Question 1 : Let $f(x)=(1+\sin x)^{\csc x}$, the value of $f(0)$ so that $f$ is a continuous function is

Question 2 : At $x = \dfrac {3}{2}$, the function $f(x) = \dfrac {|2x - 3|}{2x - 3}$ is

Question 4 : Let $f: \mathbb{R}\rightarrow (0, 1)$ be a continuous function. Then, which of the following function(s) has (have) the value zero at some point in the interval $(0, 1)$?

Question 5 : Let $f(x)$ be a continuous function on $[0,4]$ satisfying $f(x)f(4-x) = 1$ <br>The value of the definite integral $\int_{0}^{4} \frac{1}{1+f(x)} \;dx $ equals

Question 6 : $\displaystyle\lim _{ x\rightarrow { 0 }^{ + } }{ \left( \left( { x }^{ { x }^{ x } } \right) -{ x }^{ x } \right) } $ is

Question 7 : The function $f$ : $R/\{0\}\rightarrow R$ given by $f(x)=\displaystyle \frac{1}{x}-\frac{2}{e^{2x}-1}$ can be made continuous at $x=0$ by defining $f(0)$ as -<br>

Question 8 : If $f(x)=\left\{\begin{matrix}<br/>[tan(\dfrac{\pi}{4}+x)]^{1/x} &amp;&#160; x\neq 0\\ <br/>k &amp; x=0<br/>\end{matrix}\right.$ ,then for what value of $k$, $f(x)$ is continuous at $x = 0$?<br/>

Question 9 : For particle moving along $x-axis$, velocity is given as a function of time as $v=2t^{2}+\sin t$. At $t=0$, particle is at origin, if the position as a function of time is $\dfrac{2t^{3}}{m}-\cos t$, then find $m$.

Question 10 : If $ f(x) =\displaystyle\frac{x}{{\sin x}}$ and $g(x) =\displaystyle\frac{x}{{\tan x}}$ where $0 &lt; x \leq1$<span class="wysiwyg-font-size-medium">then in the interval<span class="wysiwyg-font-size-medium">

Question 11 : <span class="wysiwyg-font-size-small"><span class="wysiwyg-font-size-small">A certain radioactive material is known to decay at a rate proportional to the amount present. Initially there is 50 kg of the material present and after two hours it is observed that the material has lost 10%of its original mass, then the

Question 12 : Find the max.value of the total surface of a right circular cylinder which can be inscribed in a sphere of radius a.

Question 13 : A balloon is pumped at the rate of a cm$^{3}$/minute. The rate of increase of its surface area when the radius is b cm, is

Question 14 : A point on the parabola ${ y }^{ 2 }=18x$ at which the ordinate increases at twice the rate of the abscissa is

Question 15 : A point moves along the curve $12y=x^3$ in such a way that the rate of increase of its ordinate is more than the rate of increase of abscissa. The abscissa of the point lies in the interval

Question 16 : If $3\cos ^{ -1 }{ x } +\sin ^{ -1 }{ x } =\pi $, then $x=.....$

Question 17 : The value of $sin^{-1} x + cos^{-1} x (|x| \geq 1)$ is

Question 18 : The value of $a$ for which $\displaystyle ax^{2}+sin^{-1}(x^{2}-2x+2)+cos^{-1}(x^{2}-2x+2)=0$ has areal solution is

Question 19 : The number of solution of the equation $ 1+x^{2}+2x\:\sin \left ( \cos^{-1}y \right )= 0 $ is :

Question 20 : If the value of the determinant $\begin{vmatrix} a &amp; 1 &amp; 1 \\ 1 &amp; b &amp; 1 \\ 1 &amp; 1 &amp; c \end{vmatrix}$ is positive, then<br><br>

Question 21 : If A is a square matrix of order 3, then $|(A - A^T)^{105}|$ is equal to

Question 22 : Let n and r be two positive integers such that $n \geq r + 2$. Suppose $\Delta (n, r) =\begin{vmatrix}^nC_r &amp; ^nC_{r + 1} &amp; ^nC_{r + 2}\\ ^{n+1}C_r &amp; ^{n + 1}C_{r + 1} &amp; ^{n + 1}C_{r + 2}\\ ^{n + 2}C_r &amp; ^{n + 2}C_{r + 1}&amp; ^{n + 2} C_{r + 2}\end{vmatrix}$ Show that $\Delta(n, r) \displaystyle = \frac{^{n + 2} C_3}{^{n+ 2} C_3} \Delta (n - 2, r - 1)$ Hence or otherwise,

Question 23 : If the determinant $\displaystyle \begin{vmatrix}cos\: 2x &amp;sin^{2}x &amp;cos\: 4x \\ sin^{2}x &amp;cos2x &amp;cos^{2}x \\ cos4x &amp;cos^{2}x &amp;cos\: 2x \end{vmatrix}$ is expanded in powers of $\sin \displaystyle x$ then the constant term in the expansion is<br>

Question 24 : Let $M$ and $N$ be two $3 \times3$ matrices such that $ MN = NM$. Further, if $M \neq$ $N^2$ and $M^2= N^4$, then

Question 25 : 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 26 : Let $A =&#160;\begin{bmatrix} -2 &amp; 7 &amp; \sqrt{ 3}&#160; \\ 0 &amp; 0 &amp; -2 \\ 0 &amp; 2 &amp; 0 \end{bmatrix}&#160;$&#160; and $A^4 = \lambda$. I, then $\lambda $ is

Question 27 : If the number of elements in a matrix is $60$ then how many different order of matrix are possible&#160;

Question 28 : Let $n\ge 2$ be an integer,<br/>$A=\begin{bmatrix} \cos { \left( { \dfrac{2\pi}n} \right) &#160;} &#160;&amp; \sin { \left(\dfrac{2\pi}n \right) &#160;} &#160;&amp; 0 \\ -\sin { \left( \dfrac{2\pi}n \right) &#160;} &#160;&amp; \cos { \left(\dfrac{2\pi}n \right) &#160;} &#160;&amp; 0 \\ 0 &amp; 0 &amp; 1 \end{bmatrix}$ and $I$ is the identity matrix of order $3$., then following of which is correct

Question 29 : If $A$ is $2\times 3$ matrix and $AB$ is a $2\times 5$ matrix, then $B$ must be a

Question 30 : The function $f: [0, 3]$ $\rightarrow$ $[1, 29]$, defined by $f(x) = 2x^3-15x^2 + 36x+ 1$, is<br>

Question 31 : Which of the following functions from $Z$ to itself are bijections?

Question 32 : $f\left( x \right) =\begin{cases} x\left( \dfrac { { ae }^{ \dfrac { 1 }{ \left| x \right| } }+{ 3.e }^{ \dfrac { -1 }{ x } } }{ \left( a+2 \right) { e }^{ \dfrac { 1 }{ \left| x \right| } }-{ e }^{ \dfrac { -1 }{ x } } } \right) \\ 0 \end{cases},\begin{matrix} x\neq 0 \\ x=0 \end{matrix}$ is differentiable at $x=0$ then $[a]=$__ ([] denotes greatest integers function )

Question 33 : Let f: $X\rightarrow Y$ be a function defined by $f(x)=a \sin \left (x+\dfrac {\pi}{4}\right )+b \cos x+c$. If f is both one-one and onto, then find the sets $X$ and $Y$

Question 34 : Conclude from the following:<br/>$n^2 &gt; 10$, and n is a positive integer.A: $n^3$B: $50$

Question 35 : 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 36 : 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; $End$