List of practice Questions

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

A and B are events such that \( P(A \cup B) = 3/4 \), \( P(A) = 1/4 \), and \( P(A \cap B) = 2/3 \). Then the probability that \( P(A \cap B) \) is

Value of \[ \int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sin x + \cos x} \, dx \]

The range of the function \[ f(x) = \frac{1}{2 - \cos 3x} \] is

The area bounded by \( y = |x| \), \( y = 0 \) and \( |x| = \frac{1}{2} \) will be:

The value of \( x \) obtained from the equation \[ \frac{x + \alpha}{\gamma} = \frac{x + \beta}{\alpha} = \frac{x + \gamma}{\beta} \] will be

The solution of the differential equation \[ \log x \frac{dy}{dx} + x = \sin 2x \] is

\[ \lim_{x \to \infty} \frac{x^2}{3x - 3} = ? \]

If \( \vec{a} \times \vec{b} \) and \( \vec{c} \times \vec{d} \) are perpendicular, then which of the following is always true?

A value of \( c \) for which conclusion of Mean Value Theorem holds for the function \( f(x) = \log x \) on the interval [1, 3] is

Negation of the proposition: If we control population growth, we prosper is

The equation \( z^2 + (2 - 3i)z + (2 + 3i) = 4 = 0 \) represents a circle of radius

The function \( f(x) = \sin x - kx - c \), where \( k \) and \( c \) are constants, decreases always when

Equation \( \frac{1}{8} \sin^3 x = \cos 6x \) represents

The acceleration of a sphere falling through a liquid is \( (30 - 3y) \, \text{cm/s}^2 \) where \( y \) is speed in cm/s. The maximum possible velocity of the sphere and the time when it is achieved are

A straight line parallel to the line \( 2x + y - 5 = 0 \) is also a tangent to the curve \( y^2 = 4x + 5 \). Then the point of contact is