Question 1 :
If $A$ is $2\times 3$ matrix and $AB$ is a $2\times 5$ matrix, then $B$ must be a

Question 2 :
If $A = \bigl(\begin{smallmatrix}1 &amp; -2\\ -3 &amp; 4\end{smallmatrix}\bigr)$ and $A + B = O$, then B is<br>

Question 3 :
$\left[ \begin{matrix} x \\ 3 \end{matrix}\begin{matrix} 6 \\ 2x \end{matrix} \right]$ is a singular matrix, then $x$ is equal to

Question 4 :
If $A = \bigl(\begin{bmatrix}7 &amp;2 \\ 1 &amp; 3\end{bmatrix}\bigr)$ and $A + B = \bigl(\begin{bmatrix} -1&amp; 0\\ 2 &amp; -4\end{bmatrix}\bigr)$, then the matrix B =<br/>

Question 5 :
If $f:R\rightarrow \left [\dfrac {\pi}{6}, \dfrac {\pi}{2}\right ), f(x)=\sin^{-1}\left (\dfrac {x^2-a}{x^2+1}\right )$ is a onto function, then set of values of $a$ is

Question 6 :
If $f:A\rightarrow B$ given by ${ 3 }^{ f(x) }+{ 2 }^{ -x }=4$ is a bijection, then

Question 9 :
$\displaystyle&#160;\sum _{ k=1 }^{ k=n }{ \tan ^{ -1 }{ \frac { 2k }{ 2+{ k }^{ 2 }+{ k }^{ 4 } }&#160; }&#160; } =\tan ^{ -1 }{ \left( \dfrac { 6 }{ 7 }&#160; \right)&#160; } $, then the value of '$n$' is equal to

Question 10 :
The value of $\displaystyle \:\sin ^{-1}\left ( \cot \left ( \sin ^{-1}\sqrt{\frac{2-\sqrt{3}}{4}}+\cos ^{-1} \frac{\sqrt{12}}{4}+\sec ^{-1}\sqrt{2}\right ) \right ) $ is

Question 11 :
<b>Statement I :</b> &#160;The equation $(sin^{-1}x)^3+(cos^{-1}x)^3-a\pi^3=0$ has a solution for all $a\geqslant \dfrac {1}{32}.$<br/><b>Statement II :</b> For any $x\epsilon R, sin^{-1}x+cos^{-1}x=\dfrac {\pi}{2}$ and $0\leq (sin^{-1}x-\dfrac {\pi}{4})^2\leq \dfrac {9\pi^2}{16}$.<br/>

Question 12 :
Domain of $f(x)=\cot ^{ -1 }{ x } +\cos ^{ -1 }{ x } +co\sec ^{ -1 }{ x } $ is

Question 13 :
Let $\displaystyle f:A\rightarrow B$ be a function defined by $\displaystyle y=f(x)$ where f is a bijective function, means f is injective (one-one) as well as surjective (onto), then there exist a unique mapping $\displaystyle g:B\rightarrow A$ such that $\displaystyle f(x)=y$ if and only if $\displaystyle g(y)=x\forall x \epsilon A,y \epsilon B $ Then function g is said to be inverse of f and vice versa so we write $\displaystyle g=f^{-1}:B\rightarrow A[\left \{ f(x),x \right \}:\left \{ x,f(x) \right \}\epsilon f^{-1}] $when branch of an inverse function is not given (define) then we consider its principal value branch.<br/><br/>If $\displaystyle -1&lt;x&lt;0$,then $\displaystyle \tan^{-1}x $ equals?<br/>

Question 14 :
Value of $\displaystyle \left| \begin{matrix} 3i \\ 5 \\ i \end{matrix}\,\,\,\,\begin{matrix} 2i \\ 4 \\ 2i \end{matrix}\,\,\,\,\begin{matrix} 2i \\ -3i \\ 7 \end{matrix} \right|&#160;$ is

Question 15 :
The number of positive integral solutions of the equation $\begin{vmatrix} y^3+1 &amp; y^2z &amp; y^2x\\ yz^2 &amp; z^3+1 &amp; z^2x\\ yx^2 &amp; x^2z &amp; x^3+1\end{vmatrix}=11$ is?

Question 16 :
The solution of the differential equation $\operatorname { xdy } \left( y ^ { 2 } e ^ { x y } + e ^ { \tfrac { x } { y } } \right) = y d x \left( e ^ { \frac { x } { y } } - y ^ { 2 } e ^ { x y } \right)$ is-

Question 17 :
The order and degree of the differential equation ${ y }^{ 2 }=4a(x-a)$, where $a$ is an arbitrary constant, are respectively

Question 18 :
The degree of the differential equation of all curves having normal of constant length $c$ is:<br/><br/>

Question 19 :
lf $f (x)$ and $g (x)$ are two solutions of the differential equation $a\displaystyle \frac{\mathrm{d}^{2}\mathrm{y}}{\mathrm{d}\mathrm{x}^{2}}+\mathrm{x}^{2}\displaystyle\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}}+\mathrm{y}=\mathrm{e}^{\displaystyle\mathrm{x}}$, then $f (x) - g (x)$ is the solution of<br>

Question 20 :
The solution of differential equation $x\cos^{2}y dx = y\cos^{2} x &#160;dy$ is

Question 21 :
Forces $3 \vec { OA }$, $5 \vec { OB }$ act along OA and OB. If their resultant passes through C on AB, then :<br><br>

Question 22 :
If $a=i-j,b=i+j,c=i+3j+5k$ and $n$ is a unit vector such that $b,n=0,a,n=0$ then the value of $|c,n|$ is equal to

Question 23 :
Cosine of an angle between the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ if $|\vec{a}|=2, |\vec{b}|=1$ and $\vec{a}$ ^ $\vec{b}=60^o$ is?

Question 26 :
$\displaystyle \int \dfrac { \cot ^ { 2 } x } { \left( cosec ^ { 2 } x + cosec x \right) } d x =$

Question 28 :
The average value of pressure varying from 2 to 10&#160;atm if the pressure $p $&#160;and the volume $v$&#160;are related by $&#160;pv^{3/2} = 160 $ is

Question 29 :
Solve: $\int \dfrac{2 \sin 2\ x - \cos \ x}{6 - \cos^2 \ x - 4 \sin \ x}d\theta$.

Question 30 :
Assertion: If $\displaystyle \Delta (x)= \begin{vmatrix}f(x) &g(x) \\m_{1} &m_{2} \end{vmatrix}$ then<br><br>$\displaystyle \int \Delta (x)=\begin{vmatrix}<br><br>\int f(x)dx &\int g(x)dx \\m_{1} &m_{2}\end{vmatrix}$ Reason: $\displaystyle \int \lambda f(x)dx=\lambda\int f(x)dx$

Question 31 :
lf $\displaystyle \int f(x)\sin x\cos x\&gt;dx=\frac{1}{2(b^{2}-a^{2})}\log(f(x))+c$, then $\displaystyle f(x)$ is equal to<br>

Question 32 :
A tank initially holds 10 lit. of fresh water. At t = 0, a brine solution containing $\displaystyle \frac{1}{2}$ kg of salt per lit. is poured into tank at a rate 1 lit/min while the well-stirred mixture leaves the tank at the same rate. Find&#160;the amount of salt in a tank at a particular time

Question 34 :
If $\int { \left[ \log { \left( \log { x } \right) } +\cfrac { 1 }{ { \left( \log { x } \right) }^{ 2 } } \right] } dx=x\left[ f(x)-g(x) \right] +c$ then<br><br>

Question 35 :
The solution of $\displaystyle \frac { dy }{ dx } =\left( \frac { ax+b }{ cy+d } \right)$ represents aparabola if :-

Question 36 :
Conclude from the following:<br/>$n^2 &gt; 10$, and n is a positive integer.A: $n^3$B: $50$

Question 37 :
If $x+y \leq 2, x\leq 0, y\leq 0$ the point at which maximum value of $3x+2y$ attained will be.<br/>

Question 38 :
Find the output of the program given below if$ x = 48$<br/>and $y = 60$<br/>10&#160; $ READ x, y$<br/>20&#160; $Let x = x/3$<br/>30&#160; $ Let y = x + y + 8$<br/>40&#160; $ z = \dfrac y4$<br/>50&#160; $PRINT z$<br/>60&#160; $End$

Question 39 :
In a bolt factory, machines $A, B$ and $C$ manufacture $25 \%, 35 \%$ and $40 \%$ respectivelyof the total number of bolts. The percentage of defective bolts among the manufactured boltsis $5 \%$ for $A$ , $4 \%$ for $B$ and $2 \%$ for $C$. A bolt is drawn randomly from the manufactured products and is found to be defective then,

Question 40 :
A sample of size $4$ is drawn with replacement (without replacement )from an urn containing $12$ balls, of which $8$ are white, what is the conditional probability that the ball drawn on the third draw was white, given that the sample contains $3$ white balls ?

Question 41 :
A letter is known to have come either from TATANAGAR or from CALCUTTA. On the envelope just two Consecutive letters TA are visible. What is the probability that the letters came from TATANAGAR ?&#160;