MCQOPTIONS
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MCQOPTIONS
This section includes 8666 Mcqs, each offering curated multiple-choice questions to sharpen your Joint Entrance Exam - Main (JEE Main) knowledge and support exam preparation. Choose a topic below to get started.
| 5651. |
If \[y=\sqrt{\frac{1+x}{1-x}},\]then \[\frac{dy}{dx}=\] [AISSE 1981; RPET 1995] |
| A. | \[\frac{2}{{{(1+x)}^{1/2}}{{(1-x)}^{3/2}}}\] |
| B. | \[\frac{1}{{{(1+x)}^{1/2}}{{(1-x)}^{3/2}}}\] |
| C. | \[\frac{1}{2{{(1+x)}^{1/2}}{{(1-x)}^{3/2}}}\] |
| D. | \[\frac{1}{{{(1+x)}^{3/2}}{{(1-x)}^{1/2}}}\] |
| Answer» C. \[\frac{1}{2{{(1+x)}^{1/2}}{{(1-x)}^{3/2}}}\] | |
| 5652. |
If \[y={{e}^{x+{{e}^{x+{{e}^{x+....\infty }}}}}}\], then \[\frac{dy}{dx}=\] [AISSE 1990; UPSEAT 2002; DCE 2002] |
| A. | \[\frac{y}{1-y}\] |
| B. | \[\frac{1}{1-y}\] |
| C. | \[\frac{y}{1+y}\] |
| D. | \[\frac{y}{y-1}\] |
| Answer» B. \[\frac{1}{1-y}\] | |
| 5653. |
If \[x=\frac{1-{{t}^{2}}}{1+{{t}^{2}}}\]and \[y=\frac{2t}{1+{{t}^{2}}}\], then \[\frac{dy}{dx}=\] [Karnataka CET 2000; Pb. CET 2002] |
| A. | \[\frac{-y}{x}\] |
| B. | \[\frac{y}{x}\] |
| C. | \[\frac{-x}{y}\] |
| D. | \[\frac{x}{y}\] |
| Answer» D. \[\frac{x}{y}\] | |
| 5654. |
The first derivative of the function \[\left[ {{\cos }^{-1}}\left( \sin \sqrt{\frac{1+x}{2}} \right)+{{x}^{x}} \right]\] with respect to x at x = 1 is [MP PET 1998] |
| A. | \[\frac{3}{4}\] |
| B. | 0 |
| C. | \[\frac{1}{2}\] |
| D. | \[-\frac{1}{2}\] |
| Answer» B. 0 | |
| 5655. |
If \[y={{x}^{x}}\], then \[\frac{dy}{dx}=\] [AISSE 1984; DSSE 1982; MNR 1979; SCRA 1996; RPET 1996; Kerala (Engg.) 2002] |
| A. | \[{{x}^{x}}\log ex\] |
| B. | \[{{x}^{x}}\left( 1+\frac{1}{x} \right)\] |
| C. | \[(1+\log x)\] |
| D. | \[{{x}^{x}}\log x\] |
| Answer» B. \[{{x}^{x}}\left( 1+\frac{1}{x} \right)\] | |
| 5656. |
If \[\ln \,(x+y)=2xy,\]then \[y'(0)\]= [IIT Screening 2004] |
| A. | 1 |
| B. | ?1 |
| C. | 2 |
| D. | 0 |
| Answer» B. ?1 | |
| 5657. |
If \[x=\sin t\cos 2t\] and \[y=\cos t\sin 2t\], then at \[t=\frac{\pi }{4},\] the value of \[\frac{dy}{dx}\] is equal to [Pb. CET 2000] |
| A. | ?2 |
| B. | 2 |
| C. | \[\frac{1}{2}\] |
| D. | \[-\frac{1}{2}\] |
| Answer» D. \[-\frac{1}{2}\] | |
| 5658. |
If \[x={{\sin }^{-1}}(3t-4{{t}^{3}})\] and \[y={{\cos }^{-1}}\,\,\sqrt{(1-{{t}^{2}})}\], then \[\frac{dy}{dx}\] is equal to [Kerala (Engg.) 2002] |
| A. | ½ |
| B. | 2/5 |
| C. | 3/2 |
| D. | 1/3 |
| Answer» E. | |
| 5659. |
If \[x=a{{\cos }^{4}}\theta ,y=a{{\sin }^{4}}\theta ,\] then \[\frac{dy}{dx}\], at \[\theta =\frac{3\pi }{4}\], is [Kerala (Engg.) 2002] |
| A. | ?1 |
| B. | 1 |
| C. | \[-{{a}^{2}}\] |
| D. | \[{{a}^{2}}\] |
| Answer» B. 1 | |
| 5660. |
If \[\cos x=\frac{1}{\sqrt{1+{{t}^{2}}}}\]and \[\sin y=\frac{t}{\sqrt{1+{{t}^{2}}}}\], then \[\frac{dy}{dx}=\] [MP PET 1994] |
| A. | ?1 |
| B. | \[\frac{1-t}{1+{{t}^{2}}}\] |
| C. | \[\frac{1}{1+{{t}^{2}}}\] |
| D. | 1 |
| Answer» E. | |
| 5661. |
If\[x=a(\cos \theta +\theta \sin \theta )\], \[y=a(\sin \theta -\theta \cos \theta ),\text{ }\]then \[\frac{dy}{dx}=\] [DCE 1999] |
| A. | \[\cos \theta \] |
| B. | \[\tan \theta \] |
| C. | \[\sec \theta \] |
| D. | cosecq |
| Answer» C. \[\sec \theta \] | |
| 5662. |
If \[\tan y=\frac{2t}{1-{{t}^{2}}}\]and \[\sin x=\frac{2t}{1+{{t}^{2}}},\]then \[\frac{dy}{dx}=\] |
| A. | \[\frac{2}{1+{{t}^{2}}}\] |
| B. | \[\frac{1}{1+{{t}^{2}}}\] |
| C. | 1 |
| D. | 2 |
| Answer» D. 2 | |
| 5663. |
If \[x=\frac{2\,t}{1+{{t}^{2}}},\,\,y=\frac{1-{{t}^{2}}}{1+{{t}^{2}}},\]then \[\frac{d\,y}{d\,x}\] equals [RPET 1999] |
| A. | \[\frac{2\,t}{{{t}^{2}}+1}\] |
| B. | \[\frac{2\,t}{{{t}^{2}}-1}\] |
| C. | \[\frac{2\,t}{1-{{t}^{2}}}\] |
| D. | None of these |
| Answer» C. \[\frac{2\,t}{1-{{t}^{2}}}\] | |
| 5664. |
If \[x=a(t+\sin t)\]and \[y=a(1-\cos t)\], then \[\frac{dy}{dx}\] equals [RPET 1996; MP PET 2002] |
| A. | \[\tan (t/2)\] |
| B. | \[\cot (t/2)\] |
| C. | \[\tan 2t\] |
| D. | \[\tan t\] |
| Answer» B. \[\cot (t/2)\] | |
| 5665. |
If \[{{x}^{3}}+{{y}^{3}}-3axy=0\], then \[\frac{dy}{dx}\] equals [RPET 1996] |
| A. | \[\frac{ay-{{x}^{2}}}{{{y}^{2}}-ax}\] |
| B. | \[\frac{ay-{{x}^{2}}}{ay-{{y}^{2}}}\] |
| C. | \[\frac{{{x}^{2}}+ay}{{{y}^{2}}+ax}\] |
| D. | \[\frac{{{x}^{2}}+ay}{ax-{{y}^{2}}}\] |
| Answer» B. \[\frac{ay-{{x}^{2}}}{ay-{{y}^{2}}}\] | |
| 5666. |
If \[x=a{{\cos }^{3}}\theta ,y=a{{\sin }^{3}}\theta \], then \[\sqrt{1+{{\left( \frac{dy}{dx} \right)}^{2}}}=\] [EAMCET 1992] |
| A. | \[{{\tan }^{2}}\theta \] |
| B. | \[{{\sec }^{2}}\theta \] |
| C. | \[\sec \theta \] |
| D. | \[|\sec \theta |\] |
| Answer» E. | |
| 5667. |
If \[{{x}^{2}}{{e}^{y}}+2xy{{e}^{x}}+13=0\], then dy/dx = [RPET 1987] |
| A. | \[\frac{2x{{e}^{y-x}}+2y(x+1)}{x(x{{e}^{y-x}}+2)}\] |
| B. | \[\frac{2x{{e}^{x-y}}+2y(x+1)}{x(x{{e}^{y-x}}+2)}\] |
| C. | \[-\frac{2x{{e}^{y-x}}+2y(x+1)}{x(x{{e}^{y-x}}+2)}\] |
| D. | None of these |
| Answer» D. None of these | |
| 5668. |
If \[3\sin (xy)+4\cos (xy)=5\], then \[\frac{dy}{dx}=\] [EAMCET 1994] |
| A. | \[-\frac{y}{x}\] |
| B. | \[\frac{3\sin (xy)+4\cos (xy)}{3\cos (xy)-4\sin (xy)}\] |
| C. | \[\frac{3\cos (xy)+4\sin (xy)}{4\cos (xy)-3\sin (xy)}\] |
| D. | None of these |
| Answer» B. \[\frac{3\sin (xy)+4\cos (xy)}{3\cos (xy)-4\sin (xy)}\] | |
| 5669. |
Let \[y={{t}^{10}}+1\]and \[x={{t}^{8}}+1,\]then \[\frac{{{d}^{2}}y}{d{{x}^{2}}}\]is [UPSEAT 2004] |
| A. | \[\frac{5}{2}t\] |
| B. | \[20{{t}^{8}}\] |
| C. | \[\frac{5}{16{{t}^{6}}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 5670. |
If \[x={{t}^{2}}\], \[y={{t}^{3}}\], then \[\frac{{{d}^{2}}y}{d{{x}^{2}}}\] = [EAMCET 1994] |
| A. | \[\frac{3}{2}\] |
| B. | \[\frac{3}{(4t)}\] |
| C. | \[\frac{3}{2(t)}\] |
| D. | \[\frac{3t}{2}\] |
| Answer» C. \[\frac{3}{2(t)}\] | |
| 5671. |
If \[x=t+\frac{1}{t},y=t-\frac{1}{t},\]then \[\frac{{{d}^{2}}y}{d{{x}^{2}}}\]is equal to |
| A. | \[-4t{{({{t}^{2}}-1)}^{-2}}\] |
| B. | \[-4{{t}^{3}}{{({{t}^{2}}-1)}^{-3}}\] |
| C. | \[({{t}^{2}}+1){{({{t}^{2}}-1)}^{-1}}\] |
| D. | \[-4{{t}^{2}}{{({{t}^{2}}-1)}^{-2}}\] |
| Answer» C. \[({{t}^{2}}+1){{({{t}^{2}}-1)}^{-1}}\] | |
| 5672. |
If \[x=a\text{ }\left( \cos t+\log \tan \frac{t}{2} \right)\,,y=a\sin t,\]then \[\frac{dy}{dx}=\] [RPET 1997; MP PET 2001] |
| A. | \[\tan t\] |
| B. | \[-\tan t\] |
| C. | \[\cot t\] |
| D. | \[-\cot t\] |
| Answer» B. \[-\tan t\] | |
| 5673. |
If \[x=a\sin 2\theta (1+\cos 2\theta ),y=b\cos 2\theta (1-\cos 2\theta )\], then \[\frac{dy}{dx}=\] [Kurukshetra CEE 1998] |
| A. | \[\frac{b\tan \theta }{a}\] |
| B. | \[\frac{a\tan \theta }{b}\] |
| C. | \[\frac{a}{b\tan \theta }\] |
| D. | \[\frac{b}{a\tan \theta }\] |
| Answer» B. \[\frac{a\tan \theta }{b}\] | |
| 5674. |
If \[\sin y=x\cos (a+y),\] then \[\frac{dy}{dx}=\] |
| A. | \[\frac{{{\cos }^{2}}(a+y)}{\cos a}\] |
| B. | \[\frac{\cos (a+y)}{{{\cos }^{2}}a}\] |
| C. | \[\frac{{{\sin }^{2}}(a+y)}{\sin a}\] |
| D. | None of these |
| Answer» B. \[\frac{\cos (a+y)}{{{\cos }^{2}}a}\] | |
| 5675. |
If \[{{x}^{2}}+{{y}^{2}}=t-\frac{1}{t},\]\[{{x}^{4}}+{{y}^{4}}={{t}^{2}}+\frac{1}{{{t}^{2}}}\], then \[{{x}^{3}}y\frac{dy}{dx}=\] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» B. 2 | |
| 5676. |
Let \[f(x)={{e}^{x}}\], \[g(x)={{\sin }^{-1}}x\] and \[h(x)=f(g(x)),\] then \[h'(x)/h(x)=\] [EAMCET 2002] |
| A. | \[{{e}^{{{\sin }^{-1}}x}}\] |
| B. | \[1/\sqrt{1-{{x}^{2}}}\] |
| C. | \[{{\sin }^{-1}}x\] |
| D. | \[1/\,(1-{{x}^{2}})\] |
| Answer» C. \[{{\sin }^{-1}}x\] | |
| 5677. |
The derivative of \[F[f\{\varphi (x)\}]\] is [AMU 2001] |
| A. | \[{F}'\,[f\,\{\varphi \,(x)\}]\] |
| B. | \[F\,[f\,\{\varphi \,(x)\}\,]\,{f}'\{\varphi (x)\}\] |
| C. | \[{F}'[f\,\{\varphi \,(x)\}]\,{f}'\{\varphi (x)\}\] |
| D. | \[{F}'\,[f\,\{\varphi \,(x)\}]\,{f}'\{\varphi (x)\}\,{\varphi }'\,(x)\] |
| Answer» E. | |
| 5678. |
The differential coefficient of \[f[\log (x)]\] when \[f(x)=\log x\]is [Kurukshetra CEE 1998; DCE 2000] |
| A. | \[x\log x\] |
| B. | \[\frac{x}{\log x}\] |
| C. | \[\frac{1}{x\log x}\] |
| D. | \[\frac{\log x}{x}\] |
| Answer» D. \[\frac{\log x}{x}\] | |
| 5679. |
Let f and g be differentiable functions satisfying \[{g}'(a)=2,\] \[g(a)=b\] and \[fog=I\](identity function). Then \[f'(b)\] is equal to |
| A. | \[\frac{1}{2}\] |
| B. | 2 |
| C. | \[\frac{2}{3}\] |
| D. | None of these |
| Answer» B. 2 | |
| 5680. |
Let \[g(x)\] be the inverse of the function \[f(x)\] and \[f'(x)=\frac{1}{1+{{x}^{3}}}\]. Then \[{g}'(x)\] is equal to [Kurukshetra CEE 1996] |
| A. | \[\frac{1}{1+{{(g(x))}^{3}}}\] |
| B. | \[\frac{1}{1+{{(f(x))}^{3}}}\] |
| C. | \[1+{{(g(x))}^{3}}\] |
| D. | \[1+{{(f(x))}^{3}}\] |
| Answer» D. \[1+{{(f(x))}^{3}}\] | |
| 5681. |
Let g (x) be the inverse of an invertible function \[f(x)\] which is differentiable at x = c, then \[g'(f(c))\]equals |
| A. | \[f'(c)\] |
| B. | \[\frac{1}{f'(c)}\] |
| C. | \[f(c)\] |
| D. | None of these |
| Answer» C. \[f(c)\] | |
| 5682. |
If \[x=\frac{1-{{t}^{2}}}{1+{{t}^{2}}}\]and \[y=\frac{2at}{1+{{t}^{2}}}\], then \[\frac{dy}{dx}=\] |
| A. | \[\frac{a(1-{{t}^{2}})}{2t}\] |
| B. | \[\frac{a({{t}^{2}}-1)}{2t}\] |
| C. | \[\frac{a({{t}^{2}}+1)}{2t}\] |
| D. | \[\frac{a({{t}^{2}}-1)}{t}\] |
| Answer» C. \[\frac{a({{t}^{2}}+1)}{2t}\] | |
| 5683. |
If\[f(x)=\frac{1}{1-x}\], then the derivative of the composite function \[f[f\{f(x)\}]\] is equal to [Orissa JEE 2003] |
| A. | 0 |
| B. | \[\frac{1}{2}\] |
| C. | 1 |
| D. | 2 |
| Answer» D. 2 | |
| 5684. |
If \[y=f\left( \frac{5x+1}{10{{x}^{2}}-3} \right)\]and \[f'(x)=\cos x\], then \[\frac{dy}{dx}=\] [MP PET 1987] |
| A. | \[\cos \,\left( \frac{5x+1}{10{{x}^{2}}-3} \right)\,\frac{dy}{dx}\,\left( \frac{5x+1}{10{{x}^{2}}-3} \right)\] |
| B. | \[\frac{5x+1}{10{{x}^{2}}-3}\,\,\cos \,\left( \frac{5x+1}{10{{x}^{2}}-3} \right)\] |
| C. | \[\cos \,\left( \frac{5x+1}{10{{x}^{2}}-3} \right)\] |
| D. | None of these |
| Answer» B. \[\frac{5x+1}{10{{x}^{2}}-3}\,\,\cos \,\left( \frac{5x+1}{10{{x}^{2}}-3} \right)\] | |
| 5685. |
If \[a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0\], then \[\frac{dy}{dx}=\] |
| A. | \[-\frac{ax+hy+g}{hx-by+f}\] |
| B. | \[\frac{ax+hy+g}{hx-by+f}\] |
| C. | \[\frac{ax-hy-g}{hx-by-f}\] |
| D. | None of these |
| Answer» B. \[\frac{ax+hy+g}{hx-by+f}\] | |
| 5686. |
If \[y\sec x+\tan x+{{x}^{2}}y=0\], then \[\frac{dy}{dx}\]= [DSSE 1981; CBSE 1981] |
| A. | \[\frac{2xy+{{\sec }^{2}}x+y\sec x\tan x}{{{x}^{2}}+\sec x}\] |
| B. | \[-\frac{2xy+{{\sec }^{2}}x+\sec x\tan x}{{{x}^{2}}+\sec x}\] |
| C. | \[-\frac{2xy+{{\sec }^{2}}x+y\sec x\tan x}{{{x}^{2}}+\sec x}\] |
| D. | None of these |
| Answer» D. None of these | |
| 5687. |
If then \[\frac{dy}{dx}=\] [DSSE 1980; CBSE 1980] |
| A. | \[\frac{y[2xy-{{y}^{2}}\cos (xy)-1]}{x{{y}^{2}}\cos (xy)+{{y}^{2}}-x}\] |
| B. | \[\frac{[2xy-{{y}^{2}}\cos (xy)-1]}{x{{y}^{2}}\cos (xy)+{{y}^{2}}-x}\] |
| C. | \[-\frac{y[2xy-{{y}^{2}}\cos (xy)-1]}{x{{y}^{2}}\cos (xy)+{{y}^{2}}-x}\] |
| D. | None of these |
| Answer» B. \[\frac{[2xy-{{y}^{2}}\cos (xy)-1]}{x{{y}^{2}}\cos (xy)+{{y}^{2}}-x}\] | |
| 5688. |
If \[\tan (x+y)+\tan (x-y)=1,\]then \[\frac{dy}{dx}=\] [DSSE 1979] |
| A. | \[\frac{{{\sec }^{2}}(x+y)+{{\sec }^{2}}(x-y)}{{{\sec }^{2}}(x+y)-{{\sec }^{2}}(x-y)}\] |
| B. | \[\frac{{{\sec }^{2}}(x+y)+{{\sec }^{2}}(x-y)}{{{\sec }^{2}}(x-y)-{{\sec }^{2}}(x+y)}\] |
| C. | \[\frac{{{\sec }^{2}}(x+y)-{{\sec }^{2}}(x-y)}{{{\sec }^{2}}(x+y)+{{\sec }^{2}}(x-y)}\] |
| D. | None of these |
| Answer» C. \[\frac{{{\sec }^{2}}(x+y)-{{\sec }^{2}}(x-y)}{{{\sec }^{2}}(x+y)+{{\sec }^{2}}(x-y)}\] | |
| 5689. |
If \[x=2\cos t-\cos 2t\],\[y=2\sin t-\sin 2t\], then at \[t=\frac{\pi }{4},\frac{dy}{dx}=\] |
| A. | \[\sqrt{2}+1\] |
| B. | \[\sqrt{2+1}\] |
| C. | \[\frac{\sqrt{2+1}}{2}\] |
| D. | None of these |
| Answer» B. \[\sqrt{2+1}\] | |
| 5690. |
If \[{{y}^{2}}=a{{x}^{2}}+bx+c\], then \[{{y}^{3}}\frac{{{d}^{2}}y}{d{{x}^{2}}}\]is [DCE 1999] |
| A. | A constant |
| B. | A function of x only |
| C. | A function of y only |
| D. | A function of x and y |
| Answer» B. A function of x only | |
| 5691. |
\[\frac{d}{dx}{{\tan }^{-1}}\left[ \frac{3{{a}^{2}}x-{{x}^{3}}}{a({{a}^{2}}-3{{x}^{2}})} \right]\]at \[x=0\]is |
| A. | \[\frac{1}{a}\] |
| B. | \[\frac{3}{a}\] |
| C. | \[3a\] |
| D. | 3 |
| Answer» C. \[3a\] | |
| 5692. |
If \[y=\sin px\] and \[{{y}_{n}}\] is the nth derivative of y, then \[\left| \,\begin{matrix} y & {{y}_{1}} & {{y}_{2}} \\ {{y}_{3}} & {{y}_{4}} & {{y}_{5}} \\ {{y}_{6}} & {{y}_{7}} & {{y}_{8}} \\ \end{matrix}\, \right|\] is equal to [AMU 2002] |
| A. | 1 |
| B. | 0 |
| C. | ? 1 |
| D. | None of these |
| Answer» C. ? 1 | |
| 5693. |
\[f(x)=\left| \begin{matrix} {{x}^{3}} & {{x}^{2}} & 3{{x}^{2}} \\ 1 & -6 & 4 \\ p & {{p}^{2}} & {{p}^{3}} \\ \end{matrix} \right|\] , here p is a constant, then \[\frac{{{d}^{3}}f(x)}{d{{x}^{3}}}\] is [DCE 2000] |
| A. | Proportional to \[{{x}^{2}}\] |
| B. | Proportional to x |
| C. | Proportional to \[{{x}^{3}}\] |
| D. | A constant |
| Answer» E. | |
| 5694. |
Let \[f(x)=\left| \begin{matrix} {{x}^{3}} & \sin x & \cos x \\ 6 & -1 & 0 \\ p & {{p}^{2}} & {{p}^{3}} \\ \end{matrix} \right|\], where p is a constant. Then \[\frac{{{d}^{3}}}{d{{x}^{3}}}\left\{ f(x) \right\}\]at \[x=0\]is [IIT 1997 Cancelled] |
| A. | p |
| B. | \[p+{{p}^{2}}\] |
| C. | \[p+{{p}^{3}}\] |
| D. | Independent of p |
| Answer» E. | |
| 5695. |
If \[{{f}_{n}}(x)\], \[{{g}_{n}}(x)\], \[{{h}_{n}}(x),n=1,\,2,\,3\]are polynomials in x such that \[{{f}_{n}}(a)={{g}_{n}}(a)={{h}_{n}}(a),n=1,2,3\] and \[F(x)=\left| \begin{matrix} {{f}_{1}}(x) & {{f}_{2}}(x) & {{f}_{3}}(x) \\ {{g}_{1}}(x) & {{g}_{2}}(x) & {{g}_{3}}(x) \\ {{h}_{1}}(x) & {{h}_{2}}(x) & {{h}_{3}}(x) \\ \end{matrix} \right|\]. Then \[{F}'(a)\]is equal to |
| A. | 0 |
| B. | \[{{f}_{1}}(a){{g}_{2}}(a){{h}_{3}}(a)\] |
| C. | 1 |
| D. | None of these |
| Answer» B. \[{{f}_{1}}(a){{g}_{2}}(a){{h}_{3}}(a)\] | |
| 5696. |
\[\frac{d}{dx}{{\cos }^{-1}}\sqrt{\frac{1+{{x}^{2}}}{2}}=\] [AI CBSE 1988] |
| A. | \[\frac{-1}{2\sqrt{1-{{x}^{4}}}}\] |
| B. | \[\frac{1}{2\sqrt{1-{{x}^{4}}}}\] |
| C. | \[\frac{-x}{\sqrt{1-{{x}^{4}}}}\] |
| D. | \[\frac{x}{\sqrt{1-{{x}^{4}}}}\] |
| Answer» D. \[\frac{x}{\sqrt{1-{{x}^{4}}}}\] | |
| 5697. |
If \[f(x)={{\tan }^{-1}}\left\{ \frac{\log \left( \frac{e}{{{x}^{2}}} \right)}{\log (e{{x}^{2}})} \right\}+{{\tan }^{-1}}\left( \frac{3+2\log x}{1-6\log x} \right)\], then \[\frac{{{d}^{n}}y}{d{{x}^{n}}}\] is \[(n\ge 1)\] |
| A. | \[{{\tan }^{-1}}\{{{(\log x)}^{n}}\}\] |
| B. | 0 |
| C. | \[\frac{1}{2}\] |
| D. | None of these |
| Answer» C. \[\frac{1}{2}\] | |
| 5698. |
If \[x=A\cos 4t+B\sin 4t\],then \[\frac{{{d}^{2}}x}{d{{t}^{2}}}=\] [Karnataka CET 2004] |
| A. | ? 16x |
| B. | 16 x |
| C. | x |
| D. | ? x |
| Answer» B. 16 x | |
| 5699. |
If \[y=1-x+\frac{{{x}^{2}}}{2!}-\frac{{{x}^{3}}}{3!}+\frac{{{x}^{4}}}{4!}-\]....., then \[\frac{{{d}^{2}}y}{d{{x}^{2}}}=\] [Karnataka CET 2003] |
| A. | \[x\] |
| B. | \[-x\] |
| C. | \[-y\] |
| D. | \[y\] |
| Answer» E. | |
| 5700. |
The nth derivative of \[x{{e}^{x}}\] vanishes when [AMU 1999] |
| A. | \[x=0\] |
| B. | \[x=-1\] |
| C. | \[x=-n\] |
| D. | \[x=n\] |
| Answer» D. \[x=n\] | |