Recent questions tagged kvquestionbank2012

If $ y=log \tan \bigg(\Large\frac{\pi}{4}+\frac{\pi}{2}\bigg)$then$\Large\frac{dy}{dx}$is\[(a)\;0 \qquad(b)\;\cos x\qquad(c)\;-\sec x\qquad(d)\;\sec x\]

asked Jan 29, 2013 by meena.p

0 answers

If $ x=a(\theta-\sin \theta).y=a(1-\cos \theta) \;then\; \Large\frac{d^2y}{dx^2}\;\normalsize at\;\theta=\Large\frac{\pi}{2}\;is.$\[(a)\;\frac{1}{a}\qquad(b)\;\frac{1}{2}\qquad(c)\;-\frac{1}{a}\qquad(d)\;-\frac{1}{2a}\qquad\]

asked Jan 29, 2013 by meena.p

0 answers

The minimum value of f(x) = |3-x| + |2+x| + |5-x| is \[(a)\;0\qquad(b)\;7\qquad(c)\;8\qquad(d)\;10\]

asked Jan 28, 2013 by meena.p

0 answers

The tangent to the curve\[x=a\sqrt{\cos 2\theta} \cos\theta y=a\sqrt{cos2\theta}\sin \theta\]at the point corresponding to $ \theta =\frac{\pi}{6}$ is \begin{array}{1 1}(a)\;Parallel\; to\; the\; x-axis & \qquad (b)\;Parallel\; to\; the\; y-axis \\ (c)\;Parallel\; to\; the\; line\; y = x & \qquad(d)\;none\;of\;these\end{array}

asked Jan 28, 2013 by meena.p

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If the line ax+by+c = 0 is normal to the curve xy = 1 then \[(a)\;a>0,b>0 \qquad (b)\;a<0,b<0 \qquad(c)\;a<0,b>0\qquad(d)\;a<0,b<0.\]

asked Jan 28, 2013 by meena.p

1 answer

$f(x)=\Large\frac{x^2-1}{x^2+1} $ for every real number then minimum value of \begin{array}{1 1}(a)\;does\; not\; exist & \qquad(b)\;is\;not\;attained\;even\;though\;f\;is\;bounded\\(c)\;is\;equal\;to\;1 & \qquad(d)\;is\;equal\;to\;-1\end{array}

asked Jan 28, 2013 by meena.p

0 answers

The function $ \Large\frac{\sin(x+\alpha)}{\sin(x+\beta)}$ has no maximum or minimum value if \[(a)\;\beta-\alpha=k\pi\qquad(b)\;\beta-\alpha \neq k\pi\qquad(c)\;\beta-\alpha = 2k\pi\qquad(d)\;None\; of\; these\; where\; k\; is\; an\; integer.\]

asked Jan 28, 2013 by meena.p

0 answers

If $f(x)= \left\{ \begin{array}{1 1} 3x^2+12x-1 & \quad :\;-1\leq x \leq 2 \\37-x & \quad :\;2<x\leq 3 \end{array} \right. $ then\[\begin{array}{1 1}(a)\;f(x)\; is\; increasing\; on\; [-1,2]\\(b)\;f(x)\; is\; continuous\; on\; [-1,3]\\(c)\;f'(2) doesn't \;exist\\(d)\;f(x)\; has\; the\; maximum\; value\; at x = 2\end{array}\]

asked Jan 28, 2013 by meena.p

0 answers

If $\theta$ is the semivertical angle of a cone of maximum volume and given slant height, then $\tan \theta$ is given by \[(a)\;2\qquad(b)\;1\qquad(c)\;\sqrt 2\qquad(d)\;\sqrt 3\]

asked Jan 28, 2013 by meena.p

0 answers

If $ y=a log|x|+bx^2+x $ has its extreme values at x = 1 and x = 2 then\[(a)\;a=2,b=-1\qquad(b)\;a=2,b=-1/2\qquad(c)\;-2,b=1/2\qquad(d)\;none\;of\;these\]

asked Jan 28, 2013 by meena.p

0 answers

The difference between the greatest and the least values of the function $ f(x)=\large\cos x+\large\frac{1}{2}\cos 2x-\frac{1}{3}\cos 3x $ is \[(a)\;2/3\qquad(b)\;8/7\qquad(c)\;9/4\qquad(d)\;3/8\]

asked Jan 28, 2013 by meena.p

0 answers

The co-ordinates of the point p(x,y) in the first quadrant on the ellipse $\Large\frac{ x^2}{8}+\frac{y^2}{18}$=1 so that the area of the triangle formed by the tangent at P and the co-ordinate axes is the smallest are given by \[(a)\;(2,3)\qquad(b)\;(\sqrt 8,0)\qquad(c)\;(\sqrt {18}, 0)\qquad(d)none\;of\;these\]

asked Jan 28, 2013 by meena.p

0 answers

The value of a for which the function $ f(x)=a\sin(x)+\frac{1}{3}\sin 3x.$has an extreme at $ x=\frac{\pi}{3}$ is\[(a)\;1\qquad(b)\;-1\qquad(c)\;0 \qquad(d)\;2\]

asked Jan 28, 2013 by meena.p

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The function $y=\tan^{-1}x-x $ decreases in the interval of \[(a)\;(1,\infty)\qquad(b)\;(-1,\infty)\qquad(c)\;(-\infty,\infty)\qquad(d)\;(0,\infty)\]

asked Jan 28, 2013 by meena.p

0 answers

The function $ f(x)=2 log (x-2) - x^2+4x+1 $increases in the interval.\[(a)\;(1,2)\qquad(b)\;(2,3)\qquad(c)\;(5/2,3)\qquad(d)\;(2,4)\]

asked Jan 28, 2013 by meena.p

1 answer

The slope of the tangent to the curve represented by $ x=t^2+3t-8\;and\;y=2t^2-2t-5$at the point M(2,-1) is \[(a)\;\frac{7}{6}\qquad(b)\;\frac{2}{3}\qquad(c)\;\frac{3}{2}\qquad(d)\;\frac{6}{7}\]

asked Jan 28, 2013 by meena.p

0 answers

Verify the Rolles Theorem for the function $ f(x)=\sin x- \cos x, $in the interval $ \bigg[\Large\frac{\pi}{4},\frac{5\pi}{4}\bigg]$

asked Jan 28, 2013 by meena.p

1 answer

Using LMV Theorem, find a point on the curve $ y=(x-3)^2,$ where the tangent is parallel to the chord joining (3,0) and (5,4).

asked Jan 28, 2013 by meena.p

1 answer

If f(x) and g(x) are functions derivable in [a,b] such that f(a) = 4, f(b) = 10, g(a) =1, g(b) =3.Show that for a < c < b, we have $ f'(c)=3g'(c).$

asked Jan 28, 2013 by meena.p

1 answer

It is given that for the function $ f(x)=x^3-6x^2+px+q\;on\;[1,3].$Rolles theorem holds with $ c=2+\large\frac{1}{\sqrt 3}.$Find the values of p and q.

asked Jan 28, 2013 by meena.p

1 answer

Verify Rolles theorem for the function $ f(x)=\Large e^{1-x^2}$in the interval [-1,1]

asked Jan 28, 2013 by meena.p

1 answer

If $ x\sqrt{1+y}+y\sqrt{1+x}=0,\normalsize\; Prove\; that\; \Large\frac{dy}{dx}=\frac{-1}{(1+x)^2}$

asked Jan 28, 2013 by meena.p

1 answer

Differentiate $ \cos^{-1} \bigg[\large\frac{3 \cos x-2 \sin x}{\sqrt {13}}\bigg] w.r.t \; \sin^{-1} \bigg[\large\frac{5\sin x +4 \cos x}{\sqrt{41}}\bigg]$

asked Jan 28, 2013 by meena.p

1 answer

If $ x=\cos \theta + log \tan \Large\frac{\theta }{2},\normalsize y=\sin \theta \;find\; \Large\frac {d^2y}{dx^2}\normalsize \;at\;\theta =\Large\frac{\pi}{4}$

asked Jan 28, 2013 by meena.p

1 answer

If $ y^{ \cos x}+(\tan^{-1}x)^y=1,find \Large\frac{dy}{dx}$

asked Jan 28, 2013 by meena.p

1 answer

If $ y=f \bigg(\Large\frac{2x-1}{x^2+1}\bigg)\;\normalsize and\;f' (x)=\sin x^2,\;find\; \Large\frac{dy}{dx}$

asked Jan 28, 2013 by meena.p

1 answer

Find $\Large\frac{dy}{dx}$when$y=\tan^{-1}\bigg[\Large\frac{x^{\frac{1}{3}}+a^{\frac{1}{3}}}{1-x^{\frac{1}{3}}a^{\frac{1}{3}}}\bigg]$

asked Jan 28, 2013 by meena.p

1 answer

If $ y=|\cos x|+|\sin x|,\;find\; \Large\frac{dy}{dx}\; \normalsize at\;x=\Large\frac{2\pi}{3}$

asked Jan 28, 2013 by meena.p

1 answer

If $ y=\Large\frac{1}{\sqrt{b^2-a^2}} \normalsize log \bigg[\Large\frac{\sqrt{b+a}+\sqrt{b-a}\tan \frac{x}{2}} {\sqrt{b+a}-\sqrt{b-a}\tan \frac{x}{2}}\bigg]$ prove that $ \Large\frac {dy}{dx}=\frac{\sec^2\frac{x}{2}}{(b+a)-(b-a) \tan^2\frac{x}{2}}$

asked Jan 28, 2013 by meena.p

1 answer

If $ \sqrt { 1-x^2} +\sqrt {1-y^2}=a(x-y),\;prove\;that\;\Large\frac{dy}{dx}=\sqrt{\frac{1-y^2}{1-x^2}}$

asked Jan 25, 2013 by meena.p

1 answer

Prove that $ \Large\frac{d}{dx}\bigg[\Large\frac{1}{4 \sqrt 2} \normalsize log\bigg|\Large\frac{x^2+\sqrt 2 x+1}{x^2- \sqrt 2 x+1} \bigg|+\Large\frac {1}{2 \sqrt 2} \normalsize \tan^{-1} \Large\frac{\sqrt 2 x}{1-x^2}\bigg]=\Large\frac{1}{1+x^4}$

asked Jan 25, 2013 by meena.p

1 answer

Let $y= \tan^{-1}\bigg[\Large\frac{4x}{1+5x^2}\bigg] \normalsize +\tan^{-1}\bigg[\Large\frac{2+3x}{3-2x}\bigg],$ show that $ \Large\frac{dy}{dy}=\frac{5}{1+25x^2}$

asked Jan 25, 2013 by meena.p

1 answer

Given that $ \cos \frac{x}{2}.\cos \frac{x}{4}.\cos \frac{x}{8}.......=\Large\frac {\sin x}{x},$ \[Prove\;that\;\frac{1}{2^2}\sec^2\frac{x}{2}+\frac{1}{2^4}\sec^2\frac{x}{4}+.......=cosec^2x-\frac{1}{x^2}\]

asked Jan 25, 2013 by meena.p

1 answer

Find $\Large\frac{dy}{dx},$ when $y=\sin^{-1}\bigg[x\sqrt{1-x}-\sqrt x \sqrt{1-x^2}\bigg]$

asked Jan 25, 2013 by meena.p

1 answer

Differentiate w.r.t. x, $ y=\tan^{-1}\bigg[\Large\frac{a\cos x-b\sin x}{b\cos x+a \sin x}\bigg]$

asked Jan 25, 2013 by meena.p

1 answer

If $ y=\frac{2}{\Large\sqrt {a^2-b^2}}\tan^{-1}\bigg[\sqrt{\frac{a-b}{a+b}}\tan \frac{x}{2} \bigg],$ prove that $ \Large\frac{dy}{dx}=\frac{1}{a+b \cos x}, \normalsize a>b>0 $

asked Jan 25, 2013 by meena.p

1 answer

If $y=\sin ^{-1}\big [x^2\sqrt{1-x^2}+x\sqrt {1-x^4}\big],$ prove that $ \Large\frac{dy}{dx}=\frac{2x}{\sqrt{1-x^4}}+\frac{1}{\sqrt{1-x^2}}$

asked Jan 25, 2013 by meena.p

1 answer

Find$ \Large\frac{dy}{dx}\;\normalsize when\;y=\sin ^{-1} \bigg[\Large\frac{5x+12 \sqrt{1-x^2}}{13}\bigg]$

asked Jan 25, 2013 by meena.p

1 answer

Let $\phi(x)\;andh(x)$ be derivable at x = c. Show that necessary and sufficient condition for the function defined as :\[f(x)= \left\{ \begin{array}{1 1} \phi(x), & \quad x \leq c \\ h(x), & \quad x>c \end{array} \right. \]to be derivable at x= c are $(i)\phi(c)=h(c)\quad(ii)\phi'(c)=h'(c)$

asked Jan 25, 2013 by meena.p

1 answer

Let $f(x) = \left\{ \begin{array}{l l} \Large\frac{1-sin^3x}{3cos^2x}, & \quad if\; { x < \Large\frac{\pi}{2}}\\ a, & \quad if\; { x = \Large\frac{\pi}{2}} \\ \Large\frac{b(1-sin x)}{(\pi - 2x)^2}, & \quad if \;{ x > \Large\frac{\pi}{2}} \end{array}. \right.$ If f(x) is continuous function at x $ x = \Large\frac{\pi}{2} $, find a and b.

asked Jan 25, 2013 by meena.p

1 answer

Discuss the continuity of the following function at x = 0 $f(x)= \left\{ \begin{array}{1 1} \Large\frac{x^4+x^3+2x^2}{\tan ^{-1}x} & \quad ,\;x\neq0 \\ 10 & \quad ,\;x=0 \end{array} \right. $

asked Jan 25, 2013 by meena.p

1 answer

Let $ f(x)= \left\{ \begin{array}{1 1} |x|\cos \frac{1}{x} & \quad if\;x\neq0 \\ 0 & \quad if\;x=0 \end{array} \right. $ then discuss the continuity of f(x) at x = 0

asked Jan 25, 2013 by meena.p

1 answer

Find all the points of discontinuity of f defined by f(x) = | x | - | x+1|.

asked Jan 25, 2013 by meena.p

1 answer

The function $ \Large\frac{log(1+ax)-log(1-bx)}{x}$ is not defined at x = 0. Find the value of f(x) so that f(x) is continuous at x = 0.

asked Jan 25, 2013 by meena.p

1 answer

Show that the function $f(x) = | \sin x+\cos x| $ is continuous at $ x =\Large\pi$

asked Jan 25, 2013 by meena.p

1 answer

Determine the values of a, b and c for which the function :$f(x)= \left\{ \begin{array}{1 1} \Large\frac{\sin (a+1)x+\sin x}{x} & \quad ,\;x<0 \\ c & \quad ,\;x=0 \\ \Large\frac{\sqrt{x+bx^2}-\sqrt x}{b\sqrt{x^3}} & \quad ,\;x>0 \end{array} \right. $ may continuous at x = 0

asked Jan 25, 2013 by meena.p

1 answer

Show that the function $ f(x)= \left\{ \begin{array}{1 1} x+\lambda, & \quad x< 1 \\ \lambda x^2+1, & \quad x\geq1 \end{array} \right. $ is continuous function, regardless of the choice of $ \lambda \in R $

asked Jan 25, 2013 by meena.p

1 answer

Show that Sin | x | is continuous.

asked Jan 25, 2013 by meena.p

1 answer

Show that the function : $f(x)= \left\{ \begin{array}{1 1} |x|, & \quad x\leq2 \\ [x], & \quad x>2 \end{array} \right. $ is continuous [0,2].

asked Jan 25, 2013 by meena.p

1 answer

Find a and b if the function :$ f(x)= \left\{ \begin{array}{1 1} [1+| \sin x | ]^{\Large\frac{a}{\sin |x|}}, & \quad \Large\frac{-\pi}{6}< \normalsize x<0 \\ 6 & \quad ,x=0 \\ e^{\Large\frac{\tan 2x}{\tan 3x}}, & \quad 0<x<\Large\frac{\pi}{6} \end{array} \right. $ is continuous on $\bigg(\Large\frac{-\pi}{6},\frac{\pi}{6}\bigg).$

asked Jan 25, 2013 by meena.p

0 answers

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