List of practice Questions

The sum of the infinite series $ \cot^{-1} \left( \frac{7}{4} \right) + \cot^{-1} \left( \frac{19}{4} \right) + \cot^{-1} \left( \frac{39}{4} \right) + \cot^{-1} \left( \frac{67}{4} \right) + ... $ is:

Let $(\alpha,\beta,\gamma)$ be the foot of the perpendicular from the point $(25,2,41)$ on the line \[ \frac{x-4}{3}=\frac{y+1}{7}=\frac{z-2}{3}. \] Then $\alpha+\beta+\gamma$ is equal to:

Let $ A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} $ and $ P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta > 0. $ If $ B = P A P^T $, $ C = P^T B P $, and the sum of the diagonal elements of $ C $ is $ \frac{m}{n} $, where gcd(m, n) = 1, then $ m + n $ is:

The interior angles of a polygon with $ n $ sides, are in an A.P. with common difference 6°. If the largest interior angle of the polygon is 219°, then $ n $ is equal to:

A coin is tossed three times. Let X denote the number of times a tail follows a head. If $\mu$ and $\sigma^2$ denote the mean and variance of X, then the value of $64(\mu + \sigma^2)$ is:

Let $ M $ and $ m $ respectively be the maximum and the minimum values of $ f(x) = \begin{vmatrix} 1 + \sin^2x & \cos^2x & 4\sin4x \\ \sin^2x & 1 + \cos^2x & 4\sin4x \\ \sin^2x & \cos^2x & 1 + 4\sin4x \end{vmatrix}, \quad x \in \mathbb{R} $ for $ x \in \mathbb{R} $. Then $ M^4 - m^4 $ is equal to:

Two parabolas have the same focus $(4, 3)$ and their directrices are the $x$-axis and the $y$-axis, respectively. If these parabolas intersect at the points $A$ and $B$, then $(AB)^2$ is equal to:

The area of the region $ \{(x, y): |x - y| \le y \le 4\sqrt{x}\} $ is

Let $ \vec{a} $ and $ \vec{b} $ be the vectors of the same magnitude such that $ \frac{| \vec{a} + \vec{b} | + | \vec{a} - \vec{b} |}{| \vec{a} + \vec{b} | - | \vec{a} - \vec{b} |} = \sqrt{2} + 1. \quad \text{Then } \frac{| \vec{a} + \vec{b} |^2}{| \vec{a} |^2} \text{ is:} $

Let a random variable X take values 0, 1, 2, 3 with $ P(X = 0) = P(X = 1) = p, \, P(X = 2) = P(X = 3), \, \text{and} \, F(X^2) = 2F(X). $ Then the value of $ 8p - 1 $ is:

If $ \vec{a} $ is a non-zero vector such that its projections on the vectors $ 2\hat{i} - \hat{j} + 2\hat{k},\ \hat{i} + 2\hat{j} - 2\hat{k} $, and $ \hat{k} $ are equal, then a unit vector along $ \vec{a} $ is:

Let the coefficients of three consecutive terms $ T_r $, $ T_{r+1} $, and $ T_{r+2} $ in the binomial expansion of $ (a + b)^{12} $ be in a G.P. and let $ p $ be the number of all possible values of $ r $. Let $ q $ be the sum of all rational terms in the binomial expansion of $ \left( 4\sqrt{3} + 3\sqrt{4} \right)^{12} $. Then $ p + q $ is equal to:

Let a line pass through two distinct points $ P(-2, -1, 3) $ and $ Q $, and be parallel to the vector $ 3\hat{i} + 2\hat{j} + 2\hat{k} $. If the distance of the point $ Q $ from the point $ R(1, 3, 3) $ is 5, then the square of the area of $ \triangle PQR $ is equal to:

The length of the chord of the ellipse: \[ \frac{x^2}{4} + \frac{y^2}{2} = 1, \] whose mid-point is $ \left( 1, \frac{1}{2} \right) $, is:

If a box contains 19 unbiased coins and 1 biased coin with both faces heads, and a coin is randomly chosen from this box and tossed. If the head appears, then the probability that the unbiased coin was selected is:

There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is:

If
$ 2^m 3^n 5^k, \text{ where } m, n, k \in \mathbb{N}, \text{ then } m + n + k \text{ is equal to:} $

Each of the angles $ \beta $ and $ \gamma $ that a given line makes with the positive y- and z-axes, respectively, is half the angle that this line makes with the positive x-axis. Then the sum of all possible values of the angle $ \beta $ is

Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is:

Let for some function $ y = f(x) $, $\int_0^x t f(t) \, dt = x^2 f(x), x>0$ and $ f(2) = 3 $. Then $ f(6) $ is equal to:

If the equation of the parabola with vertex $\left(\frac{3}{2},\, 3\right)$ and the directrix $x + 2y = 0$ is
$$ ax^2 + by^2 - cxy - 30x - 60y + 225 = 0, $$ then $\alpha + \beta + \gamma$ is equal to:

Let $ S $ be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set $ S $, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is:

Which of these is aromatic in nature?

For $[ \text{NiCl}_4 ]^{2-}$, what is the charge on metal and shape of complex respectively?

The incorrect statements among the following is: