\u09cd\u0996\u09b2\u09be\u09b0 \u09aa\u09b0\u09bf\u09ae\u09be\u09aa \u09af\u09c7 \u09b8\u09bf\u09b8\u09cd\u099f\u09c7\u09ae\u09c7 \u09b6\u0995\u09cd\u09a4\u09bf \u09b6\u09c1\u09a7\u09c1\u09ae\u09be\u09a4\u09cd\u09b0 \u09b6\u09c3\u0999\u09cd\u0996\u09b2 \u0985\u09ac\u09b8\u09cd\u09a5\u09be \u09a5\u09c7\u0995\u09c7 \u098f\u0995\u099f\u09bf \u09ac\u09bf\u09b6\u09c3\u0999\u09cd\u0996\u09b2 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09b0 \u09a6\u09bf\u0995\u09c7 \u09aa\u09b0\u09bf\u09ac\u09b9\u09a8 \u0995\u09b0\u09c7 \u09a5\u09be\u0995\u09c7\u0964
\n\u09a6\u09c7\u0993\u09af\u09bc\u09be \u0986\u099b\u09c7,
\n \\mathrm{m}=4 \\mathrm{~kg}, ~~~\\mathrm{l}_{\\mathrm{v}}=2.26 \\times 10^{6} \\mathrm{~Jkg}^{-1}, \\mathrm{~~T}=100^{\\circ} \\mathrm{C}=373 \\mathrm{~K}, ~~~\\Delta \\mathrm{Q}=\\mathrm{ml_v} \\quad <\/span>
\n\u098f\u09a8\u099f\u09cd\u09b0\u09aa\u09bf\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8 \\Delta \\mathrm{S}=\\frac{\\Delta \\mathrm{Q}}{\\mathrm{T}}=\\frac{4 \\times 2.26 \\times 10^{6}}{373}=24235.924 \\mathrm{~JK}^{-1} <\/span> (Ans.)<\/p>\n

34. \"\"<\/p>\n

\u098f\u0995\u099f\u09bf \u09b2\u09ae\u09cd\u09ac\u09be \u09aa\u09b0\u09bf\u09ac\u09be\u09b9\u09c0 \u09a4\u09be\u09b0\u09c7, \\mathrm{r=0.15} <\/span> \u09ac\u09cd\u09af\u09be\u09b8\u09be\u09b0\u09cd\u09a7\u09c7\u09b0 \u098f\u0995\u099f\u09bf \u09ac\u09c3\u09a4\u09cd\u09a4\u09be\u0995\u09be\u09b0 \u0995\u09c1\u09a3\u09cd\u09a1\u09b2\u09c0 \u09a4\u09c8\u09b0\u09bf \u0995\u09b0\u09c7 \u0989\u09b9\u09be\u09b0 \u09ac\u09be\u0995\u09bf \u0985\u0982\u09b6 \u09b8\u09cb\u099c\u09be \u09b0\u09be\u0996\u09be \u09b9\u09b2\u0964 \u0985\u09a8\u09cd\u09af \u098f\u0995\u099f\u09bf \u09b2\u09ae\u09cd\u09ac\u09be \u09aa\u09b0\u09bf\u09ac\u09be\u09b9\u09c0 \u0989\u0995\u09cd\u09a4 \u0995\u09c1\u09a3\u09cd\u09a1\u09b2\u09c0\u09b0 \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0 \\mathrm{d}=0.25 <\/span>\u09a6\u09c2\u09b0\u09c7 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2\u09c7 \u09a5\u09c7\u0995\u09c7 \u098f\u0995\u0987 \u09aa\u09b0\u09bf\u09ae\u09be\u09a3 \u09ac\u09bf\u09a6\u09cd\u09af\u09c1\u09ce \u099a\u09bf\u09a4\u09cd\u09b0\u09c7\u09b0 \u09a8\u09cd\u09af\u09be\u09af\u09bc \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09a6\u09bf\u0995\u09c7 \u09aa\u09cd\u09b0\u09ac\u09be\u09b9\u09bf\u09a4 \u09b9\u099a\u09cd\u099b\u09c7\u0964 \u0995\u09c1\u09a3\u09cd\u09a1\u09b2\u09c0\u09b0 \u0995\u09c7\u09a8\u09cd\u09a6\u09cd\u09b0\u09c7 4.72 \\mu \\mathrm{T} <\/span> \u099a\u09cc\u09ae\u09cd\u09ac\u0995 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0 \u09b8\u09c3\u09b7\u09cd\u099f\u09bf \u0995\u09b0\u09a4\u09c7 \u09aa\u09cd\u09b0\u09a4\u09bf\u099f\u09bf \u09aa\u09b0\u09bf\u09ac\u09be\u09b9\u09c0\u09a4\u09c7 \u09ac\u09bf\u09a6\u09cd\u09af\u09c1\u09ce \u09aa\u09cd\u09b0\u09ac\u09be\u09b9 \u0995\u09a4 \u09b9\u09ac\u09c7?
\n \\left[\\mu_{0}=4 \\pi \\times 10^{-7} ~\\mathrm{TmA}^{-1}\\right] <\/span>
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983 \"\"<\/strong>
\n\u0995\u09c1\u09a3\u09cd\u09a1\u09b2\u09c0\u09b0 \u0995\u09be\u09b0\u09a3\u09c7 \u099a\u09cc\u09ae\u09cd\u09ac\u0995\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0 B_{1}=\\frac{\\mu_{0} I}{2 r} <\/span>; \u09a6\u09bf\u0995 \u0995\u09be\u0997\u099c \u09a4\u09b2\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u09b2\u09ae\u09cd\u09ac\u09ad\u09be\u09ac\u09c7 \u0989\u09aa\u09b0\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7
\n \\mathrm{AB}<\/span> \u09a4\u09be\u09b0\u09c7\u09b0 \u0995\u09be\u09b0\u09a3\u09c7 \u099a\u09cc\u09ae\u09cd\u09ac\u0995\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0 \\mathrm{B}_{2}=\\frac{\\mu_{0} \\mathrm{I}}{2 \\pi \\mathrm{r}} <\/span>; \u09a6\u09bf\u0995 \u0995\u09be\u0997\u099c \u09a4\u09b2\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u09b2\u09ae\u09cd\u09ac\u09ad\u09be\u09ac\u09c7 \u0989\u09aa\u09b0\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7
\n \\mathrm{CD}<\/span> \u09a4\u09be\u09b0\u09c7\u09b0 \u0995\u09be\u09b0\u09a3\u09c7 \u099a\u09cc\u09ae\u09cd\u09ac\u0995\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0, \\mathrm{B}_{3}=\\frac{\\mu_{0} \\mathrm{I}}{2 \\pi \\mathrm{d}} <\/span>; \u09a6\u09bf\u0995 \u0995\u09be\u0997\u099c \u09a4\u09b2\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u09b2\u09ae\u09cd\u09ac\u09ad\u09be\u09ac\u09c7 \u09a8\u09bf\u099a\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09a8\u09c0\u099f \u099a\u09cc\u09ae\u09cd\u09ac\u0995\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0, \\mathrm{B}=\\mathrm{B}_{1}+\\mathrm{B}_{2}-\\mathrm{B}_{3} \\Rightarrow 4.72 \\times 10^{-6}=\\frac{\\mu_{0} \\mathrm{I}}{2 \\mathrm{r}}+\\frac{\\mu_{0} \\mathrm{I}}{2 \\pi \\mathrm{r}}-\\frac{\\mu_{0} \\mathrm{I}}{2 \\pi \\mathrm{d}} <\/span>
\n \\Rightarrow 4.72 \\times 10^{-6}=\\frac{4 \\times \\pi \\times 10^{-7} \\times \\mathrm{I}}{2}\\left(\\frac{1}{0.15}+\\frac{1}{\\pi \\times 0.15}-\\frac{1}{\\pi \\times 0.25}\\right) \\Rightarrow \\mathrm{I}=1 \\mathrm{~A} <\/span> (Ans.)<\/p>\n

35. \u098f\u0995\u099f\u09bf \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09aa\u09be\u09a4 \u09a7\u09be\u09b0\u0995\u09c7\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099f\u09bf \u09aa\u09be\u09a4\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09ab\u09b2 200 \\mathrm{~cm}^{2} <\/span> \u098f\u09ac\u0982 \u09ac\u09be\u09af\u09bc\u09c1\u09a4\u09c7 \u09aa\u09be\u09a4\u09a6\u09cd\u09ac\u09af\u09bc\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af\u09ac\u09b0\u09cd\u09a4\u09c0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac 0.4 \\mathrm{~cm} <\/span> \u09b9\u09b2\u09c7 \u098f\u09b0 (i) \u09a7\u09be\u09b0\u0995\u09a4\u09cd\u09ac \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u0964 (ii) \u09af\u09a6\u09bf \u09a7\u09be\u09b0\u0995\u099f\u09bf 500 \\mathrm{~V} <\/span> \u09ac\u09c8\u09a6\u09cd\u09af\u09c1\u09a4\u09bf\u0995 \u0989\u09ce\u09b8\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u09b8\u0982\u09af\u09cb\u0997 \u0995\u09b0\u09be \u09b9\u09af\u09bc, \u09a4\u09ac\u09c7 \u09a7\u09be\u09b0\u0995\u09c7 \u0995\u09a4 \u09b6\u0995\u09cd\u09a4\u09bf \u09b8\u099e\u09cd\u099a\u09bf\u09a4 \u09b9\u09ac\u09c7?
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> A=200 \\mathrm{~cm}^{2}=200 \\times 10^{-4} \\mathrm{~m}=0.02 \\mathrm{~m}^{2} <\/span>
\n \\mathrm{d}=0.4 \\mathrm{~cm}=0.4 \\times 10^{-2} \\mathrm{~m}=4 \\times 10^{-3} \\mathrm{~m} <\/span>
\n \\mathrm{V}=500 \\mathrm{~V} <\/span>
\n(i) \u09a7\u09be\u09b0\u0995\u09a4\u09cd\u09ac, \\mathrm{C=\\frac{\\epsilon_{0} A}{d}=\\frac{8.84 \\times 10^{-12} \\times 0.02}{4 \\times 10^{-3}}=4.43 \\times 10^{-11} ~F} <\/span>
\n(ii) \\mathrm{U=\\frac{1}{2} C V^{2}=\\frac{1}{2} \\times 4.43 \\times 10^{-11} \\times 500 \\times 500=5.53 \\times 10^{-6} ~J} <\/span> (Ans.)<\/p>\n

36. \u098f\u0995\u099f\u09bf 1.5\\mathrm{~kW} <\/span> \u0987\u09b2\u09c7\u0995\u09cd\u099f\u09cd\u09b0\u09bf\u0995 \u0995\u09c7\u09a4\u09b2\u09c0\u09a4\u09c7 2 \u09b2\u09bf\u099f\u09be\u09b0 \u09aa\u09be\u09a8\u09bf \u09a8\u09bf\u09df\u09c7 \u0997\u09b0\u09ae \u0995\u09b0\u09b2\u09c7 \u09a4\u09be \\mathrm{6 ~min ~20~sec} <\/span> \u09aa\u09b0 \u09ab\u09c1\u099f\u09a4\u09c7 \u09b6\u09c1\u09b0\u09c1 \u0995\u09b0\u09c7\u0964 \u09aa\u09cd\u09b0\u09a5\u09ae\u09c7 \u0995\u09c7\u09a4\u09b2\u09c0\u09a4\u09c7 \u09aa\u09be\u09a8\u09bf\u09b0 \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be \u0995\u09a4 \u099b\u09bf\u09b2? \u0995\u09c7\u09a4\u09b2\u09c0\u09a4\u09c7 \u09aa\u09be\u09a8\u09bf \u09ab\u09cb\u099f\u09be\u09a4\u09c7 \u0995\u09a4 unit \u09ac\u09bf\u09a6\u09cd\u09af\u09c1\u09ce \u0996\u09b0\u099a \u09b9\u09af\u09bc\u09c7\u099b\u09c7? (\u09a4\u09be\u09aa\u0995\u09cd\u09b7\u09df \u09a8\u0997\u09a3\u09cd\u09af \u09a7\u09b0\u09be \u09af\u09c7\u09a4\u09c7 \u09aa\u09be\u09b0\u09c7)
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong>
\n\n\n\\begin{array}{l}\\begin{array}{l|l}\n\n\\mathrm{pt}=\\mathrm{ms} \\Delta \\theta \\Rightarrow \\mathrm{pt}=\\mathrm{ms}\\left(\\theta-\\theta_{0}\\right) & \\mathrm{V}=2 \\mathrm{~L}=2 \\times 10^{-3} \\mathrm{~m}^{3} \\\\\n\n\\Rightarrow \\theta-\\theta_{0}=\\frac{\\mathrm{pt}}{\\mathrm{ms}}=\\frac{1.5 \\times 10^{3} \\times 380}{2 \\times 4200}=67.86 & \\mathrm{m}=\\mathrm{V} \\rho=2 \\times 10^{-3} \\times 10^{3}=2 \\mathrm{~kg} \\\\\n\n\\Rightarrow \\theta_{0}=\\theta-67.86=32.14^{\\circ} \\mathrm{C} & \\mathrm{p}=1.5 \\times 10^{3} \\mathrm{~W} \\\\\n\n\\text { So, Initial Temperature of water, } \\theta_{0}=32.14^{\\circ} \\mathrm{C} & \\mathrm{t}=(6 \\times 60+20) \\mathrm{s}=380 \\mathrm{~s}=\\frac{380}{3600} \\mathrm{hr} \\\\\n\n\\mathrm{E}=\\mathrm{Pt}=1.5 \\times \\frac{380}{3600} \\text { unit }=0.158 \\text { unit } & \\theta_{0}=? \\\\\n\n\\text { Therefore, total electricity needed }=0.158 \\text { unit } & \\theta=100^{\\circ} \\mathrm{C} \\\\\n\n& \\mathrm{s}=4200 \\mathrm{Jkg}^{-1} \\mathrm{~K}^{-1}\n\n\\end{array}\n\n\\end{array}\n\n<\/span><\/p>\n

37. \u098f\u0995\u099f\u09bf \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09c0 \u09aa\u09cd\u09b0\u09ac\u09be\u09b9\u0995\u09c7 \\mathrm{I}=100 \\sin 629 \\mathrm{t} <\/span> \u0985\u09cd\u09af\u09be\u09ae\u09cd\u09aa\u09bf\u09df\u09be\u09b0 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09b2\u09c7, \u09a4\u09a1\u09bc\u09bf\u09ce \u09aa\u09cd\u09b0\u09ac\u09be\u09b9\u09c7\u09b0 \u09b6\u09c0\u09b0\u09cd\u09b7\u09ae\u09be\u09a8, \u0995\u09ae\u09cd\u09aa\u09be\u0999\u09cd\u0995 \u098f\u09ac\u0982 \u09ac\u09b0\u09cd\u0997\u09ae\u09c2\u09b2\u09c0\u09df \u0997\u09a1\u09bc \u09ae\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u09a6\u09c7\u0993\u09af\u09bc\u09be \u0986\u099b\u09c7, \\mathrm{I}=100 \\sin 629 \\mathrm{t}\\ldots \\mathrm{(i);}~~~\\mathrm{I}=\\mathrm{I}_{0} \\sin \\omega \\mathrm{t}\\ldots\\mathrm{(ii)}<\/span>
\n\\mathrm{(i),~(ii)}<\/span> \u09a4\u09c1\u09b2\u09a8\u09be \u0995\u09b0\u09c7 \u09aa\u09be\u0987, \\mathrm { I } _ { 0 } = 1 0 0 \\mathrm { A } ~~\\text { (Ans.) } <\/span>
\n \\mathrm{ \\omega=629 \\Rightarrow 2 \\pi \\mathrm{f}=629 \\Rightarrow \\mathrm{f}=100 \\mathrm{~Hz}~~\\text{(Ans.)}} <\/span>
\n \\mathrm{I}_{\\mathrm{rms}}=\\frac{\\mathrm{I}_{0}}{\\sqrt{2}}=70.71 \\mathrm{~A}~~~~\\text{(Ans.)} <\/span><\/p>\n

38. 0.40 \\mathrm{~nm} <\/span> \u09a4\u09b0\u0999\u09cd\u0997\u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af\u09c7\u09b0 \u098f\u0995\u099f\u09bf \u09ab\u09cb\u099f\u09a8 \u09b8\u09cd\u09a5\u09bf\u09b0\u09be\u09ac\u09b8\u09cd\u09a5\u09be\u09af\u09bc \u09a5\u09be\u0995\u09be \u098f\u0995\u099f\u09bf \u0987\u09b2\u09c7\u0995\u099f\u09cd\u09b0\u09a8\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u09b8\u0982\u0998\u09b0\u09cd\u09b7\u09c7\u09b0 \u09aa\u09b0 \u09ab\u09cb\u099f\u09a8\u099f\u09bf \u09aa\u09c2\u09b0\u09cd\u09ac\u09c7\u09b0 \u0997\u09a4\u09bf\u09aa\u09a5\u09c7\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 \\mathrm{150^{o}} <\/span> \u0995\u09cb\u09a3\u09c7 \u09ac\u09bf\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u09b9\u09df\u0964 \u09ac\u09bf\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u09ab\u09cb\u099f\u09a8\u09c7\u09b0 \u09ac\u09c7\u0997 \u0993 \u09a4\u09b0\u0999\u09cd\u0997\u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983 <\/strong>\u09a6\u09c7\u0993\u09df\u09be \u0986\u099b\u09c7,
\n\\lambda_{0}=0.4 \\mathrm{~nm}=4 \\times 10^{-10} \\mathrm{~m}<\/span>
\n\\mathrm{m}_{0}=9.1 \\times 10^{-31} \\mathrm{~kg}<\/span>
\n\\mathrm{h}=6.626 \\times 10^{-34} \\mathrm{Js}<\/span>
\n\\theta=150^{\\circ}<\/span><\/p>\n

\u09a7\u09b0\u09bf, \u09ac\u09bf\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u09ab\u09cb\u099f\u09a8\u09c7\u09b0 \u09a4\u09b0\u0999\u09cd\u0997\u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af \\mathrm{=\\lambda^{\\prime}}<\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \\mathrm{\\lambda^{\\prime}-\\lambda_{0}=\\frac{\\mathrm{h}}{\\mathrm{m}_{0} \\mathrm{c}}(1-\\cos \\theta) \\Rightarrow \\lambda^{\\prime}=\\lambda_{0}+\\frac{\\mathrm{h}}{\\mathrm{m}_{0} \\mathrm{c}}(1-\\cos \\theta)}<\/span>
\n \\mathrm{=4 \\times 10^{-10}+\\frac{6.626 \\times 10^{-34}}{9.1 \\times 10^{-31} \\times 3 \\times 10^{8}}\\left(1-\\cos 150^{\\circ}\\right)}<\/span>
\n \\mathrm{=4.045 \\times 10^{-10} \\mathrm{~m}=0.4045 \\~nm~~~~\\text{(Ans.)}}<\/span>
\n\u09ac\u09bf\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u09ab\u09cb\u099f\u09a8\u09c7\u09b0 \u09ac\u09c7\u0997 =\\mathrm{c}=3 \\times 10^{8} \\mathrm{~ms}^{-1}~~~\\text{(Ans.)}<\/span><\/p>\n

39. \u098f\u0995\u099f\u09bf \u0987\u09b2\u09c7\u0995\u099f\u09cd\u09b0\u09a8 (\u09a8\u09bf\u09b6\u09cd\u099a\u09b2 \u09ad\u09b0 9.1 \\times 10^{-31} \\mathrm{~kg} <\/span> ) \u0986\u09b2\u09cb\u09b0 \u09a6\u09cd\u09b0\u09c1\u09a4\u09bf\u09b0 90 \\% <\/span> \u09a6\u09cd\u09b0\u09a4\u09bf\u09a4\u09c7 \u099a\u09b2\u099b\u09c7\u0964 \u0986\u0987\u09a8\u09b8\u09cd\u099f\u09be\u0987\u09a8\u09c7\u09b0 \u0986\u09aa\u09c7\u0995\u09cd\u09b7\u09bf\u0995 \u09a4\u09a4\u09cd\u09a4\u09cd\u09ac \u0985\u09a8\u09c1\u09b8\u09be\u09b0\u09c7 \u0987\u09b2\u09c7\u0995\u099f\u09cd\u09b0\u09a8\u099f\u09bf\u09b0 \u0997\u09a4\u09bf\u09b6\u0995\u09cd\u09a4\u09bf \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u0987\u09b2\u09c7\u0995\u099f\u09cd\u09b0\u09a8\u09c7\u09b0 \u09a8\u09bf\u09b6\u09cd\u099a\u09b2 \u09ad\u09b0, \\mathrm{m}_{0}=9.1 \\times 10^{-31} \\mathrm{~kg} <\/span>
\n\u0987\u09b2\u09c7\u0995\u099f\u09cd\u09b0\u09a8\u09c7\u09b0 \u09ac\u09c7\u0997, \\mathrm{v}=90 \\times \\frac{1}{100} \\times \\mathrm{c} \\Rightarrow \\mathrm{v}=0.9 \\mathrm{c} <\/span>
\n\u09a7\u09b0\u09bf, \u0987\u09b2\u09c7\u0995\u099f\u09cd\u09b0\u09a8\u09c7\u09b0 \u0997\u09a4\u09bf\u09b6\u0995\u09cd\u09a4\u09bf= \\mathrm{E_k} <\/span>
\n \\begin{array}{l}\n\n\\therefore \\mathrm{E_k}=\\mathrm{mc}^{2}-\\mathrm{m}_{0} \\mathrm{c}^{2}=\\frac{\\mathrm{m}_{0}}{\\sqrt{1-\\frac{\\mathrm{v}^{2}}{c^{2}}}} \\cdot \\mathrm{c}^{2}-\\mathrm{m}_{0} \\mathrm{c}^{2}\\left[\\because \\mathrm{m}=\\frac{\\mathrm{m}_{0}}{\\sqrt{1-\\frac{\\mathrm{v}^{2}}{\\mathrm{c}^{2}}}}\\right] \\\\\n\n=\\mathrm{m}_{0} \\mathrm{c}^{2}\\left\\{\\frac{1}{\\sqrt{1-\\frac{\\mathrm{v}^{2}}{\\mathrm{c}^{2}}}}-1\\right\\}=1.2941 \\times 9.1 \\times 10^{-31} \\times\\left(3 \\times 10^{8}\\right)^{2} \\\\\n\n=1.0599 \\times 10^{-13} \\mathrm{~J}=0.66244 ~\\mathrm{MeV} \\text { (Ans.) }\n\n\\end{array}\n\n<\/span><\/p>\n

40. \u0995\u09cb\u09a8 \u09a4\u09c7\u099c\u09b8\u09cd\u0995\u09cd\u09b0\u09bf\u09df \u09aa\u09a6\u09be\u09b0\u09cd\u09a5\u09c7\u09b0 \u0985\u09b0\u09cd\u09a7\u09be\u09df\u09c1 1000 <\/span> \u09ac\u099b\u09b0\u0964 \u0995\u09a4 \u09ac\u099b\u09b0 \u09aa\u09b0 \u0989\u09b9\u09be\u09b0 \u09a4\u09c7\u099c\u09b8\u09cd\u0995\u09cd\u09b0\u09bf\u09df\u09a4\u09be \u0995\u09cd\u09b7\u09df\u09aa\u09cd\u09b0\u09be\u09aa\u09cd\u09a4 \u09b9\u09df\u09c7 1\/10 \\mathrm{~th} <\/span> \u09b9\u09ac\u09c7? \u0990 \u09a4\u09c7\u099c\u09b8\u09cd\u0995\u09cd\u09b0\u09bf\u09df \u09aa\u09a6\u09be\u09b0\u09cd\u09a5\u09c7\u09b0 \u0997\u09a1\u09bc \u0986\u09df\u09c1 \u0995\u09a4 \u09b9\u09ac\u09c7?
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u09a6\u09c7\u0993\u09af\u09bc\u09be \u0986\u099b\u09c7, \\mathrm{T}_{\\frac{1}{2}}=1000~\\mathrm{years} <\/span><\/p>\n

\u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf, \u0997\u09dc \u0986\u09df\u09c1, \\tau=\\frac{\\mathrm{T}_{\\frac{1}{2}}}{\\ln (2)}=\\frac{1000}{0.693}=1442.69~\\mathrm{years~~~(Ans.)} <\/span><\/p>\n

\u0995\u09cd\u09b7\u09df\u09a7\u09cd\u09b0\u09c1\u09ac\u0995, \\lambda=\\mathrm{\\frac{ln2}{T_\\frac{1}{2}};~~N=N_0~\\times~\\frac{1}{10}} <\/span>
\n\n\n\\begin{array}{l}\n\n\\mathrm{N}=\\mathrm{N}_{0} \\cdot \\mathrm{e}^{-\\lambda \\mathrm{t}} \\Rightarrow \\frac{\\mathrm{N}_{0}}{10}=\\mathrm{N}_{0} \\cdot \\mathrm{e}^{-\\frac{\\ln 2}{\\mathrm{~T}_\\frac{1}{2}} \\mathrm{t}} \\Rightarrow-\\ln (10)=\\frac{-\\ln (2)}{1000} \\mathrm{t} \\times \\ln (\\mathrm{e}) \\\\\n\n\\Rightarrow \\mathrm{t}=1000 \\times \\frac{\\ln (10)}{\\ln (2)}=3321.928 \\text { years (Ans.) }\n\n\\end{array}\n\n<\/span><\/p>\n

\u09b0\u09b8\u09be\u09df\u09a8 (Chemistry)<\/strong><\/h3>\n

41. \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u0986\u09df\u09a4\u09a8\u09c7\u09b0 \u09ac\u09bf\u09b6\u09c1\u09a6\u09cd\u09a7 \u0985\u0995\u09cd\u09b8\u09bf\u099c\u09c7\u09a8 \u0997\u09cd\u09af\u09be\u09b8 \u098f\u0995\u099f\u09bf \u099b\u09cb\u099f \u099b\u09bf\u09a6\u09cd\u09b0 \u09a6\u09bf\u09df\u09c7 \u09a8\u09bf\u0983\u09b8\u09b0\u09bf\u09a4 \u09b9\u09a4\u09c7 \\mathrm{80~seconds} <\/span> \u09b8\u09ae\u09df \u09b2\u09be\u0997\u09c7\u0964 \u098f\u0995\u0987 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09df \u09b8\u09ae\u09be\u09a8 \u0986\u09df\u09a4\u09a8\u09c7\u09b0 20~\\% <\/span> \u0985\u099c\u09be\u09a8\u09be \u0997\u09cd\u09af\u09be\u09b8 \u09ae\u09bf\u09b6\u09cd\u09b0\u09bf\u09a4 \u0985\u0995\u09cd\u09b8\u09bf\u099c\u09c7\u09a8 \u09a8\u09bf\u0983\u09b8\u09b0\u09a3\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \\mathrm{85~seconds} <\/span> \u09b8\u09ae\u09df \u09b2\u09be\u0997\u09c7\u0964 \u0985\u099c\u09be\u09a8\u09be \u0997\u09cd\u09af\u09be\u09b8\u099f\u09bf\u09b0 \u0986\u09a3\u09ac\u09bf\u0995 \u09ad\u09b0 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u09a6\u09c7\u0993\u09af\u09bc\u09be \u0986\u099b\u09c7, \\mathrm{M}_{\\mathrm{O}_{2}}=32 \\mathrm{~g}, \\mathrm{t}_{\\mathrm{O}_{2}}=80 \\mathrm{sec}, \\mathrm{t}_{\\text {mixture }}=85 \\mathrm{sec} <\/span>
\n \\mathrm{X}_{\\mathrm{O}_{2}}=0.8, \\mathrm{X}_{\\text {unknown}}=0.2, \\mathrm{M}_{\\text {unknown}}=\\mathrm{W} <\/span>
\n \\mathrm{M}_{\\text {mixture}}=\\mathrm{X}_{\\mathrm{O}_{2}} \\times \\mathrm{M}_{\\mathrm{O}_{2}}+\\mathrm{X}_{\\text {unknown}} \\times \\mathrm{M}_{\\text {unknown}}=0.8 \\times 32+0.2 \\times \\mathrm{W}=25.6+0.2 \\times \\mathrm{W} <\/span>
\n\u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf, \\mathrm{\\sqrt{\\frac{M_{O_2}}{M_{mixture}}}=\\frac{t_{O_2}}{t_{mixture}} \\Rightarrow\\frac{M_{O_2}}{M_{mixture}}={(\\frac{t_{O_2}}{t_{mixture}})}^{2}\\Rightarrow \\frac{32}{0.2~W+25.6}={(\\frac{80}{85})}^{2}} <\/span>
\n \\Rightarrow 0.2 \\mathrm{~W}+25.6=36.125 \\Rightarrow 0.2 \\mathrm{~W}=10.525 \\Rightarrow \\mathrm{W}=52.625 \\mathrm{~g~~~(Ans.)} <\/span><\/p>\n

42. \u0985\u09cd\u09af\u09be\u09ae\u09cb\u09a8\u09bf\u09af\u09bc\u09be \u0985\u0995\u09cd\u09b8\u09bf\u099c\u09c7\u09a8\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u09a8\u09bf\u09ae\u09cd\u09a8\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u0985\u09a8\u09c1\u09af\u09be\u09af\u09bc\u09c0 \u09ac\u09bf\u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be \u0995\u09b0\u09c7\u0964
\n 4 \\mathrm{NH}_{3}(\\mathrm{~g})+5 \\mathrm{O}_{2}(\\mathrm{~g})=4 \\mathrm{NO}(\\mathrm{g})+6 \\mathrm{H}_{2} \\mathrm{O}(\\mathrm{g}) <\/span>
\n\u0995\u09cb\u09a8\u09cb \u09ae\u09c1\u09b9\u09c1\u09b0\u09cd\u09a4\u09c7 \u0985\u09cd\u09af\u09be\u09ae\u09cb\u09a8\u09bf\u09af\u09bc\u09be 0.24 \\mathrm{~mol} \\mathrm{~L}^{-1} \\mathrm{~s}^{-1} <\/span> \u09b9\u09be\u09b0\u09c7 \u09ac\u09bf\u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be \u0995\u09b0\u09b2\u09c7 \u09ac\u09bf\u0995\u09cd\u09b0\u09bf\u09df\u09be\u099f\u09bf\u09b0 \u09b9\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u099f\u09bf \u09b2\u09bf\u0996 \u098f\u09ac\u0982 \\mathrm{H}_{2} \\mathrm{O}(\\mathrm{g}) <\/span> \u0989\u09ce\u09aa\u09be\u09a6\u09a8\u09c7\u09b0 \u09b9\u09be\u09b0 \u09ac\u09c7\u09b0 \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong>
\n\u09ac\u09bf\u0995\u09cd\u09b0\u09bf\u09df\u09be\u099f\u09bf \u09b9\u09b2\u09cb\u0983 4 \\mathrm{NH}_{3}(\\mathrm{~g})+5 \\mathrm{O}_{2}(\\mathrm{~g})=4 \\mathrm{NO}(\\mathrm{g})+6 \\mathrm{H}_{2} \\mathrm{O}(\\mathrm{g}) <\/span>
\n\u09ac\u09bf\u0995\u09cd\u09b0\u09bf\u09df\u09be\u099f\u09bf\u09b0 \u09b9\u09be\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\u0983
\n\u09ac\u09bf\u0995\u09cd\u09b0\u09bf\u09df\u09be\u09b0 \u09b9\u09be\u09b0 =\\frac{-1}{4} \\frac{\\Delta\\left[\\mathrm{NH}_{3}\\right]}{\\Delta \\mathrm{t}}=\\frac{-1}{5} \\frac{\\Delta\\left[\\mathrm{O}_{2}\\right]}{\\Delta \\mathrm{t}}=\\frac{1}{4} \\frac{\\Delta[\\mathrm{NO}]}{\\Delta \\mathrm{t}}=\\frac{1}{6} \\frac{\\Delta\\left[\\mathrm{H}_{2} \\mathrm{O}\\right]}{\\Delta \\mathrm{t}} <\/span><\/p>\n

\u09a6\u09c7\u0993\u09af\u09bc\u09be \u0986\u099b\u09c7, \\frac{-\\Delta\\left[\\mathrm{NH}_{3}\\right]}{\\Delta \\mathrm{t}}=0.24 ; \\frac{\\Delta\\left[\\mathrm{H}_{2} \\mathrm{O}\\right]}{\\Delta \\mathrm{t}}=? <\/span><\/p>\n

\u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a5\u09c7\u0995\u09c7 \u09aa\u09be\u0987,
\n \\frac{-1}{4} \\frac{\\Delta\\left[\\mathrm{NH}_{3}\\right]}{\\Delta \\mathrm{t}}=\\frac{1}{6} \\cdot \\frac{\\Delta\\left[\\mathrm{H}_{2} \\mathrm{O}\\right]}{\\Delta \\mathrm{t}} \\Rightarrow \\frac{\\Delta\\left[\\mathrm{H}_{2} \\mathrm{O}\\right]}{\\Delta \\mathrm{t}}=\\frac{6}{4}\\left(\\frac{-\\Delta\\left[\\mathrm{NH}_{3}\\right]}{\\Delta \\mathrm{t}}\\right)=\\frac{6}{4} \\times 0.24=0.36 \\mathrm{~mol} \\mathrm{~L}^{-1} \\mathrm{~s}^{-1} <\/span><\/p>\n

43. 400 ~\\mathrm{ml} ~0.1 ~\\mathrm{M} ~~\\mathrm{NaOH} <\/span> \u098f\u09b0 \u09b8\u09be\u09a5\u09c7 600 ~\\mathrm{ml} ~0.2 ~\\mathrm{M} <\/span> \u098f\u09b8\u09bf\u099f\u09bf\u0995 \u098f\u09b8\u09bf\u09a1 \u09af\u09cb\u0997 \u0995\u09b0\u09c7 \u098f\u0995\u099f\u09bf \u09ac\u09be\u09ab\u09be\u09b0 \u09a6\u09cd\u09b0\u09ac\u09a3 \u09aa\u09cd\u09b0\u09b8\u09cd\u09a4\u09c1\u09a4 \u0995\u09b0\u09be \u09b9\u09b2\u0964 \u09ac\u09be\u09ab\u09be\u09b0 \u09a6\u09cd\u09b0\u09ac\u09a3\u09c7\u09b0 \\mathrm{pH} <\/span> \u0995\u09a4? (\u098f\u09b8\u09bf\u099f\u09bf\u0995 \u098f\u09b8\u09bf\u09a1\u09c7\u09b0 \u099c\u09a8\u09cd\u09af\u0983 \\mathrm{pK}_{\\mathrm{a}}=4.76 <\/span> )\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \\mathrm{V}_{\\mathrm{NaOH}}=400 \\mathrm{~mL}=0.4 \\mathrm{~L}, \\mathrm{~S}_{\\mathrm{NaOH}}=0.1 ~\\mathrm{M} <\/span>
\n \\mathrm{V}_{\\mathrm{CH}_{3} \\mathrm{COOH}}=600 \\mathrm{~mL}=0.6 \\mathrm{~L}, \\mathrm{~S}_{\\mathrm{CH}_{3} \\mathrm{COOH}}=0.2 \\mathrm{M} <\/span>
\n \\mathrm{CH}_{3} \\mathrm{COOH}+\\mathrm{NaOH} \\rightarrow \\mathrm{CH}_{3} \\mathrm{COONa}+\\mathrm{H}_{2} \\mathrm{O} <\/span>
\n\u098f\u0996\u09be\u09a8\u09c7, \\mathrm{V}_{\\mathrm{CH}_{3} \\mathrm{COOH}} \\times \\mathrm{S}_{\\mathrm{CH}_{3} \\mathrm{COOH}}>\\mathrm{V}_{\\mathrm{NaOH}} \\times \\mathrm{S}_{\\mathrm{NaOH}} \u09b8\u09c1\u09a4\u09b0\u09be\u0982 [katex] \\mathrm{NaOH} <\/span> \u09b2\u09bf\u09ae\u09bf\u099f\u09bf\u0982 \u09ac\u09bf\u0995\u09cd\u09b0\u09bf\u09df\u0995\u0964
\n\u09a6\u09cd\u09b0\u09ac\u09a3\u09c7 \\mathrm{CH}_{3} \\mathrm{COONa} <\/span> \u098f\u09b0,
\n\u09ae\u09cb\u09b2\u09b8\u0982\u0996\u09cd\u09af\u09be \\mathrm{n}_{\\mathrm{CH}_{2}\\mathrm{COONa}} =\\mathrm{V}_{\\mathrm{Na} \\mathrm{OH}} \\times \\mathrm{S}_{\\mathrm{Na} \\mathrm{OH}}=0.4 \\times 0.1=0.04 <\/span>
\n\u09a6\u09cd\u09b0\u09ac\u09a3\u09c7 \\mathrm{CH}_{3} \\mathrm{COOH} <\/span> \u098f\u09b0,
\n\u09ae\u09cb\u09b2\u09b8\u0982\u0996\u09cd\u09af\u09be \\mathrm{n}_{\\mathrm{CH}_{3} \\mathrm{COOH}}=\\mathrm{V}_{\\mathrm{CH}_{3} \\mathrm{COOH}} \\times \\mathrm{S}_{\\mathrm{CH}_{3} \\mathrm{COOH}}-\\mathrm{V}_{\\mathrm{NaOH}} \\times \\mathrm{S}_{\\mathrm{NaOH}} <\/span>
\n =0.6 \\times 0.2-0.4 \\times 0.1=0.08 <\/span>
\n \\therefore \\mathrm{pH}=\\mathrm{pk}_{\\mathrm{a}}+\\log \\frac{\\mathrm{n}_{\\mathrm{CH}_{3} \\text {COONa }}}{\\mathrm{n}_{\\mathrm{CH}_{3} \\mathrm{COOH}}}=4.76+\\log \\left(\\frac{0.04}{0.08}\\right)=4.46~~~~~\\text{(Ans.)} <\/span><\/p>\n

44. (a) \u098f\u0995\u099f\u09bf \u09b2\u09c7\u09a1-\u098f\u09b8\u09bf\u09a1 \u09ac\u09cd\u09af\u09be\u099f\u09be\u09b0\u09c0 \u099a\u09be\u09b0\u09cd\u099c \u0995\u09b0\u09be \u09b9\u099a\u09cd\u099b\u09c7\u0964 \u09aa\u099c\u09bf\u099f\u09bf\u09ad \u09aa\u09be\u09a4 \u0993 \u09a8\u09c7\u0997\u09c7\u099f\u09bf\u09ad \u09aa\u09be\u09a4\u09c7\u09b0 \u09ac\u09bf\u0995\u09cd\u09b0\u09bf\u09df\u09be\u0997\u09c1\u09b2\u09bf \u09b2\u09bf\u0996\u0964 \u0995\u09cb\u09a8 \u09aa\u09be\u09a4\u099f\u09bf \u0985\u09cd\u09af\u09be\u09a8\u09cb\u09a1?
\n(b) \u098f\u0995\u099f\u09bf \\mathrm{5~ litre} <\/span> \u0986\u09af\u09bc\u09a4\u09a8\u09c7\u09b0 \u09aa\u09be\u09a4\u09cd\u09b0 4 ~\\mathrm{gm} <\/span> \u09b9\u09be\u0987\u09a1\u09cd\u09b0\u09cb\u099c\u09c7\u09a8 \u0993 7 ~\\mathrm{gm} <\/span> \u09a8\u09be\u0987\u099f\u09cd\u09b0\u09cb\u099c\u09c7\u09a8 \u0997\u09cd\u09af\u09be\u09b8 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09ad\u09b0\u09cd\u09a4\u09bf \u0995\u09b0\u09be \u09b9\u09b2 \u0964 \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be 50^{\\circ} \\mathrm{C} <\/span> \u09b9\u09b2\u09c7, \u09aa\u09be\u09a4\u09cd\u09b0\u09c7 \u099a\u09be\u09aa \u0995\u09a4?
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong>
\n(a) \u09a8\u09c7\u0997\u09c7\u099f\u09bf\u09ad \u09aa\u09be\u09a4\u099f\u09bf \u0985\u09cd\u09af\u09be\u09a8\u09cb\u09a1\u0964 \\left(\\mathrm{PbO}_{2}\\right. <\/span> \u09ac\u09bf\u09b9\u09c0\u09a8 \\mathrm{Pb} <\/span> \u09aa\u09be\u09a4 \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09a8\u09c7\u0997\u09c7\u099f\u09bf\u09ad \u09aa\u09be\u09a4\u099f\u09bf \u0985\u09cd\u09af\u09be\u09a8\u09cb\u09a1\u0964)<\/p>\n

\u09aa\u099c\u09bf\u099f\u09bf\u09ad \u09aa\u09be\u09a4\u09c7 \u09ac\u09bf\u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u0997\u09c1\u09b2\u09bf \u09b9\u09b2\u09cb\u0983
\n\n\n\\begin{array}{l}\n\n\\mathrm{Pb} \\longrightarrow \\mathrm{Pb}^{2+}+2 \\mathrm{e}^{-} \\\\\n\n\\mathrm{Pb}^{2+}+\\mathrm{SO}_{4}^{2-} \\longrightarrow \\mathrm{PbSO}_{4}(\\mathrm{~s}) \\\\\n\n\\rule{7cm}{0.025cm}\\\\\n\n\\mathrm{Pb}+\\mathrm{SO}_{4}^{2-} \\longrightarrow \\mathrm{PbSO}_{4}(\\mathrm{~s})+2 \\mathrm{e}^{-}\n\n\\end{array}\n\n<\/span><\/p>\n

\u09a8\u09c7\u0997\u09c7\u099f\u09bf\u09ad \u09aa\u09be\u09a4\u09c7 \u09ac\u09bf\u0995\u09cd\u09b0\u09bf\u09df\u09be\u0997\u09c1\u09b2\u09bf \u09b9\u09b2\u09cb\u0983
\n\n\n\\begin{array}{l}\n\n\\mathrm{PbO}_{2}(\\mathrm{~s})+2 \\mathrm{H}_{2} \\mathrm{O} \\rightleftharpoons \\mathrm{Pb}^{4+}+4 \\mathrm{OH}^{-} \\\\\n\n\\mathrm{Pb}^{4+}+2 \\mathrm{e}^{-} \\longrightarrow \\mathrm{Pb}^{2+} \\\\\n\n\\mathrm{Pb}^{2+}+\\mathrm{SO}_{4}^{2-} \\longrightarrow \\mathrm{PbSO}_{4}(\\mathrm{~s}) \\\\\n\n4 \\mathrm{OH}^{-}+4 \\mathrm{H}^{+} \\longrightarrow 4 \\mathrm{H}_{2} \\mathrm{O} \\\\\n\n\\rule{11cm}{0.025cm}\\\\\n\n\\mathrm{PbO}_{2}(\\mathrm{~s})+4 \\mathrm{H}^{+}+\\mathrm{SO}_{4}^{2-}+2 \\mathrm{e}^{-} \\longrightarrow \\mathrm{PbSO}_{4}(\\mathrm{~s})+2 \\mathrm{H}_{2} \\mathrm{O}\n\n\\end{array}\n\n<\/span><\/p>\n

(b)
\n\n\n\\begin{array}{l}\n\n\\mathrm{n}_{\\mathrm{H}_{2}}=\\frac{\\mathrm{W}_{\\mathrm{H}_{2}}}{\\mathrm{M}_{\\mathrm{H}_{2}}}=\\frac{4}{2}=2 \\text { mole }, \\mathrm{n}_{\\mathrm{N}_{2}}=\\frac{\\mathrm{w}_{\\mathrm{N}_{2}}}{\\mathrm{M}_{\\mathrm{N}_{2}}}=\\frac{7}{28}=0.25 \\text { mole } \\\\\n\n\\mathrm{n}=\\mathrm{n}_{\\mathrm{N}_{2}}+\\mathrm{n}_{\\mathrm{H}_{2}}=0.25+2=2.25 \\text { mole }, \\mathrm{V}=5 \\mathrm{~L} \\\\\n\n\\mathrm{~T}=50^{\\circ} \\mathrm{C}=323 \\mathrm{~K}, \\mathrm{R}=0.0821 \\text { Latm } \\mathrm{mole}^{-1} \\mathrm{~K}^{-1} \\\\\n\n\\mathrm{PV}=\\mathrm{nRT} \\Rightarrow \\mathrm{P}=\\frac{\\mathrm{nRT}}{\\mathrm{V}}=\\frac{2.25 \\times 0.0821 \\times 323}{5}=11.933 \\text { atm (Ans.) }\n\n\\end{array}\n\n<\/span><\/p>\n

45. \\mathrm{Au} <\/span> \u09a4\u09dc\u09bf\u09ce\u09a6\u09cd\u09ac\u09be\u09b0\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af \u09a6\u09bf\u09af\u09bc\u09c7 \\mathrm{AuCl}_{4}^{-} <\/span> \u0986\u09af\u09bc\u09a8\u09c7\u09b0 \u09a6\u09cd\u09b0\u09ac\u09a3\u09c7 \u0985\u09aa\u09b0\u09bf\u09ac\u09a4\u09a8\u09c0\u09af\u09bc \u09a4\u09dc\u09bf\u09ce \u09aa\u09cd\u09b0\u09ac\u09be\u09b9\u09bf\u09a4 \u0995\u09b0\u09be \u09b9\u09b2\u0964 10.0 <\/span> \u09ae\u09bf\u09a8\u09bf\u099f \u09aa\u09b0 \u0995\u09cd\u09af\u09be\u09a5\u09cb\u09a1\u09c7\u09b0 \u09ad\u09b0 1.314 \\mathrm{~gm} <\/span> \u09ac\u09c3\u09a6\u09cd\u09a7\u09bf \u09aa\u09c7\u09b2\u09c7 \u09a4\u09a1\u09bc\u09bf\u09ce \u09aa\u09cd\u09b0\u09ac\u09be\u09b9\u09c7\u09b0 \u09ae\u09be\u09a8 \u09ac\u09be\u09b9\u09bf\u09b0 \u0995\u09b0\u0964 (Au \u098f\u09b0 \u09aa\u09be\u09b0\u09ae\u09be\u09a3\u09ac\u09bf\u0995 \u09ad\u09b0 = 197)
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u0995\u09cd\u09af\u09be\u09a5\u09cb\u09a1 \u09ac\u09bf\u0995\u09cd\u09b0\u09bf\u09df\u09be\u0983 \\mathrm{A}^{3+}+3 \\mathrm{e}^{-} \\longrightarrow \\mathrm{Au} <\/span>
\n\u09ac\u09bf\u0995\u09cd\u09b0\u09bf\u09df\u09be \u09a5\u09c7\u0995\u09c7 \u09a6\u09c7\u0996\u09be \u09af\u09be\u09df \\mathrm{e}_{\\mathrm{Au}}=3, \u09ae\u09cb\u09b2\u09b8\u0982\u0996\u09cd\u09af\u09be [katex] \\mathrm{n}_{\\mathrm{Au}}=\\frac{\\mathrm{W}}{\\mathrm{m}}=\\frac{1.314}{197} \\mathrm{~mole} <\/span>
\n\u0985\u09a4\u09bf\u0995\u09cd\u09b0\u09be\u09a8\u09cd\u09a4 \u09b8\u09ae\u09af\u09bc \\mathrm{t}=10 \\mathrm{~min}=10 \\times 60 \\mathrm{sec}, \u09a4\u09a1\u09bc\u09bf\u09ce \u09aa\u09cd\u09b0\u09ac\u09be\u09b9 [katex] =\\mathrm{I} <\/span>
\n\u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf, \\mathrm{It}=\\mathrm{n}_{\\mathrm{Au}} \\mathrm{e}_{\\mathrm{Au}} \\mathrm{F} \\Rightarrow \\mathrm{I} \\times 10 \\times 60=\\frac{1.314}{197} \\times 3 \\times 96500 \\Rightarrow \\mathrm{I}=3.22 \\mathrm{~A~~~~~(Ans.)} <\/span><\/p>\n

46. \u099f\u09b2\u09c1\u0987\u09a8\u0995\u09c7 \u0995\u09bf\u09ad\u09be\u09ac\u09c7 \u09ac\u09c7\u09a8\u099c\u09bf\u09a8\u09c7 \u098f\u09ac\u0982 \u09ac\u09c7\u09a8\u099c\u09bf\u09a8\u0995\u09c7 \u099f\u09b2\u09c1\u0987\u09a8\u09c7 \u09aa\u09b0\u09bf\u09a3\u09a4 \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc? \u09b0\u09be\u09b8\u09be\u09af\u09bc\u09a8\u09bf\u0995 \u09ac\u09bf\u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b0 \u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09c7 \u09a6\u09c7\u0996\u09be\u0993\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983 \"\"<\/strong><\/p>\n

\"\"<\/p>\n

47. \u0995\u09cb\u09a8 \u09af\u09cc\u0997\u09c7 \\ell <\/span> \u09b6\u09cb\u09b7\u09a3 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af\u09c7\u09b0 \u0995\u09cb\u09a8\u09cb \u09b8\u09c7\u09b2\u09c7 \u0986\u09aa\u09a4\u09bf\u09a4 \u09b0\u09b6\u09cd\u09ae\u09bf\u09b0 10 \u09b6\u09a4\u09be\u0982\u09b6 \u0986\u09b2\u09cb \u09b6\u09cb\u09b7\u09a3 \u0995\u09b0\u09c7\u0964 \u09b8\u09c7\u09b2\u09c7\u09b0 \u09b6\u09cb\u09b7\u09a8 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af \u09aa\u09be\u0981\u099a \u0997\u09c1\u09a3 \u09ac\u09c7\u09b6\u09c0 \u09b9\u09b2\u09c7 \u0986\u09aa\u09a4\u09bf\u09a4 \u09b0\u09b6\u09cd\u09ae\u09bf\u09b0 \u0995\u09a4 \u0985\u0982\u09b6 \u09b6\u09cb\u09b7\u09bf\u09a4 \u09b9\u09ac\u09c7?
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf, \\log \\frac{\\mathrm{I}_{0}}{\\mathrm{I}} \\propto \\ell <\/span>
\n\u09e7\u09ae \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7, \\mathrm{I}_{\\mathrm{i}}=0.9 ~\\mathrm{I}_{0_{\\mathrm{i}}} ~~~~\\& ~~~~\\ell_{\\mathrm{i}}=\\ell <\/span>
\n\u09e8\u09af\u09bc \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7, \\ell_{\\mathrm{f}}=\\ell_{\\mathrm{i}}+5 \\times \\ell_{\\mathrm{i}}=6 \\ell_{\\mathrm{i}}=6 \\ell <\/span>
\n\u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf,
\n \\frac{\\log \\left(\\frac{\\mathrm{I}_{0_{\\mathrm{i}}}}{\\mathrm{I}_{\\mathrm{i}}}\\right)}{\\log \\left(\\frac{\\mathrm{I}_{0_{\\mathrm{f}}}}{\\mathrm{I}_{\\mathrm{f}}}\\right)}=\\frac{\\ell_{\\mathrm{i}}}{\\ell_{\\mathrm{f}}} \\Rightarrow \\log \\left(\\frac{\\mathrm{I}_{0_{\\mathrm{f}}}}{\\mathrm{I}_{\\mathrm{f}}}\\right)=\\frac{\\ell_{\\mathrm{f}}}{\\ell_{\\mathrm{i}}} \\times \\log \\left(\\frac{\\mathrm{I}_{\\mathrm{0}_{\\mathrm{i}}}}{\\mathrm{I}_{\\mathrm{i}}}\\right)=\\frac{6 \\ell}{\\ell} \\times \\log \\left(\\frac{\\mathrm{I}_{0_{\\mathrm{i}}}}{0.9 \\times \\mathrm{I}_{0_{\\mathrm{i}}}}\\right)=6 \\times \\log \\left(\\frac{1}{0.9}\\right) <\/span>
\n \\log \\left(\\frac{\\mathrm{I}_{\\mathrm{0}_{\\mathrm{f}}}}{\\mathrm{I}_{\\mathrm{f}}}\\right)=0.2745 \\Rightarrow \\frac{\\mathrm{I}_{0_{\\mathrm{f}}}}{\\mathrm{I}_{\\mathrm{f}}}=1.8816 \\Rightarrow \\mathrm{I}_{\\mathrm{f}}=0.5314 ~\\mathrm{I}_{0_{\\mathrm{f}}} <\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09bf\u09a4 \u09b6\u09a4\u0995\u09b0\u09be \u09b6\u09cb\u09b7\u09a3 =\\left(\\frac{\\mathrm{I}_{0_ \\mathrm{f}}-\\mathrm{I}_{\\mathrm{f}}}{\\mathrm{I}_{0_{\\mathrm{f}}}}\\right) \\times 100 \\%=\\left(\\frac{\\mathrm{I}_{0_\\mathrm{f}}-0.5314 \\mathrm{I}_{0_\\mathrm{f}}}{\\mathrm{I}_{0_\\mathrm{f}}}\\right) \\times 100 \\% =(1-0.5314) \\times 100 \\%=46.86 \\%~~~~~\\text{(Ans.)} <\/span><\/p>\n

48. \\mathrm{Ca}^{2+} <\/span> \u0986\u09af\u09bc\u09a8\u0995\u09c7 \\mathrm{CaCO}_{3} <\/span> \u09b9\u09bf\u09b8\u09be\u09ac\u09c7 \u09b8\u09ae\u09cd\u09aa\u09c2\u09b0\u09cd\u09a3\u09b0\u09c2\u09aa\u09c7 \u0985\u09a7\u0983\u0995\u09cd\u09b7\u09bf\u09aa\u09cd\u09a4 \u0995\u09b0\u09a4\u09c7 200 \\mathrm{~mL} <\/span> \u0986\u09af\u09bc\u09a4\u09a8\u09c7\u09b0 \u098f\u0995\u099f\u09bf \u09aa\u09be\u09a8\u09bf\u09b0 \u09a8\u09ae\u09c1\u09a8\u09be\u09af\u09bc 0.025 ~\\mathrm{M} <\/span> \u0998\u09a8\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09b0 5.0 \\mathrm{~mL} ~~\\mathrm{Na}_{2} \\mathrm{CO}_{3} <\/span> \u09a6\u09cd\u09b0\u09ac\u09a3 \u09aa\u09cd\u09b0\u09af\u09bc\u09cb\u099c\u09a8 \u09b9\u09b2\u0964 \u09aa\u09be\u09a8\u09bf\u09b0 \u09a8\u09ae\u09c1\u09a8\u09be\u099f\u09bf\u09b0 \u0995\u09cd\u09b7\u09b0\u09a4\u09be \\text{ppm -} <\/span> \u098f \u09b9\u09bf\u09b8\u09be\u09ac \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \\mathrm{Na}_{2} \\mathrm{CO}_{3} \\longrightarrow 2 \\mathrm{Na}^{+}+\\mathrm{CO}_{3}^{2-} <\/span>
\n \\mathrm{CO}_{3}^{2-}+\\mathrm{Ca}^{2+} \\longrightarrow \\mathrm{CaCO}_{3} <\/span>
\n\u09ac\u09bf\u0995\u09cd\u09b0\u09bf\u09df\u09be \u09a5\u09c7\u0995\u09c7 \u09b2\u09bf\u0996\u09be \u09af\u09be\u09df \u09af\u09c7,
\n \\mathrm{n}_{\\mathrm{Na}_{2} \\mathrm{CO}_{3}}=\\mathrm{n}_{\\mathrm{CO}_{3}^{2-}}=\\mathrm{n}_{\\mathrm{CaCO}_{3}} <\/span>
\n\n\n\\begin{array}{l}\n\n\\therefore \\mathrm{n}_{\\mathrm{CaCO}_{3}}=\\mathrm{V}_{\\mathrm{Na}_{2} \\mathrm{CO}_{3}} \\times \\mathrm{S}_{\\mathrm{Na}_{2} \\mathrm{CO}_{3}}=5 \\times 10^{-3} \\times 0.025=1.25 \\times 10^{-4} \\mathrm{~mole} \\\\\n\n\\therefore\\mathrm{s}=\\frac{\\mathrm{n}_{~\\mathrm{CaCO}_{3}}}{\\mathrm{~V}_{\\mathrm{H}_{2} \\mathrm{O}}}=\\frac{1.25 \\times 10^{-4}}{200 \\times 10^{-3}}=6.25 \\times 10^{-4} \\mathrm{M} \\\\\n\n=6.25 \\times 10^{-4} \\mathrm{~mole} \\mathrm{~L}^{-1}=6.25 \\times 10^{-4} \\times 100 ~\\mathrm{gL}^{-1}=0.0625 ~\\mathrm{gL}^{-1} \\\\\n\n=0.0625 \\times 10^{3} ~\\mathrm{mgL}^{-1}=62.5 ~\\mathrm{mgL}^{-1}=62.5 ~\\mathrm{ppm} ~~~~\\text { (Ans.) }\n\n\\end{array}\n\n<\/span><\/p>\n

49. \u098f\u0995\u099f\u09bf \\text{1.0 L} <\/span> \u0986\u09df\u09a4\u09a8\u09c7\u09b0 \u09b6\u09c1\u09b7\u09cd\u0995 \u09ac\u09be\u09a4\u09be\u09b8\u09c7\u09b0 \u09a8\u09ae\u09c1\u09a8\u09be\u09df \\mathrm{25^{\\circ}} <\/span> \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09df \u0993 \\mathrm{786~mm~Hg} <\/span> \u099a\u09be\u09aa\u09c7 \\mathrm{0.925~gm~~N_2} <\/span> \u098f\u09ac\u0982 \u0985\u099c\u09be\u09a8\u09be \u09aa\u09b0\u09bf\u09ae\u09be\u09a3 \u0985\u0995\u09cd\u09b8\u09bf\u099c\u09c7\u09a8, \u0986\u09b0\u09cd\u0997\u09a8 \u0993 \u0995\u09be\u09b0\u09cd\u09ac\u09a8 \u09a1\u09be\u0987\u0985\u0995\u09cd\u09b8\u09be\u0987\u09a1 \u09b8\u09b9 \u0985\u09a8\u09cd\u09af\u09be\u09a8\u09cd\u09af \u0997\u09cd\u09af\u09be\u09b8 \u0986\u099b\u09c7\u0964 \u098f\u0987 \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09af\u09bc \u09ac\u09be\u09a4\u09be\u09b8\u09c7\u09b0 \u09a8\u09ae\u09c1\u09a8\u09be\u09df \\mathrm{N}_{2} <\/span> \u098f\u09b0 \u0986\u0982\u09b6\u09bf\u0995 \u099a\u09be\u09aa \u0995\u09a4?
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \\mathrm{P}=\\frac{786}{760} \\mathrm{~atm} ; \\mathrm{T}=25^{\\circ} \\mathrm{C}=298 \\mathrm{~K} ; \\mathrm{V}=1 \\mathrm{~L} <\/span>
\n\n\n\\begin{array}{l}\n\n\\mathrm{n}_{\\text {total }}=\\frac{\\mathrm{PV}}{\\mathrm{RT}}=\\frac{786 ~\\times~ 1}{760~ \\times~ 0.0821 ~\\times ~298}=0.04227 \\mathrm{~mol} \\\\\n\n\\mathrm{n}_{\\mathrm{N}_{2}}=\\frac{\\mathrm{w}}{\\mathrm{m}}=\\frac{0.925}{28}=0.03304 \\mathrm{~mol} \\\\\n\n\\text{Partial pressure of}\\mathrm{~N}_{2}, ~\\mathrm{P}_{\\mathrm{N}_{2}}=\\frac{\\mathrm{n}_{\\mathrm{N}_{2}}}{\\mathrm{n}_{\\text {total }}} \\times \\mathrm{P}=\\frac{0.03304}{0.04227} \\times \\frac{786}{760} \\text { atm }=0.81 \\mathrm{~atm} \\text { (Ans.) }\n\n\\end{array}\n\n<\/span><\/p>\n

50. (a) \u09a8\u09bf\u099a\u09c7\u09b0 \u0985\u09cd\u09af\u09be\u09a8\u09be\u09df\u09a8 \u0997\u09c1\u09b2\u09cb\u09b0 \u09b8\u0982\u0995\u09c7\u09a4 \u09b2\u09bf\u0996\u0964
\n(i) \u098f\u09b8\u09bf\u099f\u09c7\u099f (ii) \u0986\u09af\u09bc\u09a1\u09c7\u099f (iii) \u09aa\u09be\u09b0\u0995\u09cd\u09b2\u09cb\u09b0\u09c7\u099f (iv) \u0985\u0995\u09cd\u09b8\u09be\u09b2\u09c7\u099f (v) \u0986\u09b0\u09cd\u09b8\u09bf\u09a8\u09c7\u099f<\/p>\n

(b) \u09a8\u09bf\u099a\u09c7\u09b0 \u09b8\u0999\u09cd\u0995\u09b0 \u09a7\u09be\u09a4\u09c1\u0997\u09c1\u09b2\u09cb\u09a4\u09c7 \u0989\u09aa\u09b8\u09cd\u09a5\u09bf\u09a4 \u09a7\u09be\u09a4\u09c1\u09b8\u09ae\u09c2\u09b9\u09c7\u09b0 \u09a8\u09be\u09ae \u09b2\u09bf\u0996\u0964
\n(i) \u09b8\u09cd\u099f\u09c7\u0987\u09a8\u09b2\u09c7\u09b8 \u09b8\u09cd\u099f\u09c0\u09b2 (ii) \u09a8\u09bf\u099f\u09bf\u09a8\u09cb\u09b2 (iii) \u09ac\u09cd\u09b0\u09be\u09b8 (iv) \u09ac\u09cd\u09b0\u09cb\u099e\u09cd\u099c (v) \u09b8\u09cb\u09b2\u09cd\u09a1\u09be\u09b0<\/p>\n

\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong><\/p>\n

(a) \"\"<\/p>\n

(b) (i) \u0986\u09df\u09b0\u09a8, \u0995\u09cd\u09b0\u09cb\u09ae\u09bf\u09df\u09be\u09ae, \u09a8\u09bf\u0995\u09c7\u09b2 \u09ae\u09b2\u09bf\u09ac\u09a1\u09c7\u09a8\u09be\u09ae, \u099f\u09be\u0987\u099f\u09c7\u09a8\u09bf\u09df\u09be\u09ae (ii) \u09a8\u09bf\u0995\u09c7\u09b2, \u099f\u09be\u0987\u099f\u09c7\u09a8\u09bf\u09df\u09be\u09ae (iii) \u0995\u09aa\u09be\u09b0, \u099c\u09bf\u0982\u0995 (iv) \u0995\u09aa\u09be\u09b0, \u099f\u09bf\u09a8 (v) \u09b2\u09c7\u09a1, \u099f\u09bf\u09a8, \u0987\u09a8\u09a1\u09bf\u09df\u09be\u09ae<\/p>\n

51. \u09a4\u09bf\u09a8\u099f\u09bf \u09af\u09cc\u0997 \\mathrm{E}, \\mathrm{F} <\/span> \u098f\u09ac\u0982 \\mathrm{G} <\/span> \u098f\u09b0 \u0986\u09a3\u09ac\u09bf\u0995 \u09b8\u0982\u0995\u09c7\u09a4 \\mathrm{C}_{3} \\mathrm{H}_{6} \\mathrm{O} <\/span> \u0964 \\mathrm{E} <\/span> \u0985\u09cd\u09af\u09be\u09b2\u0995\u09cb\u09b9\u09b2, \\mathrm{F} <\/span> \u0995\u09bf\u099f\u09cb\u09a8 \u098f\u09ac\u0982 \\mathrm{G} <\/span> \u0985\u09cd\u09af\u09be\u09b2\u09a1\u09bf\u09b9\u09be\u0987\u09a1\u0964 \\mathrm{E}, \\mathrm{F} <\/span> \u098f\u09ac\u0982 \\mathrm{G} <\/span> \u09af\u09cc\u0997\u09c7\u09b0 \u09b8\u09ae\u09cd\u09ad\u09be\u09ac\u09cd\u09af \u0997\u09be\u09a0\u09a8\u09bf\u0995 \u09b8\u0982\u0995\u09c7\u09a4 \u09b8\u09ae\u09c2\u09b9 \u09b2\u09bf\u0996\u0964<\/p>\n

\"\"\"\"<\/p>\n

52.
\n(a) \u098f\u0995\u099f\u09bf \u09a8\u09be\u0987\u099f\u09cd\u09b0\u09cb\u099c\u09c7\u09a8\u09af\u09c1\u0995\u09cd\u09a4 \u09b8\u09be\u09b0 \u0993 \u098f\u0995\u099f\u09bf \u09ab\u09b8\u09ab\u09b0\u09be\u09b8\u09af\u09c1\u0995\u09cd\u09a4 \u09b8\u09be\u09b0\u09c7\u09b0 \u09a8\u09be\u09ae \u0997\u09be\u09a0\u09a8\u09bf\u0995 \u09b8\u0982\u0995\u09c7\u09a4\u09b8\u09b9 \u09b2\u09bf\u0996\u0964
\n(b) \u09b2\u09bf\u09aa\u09b8\u09cd\u099f\u09bf\u0995 \u098f \u09ac\u09cd\u09af\u09ac\u09b9\u09c3\u09a4 \u09ae\u09af\u09bc\u09b6\u09cd\u099a\u09be\u09b0\u09be\u0987\u099c\u09be\u09b0 \u098f\u09b0 \u0997\u09be\u09a0\u09a8\u09bf\u0995 \u09b8\u0982\u0995\u09c7\u09a4 \u09b2\u09bf\u0996\u0964
\n(c) \u099a\u09bf\u09aa\u09b8, \u099a\u09be\u09a8\u09be\u099a\u09c1\u09b0 \u0987\u09a4\u09cd\u09af\u09be\u09a6\u09bf \u09a4\u09c8\u09b0\u09bf\u09a4\u09c7 \u09ac\u09cd\u09af\u09ac\u09b9\u09c3\u09a4 \u098f\u0995\u099f\u09bf \u09aa\u09cd\u09b0\u09bf\u099c\u09be\u09b0\u09ad\u09c7\u099f\u09bf\u09ad \u098f\u09b0 \u0997\u09be\u09a0\u09a8\u09bf\u0995 \u09b8\u0982\u0995\u09c7\u09a4 \u09b2\u09bf\u0996\u0964
\n(d) \u0997\u09cd\u09b2\u09be\u09b8 \u0995\u09cd\u09b2\u09bf\u09a8\u09be\u09b0\u09c7 \u0995\u09cb\u09a8 \u0995\u09cd\u09b7\u09be\u09b0 \u09ac\u09cd\u09af\u09ac\u09b9\u09be\u09b0 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964<\/p>\n

\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983\u00a0<\/strong><\/p>\n

(a) \u09a8\u09be\u0987\u099f\u09cd\u09b0\u09cb\u099c\u09c7\u09a8\u09af\u09c1\u0995\u09cd\u09a4 \u09b8\u09be\u09b0 - \u0987\u0989\u09b0\u09bf\u09df\u09be\u0964 \"\"<\/p>\n

\u09ab\u09b8\u09ab\u09b0\u09be\u09b8\u09af\u09c1\u0995\u09cd\u09a4 \u09b8\u09be\u09b0 -\u00a0 Sigma-Aldrich<\/p>\n

\"\"<\/p>\n

(b)\u00a0 \u09b2\u09bf\u09aa\u09b8\u09cd\u099f\u09bf\u0995 \u098f \u09ac\u09cd\u09af\u09ac\u09b9\u09c3\u09a4 \u09ae\u09af\u09bc\u09b6\u09cd\u099a\u09be\u09b0\u09be\u0987\u099c\u09be\u09b0 \u0997\u09cd\u09b2\u09bf\u09b8\u09be\u09b0\u09b2\u0964<\/p>\n

\"\"<\/p>\n

(c) \u099a\u09bf\u09aa\u09b8, \u099a\u09be\u09a8\u09be\u099a\u09c1\u09b0 \u0987\u09a4\u09cd\u09af\u09be\u09a6\u09bf \u09a4\u09c8\u09b0\u09bf\u09a4\u09c7 \u09ac\u09cd\u09af\u09ac\u09b9\u09c3\u09a4 \u098f\u0995\u099f\u09bf \u09aa\u09cd\u09b0\u09bf\u099c\u09be\u09b0\u09ad\u09c7\u099f\u09bf\u09ad- \u09b8\u09cb\u09a1\u09bf\u09df\u09be\u09ae \u09ac\u09c7\u09a8\u099c\u09df\u09c7\u099f<\/p>\n

\"\"<\/p>\n

(d) \u0985\u09cd\u09af\u09be\u09ae\u09cb\u09a8\u09bf\u09df\u09be\u09ae \u09b9\u09be\u0987\u09a1\u09cd\u09b0\u09cb\u0995\u09cd\u09b8\u09be\u0987\u09a1 Ammonium Hydroxide (NH_4OH<\/span>)<\/p>\n

53. \u098f\u0995 \u09b2\u09bf\u099f\u09be\u09b0 2.10 <\/span> \u0986\u09aa\u09c7\u0995\u09cd\u09b7\u09bf\u0995 \u0997\u09c1\u09b0\u09c1\u09a4\u09cd\u09ac \u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \\mathrm{H}_{2} \\mathrm{SO}_{4} <\/span> \u098f \u0995\u09bf \u09aa\u09b0\u09bf\u09ae\u09be\u09a3 \u09aa\u09be\u09a4\u09bf\u09a4 \u09aa\u09be\u09a8\u09bf \u09af\u09cb\u0997 \u0995\u09b0\u09b2\u09c7 \u0989\u09b9\u09be 1.40 <\/span> \u0986\u09aa\u09c7\u0995\u09cd\u09b7\u09bf\u0995 \u0997\u09c1\u09b0\u09c1\u09a4\u09cd\u09ac \u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f \\mathrm{H}_{2} \\mathrm{SO}_{4} <\/span> \u098f \u09aa\u09b0\u09bf\u09a3\u09a4 \u09b9\u09ac\u09c7?
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong>
\n\u0986\u09aa\u09c7\u0995\u09cd\u09b7\u09bf\u0995 \u0997\u09c1\u09b0\u09c1\u09a4\u09cd\u09ac = \u09a6\u09cd\u09b0\u09ac\u09a3\u09c7\u09b0 \u0998\u09a8\u09a4\u09cd\u09ac \/ <\/span> 4^{\\circ}\\mathrm{~C} <\/span> \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09df \u09aa\u09be\u09a8\u09bf\u09b0 \u0998\u09a8\u09a4\u09cd\u09ac = \u09a6\u09cd\u09b0\u09ac\u09a3\u09c7\u09b0 \u09ad\u09b0 \/ <\/span> \u09a6\u09cd\u09b0\u09ac\u09a3\u09c7\u09b0 \u0986\u09df\u09a4\u09a8 ~\\times 4^{\\circ}\\mathrm{~C} <\/span> \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09df \u09aa\u09be\u09a8\u09bf\u09b0 \u0998\u09a8\u09a4\u09cd\u09ac<\/p>\n

\u098f\u0996\u09be\u09a8\u09c7,
\n V_{\\text {Solution}}=1 \\mathrm{~L}=10^{-3} \\mathrm{~m}^{3}, \\mathrm{~d}_{1}=2.1, \\mathrm{~d}_{2}=1.4, \\mathrm{~V}_{\\text {Extra Water}}=? <\/span><\/p>\n

\u098f\u0996\u09a8,
\n \\mathrm{d_1=\\frac{M_{Soluion}}{V_{Solution}\\times1000}}~~~~~~~~~~\\therefore \\mathrm{M_{Solution}=2.1~kg} <\/span><\/p>\n

\u098f\u0996\u09a8,
\n \\mathrm{M_{Solution}^{\\prime}=\\mathrm{M}_{Solution}+\\mathrm{V}_{Extra~ Water} \\times \\rho_{~\\text {Water}}=2.1+\\mathrm{V}_{\\text {Extra Water}} \\times 1000} <\/span>
\n \\therefore \\mathrm{d_2=\\frac{{M^{~\\prime}_{Solution}}}{V_{Solution}+V_{Extra~Water}}\\Rightarrow V_{Extra~Water}=1.75\\times10^{-3}~m^3=1.75~L~~(Ans.)} <\/span><\/p>\n

54. \u09a8\u09bf\u09ae\u09cd\u09a8\u09b2\u09bf\u0996\u09bf\u09a4\u09ad\u09be\u09ac\u09c7 \u098f\u0995\u099f\u09bf \u09ae\u09bf\u09b6\u09cd\u09b0\u09a3 \u09a4\u09c8\u09b0\u09bf \u0995\u09b0\u09be \u09b9\u09b2\u0983 100 ~\\mathrm{ml} ~0.05 ~\\mathrm{M} ~~\\mathrm{Na}_{2} \\mathrm{CO}_{3} <\/span> \u09a6\u09cd\u09b0\u09ac\u09a3 +~100~ \\mathrm{ml} ~0.10 \\mathrm{~N} ~~\\mathrm{H}_{2} \\mathrm{SO}_{4} <\/span> \u09a6\u09cd\u09b0\u09ac\u09a3 +~50 ~\\mathrm{ml} ~0.1 \\mathrm{~N} ~~\\mathrm{NaOH} <\/span> \u09a6\u09cd\u09b0\u09ac\u09a3 +~50~ \\mathrm{ml} ~0.2 \\mathrm{~N}~~ \\mathrm{HCl} <\/span> \u09a6\u09cd\u09b0\u09ac\u09a3\u0964 \u09ae\u09bf\u09b6\u09cd\u09b0\u09a3\u099f\u09bf\u09b0 \u09b6\u0995\u09cd\u09a4\u09bf\u09ae\u09be\u09a4\u09cd\u09b0\u09be \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u0964
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \u09ae\u09bf\u09b6\u09cd\u09b0\u09a3\u09c7,
\n\n\n\\begin{array}{l}\n\n\\left[\\mathrm{H}^{+}\\right]=\\frac{\\sum\\left(\\mathrm{e}_{\\mathrm{Acid}} \\times \\mathrm{S}_{\\mathrm{acid}} \\times \\mathrm{V}_{\\mathrm{Acid}}\\right)-\\sum\\left(\\mathrm{e}_{\\mathrm{Base}} \\times \\mathrm{S}_{\\mathrm{Base}} \\times \\mathrm{V}_{\\mathrm{Base}}\\right)}{\\sum \\mathrm{V}_{\\mathrm{acid}}+\\sum \\mathrm{V}_{\\mathrm{Base}}} \\\\\n\n=\\frac{\\mathrm{e}_{\\mathrm{H}_{2} \\mathrm{SO}_{4}} \\times \\mathrm{S}_{\\mathrm{H}_{2} \\mathrm{SO}_{4}} \\times \\mathrm{V}_{\\mathrm{H}_{2} \\mathrm{SO}_{4}}+\\mathrm{e}_{\\mathrm{HCl}} \\times \\mathrm{S}_{\\mathrm{HCl}} \\times \\mathrm{V}_{\\mathrm{HCl}}-\\mathrm{e}_{\\mathrm{Na}_{2} \\mathrm{CO}_{3}} \\times \\mathrm{V}_{\\mathrm{Na}_{2} \\mathrm{CO}_{3}} \\times \\mathrm{S}_{\\mathrm{Na}_{2} \\mathrm{CO}_{3}}-\\mathrm{e}_{\\mathrm{NaOH}} \\times \\mathrm{S}_{\\mathrm{NaOH}} \\times \\mathrm{V}_{\\mathrm{NaOH}}}{\\mathrm{V}_{\\mathrm{H}_{2} \\mathrm{SO}_{4}}+\\mathrm{V}_{\\mathrm{HCl}}+\\mathrm{V}_{\\mathrm{Na}_{2} \\mathrm{CO}_{3}}+\\mathrm{V}_{\\mathrm{NaOH}}} \\\\\n\n=\\frac{\\mathrm{N}_{\\mathrm{H}_{2} \\mathrm{SO}_{4}} \\times \\mathrm{V}_{\\mathrm{H}_{2} \\mathrm{SO}_{4}}+\\mathrm{N}_{\\mathrm{HCl}} \\times \\mathrm{V}_{\\mathrm{HCl}}-\\mathrm{e}_{\\mathrm{Na}_{2} \\mathrm{CO}_{3}} \\times \\mathrm{V}_{\\mathrm{Na}_{2} \\mathrm{CO}_{3}} \\times \\mathrm{S}_{\\mathrm{Na}_{2} \\mathrm{CO}_{3}}-\\mathrm{N}_{\\mathrm{NaOH}} \\times \\mathrm{V}_{\\mathrm{NaOH}}}{\\mathrm{V}_{\\mathrm{H}_{2} \\mathrm{SO}_{4}}+\\mathrm{V}_{\\mathrm{HCl}}+\\mathrm{V}_{\\mathrm{Na}_{2} \\mathrm{CO}_{3}}+\\mathrm{V}_{\\mathrm{NaOH}}} \\\\\n\n=\\frac{0.1 \\times 100+50 \\times 0.2-2 \\times 0.05 \\times 100-0.1 \\times 50}{100+100+50+50}=\\frac{1}{60} \\mathrm{M} \\\\\n\n\\end{array}\n\n<\/span>
\n\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u09b6\u0995\u09cd\u09a4\u09bf\u09ae\u09be\u09a4\u09cd\u09b0\u09be, \\mathrm{pH}=-\\log \\left[\\mathrm{H}^{+}\\right]=-\\log \\left(\\frac{1}{60}\\right)=1.778 \\text { (Ans.) } <\/span><\/p>\n

55. \u09af\u09a6\u09bf \u09aa\u09be\u09a8\u09bf\u09b0 \u0997\u09a0\u09a8 \u09a4\u09be\u09aa 25^{\\circ} \\mathrm{C} <\/span> \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09af\u09bc -68370 ~\\mathrm{Cal} <\/span> \u09b9\u09af\u09bc \u09a4\u09ac\u09c7 60^{\\circ} \\mathrm{C} <\/span> \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09af\u09bc \u09a4\u09be \u0995\u09a4 \u09b9\u09ac\u09c7? \u09a6\u09c7\u09af\u09bc\u09be \u0986\u099b\u09c7, \\mathrm{C}_{\\mathrm{P}}\\left(\\mathrm{H}_{2}\\right)=6.90 ~\\mathrm{Cal} ~\\mathrm{mole}^{-1} \\mathrm{deg}^{-1},~\\mathrm{C}_{\\mathrm{P}}\\left(\\mathrm{O}_{2}\\right)=7.05 ~\\mathrm{Cal ~mole ^{-1}} \\mathrm{deg}^{-1},~\\mathrm{C}_{\\mathrm{P}}\\left(\\mathrm{H}_{2} \\mathrm{O}\\right)=18.0~\\mathrm{Cal mole^{-1}} \\mathrm{deg}^{-1} <\/span>
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \\mathrm{H}_{2}(\\mathrm{~g})+\\frac{1}{2} \\mathrm{O}_{2}(\\mathrm{~g})=\\mathrm{H}_{2} \\mathrm{O}(~\\mathrm{l}) <\/span>
\n 25^{\\circ} \\mathrm{C} <\/span> \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09df \u0997\u09a0\u09a8 \u09a4\u09be\u09aa -68370 ~\\mathrm{Cal} <\/span>
\n\u0985\u09b0\u09cd\u09a5\u09be\u09ce 1 mole \u09aa\u09be\u09a8\u09bf \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09a4\u09c7 68370 ~\\mathrm{Cal} <\/span> \u09a4\u09be\u09aa \u0989\u09ce\u09aa\u0928\u094d\u0928 \u09b9\u09af\u09bc\u0964
\n 60^{\\circ} \\mathrm{C} <\/span> \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09af\u09bc hydrogen, oxygen \u0995\u09b0\u09cd\u09a4\u09c3\u0995 \u0985\u09a4\u09bf\u09b0\u09bf\u0995\u09cd\u09a4 \u0997\u09c3\u09b9\u09c0\u09a4 \u09a4\u09be\u09aa,
\n =\\left(6.90+\\frac{1}{2} \\times 7.05\\right) \\times 35 ~\\mathrm{Cal}=364.875 ~\\mathrm{Cal} <\/span>
\n 60^{\\circ} \\mathrm{C} \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09df \u09aa\u09be\u09a8\u09bf \u0995\u09b0\u09cd\u09a4\u09c3\u0995 \u0997\u09c3\u09b9\u09c0\u09a4 \u0985\u09a4\u09bf\u09b0\u09bf\u0995\u09cd\u09a4 \u09a4\u09be\u09aa = [katex] \\mathrm{18.0\\times35~Cal=630~Cal} <\/span>
\n \\therefore \u0985\u09a4\u09bf\u09b0\u09bf\u0995\u09cd\u09a4 \u0997\u09c3\u09b9\u09c0\u09a4 \u09a4\u09be\u09aa\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ae\u09be\u09a3 [katex] =(630-364.875) \\mathrm{Cal}=265.125 ~\\mathrm{Cal} <\/span>
\n \\therefore 60^{\\circ} \\mathrm{C} \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09df \u0997\u09a0\u09a8 \u09a4\u09be\u09aa [katex] =(-68370+265.125) \\mathrm{Cal}=-68104.875 ~\\mathrm{Cal} <\/span><\/p>\n

56. \u09ab\u09be\u0981\u0995\u09be \u09b8\u09cd\u09a5\u09be\u09a8\u09c7 \u0995\u09c0 \u09ac\u09b8\u09ac\u09c7?<\/p>\n

\"\"<\/p>\n

\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong> \"\"<\/p>\n

57. \"\"<\/p>\n

\u09b8\u09ae\u09be\u09a7\u09be\u09a8: \"\"<\/strong><\/p>\n

58. \u09ac\u09bf\u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be\u09b0 \u09b8\u09be\u09b9\u09be\u09af\u09cd\u09af\u09c7 \u09a6\u09c7\u0996\u09be\u0993 \u0995\u09bf \u0998\u099f\u09c7 \u09af\u0996\u09a8\u0983
\n(i) \u0995\u0995\u09cd\u09b7 \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09af\u09bc \u0995\u09aa\u09be\u09b0 \u09b8\u09be\u09b2\u09ab\u09c7\u099f \u09a6\u09cd\u09b0\u09ac\u09a3\u09c7 \u09a7\u09c0\u09b0\u09c7 \u09a7\u09c0\u09b0\u09c7 \u0985\u09cd\u09af\u09be\u09ae\u09cb\u09a8\u09bf\u09af\u09bc\u09be \u09a6\u09cd\u09b0\u09ac\u09a3 \u09af\u09cb\u0997 \u0995\u09b0\u09b2\u09c7\u0983
\n(ii) \\mathrm{ZnCl}_{2} <\/span> (aq) \u098f\u09ac\u0982 \\mathrm{CuCl}_{2} <\/span> (aq) \u0985\u09ae\u09cd\u09b2\u09c0\u09af\u09bc \u09a6\u09cd\u09b0\u09ac\u09a3\u09c7 \u09b9\u09be\u0987\u09a1\u09cd\u09b0\u09cb\u099c\u09c7\u09a8 \u09b8\u09be\u09b2\u09ab\u09be\u0987\u09a1 \u0997\u09cd\u09af\u09be\u09b8 \u099a\u09be\u09b2\u09a8\u09be \u0995\u09b0\u09b2\u09c7\u0983
\n(iii) \u09b2\u09cb\u09b9\u09bf\u09a4 \u09a4\u09aa\u09cd\u09a4 \u09b2\u09cb\u09b9\u09be\u09b0 \u09a8\u09b2\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af \u09a6\u09bf\u09af\u09bc\u09c7 400^{\\circ} \\mathrm{C} <\/span> \u09a4\u09be\u09aa\u09ae\u09be\u09a4\u09cd\u09b0\u09be\u09df \u0989\u09a4\u09cd\u09a4\u09aa\u09cd\u09a4 \u0987\u09a5\u09be\u0987\u09a8 \u0997\u09cd\u09af\u09be\u09b8 \u099a\u09be\u09b2\u09a8\u09be \u0995\u09b0\u09b2\u09c7\u0983
\n(iv) \u0985\u09ae\u09cd\u09b2\u09c0\u09af\u09bc \u09ab\u09c7\u09b0\u09bf\u0995 \u09b8\u09be\u09b2\u09ab\u09c7\u099f \u09b2\u09ac\u09a3\u09c7\u09b0 \u09a6\u09cd\u09b0\u09ac\u09a3\u09c7 \u0985\u09a4\u09bf\u09b0\u09bf\u0995\u09cd\u09a4 \u0985\u09cd\u09af\u09be\u09ae\u09cb\u09a8\u09bf\u09af\u09bc\u09be\u09ae \u09a5\u09be\u09df\u09cb\u09b8\u09be\u09af\u09bc\u09be\u09a8\u09be\u0987\u09a1 \u09a6\u09cd\u09b0\u09ac\u09a3 \u09af\u09cb\u0997 \u0995\u09b0\u09b2\u09c7\u0983\"
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong><\/p>\n

\"\"
\n(i) \\mathrm{CuSO}_{4} <\/span> (aq) \\stackrel{\\mathrm{NH}_{3} \\text { (g) }}{\\longrightarrow}\\left[\\mathrm{Cu}\\left(\\mathrm{NH}_{3}\\right)_{4}\\left(\\mathrm{H}_{2} \\mathrm{O}\\right)\\right] \\mathrm{SO}_{4} <\/span>
\nOr, \\mathrm{CuSO}_{4}+\\mathrm{NH}_{4} \\mathrm{OH}(\\mathrm{aq}) \\rightarrow\\left(\\mathrm{NH}_{4}\\right)_{2} \\mathrm{SO}_{4}+\\mathrm{CuSO}_{4} . \\mathrm{Cu}(\\mathrm{OH})_{2} <\/span> [\u0995\u09cd\u09b7\u09be\u09b0\u09c0\u09df \u0995\u09aa\u09be\u09b0 \u09b8\u09be\u09b2\u09ab\u09c7\u099f]
\n(ii) \\mathrm{ZnCl}_{2}(\\mathrm{aq})+\\mathrm{CuCl}_{2}(\\mathrm{aq})+2 \\mathrm{H}_{2} \\mathrm{~S}(\\mathrm{~g}) \\stackrel{\\text { acidic }}{\\longrightarrow} \\mathrm{ZnS}+\\mathrm{CuS}+4 \\mathrm{HCl} <\/span>
\n(iii) 3 \\mathrm{CH} \\equiv \\mathrm{CH} \\frac{\\mathrm{Fe}}{400^{\\circ} \\mathrm{C}} <\/span>
\n(iv) \\mathrm{Fe}_{2}\\left(\\mathrm{SO}_{4}\\right)_{3}+6 \\mathrm{NH}_{4} \\mathrm{SCN} \\stackrel{\\text { acidic }}{\\longrightarrow} \\mathrm{Fe}\\left[\\mathrm{Fe}(\\mathrm{SCN})_{6}\\right]+3\\left(\\mathrm{NH}_{4}\\right)_{2} \\mathrm{SO}_{4} <\/span><\/p>\n

59. \u09aa\u09b0\u09cd\u09af\u09be\u09af\u09bc \u09b8\u09be\u09b0\u09a3\u09c0\u09b0 \u09a4\u09c3\u09a4\u09c0\u09af\u09bc \u09aa\u09b0\u09cd\u09af\u09be\u09af\u09bc\u09c7\u09b0 \u09ae\u09cc\u09b2\u0997\u09b2\u09cb\u0983 \\mathrm{Na}, \\mathrm{Mg}, \\mathrm{Al}, \\mathrm{Si}, \\mathrm{P}, \\mathrm{S}, \\mathrm{Cl}, \\mathrm{Ar} <\/span> \u0964 \u098f\u0987 \u09a4\u09a5\u09cd\u09af \u09ac\u09cd\u09af\u09ac\u09b9\u09be\u09b0 \u0995\u09b0\u09c7 \u09a8\u09bf\u09ae\u09cd\u09b2\u09c7\u09b0 \u09aa\u09cd\u09b0\u09b6\u09cd\u09a8\u0997\u09c1\u09b2\u09bf\u09b0 \u0989\u09a4\u09cd\u09a4\u09b0 \u09a6\u09be\u0993\u0964
\n(a) \u09af\u09c7 \u09ae\u09cc\u09b2\u0997\u09c1\u09b2\u09cb \u09a8\u09be\u0987\u099f\u09cd\u099f\u09c7\u099f \u0997\u09a0\u0928 \u0995\u09b0\u09c7 \u09a4\u09be\u09b0 \u09a4\u09be\u09b2\u09bf\u0995\u09be \u09a6\u09be\u0993\u0964
\n(b) \u098f\u09a6\u09c7\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 \u0995\u09cb\u09a8 \u09ae\u09cc\u09b2\u09a6\u09cd\u09ac\u09df \u09b8\u099a\u09b0\u09be\u099a\u09b0 \u09aa\u09cd\u09b0\u09be\u09aa\u09cd\u09a4 \u0986\u09af\u09bc\u09a8\u09bf\u0995 \u09af\u09cc\u0997 \u0989\u09ce\u09aa\u09a8\u09cd\u09a8 \u0995\u09b0\u09c7?
\n(c) \u0995\u09cb\u09a8 \u09ae\u09cc\u09b2\u099f\u09bf \u09b8\u09be\u09a7\u09be\u09b0\u09a3\u09a4 \u0985\u09a3\u09c1 \u09b9\u09bf\u09b8\u09be\u09ac\u09c7 \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09af\u09bc?
\n(d) \u0995\u09cb\u09a8 \u09ae\u09cc\u09b2\u09c7\u09b0 \u09a7\u09be\u09a4\u09ac \u0993 \u0985\u09a7\u09be\u09a4\u09ac \u0989\u09ad\u09df \u09ac\u09c8\u09b6\u09bf\u09b7\u09cd\u099f\u09cd\u09af \u0986\u099b\u09c7?
\n\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983<\/strong>
\n (a)~ \\mathrm{Na, ~Mg,~Al~~~(b) ~Na,~Mg~~~(c) ~Cl,~ P,~ S~~~(d) ~Si} <\/span><\/p>\n

60. (a) \u0997\u09be\u09a0\u09a8\u09bf\u0995 \u09b8\u0982\u0995\u09c7\u09a4 \u09b2\u09bf\u0996\u0983 \\mathrm{4,4^{\\prime}-dichlorodiphenyl~ 1, 1, 1 -trichloroethane} <\/span>
\n(b) IUPAC \u09a8\u09be\u09ae \u09b2\u09bf\u0996\u0983 \\left(\\mathrm{CH}_{3}\\right)_{2} \\mathrm{CH} \\mathrm{CO} \\mathrm{CH}_{3} <\/span>
\n(c) \u0995\u09be\u09b0\u09cd\u09af\u0995\u09b0\u09c0 \u09ae\u09c2\u09b2\u0995\u09c7\u09b0 \u0997\u09be\u09a0\u09a8\u09bf\u0995 \u09b8\u0982\u0995\u09c7\u09a4 \u09b2\u09bf\u0996\u0964
\n(i) \u098f\u09b8\u09cd\u099f\u09be\u09b0 (ii) \u0985\u09cd\u09af\u09be\u09ae\u09be\u0987\u09a1 (iii) \u0985\u09cd\u09af\u09be\u09a8\u09b9\u09be\u0987\u09a1\u09cd\u09b0\u09be\u0987\u09a1<\/p>\n

\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983 \"\"<\/strong><\/p>\n

\"\"<\/p>\n","protected":false},"excerpt":{"rendered":"

\u09b2\u09bf\u0996\u09bf\u09a4 \u0985\u0982\u09b6 (Written Part) \u0989\u099a\u09cd\u099a\u09a4\u09b0 \u0997\u09a3\u09bf\u09a4 (Higher Mathematics) 1. \u09af\u09a6\u09bf \u09b9\u09af\u09bc, \u09a4\u09be\u09b9\u09b2\u09c7 \u09a1\u09cb\u09ae\u09c7\u09a8 \u098f\u09ac\u0982 \u09b0\u09c7\u099e\u09cd\u099c\u09b8\u09b9 \u09ac\u09c7\u09b0 \u0995\u09b0\u0964 \u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983 \u09a7\u09b0\u09bf, \u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u098f\u0996\u09a8, \u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u098f\u09b0 \u09a1\u09cb\u09ae\u09c7\u09a8 \u098f\u09b0 \u09b0\u09c7\u099e\u09cd\u099c = R-{-1} (Ans) 2. \u09af\u09a6\u09bf , \u09b9\u09af\u09bc, \u09a4\u09be\u09b9\u09b2\u09c7 A \u09ae\u09cd\u09af\u09be\u099f\u09cd\u09b0\u09bf\u0995\u09cd\u09b8\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964 \u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983 \u09a7\u09b0\u09bf,<\/p>\n

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