List of practice Questions

The point \( (-3, -2) \) lies

The probability of choosing randomly a number \( c \) from the set \( \{1, 2, 3, \dots, 9\} \) such that the quadratic equation \( x^2 + 4x + c = 0 \) has real roots, is

The maximum of \( Z \) is where, \( Z = 4x + 2y \) subject to constraints \[ 4x + 2y \geq 46, \quad x + 3y \leq 24, \quad x, y \geq 0 \]

If \( \theta \) be the angle between the vectors \( a = 2\hat{i} + 2\hat{j} - \hat{k} \) and \( b = 6\hat{i} - 3\hat{j} + 2\hat{k} \), then

If \( x, y \) and \( z \) are non-zero real numbers and \( a = x\hat{i} + 2\hat{j}, b = y\hat{j} + 3\hat{k} \) and \( c = x\hat{i} + y\hat{j} + z\hat{k} \) are such that \( a \times b = z\hat{i} - 3\hat{j} + \hat{k} \), then \( [abc] \) is equal to

Maximum value of \( z = 12x + 3y \), subject to constraints \( x \geq 0, y \geq 0, x + y \leq 5 \) and \( 3x + y \leq 9 \) is

If \( p = \hat{i} + \hat{j}, q = 4\hat{k} - \hat{j} \) and \( r = \hat{i} + \hat{k} \), then the unit vector in the direction of \( 3p + q - 2r \) is

The line \( \frac{x-3}{4} = \frac{y-4}{5} = \frac{z-5}{6} \) is parallel to the plane.

Evaluate the integral \( \int \frac{1}{x \sqrt{ax^2 - x^2}} \, dx \)

Evaluate the integral \( 3 \int \frac{3^x}{\sqrt{1 - 9^x}} \, dx \)

Evaluate the integral \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x \, dx \)

The angle between the lines \( \frac{x+4}{3} = \frac{y-1}{5} = \frac{z+3}{4} \) and \( \frac{x+1}{1} = \frac{y-4}{1} = \frac{z-5}{2} \)

The point of intersection of the lines \( \frac{x - 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} \) and \( \frac{x - 5}{2} = \frac{y - 2}{1} = \frac{z - 5}{2} \)

Five persons A, B, C, D and E are in queue at a shop. The probability that A and B are always together is.

If the probability for A to fail in an examination is 0.2 and that for B is 0.3, then the probability that either A or B fail is.

Three vertices are chosen randomly from the seven vertices of a regular 7-sided polygon. The probability that they form the vertices of an isosceles triangle is.

If A, B, and C are three mutually exclusive and exhaustive events such that \( P(A) = 2P(B) = 3P(C) \). What is \( P(B) \)?

If \( U_{n+1} = 3U_n - 2U_{n-1} \) and \( U_0 = 2, U_1 = 3 \), then \( U_n \) is equal to

If \( 4^n + 15n + P \) is divisible by 9 for all \( n \in \mathbb{N} \), then the least negative integral value of \( P \) is

\( (2^{3n} - 1) \) is divisible by

If \( S = \frac{2^2 - 1}{2} + \frac{3^2 - 2}{6} + \frac{4^2 - 3}{12} + \ldots\) \text{ upto 10 terms, then S is equal to}

\( \sum_{n=1}^{m} n \cdot n! \) is equal to

The first and fifth terms of an A.P. are -14 and 2 respectively and the sum of its $n$ terms is 40. The value of $n$ is

The solution of the differential equation \( \frac{d^2y}{dx^2} = 0 \) represents

The solution of the differential equation \( \frac{dy}{dx} + \sqrt{\frac{1 - y^2}{1 - x^2}} = 0 \) is