List of practice Questions

Choose the best pronunciation of the word, Sorbet from the following options.

The set of points of discontinuity of the function \[ f(x) = \sin (2 \sin x) \sin^2 x \] is given by

The volume \( V \) and depth \( x \) of water in a vessel are connected by the relation \[ V = 5x - \frac{x^2}{6} \] and the volume of water is increasing, at the rate of 5 cm\(^3\)/sec, when \( x = 2 \) cm. The rate at which the depth of water is increasing is

If vectors \( \mathbf{i} + \mathbf{j} + \mathbf{k} \) and \( \mathbf{i} + \mathbf{j} + \mathbf{k} \) are coplanar, then find

If matrix \( A = \begin{pmatrix} 3 & 2 & 4 \\1 & 2 & -1 \end{pmatrix} \) and

The angle between a pair of tangents drawn from a point \( T \) to the curve \[ x^2 + y^2 + 4x - 6y - 9 = 3x + 1 \] is

The parabola having its focus at (3, 2) and directrix along the y-axis has its vertex at

The rank of the matrix \[ \begin{pmatrix} 1 & 2 & 5 \\ 2 & 4 & 4 \\ 3 & 6 & 6 \end{pmatrix} \] is

If \( f(x) = \cos x + 1 \) and \( f'(x) = 2 \cos x \), then \[ \int_0^\frac{\pi}{2} f(x)dx \] is equal to

The distance of the point \( (1, -2, 3) \) from the plane \( x - y + z = 5 \) measured parallel to the line \[ \frac{x}{2} = \frac{y}{3} = \frac{z - 1}{6} \] is

The tangent lines to the curve \( y^2 = 4ax \) at points where \( x = a \), are

If the eccentricity of the hyperbola \[ x^2 - y^2 \cos^2 \alpha = 25 \] is \( \sqrt{5} \), then the eccentricity of the ellipse \[ x^2 \cos^2 \alpha + y^2 = 5 \] is equal to:

The conditional \( (P \land Q) \Rightarrow P \) is

The line which passes through the origin and intersects the two lines \[ \frac{x - 1}{2} = \frac{y - 3}{4} = \frac{z - 14}{4} \] is

If \( u_n = \int_0^{\frac{\pi}{4}} \tan \theta d\theta \), then \( u_{n+2} \) is:

Ten different letters of an alphabet are given, words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated is

The area bounded by \( f(x) = x^2, 0 \leq x \leq 1 \), and \( g(x) = x + 2, 1 \leq x \leq 2 \) and x-axis is

The condition that the line \( \frac{x}{p} + \frac{y}{q} = 1 \) be normal to the parabola \( y^2 = 4ax \) is

A random variable \( X \) has the probability distribution \[ P(X) = \{ 0.15, 0.23, 0.12, 0.10, 0.20, 0.07, 0.06, 0.08 \} \] For the events \( E = \{ X \text{ is a prime number} \} \) and \( F = \{ X<4 \} \), then \( P(E \cup F) \) is

The value of \( \tan^{-1} \left( \frac{1}{2} \right) + \tan^{-1} \left( \frac{1}{3} \right) \) is

Let A be the centre of the circle \( x^2 + y^2 - 2x - 4y - 20 = 0 \), and B(1, 7) and D(4, -2) are points on the circle then, if tangents are drawn at B and D, which meet at C, then area of quadrilateral ABCD is

\[ \int \left[ f'(x) \left( g(x) \right)^2 \right] dx \] is equal to:

If \( x = -\frac{7}{3} \), then \( x^4 \) is

The difference between greatest and least value of \[ f(x) = 2 \sin x + \sin 2x, \, x \in \left[ 0, \frac{3\pi}{2} \right] \] is

A and B are two independent witnesses (i.e., there is no collision between them) in a case. The probability that A will speak the truth is \( x \) and the probability that B will speak the truth is \( y \). A and B agree in a certain statement. The probability that the statement is true is