Show that $19^{93}-13^{99}$ is divisible by 162.

If $\alpha ,\beta$ and $\gamma$ are the roots of the equation $x^3+3x^2-8x+1=0$, then an equation whose roots are $\alpha +1,\beta +1$ and $\gamma +1$ is given by

The number of people in a small country increases by 2% per year. If the population at the start of 1973 was 12,500, what is the predicted population at the start of the year 2013?

Let $f$ be a real valued function defined on the interval $(-1,1)$ such that $e^{-x}f(x)=2+{\int }_0^1\sqrt{1+t^4}dt$, for all $x\in (-1,1)$. And let $f^{-1}$ be the inverse function of $f$. Then find the value of $(f^{-1}{)}^{\prime }(2)$, where $(f^{-1}{)}^{\prime }(2)$ is the value of first order derivative of $f^{-1}$ at 2.

Evaluate $\int \frac{x+3}{\sqrt{5-4x-x^2}}dx$.

Consider $f(x)=4x^3-12x$. Find the image of the interval $[-1,3]$ under the mapping $f$.

If $A$ is a $2\times 2$ matrix such that trace $A=\det A=3$, then what is the trace of $A^{-1}$?

The diagonal elements of a square matrix $M$ are odd integers while the off-diagonals are even integers. Then

In a class of 80 students, 40 are girls and 40 are boys. Also, exactly 50 students wear glasses. Then the set of all possible numbers of boys without glasses is

Let $X_1,X_2$ and $X_3$ be chosen independently from the set $\{0,1,2,3,4\}$, each value being equally likely. What is the probability that the arithmetic mean of $X_1,X_2$ and $X_3$ is the same as their geometric mean?

Suppose that $X$ is chosen uniformly from $\{1,2,...,100\}$ and given $X=x$, $Y$ is chosen uniformly from $\{1,2,...,x\}$. Then $P(Y=30)=$

The area lying in the first quadrant and bounded by the circle

$$x^2+y^2=4$$

and lines

$$x=0\text{ and }x=1$$

is given by

If $(x_1,y_1)$ and $(x_2,y_2)$ are the opposite endpoints of a diameter of a circle, then the equation of the circle is given by

Sketch the curve of $y^2=x^2(3-x)$ and find the area of its loop.