Recent questions tagged kvquestionbank2012

Is the function $ f(x)=\Large\frac{3x+4\tan x}{x}$ continuous at x = 0? If not, how may the function be defined to make it continuous at this point.

asked Jan 25, 2013 by meena.p

1 answer

Show that the function :$f(x)= \left\{ \begin{array}{1 1} \Large\frac{e^{\frac{1}{x}}-1}{e^{\frac{1}{x}}+1}, & \quad when\;x\neq0 \\ 0, & \quad when\; x=0 \end{array} \right. $ is discontinuous at x = 0.

asked Jan 25, 2013 by meena.p

1 answer

Given that $ f(x)= \left\{ \begin{array}{1 1} \Large\frac{1- \cos 4x}{x^2}, & \quad if\;x<0 \\ a & \quad ,if\;x=0 \\ \Large\frac{\sqrt x}{\sqrt{16+\sqrt{x}}-4}, & \quad if\;x>0 \end{array}. \right. $ If f(x) is continuous at x = 0, find the value of a.

asked Jan 25, 2013 by meena.p

1 answer

Find $\Large \frac{dy}{dx}$ when $y=\tan^{-1}\Large\frac{1+x^2}{1-x^2}$

asked Jan 25, 2013 by meena.p

1 answer

Find $\Large \frac{dy}{dx}$ when $ y=a^x.x^a$

asked Jan 25, 2013 by meena.p

1 answer

Verify Rolles theorem for the function $f(x)=\sin 2x \;in\;\bigg[0,\Large\frac{\pi}{2}\bigg]$

asked Jan 25, 2013 by meena.p

1 answer

Let $y=y=\sqrt{e^\sqrt x },x>0\;find\;\large\frac{dy}{dx}.$

asked Jan 25, 2013 by meena.p

1 answer

If $ f(x)=\begin{vmatrix} x & x^2 & x^3 \\ 1 & 2x & 3x^2 \\ 0 & 2 & 6x \end{vmatrix}$ than find $ f' (x)$

asked Jan 25, 2013 by meena.p

1 answer

If $ f^1(x)=g(x)\;and\;g^1(x)=-f(x)$ for all n and $f(2)=f^1(2)=4,$ find the value of $f^2(24)+g^2(24)$

asked Jan 24, 2013 by meena.p

1 answer

Find the set of all points where the function f(x) = 2x |x| is differentiable.

asked Jan 24, 2013 by meena.p

1 answer

Find $ \frac{dy}{dx}\;when\;y=\Large x^{x^{x^{x.....}}}$

asked Jan 24, 2013 by meena.p

1 answer

Let f be a non zero continuous function satisfying f(x+y) = f(x).f(y) for all x, y $ \epsilon$ R. If f(2) = 9 then find the value of f(3).

asked Jan 24, 2013 by meena.p

1 answer

Let f be continuous function on [1,3]. If f takes only rational values for all x and f(2) = 10 then find the value of f(1.5)

asked Jan 24, 2013 by meena.p

1 answer

For any square matrix verify A(adj A) = |A|I

asked Jan 24, 2013 by meena.p

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If $ A=\begin{bmatrix} a & b \\ 0 & 1 \end{bmatrix}\;prove\;that\;A^n=\begin{bmatrix} \Large a^n & \Large\frac{b(a^n-1)}{a-1} \\ \normalsize 0 & 1 \end{bmatrix},n \epsilon N$

asked Jan 24, 2013 by meena.p

0 answers

Let $\begin{vmatrix} 3 & y \\ x & 1 \end{vmatrix} =\begin{vmatrix} 3 & 2 \\ 4 & 1 \end{vmatrix} $ find the possible values of x and y, find the values if x = y.

asked Jan 24, 2013 by meena.p

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If the points (2,0), (0,5) and (x,y) are collinear give a relation between x and y.

asked Jan 24, 2013 by meena.p

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If P and Q are invertible matrices of same order, then show that PQ is also invertible.

asked Jan 24, 2013 by meena.p

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Find x, if $ \begin{bmatrix} x & -5 & -1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix} \begin{bmatrix} x \\ 4 \\ 1 \end{bmatrix} =0$

asked Jan 24, 2013 by meena.p

0 answers

If $ A=\begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix} \;find\;A^2\;Hence\;find\;A^6.$

asked Jan 24, 2013 by meena.p

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If $A=\begin{bmatrix} \cos \theta & \sin \theta & \\ -\sin \theta & \cos \theta \end{bmatrix} $ then show that $A^n=\begin{bmatrix} \cos 2\theta & \sin 2\theta & \\ -\sin 2\theta & \cos 2\theta \end{bmatrix} $

asked Jan 24, 2013 by meena.p

0 answers

If a, b, c are in A.P. then find the value of: $\begin{vmatrix} 2y+4 & 5y+7 & 8y+a \\ 3y+5 & 6y+8 & 9y+b \\ 4y+6 & 7y+9 & 10y+c \end{vmatrix}$

asked Jan 24, 2013 by meena.p

0 answers

If x, y and z all positive, are $p^{th},q^{th}and\;r^{th}$ terms of a G.P. Prove that $\begin{vmatrix} logx & p & 1 \\ logy & q & 1 \\ logz & r & 1 \end{vmatrix}=0$

asked Jan 24, 2013 by meena.p

0 answers

Find the quadratic function defined by the equation $ f(x)=ax^2+bx+c\;if\;f(0)=6,f(2)=11,f(-3)=6,$ using determinants.

asked Jan 24, 2013 by meena.p

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If $ A+B+C=\large\pi,\;\normalsize show\;that\;\begin{vmatrix} \sin (A+B+C) & \sin B & \cos C \\ -\sin B & 0 & \tan A \\ \cos (A+B) & -\tan A & 0 \end{vmatrix}=0$

asked Jan 24, 2013 by meena.p

0 answers

If x, y, z are positive and unequal show that the value of determinant $ \begin{vmatrix} x & y & z \\ y & z & x \\ z & x & y \end{vmatrix}$ is negative.

asked Jan 24, 2013 by meena.p

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If $ A=\begin{bmatrix} 3 & -4 \\ -1 & 2 \end{bmatrix}$ find Matrix B such that AB= 1

asked Jan 24, 2013 by meena.p

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If $A=\begin{bmatrix} \cos \theta & i\sin \theta \\ i\sin \theta & \cos \theta \end{bmatrix}$ then prove by Mathematical Induction that :$A^n=\begin{bmatrix} \cos n\theta & i\sin n\theta \\ i\sin n\theta & \cos n\theta \end{bmatrix}$

asked Jan 24, 2013 by meena.p

0 answers

If $ p(\alpha)=\begin{bmatrix} \cos \alpha & i\sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix}\;show\;that\;p(\alpha) \times p(\beta)=p(\alpha+\beta)$

asked Jan 24, 2013 by meena.p

0 answers

Find the values of a and b such that $ A^2+aI=bA\;where\; A=\begin{bmatrix} 3 & 1 \\ 7 & 5 \end{bmatrix}$

asked Jan 24, 2013 by meena.p

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If $ A=\begin{bmatrix} 1 & -2 & 1 \\ -2 & 3 & 1 \\ 1 & 1 & 5 \end{bmatrix}$ verify $(AdjA)^{-1}=adj(A^{-1})$

asked Jan 24, 2013 by meena.p

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If two matrices $ A^{-1}\;and\;B$ are given how to find $(AB)^{-1}$verify with an example.\[(Find\; B^{-1} then\; find\; B^{-1} \times A^{-1})\]

asked Jan 24, 2013 by meena.p

0 answers

If $P(a)=\begin{bmatrix} \cos a & -\sin a & 0 \\ \sin a & \cos a & 0 \\ 0 & 0 & 1 \end{bmatrix}$then show that $[P(x)]^{-1}=[P(-x)]$

asked Jan 24, 2013 by meena.p

0 answers

Find the matrix x if $ \begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix} \times \begin{bmatrix} 0 & 2 \\ 1 & 3 \end{bmatrix}=\begin{bmatrix} 1 & 6 \\ 3 & -2 \end{bmatrix}$\[X=-\frac{1}{4}\begin{bmatrix} -53 & 18 \\ 25 & -10 \end{bmatrix}\]

asked Jan 24, 2013 by meena.p

0 answers

If $A=\begin{bmatrix} 2 & -3 \\ 3 & 4 \end{bmatrix} $satisfies the equation : $ x^2+6x+17=0\;find\;A^{-1}$

asked Jan 24, 2013 by meena.p

0 answers

If $\begin{vmatrix} x & b & c \\ a & y & c \\ a & b & z \end{vmatrix}=0$ then find the value of $ \large\frac{x}{x-a}+\frac{y}{y-b}+\frac{z}{z-c}$

asked Jan 24, 2013 by meena.p

0 answers

Given $ A=\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$ such that $ |A|=-10.\;Find\;a_{11}c_{11}+a_{12}c_{12}$

asked Jan 24, 2013 by meena.p

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If A is a non-singular matrix of order 3 and |A| = - 4 Find |adj A|

asked Jan 24, 2013 by meena.p

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For what value of 'K' the matrix $ \begin{bmatrix} k & 2 \\ 3 & 4 \end{bmatrix}$has no inverse.

asked Jan 24, 2013 by meena.p

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If $ A=\begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix},$ show that $ A^2-5A+7I=0,$ use the result to find $A^4$.

asked Jan 24, 2013 by meena.p

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Let A and B be 3 x 3 matrices such that $ A^T=-A\;and\;B^T=B.$Then the matrix $\lambda(AB+3BA)$ is skew symmetric matrix for $\lambda$.

asked Jan 24, 2013 by meena.p

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Assume X, Y, Z, W and P are matrices of order (2 x n), (3xk), (n x 3) and (p x k) respectively, then the restriction on n, k and p so that py + my will be defined are : (k=3, p=n)

asked Jan 24, 2013 by meena.p

0 answers

Find the values of a, b, c if the matrix $\begin{bmatrix} 0 & 2b & c \\ a & b & -c \\ a & -b & c \end{bmatrix}$ satisfy the equation $ A^1A=I_3$

asked Jan 24, 2013 by meena.p

0 answers

If $ A=\begin{bmatrix} 1& 1 & 1 \\ 1& 1 & 1 \\ 1& 1 & 1 \end{bmatrix}$ then prove that $ A^n=\begin{bmatrix} 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \end{bmatrix}\;n\in N$

asked Jan 24, 2013 by meena.p

0 answers

If A and B are two matrices such that AB = B and BA = A then $A^2+B^2$=_?_\

asked Jan 24, 2013 by meena.p

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If $ A=\begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix},\;then\;A^5=?$

asked Jan 24, 2013 by meena.p

1 answer

If P and Q are equal matrices of same order such that PQ = QP, then prove by induction that $ PQ^n=Q^nP.$ Further, show that $(PQ)^n=P^n\times Q^n.$

asked Jan 24, 2013 by meena.p

0 answers

If $ A=\begin{bmatrix} 3 & 1 \\ 7 & 5 \end{bmatrix}$ find x and y such that $ A^2+xl=yA.\;Find\;A^{-1}.$

asked Jan 24, 2013 by meena.p

1 answer

Give that x = -9 is a root of $ \begin{vmatrix} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{vmatrix}=0$ find the other roots. Hint: Evaluate, find other roots.

asked Jan 24, 2013 by meena.p

0 answers

If $ \begin{bmatrix} \cos^2A & \cos A\sin A \\ \cos A\sin A & \sin^2A \end{bmatrix},Y=\begin{bmatrix} \cos^2B & \cos B\sin B \\ \cos B\sin B & \sin^2B \end{bmatrix}$then show that XY is a zero matrix, provided (A-B) is an odd multiple of $\large\pi/2$

asked Jan 24, 2013 by meena.p

0 answers

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