List of practice Questions

The expression \[ \left| (a + 1)^2 \, (b + 1)^2 \, (c + 1)^2 \right| \] is equal to

For all $ n \in \mathbb{N}, 72n - 48n - 1 $ is divisible by

The inverse of matrix \[ \begin{pmatrix} 0 & 1 & -1 \\ 4 & -3 & 4 \\3 & -3 & 4 \end{pmatrix} \] is

The solution of the differential equation \[ \frac{d^2y}{dx^2} + 3y = -2x \] is

Let $x_1$ and $x_2$ be the roots of the equation $ax^2 + bx + c = 0$ ($ac \neq 0$). Find the value of $\frac{1}{x_1} + \frac{1}{x_2}$.

A line makes the same angle $\theta$ with each of the X and Z-axes. If the angle $\beta$ which it makes with Y-axis is such that $\sin^2 \beta = 3 \sin^2 \theta$, then $\cos^2 \theta$ equals

If the tangent at (1,1) on $ y^2 = x(2 - x^2) $ meets the curve again at P, then P is:

Choose the most appropriate options.
The degree of the differential equation \[ x = 1 + \frac{dy}{dx} + \frac{1}{2!} \left( \frac{d^2y}{dx^2} \right) + \frac{1}{3!} \left( \frac{d^3y}{dx^3} \right) + ... \]

Choose the most appropriate options.
If $ A(2,3) $ and $ B(-2,1) $ are two vertices of a triangle and the third vertex moves on the line $ 2x + 3y = 9 $, then the locus of the centroid of the new set of observations will be the triangle is

Consider line segments of lengths 1, 2, 3, ..., 10, what is the number of triangles that can be formed from them?

How many paths are there from the point A to the point B in the figure below, if no point in a path is to be traversed more than once?

The maximum number of points into which 4 circles and 4 straight lines intersect, is

Given that $ \alpha_1, \alpha_2, \alpha_3 $ are the roots of $ 3x^3 - x^2 - 10x + 8 = 0 $, then the value of $ \alpha_1^2 + \alpha_2^2 + \alpha_3^2 $ is

The value of the integral \[ \int_0^0 9 [x - 2\lfloor x \rfloor] \, dx \] where $ [.] $ denotes the greatest integer function, is

The number of solutions of the equation \[ 3 \sin^2 x - 7 \sin x + 2 = 0 \] in the interval $ [0, 5\pi] $ is

If \[ a = \gamma \hat{i} + \delta \hat{j} + 2\zeta \hat{k}, \, b = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}, \, r \times a = b \times a \, r \times b = a \times b \] then a unit vector in the direction of $ r $ is

From the bottom of a pole of height h, the angle of elevation of the top of a tower is $ \alpha $. The pole subtends an angle $ \beta $ at the top of the tower. The height of the tower is

If the numbers $a_1, a_2, \ldots, a_n$ are different from zero and form an arithmetic progression, then $$ \frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} + \dots + \frac{1}{a_n} $$ is equal to

If $n$ is an integer, compute the value of the fraction \[ \frac{(1 + \theta)^n}{(1 - \theta)^{n-2}} \]

Choose the most appropriate option.
If $ A $ is a square matrix such that $ A^2 = A $ and $ B = I $, then $ AB + BA + I - (I - A)^2 $ is equal to:

Calculate

\[ \begin{vmatrix} x & y & x + y \\ y & x + y & x \\ x + y & x & y \end{vmatrix} \]

Let $ f(x) = (x^3 + 2)^{30} $. If $ f^{(n)}(x) $ is a polynomial of degree $ 20 $, where $ f^{(n)}(x) $ denotes the $ n + h $-order derivative of $ f(x) $, then the value of $ n $ is:

The set of values of $ x $ satisfying the system of inequalities $ 5x + 2 < 3x + 8 $ and $ \frac{x^2}{x-1} < 4 $ is:

Choose the most appropriate options.
If $ |z^2 - 1| = |z|^2 + 1 $, then $ z $ lies on a

Choose the most appropriate options.
If $ f(x) = [x \sin n\pi x] $, then which of the following is incorrect?