Question 1 :
Which one of the following relations on R (set of real numbers) is an equivalence relation
Question 2 :
On the set $N$ of all natural numbers define the relation $R$ by $a R b$ if and only if the G.C.D. of $a$ and $b$ is $2$. Then $R$ is:
Question 3 :
$\displaystyle x^{2} = xy$ is a relation (defined on set R) which is<br/> <br/>
Question 4 :
Let $A=\left\{ 1,2,3 \right\} $ and $R=\left\{ \left( 1,2 \right) ,\left( 2,3 \right) ,\left( 1,3 \right) \right\} $ be a relation on $A$. Then $R$ is
Question 5 :
Which of the following is not an equivalence relation on $Z$?
Question 6 :
If $A= [ 1 \ 2\  3 ]$, then the set of elements of A is
Question 7 :
If$\displaystyle \begin{vmatrix} 2 & 3 \\ 4 & 4 \end{vmatrix} $+$\displaystyle \begin{vmatrix} x & 3 \\ y & 1 \end{vmatrix} $=$\displaystyle \begin{vmatrix} 10 & 6 \\ 8 & 5 \end{vmatrix} $,then (x,y)=
Question 8 :
The entries of a matrix are integers. Adding an integer to all entries on a row or on a column is called an operation. It is given that for infinitely many integers N one can obtain, after a finite number of operations, a table with all entries divisible by N. Prove that one can obtain, after a finite number of operations, the zero matrix.
Question 9 :
If $A=\begin{bmatrix} { a }^{ 2 } & ab & ac \\ ab & { b }^{ 2 } & bc \\ ac & bc & { c }^{ 2 } \end{bmatrix}$ and ${a}^{2}+{b}^{2}+{c}^{2}=1$ then ${A}^{2}=$
Question 10 :
The number of all possible matrices of order $3 \times 3$ with each entry $0$ or $1$ is:
Question 12 :
${ tan }^{ -1 }x+{ tan }^{ -1 }y={ tan }^{ -1 }\dfrac { x+y }{ 1-xy } $,      $xy<1$<br/>                                    $=\pi +{ tan }^{ -1 }\dfrac { x+y }{ 1-xy } $,      $xy>1$.<br/> Evaluate:  ${ tan }^{ -1 }\dfrac { 3sin2\alpha  }{ 5+3cos2\alpha  } +{ tan }^{ -1 }\left( \dfrac { tan\alpha  }{ 4 }  \right) $<br/>                                  where $-\dfrac { \pi  }{ 2 } <\alpha <\dfrac { \pi  }{ 2 } $
Question 13 :
Solve $\cos { \left[ \tan ^{ -1 }{ \left[ \sin { \left( \cot ^{ -1 }{ x }  \right)  }  \right]  }  \right]  } $
Question 14 :
The value of $\sin^{-1} \left( \dfrac{2 \sqrt 2}{3} \right ) + \sin^{-1} \left( \dfrac{1}{3}\right )$ is equal to
Question 15 :
The number of real values of x satisfying the equation $\tan^{-1}\left(\dfrac{x}{1-x^2}\right)+\tan^{-1}\left(\dfrac{1}{x^3}\right)=\dfrac{3\pi}{4}$, is?
Question 16 :
$x = \left| \begin{gathered}   - 1\,\,\,\,\,\, - 2\,\,\,\,\,\,\, - 2 \hfill \\  \,\,\,2\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\, - 2 \hfill \\  \,\,\,2\,\,\,\,\,\, - 2\,\,\,\,\,\,\,\,\,\,1 \hfill \\ \end{gathered}  \right|$, then $x=$
Question 17 :
If the trivial solution is the only solution of the system of equations$x-ky+z=0,kx+3y-kz=0, 3x+y-z=0$. Then the set of all values of k is:<br>
Question 18 :
Let a, b, c be three complex numbers, and let<br>$z=\begin{vmatrix}<br>0 & -b & -c\\ <br>b & 0 & -a\\ <br>c & a & 0<br>\end{vmatrix}$<br>then z equal<br>
Question 19 :
The product of a matrix and its transpose is an identity matrix. The value of determinant of this matrix is
Question 20 :
If $\Delta_1=\begin{vmatrix} 1 & 0\\ a & b\end{vmatrix}$ and $\Delta_2=\begin{vmatrix} 1 & 0\\ c & d\end{vmatrix}$ then $\Delta_2 \Delta_1$ is equal to<br>
Question 21 :
If $f\left( x \right) =\dfrac { 3sinx-sin\left( 3x \right) }{ { 2x }^{ 3 } } ,x\neq 0,f\left( 0 \right) =2,$ at $x=0,f$ is
Question 22 :
If $f(x) = (1 + 2x)^{\frac{1}{x}}$, for $x \neq 0$ is continuous at $x = 0$, then $f(0) = $_______.
Question 23 :
The function $f(x)=\dfrac{1-\sin x+\cos x}{1+\sin x+\cos x}$ is not defined at $x=\pi$. The value of $f(\pi)$ so that $f(x)$ is continuous at $x=\pi$ is
Question 24 :
If $f(x)=\displaystyle\frac{\log(1+ax)-log(1-bx)}{x}$ for $x\neq 0$ and $f(0)=k$ and $f(x)$ is continuous at $x=0$, then $k$ is equal to:<br>
Question 25 :
If $ f\left( x \right) =\begin{cases} 1 & x<0 \\ { x }^{ 2 } & x\ge 0 \end{cases}$ then at $x=0$
