Recent questions tagged kvquestionbank2012

Prove that $\begin{bmatrix} (a+1) & (a+2) & (a+2) & 1 \\ (a+2) & (a+3) & (a+3) & 1 \\(a+3) & (a+4) & (a+4) & 1 \end{bmatrix}=-2$

asked Jan 24, 2013 by meena.p

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If $f(x)=3x^2-9x+7,$ then for a square matrix A, write f(A)

asked Jan 24, 2013 by meena.p

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Find the matrix X such that :$ \begin{bmatrix} 2 & -1 \\ 0 & 1 \\ -2 & 4 \end{bmatrix}X=\begin{bmatrix} -1 & -8 & -10 \\ 3 & 4 & 0 \\10 & 20 & 10 \end{bmatrix}$

asked Jan 24, 2013 by meena.p

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The total number of elements in a matrix represents a prime number. How many possible orders a matrix can have.

asked Jan 24, 2013 by meena.p

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Prove the determinant $\begin{vmatrix} x & -\sin\alpha & \cos\alpha \\ \sin\alpha & -x & 1 \\ \cos\alpha & 1 & x \end{vmatrix}$ is independent as $\alpha$

asked Jan 24, 2013 by meena.p

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Prove that $ \begin{vmatrix} (b+c)^2 & ab & ca \\ ab & (a+c)^2 & bc \\ ca & bc & (a+b)^2 \end{vmatrix}=2abc(a+b+c)^2$ (Use properties to prove the above)

asked Jan 24, 2013 by meena.p

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Express $\begin{bmatrix} 2 & -3 \\ 1 & 4 \end{bmatrix}$ as sum of the symmetric and skew symmetric matrices.

asked Jan 24, 2013 by meena.p

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Using properties of Determinants prove that : $ \begin{vmatrix} a^2 & bc & ac+c^2 \\ a^2+ab & b^2 & ac \\ab & b^2+ac & c^2 \end{vmatrix}=4a^2b^2c^2$

asked Jan 24, 2013 by meena.p

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Prove that $\begin{vmatrix} x-3 & x-4 & x-a \\ x-2 & x-3 & x-b \\ x-1 & x-2 & x-c \end{vmatrix}=0,$ where a,b,c are in A.P

asked Jan 24, 2013 by meena.p

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Inverse of a square matrix is unique. Give an example to prove it?

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}\;and\;A+A^T=1.$Find the possible values of $\theta=\large\pi/3.$

asked Jan 24, 2013 by meena.p

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If $ \begin{bmatrix} p & q \\ r & -p \end{bmatrix}$ is such that $A^2=1$ then find the value of $1-p^2+qr$

asked Jan 24, 2013 by meena.p

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If a+b+c=0 anb $ \begin{vmatrix} a-x & a & b \\ c & b-x & a \\ b & a & c-x \end{vmatrix}$ then prove that either x=0 or $x= \pm\sqrt{\frac{3}{2}(a^2+b^2+c^2)}$

asked Jan 24, 2013 by meena.p

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If $ \cos^{-1}\Large\frac{x}{2}+\normalsize\cos^{-1}\large\frac{y}{3}=\theta ,\;\normalsize prove\;that\;9x^2-12xy\cos\theta +4y^2=36\sin^2\theta $

asked Jan 24, 2013 by meena.p

1 answer

Solve: $\sin\lfloor2\cos^{-1}\{\cot(2\tan^{-1}x)\}\rfloor=0$

asked Jan 24, 2013 by meena.p

1 answer

Solve for x:$\tan^{-1}(x+1)+\tan^{-1}x+\tan^{-1}(x-1)=\tan^{-1}3.$

asked Jan 24, 2013 by meena.p

1 answer

Prove that $ \tan^{-1}\large\frac{yz}{xr}+ \tan^{-1}\frac{zx}{yr}+ \tan^{-1}\frac{xy}{zr}=\frac{\pi}{2}\;where\;x^2+y^2+z^2=r^2$

asked Jan 24, 2013 by meena.p

1 answer

If $ \tan^{-1}a+\tan^{-1}b+\tan^{-1}c=\pi,\;prove\;that\;a+b+c=abc.$

asked Jan 24, 2013 by meena.p

1 answer

Prove: $\cos^{-1}x-\cos^{-1}y=\cos^{-1}\bigg[xy+\sqrt{1-x^2}.\sqrt{1-y^2}\bigg]$

asked Jan 24, 2013 by meena.p

1 answer

Solve for x $ \sin^{-1}\frac{2\alpha}{1+\alpha^2}+ \sin^{-1}\frac{2\beta}{1+\beta^2}=2\tan^{-1}x.$

asked Jan 24, 2013 by meena.p

1 answer

Prove that :$\cot^{-1}\bigg[\frac{ab+1}{a-b}\bigg]+\cot^{-1}\bigg[\frac{bc+1}{b-c}\bigg]+\cot^{-1}\bigg[\frac{ca+1}{c-a}\bigg]=0$

asked Jan 24, 2013 by meena.p

1 answer

If $ x= cosec \lfloor\tan^{-1}\{\cos(\cot^{-1}(\sec(\sin^{-1}a)))\}\rfloor\; $ and $ y=\sec\lfloor\cot^{-1}\{\sin(\tan^{-1}(cosec(\cos^{-1}a)))\}\rfloor\;$ where $\;a\;\in [0,1]$ Find the relationship between $x$ and $y$ in terms of $a$

asked Jan 24, 2013 by meena.p

1 answer

In any Triangle ABC, if $ A=\tan^{-1}2\;and\;B=\tan^{-1}3,$prove that $C=\frac{\pi}{4}$

asked Jan 24, 2013 by meena.p

1 answer

Prove that $\sin2[\cot^{-1}\{\cos(\tan^{-1}x)\}]=2\frac{\sqrt{x^2+1}}{x^2+2}$

asked Jan 24, 2013 by meena.p

1 answer

If $ \sin^{-1}x+\sin^{-1}y+\sin^{-1}z=\pi,$ prove that $x\sqrt{1-x^2}+y\sqrt{1-y^2}+z\sqrt{1-z^2}=2xyz$

asked Jan 23, 2013 by meena.p

1 answer

If $ \cos^{-1}\frac{x}{a}+\cos^{-1}\frac{y}{b}=\alpha,$prove that $\frac{x^2}{a^2}-\frac{2xy}{ab}\cos \alpha+\frac{y^2}{b^2}=\sin^2\alpha.$

asked Jan 23, 2013 by meena.p

1 answer

If $x,y,z \in [-1,1]\;$ such that $\;\sin^{-1}x+\sin^{-1}y+\sin^{-1}z=\frac{3\pi}{2},$ find the value of $x^{2006}+y^{2007}+z^{2008}-\Large \frac{9}{x^{2006}+y^{2007}+z^{2008}}$

asked Jan 23, 2013 by meena.p

1 answer

Prove that $4\tan^{-1}\large\frac{1}{5}-\tan^{-1}\frac{1}{70}+\tan^{-1}\frac{1}{99}=\large\frac{\pi}{4}$

asked Jan 23, 2013 by meena.p

1 answer

Prove that $\cos\bigg\{2\cot^{-1}\sqrt{\large\frac{1-x}{1+x}}\bigg\}+x=0$

asked Jan 23, 2013 by meena.p

1 answer

If $x>0$ and $\sin^{-1}\bigg(\large{\frac{5}{x}}\bigg)+\sin^{-1}\bigg(\frac{12}{x}\bigg)=\large\frac{\pi}{2},$ than find the value of $x$.

asked Jan 23, 2013 by meena.p

1 answer

If $\cos^{-1}x=\tan^{-1}x,\;show\;that\;\sin(\cos^{-1}x)=x^2$

asked Jan 23, 2013 by meena.p

1 answer

Find the value of x if $cosec^{-1}x=2\cot^{-1}7+\cot^{-1}\frac{3}{4}$

asked Jan 23, 2013 by meena.p

1 answer

Find the principal value of $\sin^{-1}\bigg(\frac{-\sqrt {3}}{2}\bigg)+\cos^{-1}\bigg(\frac{-\sqrt {3}}{2}\bigg)$

asked Jan 23, 2013 by meena.p

1 answer

Considering the principal solutions, find the number of solutions of $ \tan^{-1}2x+\tan^{-1}3x=\frac{\pi}{4}$

asked Jan 23, 2013 by meena.p

1 answer

Simplify $\tan^{-1}\big(\large \frac{\cos x}{1+\sin x}\big)$

asked Jan 23, 2013 by meena.p

1 answer

Solve for $x$: $\sin\bigg[\sin^{-1}\frac{1}{3}+\cos^{-1}x\bigg]=1$

asked Jan 23, 2013 by meena.p

1 answer

Find the value of $\tan^{-1}1+\tan^{-1}2+\tan^{-1}3.$

asked Jan 23, 2013 by meena.p

1 answer

Find the value of $\cos\bigg\{\cos^{-1}\bigg(\frac{-\sqrt {3}}{2}\bigg)+\frac{\pi}{6}\bigg\}$

asked Jan 23, 2013 by meena.p

1 answer

On the set N of natural numbers, define the operation * on N by mn=gcd(m,n) for all m, n$ \varepsilon$ N. Show that is commutative as well as associative.

asked Jan 23, 2013 by meena.p

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Determine which of the following functions \[f:R \rightarrow R\;are\;(a)One-One\quad(b)Onto\]\[(i)f(x)=|x|+x\]\[(ii)f(x)=x-[x]\]

asked Jan 23, 2013 by meena.p

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Let X = {1,2,3,4,5,6,7,8,9}. Let $ R_1\;andR_2$ be relations in x given by \[R_1=\{(x,y):x-y\;is\;divisible\;by\;3\}\;and\]\[R_2=\{(x,y)!\{x,y\} \varepsilon \{1,4,7\}or\{x,y\} \varepsilon \{2,5,8\}\]\[or \{x,y\} \varepsilon \{3,6,9\}\}.\; Show\; that \;R_1=R_2.\]

asked Jan 23, 2013 by meena.p

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Prove that relation R defined on the set N of natural numbers by \[x\;R\;y \Leftrightarrow 2x^2-3xy+y^2=0\] is not symmetric but it is reflexive.

asked Jan 23, 2013 by meena.p

1 answer

Show that the function $ f:N \rightarrow N$ given by $f(x) = x -(-1)^x\;for\;all\;x\; \varepsilon\;N $ is a bijection.

asked Jan 23, 2013 by meena.p

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Define a binary operation $^$ on the set A = {0,1,2,3,4,5} as \[ab=\left \{ \begin{array} {1 1} a+b, & \quad \text{ if a+b<6} \\ a+b-6, & \quad { if a+b \geq 6} \end{array} \right.\]Show that zero is an identity element for this operation and each element a of the set is invertiable with 6-a being the inverse of a.

asked Jan 23, 2013 by meena.p

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Let f and g be real valued functions, such that $(fog)(x)=\cos x^3\; and\;(gof)(x)=\cos^3x,$ find the functions f and g.

asked Jan 23, 2013 by meena.p

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Let $f : A \rightarrow B$ be a given function. A relation R in the set A is given by $R = {(a,b) \varepsilon A \times A : f(a) = f(b)}.$ Check, if R is an equivalence relation.

asked Jan 23, 2013 by meena.p

1 answer

Prove that the inverse of an equivalence relation is an equivalence relation.

asked Jan 23, 2013 by meena.p

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Let N denote the set of all natural numbers and R be the relation on N x N defined by $ (a,b)R(c,d) \Leftrightarrow ad(b + c) = bc(a + d).$ Show that R is an equivalence relation on N x N.

asked Jan 23, 2013 by meena.p

1 answer

Let a relation R on the set R of real numbers defined as $(x,y)\; \varepsilon\;R \Leftrightarrow x^24xy+3y^2\;=\;0.$Show that R is reflexive but neither symmetric nor transitive.

asked Jan 23, 2013 by meena.p

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Let a relation R on the set R of real numbers be defined as $ (a,b) \varepsilon R \Leftrightarrow 1+ ab > 0 for\; all\; a, b\; \varepsilon\; R.$show that R is reflexive and symmetric but not transitive.

asked Jan 23, 2013 by meena.p

0 answers

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