-
question_answer1)
\[\frac{d}{dx}{{\tan }^{-1}}\left[ \frac{\cos x-\sin x}{\cos x+\sin x} \right]=\] [AISSE 1985, 87; DSSE 1982,84; MNR 1985; Karnataka CET 2002; RPET 2002, 03]
A)
\[\frac{1}{2\,\,(1+{{x}^{2}})}\] done
clear
B)
\[\frac{1}{1+{{x}^{2}}}\] done
clear
C)
1 done
clear
D)
- 1 done
clear
View Solution play_arrow
-
question_answer2)
If \[y=\frac{x}{2}\sqrt{{{a}^{2}}+{{x}^{2}}}+\frac{{{a}^{2}}}{2}\log (x+\sqrt{{{x}^{2}}+{{a}^{2}}})\],then \[\frac{dy}{dx}=\] [AISSE 1983]
A)
\[\sqrt{{{x}^{2}}+{{a}^{2}}}\] done
clear
B)
\[\frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}\] done
clear
C)
\[2\sqrt{{{x}^{2}}+{{a}^{2}}}\] done
clear
D)
\[\frac{2}{\sqrt{{{x}^{2}}+{{a}^{2}}}}\] done
clear
View Solution play_arrow
-
question_answer3)
If \[y={{\cot }^{-1}}{{(\cos 2x)}^{1/2}}\], then the value of \[\frac{dy}{dx}\]at \[x=\frac{\pi }{6}\]will be [IIT 1992]
A)
\[{{\left( \frac{2}{3} \right)}^{1/2}}\] done
clear
B)
\[{{\left( \frac{1}{3} \right)}^{1/2}}\] done
clear
C)
\[{{(3)}^{1/2}}\] done
clear
D)
\[{{(6)}^{1/2}}\] done
clear
View Solution play_arrow
-
question_answer4)
If \[f(x+y)=f(x).f(y)\]for all x and y and \[f(5)=2\], \[f'(0)=3\], then \[f'(5)\]will be [IIT 1981; Karnataka CET 2000; UPSEAT 2002; MP PET 2002; AIEEE 2002]
A)
2 done
clear
B)
4 done
clear
C)
6 done
clear
D)
8 done
clear
View Solution play_arrow
-
question_answer5)
If \[x{{e}^{xy}}=y+{{\sin }^{2}}x\], then at \[x=0,\frac{dy}{dx}=\] [IIT 1996]
A)
-1 done
clear
B)
-2 done
clear
C)
1 done
clear
D)
2 done
clear
View Solution play_arrow
-
question_answer6)
If \[u(x,y)=y\log x+x\,\log y,\] then \[{{u}_{x}}{{u}_{y}}-{{u}_{x}}\log x-{{u}_{y}}\log y+\log x\,\,\log y=\] [EAMCET 2003]
A)
0 done
clear
B)
-1 done
clear
C)
1 done
clear
D)
2 done
clear
View Solution play_arrow
-
question_answer7)
If \[y=f\left( \frac{2x-1}{{{x}^{2}}+1} \right)\]and \[{f}'(x)=\sin {{x}^{2}},\]then \[\frac{dy}{dx}=\] [IIT 1982]
A)
\[\frac{6{{x}^{2}}-2x+2}{{{({{x}^{2}}+1)}^{2}}}\sin {{\left( \frac{2x-1}{{{x}^{2}}+1} \right)}^{2}}\] done
clear
B)
\[\frac{6{{x}^{2}}-2x+2}{{{({{x}^{2}}+1)}^{2}}}{{\sin }^{2}}\left( \frac{2x-1}{{{x}^{2}}+1} \right)\] done
clear
C)
\[\frac{-2{{x}^{2}}+2x+2}{{{({{x}^{2}}+1)}^{2}}}{{\sin }^{2}}\left( \frac{2x-1}{{{x}^{2}}+1} \right)\] done
clear
D)
\[\frac{-2{{x}^{2}}+2x+2}{{{({{x}^{2}}+1)}^{2}}}\sin {{\left( \frac{2x-1}{{{x}^{2}}+1} \right)}^{2}}\] done
clear
View Solution play_arrow
-
question_answer8)
If \[x=\sec \theta -\cos \theta \]and \[y={{\sec }^{n}}\theta -{{\cos }^{n}}\theta \], then [IIT 1989]
A)
\[({{x}^{2}}+4)\text{ }{{\left( \frac{dy}{dx} \right)}^{2}}={{n}^{2}}({{y}^{2}}+4)\] done
clear
B)
\[({{x}^{2}}+4)\text{ }{{\left( \frac{dy}{dx} \right)}^{2}}={{x}^{2}}({{y}^{2}}+4)\] done
clear
C)
\[({{x}^{2}}+4)\text{ }{{\left( \frac{dy}{dx} \right)}^{2}}=({{y}^{2}}+4)\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer9)
If \[y={{x}^{{{x}^{x......\infty }}}}\], then \[\frac{dy}{dx}=\] [UPSEAT 2004; DCE 2000]
A)
\[\frac{{{y}^{2}}}{x(1+y\log x)}\] done
clear
B)
\[\frac{{{y}^{2}}}{x(1-y\log x)}\] done
clear
C)
\[\frac{y}{x(1+y\log x)}\] done
clear
D)
\[\frac{y}{x(1-y\log x)}\] done
clear
View Solution play_arrow
-
question_answer10)
If \[y={{(x\log x)}^{\log \,\log x}}\], then \[\frac{dy}{dx}=\] [Roorkee 1981]
A)
\[{{(x\log x)}^{\log \log x}}\left\{ \frac{1}{x\log x}(\log x+\log \log x)+(\log \,\,\log x)\text{ }\left( \frac{1}{x}+\frac{1}{x\log x} \right)\text{ } \right\}\] done
clear
B)
\[{{(x\log x)}^{x\log x}}\log \log x\left[ \frac{2}{\log x}+\frac{1}{x} \right]\] done
clear
C)
\[{{(x\log x)}^{x\log x}}\log \log x\left[ \frac{2}{\log x}+\frac{1}{x} \right]\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer11)
\[\frac{d}{dx}\left[ {{\tan }^{-1}}\frac{\sqrt{1+{{x}^{2}}}+\sqrt{1-{{x}^{2}}}}{\sqrt{1+{{x}^{2}}}-\sqrt{1-{{x}^{2}}}} \right]=\] [Roorkee 1980; Karnataka CET 2005]
A)
\[\frac{-x}{\sqrt{1-{{x}^{4}}}}\] done
clear
B)
\[\frac{x}{\sqrt{1-{{x}^{4}}}}\] done
clear
C)
\[\frac{-1}{2\sqrt{1-{{x}^{4}}}}\] done
clear
D)
\[\frac{1}{2\sqrt{1-{{x}^{4}}}}\] done
clear
View Solution play_arrow
-
question_answer12)
If \[\sqrt{(1-{{x}^{6}})}+\sqrt{(1-{{y}^{6}})}={{a}^{3}}({{x}^{3}}-{{y}^{3}})\], then \[\frac{dy}{dx}=\] [Roorkee 1994]
A)
\[\frac{{{x}^{2}}}{{{y}^{2}}}\sqrt{\frac{1-{{x}^{6}}}{1-{{y}^{6}}}}\] done
clear
B)
\[\frac{{{y}^{2}}}{{{x}^{2}}}\sqrt{\frac{1-{{y}^{6}}}{1-{{x}^{6}}}}\] done
clear
C)
\[\frac{{{x}^{2}}}{{{y}^{2}}}\sqrt{\frac{1-{{y}^{6}}}{1-{{x}^{6}}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer13)
If \[y={{\sec }^{-1}}\frac{2x}{1+{{x}^{2}}}+{{\sin }^{-1}}\frac{x-1}{x+1}\],then \[\frac{dy}{dx}\]is equal to [Pb. CET 2000]
A)
1 done
clear
B)
\[\frac{x-1}{x+1}\] done
clear
C)
Does not exist done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer14)
The derivative of \[{{\tan }^{-1}}\left( \frac{\sqrt{1+{{x}^{2}}}-1}{x} \right)\]with respect to \[{{\tan }^{-1}}\left( \frac{2x\sqrt{1-{{x}^{2}}}}{1-2{{x}^{2}}} \right)\]at \[x=0\], is
A)
\[\frac{1}{8}\] done
clear
B)
\[\frac{1}{4}\] done
clear
C)
\[\frac{1}{2}\] done
clear
D)
1 done
clear
View Solution play_arrow
-
question_answer15)
If \[{{y}^{2}}=p(x)\]is a polynomial of degree three, then \[2\frac{d}{dx}\left\{ {{y}^{3}}.\frac{{{d}^{2}}y}{d{{x}^{2}}} \right\}\]= [IIT 1988; RPET 2000]
A)
\[{p}'''(x)+p'(x)\] done
clear
B)
\[{p}''(x).{p}'''(x)\] done
clear
C)
\[p(x).{p}'''(x)\] done
clear
D)
Constant done
clear
View Solution play_arrow
-
question_answer16)
Let \[f(x)\]and \[g(x)\]be two functions having finite non-zero 3rd order derivatives \[{f}'''(x)\]and \[{g}'''(x)\] for all, \[x\in R\]. If \[f(x)g(x)=1\]for all \[x\in R\], then \[\frac{{{f}'''}}{{{f}'}}-\frac{{{g}'''}}{{{g}'}}\]is equal to
A)
\[3\text{ }\left( \frac{{{f}''}}{g}-\frac{{{g}''}}{f} \right)\] done
clear
B)
\[3\text{ }\left( \frac{{{f}''}}{f}-\frac{{{g}''}}{g} \right)\] done
clear
C)
\[3\text{ }\left( \frac{g''}{g}-\frac{f''}{g} \right)\] done
clear
D)
\[3\text{ }\left( \frac{{{f}''}}{f}-\frac{{{g}''}}{f} \right)\] done
clear
View Solution play_arrow
-
question_answer17)
If \[{{I}_{n}}=\frac{{{d}^{n}}}{d{{x}^{n}}}({{x}^{n}}\log x),\]then \[{{I}_{n}}-n{{I}_{n-1}}=\] [EAMCET 2003]
A)
\[n\] done
clear
B)
\[n-1\] done
clear
C)
\[n!\] done
clear
D)
\[(n-1)!\] done
clear
View Solution play_arrow
-
question_answer18)
If \[x=\sin t\] and \[y=\sin pt\], then the value of \[\left( 1-{{x}^{2}} \right)\frac{{{d}^{2}}y}{d{{x}^{2}}}-x\frac{dy}{dx}+{{p}^{2}}y\]is equal to [Pb. CET 2002]
A)
0 done
clear
B)
1 done
clear
C)
-1 done
clear
D)
\[\sqrt{2}\] done
clear
View Solution play_arrow
-
question_answer19)
Let \[f:(0,\,+\infty )\to R\] and \[F(x)=\int_{0}^{x}{f(t)\,dt}\]. If \[F({{x}^{2}})={{x}^{2}}(1+x)\], then \[f(4)\] equals [IIT Screening 2001]
A)
\[\frac{5}{4}\] done
clear
B)
7 done
clear
C)
4 done
clear
D)
2 done
clear
View Solution play_arrow
-
question_answer20)
The volume of a spherical balloon is increasing at the rate of 40 cubic centrimetre per minute. The rate of change of the surface of the balloon at the instant when its radius is 8 centimetre, is [Roorkee 1983]
A)
\[\frac{5}{2}\] sq cm/min done
clear
B)
5 sq cm/min done
clear
C)
10 sq cm/min done
clear
D)
20 sq cm/min done
clear
View Solution play_arrow
-
question_answer21)
A man of height 1.8 metre is moving away from a lamp post at the rate of 1.2 \[m/\sec .\] If the height of the lamp post be 4.5 metre, then the rate at which the shadow of the man is lengthening is [MP PET 1989]
A)
\[0.4\,\,m/\sec \] done
clear
B)
\[0.8\,\,m/\sec \] done
clear
C)
\[1.2\,\,m/\sec \] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer22)
The radius of the cylinder of maximum volume, which can be inscribed in a sphere of radius R is [AMU 1999]
A)
\[\frac{2}{3}R\] done
clear
B)
\[\sqrt{\frac{2}{3}}R\] done
clear
C)
\[\frac{3}{4}R\] done
clear
D)
\[\sqrt{\frac{3}{4}}R\] done
clear
View Solution play_arrow
-
question_answer23)
The distance travelled s (in metre) by a particle in t seconds is given by, \[s={{t}^{3}}+2{{t}^{2}}+t.\]The speed of the particle after 1 second will be [UPSEAT 2003]
A)
8 cm/sec done
clear
B)
6 cm/sec done
clear
C)
2 cm/sec done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer24)
If \[y=4x-5\] is tangent to the curve \[{{y}^{2}}=p{{x}^{3}}+q\] at (2, 3), then [IIT 1994; UPSEAT 2001]
A)
\[p=2,q=-7\] done
clear
B)
\[p=-2,q=7\] done
clear
C)
\[p=-2,q=-7\] done
clear
D)
\[p=2,q=7\] done
clear
View Solution play_arrow
-
question_answer25)
At what points of the curve \[y=\frac{2}{3}{{x}^{3}}+\frac{1}{2}{{x}^{2}},\]tangent makes the equal angle with axis [UPSEAT 1999]
A)
\[\left( \frac{1}{2},\,\frac{5}{24} \right)\] and \[\left( -1,\,-\frac{1}{6} \right)\] done
clear
B)
\[\left( \frac{1}{2},\,\frac{4}{9} \right)\] and \[\left( -1,\,0 \right)\] done
clear
C)
\[\left( \frac{1}{3},\,\frac{1}{7} \right)\] and \[\left( -3,\,\frac{1}{2} \right)\] done
clear
D)
\[\left( \frac{1}{3},\,\frac{4}{47} \right)\] and \[\left( -1,\,-\frac{1}{3} \right)\] done
clear
View Solution play_arrow
-
question_answer26)
If the normal to the curve \[y=f(x)\] at the point \[(3,\,4)\] makes an angle \[\frac{3\pi }{4}\]with the positive x-axis then \[f'(3)\] is equal to [IIT Screening 2000; DCE 2001]
A)
\[-1\] done
clear
B)
\[-\frac{3}{4}\] done
clear
C)
\[\frac{4}{3}\] done
clear
D)
\[1\] done
clear
View Solution play_arrow
-
question_answer27)
The point(s) on the curve \[{{y}^{3}}+3{{x}^{2}}=12y\] where the tangent is vertical (parallel to y-axis), is (are) [IIT Screening 2002]
A)
\[\left( \pm \frac{4}{\sqrt{3}},-2 \right)\] done
clear
B)
\[\left( \pm \frac{\sqrt{11}}{3},1 \right)\] done
clear
C)
\[(0,\,0)\] done
clear
D)
\[\left( \pm \frac{4}{\sqrt{3}},2 \right)\] done
clear
View Solution play_arrow
-
question_answer28)
Let \[f(x)=\int\limits_{0}^{x}{\frac{\cos t}{t}dt,\,\,x>0}\] then \[f(x)\] has [Kurukshetra CEE 2002]
A)
Maxima when \[n=-2,\,-4,\,-6,\,.....\] done
clear
B)
Maxima when \[n=-1,\,-3,\,-5,\,....\] done
clear
C)
Minima when \[n=0,\,2,\,4,....\] done
clear
D)
Minima when \[n=1,3,5....\] done
clear
View Solution play_arrow
-
question_answer29)
If \[f(x)={{x}^{2}}+2bx+2{{c}^{2}}\]and \[g(x)=-{{x}^{2}}-2cx+{{b}^{2}}\] such that min \[f(x)>\] max \[g(x)\], then the relation between b and c is [IIT Screening 2003]
A)
No real value of b and c done
clear
B)
\[0<c<b\sqrt{2}\] done
clear
C)
\[|c|<\,|b|\sqrt{2}\] done
clear
D)
\[|c|\,>\,|b|\sqrt{2}\] done
clear
View Solution play_arrow
-
question_answer30)
N characters of information are held on magnetic tape, in batches of x characters each; the batch processing time is \[\alpha +\beta {{x}^{2}}\] seconds; \[\alpha \]and \[\beta \] are constants. The optimal value of x for fast processing is [MNR 1986]
A)
\[\frac{\alpha }{\beta }\] done
clear
B)
\[\frac{\beta }{\alpha }\] done
clear
C)
\[\sqrt{\frac{\alpha }{\beta }}\] done
clear
D)
\[\sqrt{\frac{\beta }{\alpha }}\] done
clear
View Solution play_arrow
-
question_answer31)
On the interval [0, 1], the function \[{{x}^{25}}{{(1-x)}^{75}}\] takes its maximum value at the point [IIT 1995]
A)
0 done
clear
B)
1/2 done
clear
C)
1/3 done
clear
D)
¼ done
clear
View Solution play_arrow
-
question_answer32)
The function \[f(x)=\int\limits_{-1}^{x}{t({{e}^{t}}-1)(t-1){{(t-2)}^{3}}{{(t-3)}^{5}}}dt\] has a local minimum at x = [IIT 1999]
A)
0 done
clear
B)
1 done
clear
C)
2 done
clear
D)
3 done
clear
View Solution play_arrow
-
question_answer33)
The maximum value of exp \[(2+\sqrt{3}\cos x+\sin x)\] is [AMU 1999]
A)
\[\exp (2)\] done
clear
B)
\[\exp (2-\sqrt{3})\] done
clear
C)
\[\exp (4)\] done
clear
D)
1 done
clear
View Solution play_arrow
-
question_answer34)
If the function \[f(x)=2{{x}^{3}}-9a{{x}^{2}}\] \[+12{{a}^{2}}x+1,\]where \[a>0\] attains its maximum and minimum at p and q respectively such that \[{{p}^{2}}=q\], then a equals [AIEEE 2003]
A)
3 done
clear
B)
1 done
clear
C)
2 done
clear
D)
\[\frac{1}{2}\] done
clear
View Solution play_arrow
-
question_answer35)
The function \[f(x)=\frac{\text{ln}(\pi +x)}{\text{ln}(e+x)}\] is [IIT 1995]
A)
Increasing on \[\left[ 0,\,\infty \right)\] done
clear
B)
Decreasing on \[\left[ 0,\,\infty \right)\] done
clear
C)
Decreasing on \[\left[ 0,\frac{\pi }{e} \right)\]and increasing on \[\left[ \frac{\pi }{e},\infty \right)\] done
clear
D)
Increasing on \[\left[ 0,\frac{\pi }{e} \right)\] and decreasing on \[\left[ \frac{\pi }{e},\infty \right)\] done
clear
View Solution play_arrow
-
question_answer36)
The function \[f(x)={{\sin }^{4}}x+{{\cos }^{4}}x\] increases, if [IIT 1999; Pb. CET 2001]
A)
\[0<x<\frac{\pi }{8}\] done
clear
B)
\[\frac{\pi }{4}<x<\frac{3\pi }{8}\] done
clear
C)
\[\frac{3\pi }{8}<x<\frac{5\pi }{8}\] done
clear
D)
\[\frac{5\pi }{8}<x<\frac{3\pi }{4}\] done
clear
View Solution play_arrow
-
question_answer37)
Let \[h(x)=f(x)-{{(f(x))}^{2}}+{{(f(x))}^{3}}\] for every real number x. Then [IIT 1998]
A)
h is increasing whenever f is increasing done
clear
B)
h is increasing whenever f is decreasing done
clear
C)
h is decreasing whenever f is decreasing done
clear
D)
Nothing can be said in general done
clear
View Solution play_arrow
-
question_answer38)
In [0, 1] Lagrange's mean value theorem is NOT applicable to [IIT Screening 2003]
A)
\[f(x)=\left\{ \begin{align} & \frac{1}{2}-x,\,\,\,\,\,\,\,x<\frac{1}{2} \\ & {{\left( \frac{1}{2}-x \right)}^{2}},\,\,\,x\ge \frac{1}{2} \\ \end{align} \right.\] done
clear
B)
\[f(x)=\left\{ \begin{align} & \frac{\sin x}{x},\,\,\,x\ne 0 \\ & \,\,\,\,\,1\,\,\,,\,\,\,x=0 \\ \end{align} \right.\] done
clear
C)
\[f(x)=x|x|\] done
clear
D)
\[f(x)=|x|\] done
clear
View Solution play_arrow
-
question_answer39)
If the function \[f(x)={{x}^{3}}-6a{{x}^{2}}+5x\]satisfies the conditions of Lagrange's mean value theorem for the interval [1, 2] and the tangent to the curve \[y=f(x)\]at \[x=\frac{7}{4}\]is parallel to the chord that joins the points of intersection of the curve with the ordinates \[x=1\] and \[x=2\]. Then the value of \[a\]is [MP PET 1998]
A)
\[\frac{35}{16}\] done
clear
B)
\[\frac{35}{48}\] done
clear
C)
\[\frac{7}{16}\] done
clear
D)
\[\frac{5}{16}\] done
clear
View Solution play_arrow
-
question_answer40)
Let \[f(x)=\left\{ \begin{align} & {{x}^{\alpha }}\ln x,x>0 \\ & 0,\,\,\,\,\,\,\,\,\,\,\,\,x=0 \\ \end{align} \right\}\], Rolle?s theorem is applicable to f for \[x\in [0,1]\], if \[\alpha =\] [IIT Screening 2004]
A)
- 2 done
clear
B)
- 1 done
clear
C)
0 done
clear
D)
\[\frac{1}{2}\] done
clear
View Solution play_arrow