Let the population of rabbits surviving at time $t$ be governed by the differential equation $\dfrac{dp(t)}{dt}=\dfrac{1}{2}p(t)-200$. If $p(0)=100$, then $p(t)$ equals:

1

$\displaystyle \lim_{x\to 0}\frac{\sin(\pi \cos^{2}x)}{x^{2}}$ is equal to :

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2

The variance of first $50$ even natural numbers is :

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1

The integral $\displaystyle \int_{0}^{\pi}\sqrt{1+4\sin^{2}\frac{x}{2}-4\sin\frac{x}{2}}\,dx$ equals:

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2

The area of the region described by $A=\{(x,y):x^{2}+y^{2}\le 1 \text{ and } y^{2}\le 1-x\}$ is :

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3

The integral $\displaystyle \int (1+x-\frac{1}{x})e^{x+\frac{1}{x}}\,dx$ is equal to

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4

If $A$ is a $3\times 3$ non-singular matrix such that $AA' = A'A$ and $B = A^{-1}A'$, then $BB'$ equals :

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5

If $\alpha,\beta \ne 0$, and $f(n) = \alpha^{n} + \beta^{n}$ and $\begin{vmatrix} 3 & 1 + f(1) & 1 + f(2) \\ 1+…

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6

If $g$ is the inverse of a function $f$ and $f'(x)=\dfrac{1}{1+x^{5}}$, then $g'(x)$ is equal to:

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7

If $x=-1$ and $x=2$ are extreme points of $f(x)=\alpha\log|x|+\beta x^{2}+x$ then

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8

Let $A$ and $B$ be two events such that $P(\overline{A\cup B})=\dfrac{1}{6}$, $P(A\cap B)=\dfrac{1}{4}$ and $P(\overlin…

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9

The locus of the foot of perpendicular drawn from the centre of the ellipse $x^{2}+3y^{2}=6$ on any tangent to it is :

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10

Let $a,b,c$ and $d$ be non-zero numbers. If the point of intersection of the lines $4ax+2ay+c=0$ and $5bx+2by+d=0$ lies…

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