List of practice Questions

The probability distribution of a discrete random variable \( X \) is given as \[ \begin{array}{|c|c|c|c|c|c|} \hline X & 1 & 2 & 4 & 2A & 3A & 5A \\ \hline P(X) & \frac{1}{2} & \frac{1}{5} & \frac{3}{25} & K & \frac{1}{25} & \frac{1}{25} \\ \hline \end{array} \] Then the value of \( A \) if \( E(X) = 2.94 \) is

18 Points are indicated on the perimeter of a triangle ABC as shown below. If three points are chosen then probability that it will form a triangle is

If \( \vec{a} \) and \( \vec{b} \) are unit vectors, then the angle between \( \vec{a} \) and \( \vec{b} \) for which \( \vec{a} - \vec{b} = 0 \)

If the volume of a sphere is increasing at a constant rate, then the rate at which its radius is increasing is

What can be said regarding a line if its slope is negative?

The altitude of a cone is 20 cm and its semi vertical angle is 30°. If the semi vertical angle is increasing at the rate of 2° per second, then the radius of the base is increasing at the rate of

If the position vector of a point A is \( \vec{A} = \vec{a} + 2\vec{b} \) and \( \vec{a} \) divides \( AB \) in the ratio 2 : 3, then the position vector of B is

Solution of \[ x - y + z = 4, \quad x - 2y + 2z = 9, \quad 2x + y + 3z = 1 \] is

Solution of \[ x - y + z = 4, \quad x - 2y + 2z = 9, \quad 2x + y + 3z = 1 \] is

The integral \[ \int e^x \left( 1 + \tan x + \tan^2 x \right) dx \] is equal to

In a \( \triangle ABC \), if coordinates of point A are \( (1, 2) \) and the equations of the medians through B and C are \[ x + y = 5 \quad \text{and} \quad x = 4 \quad \text{respectively, then the coordinates of B are} \]

The distance of the point \( (2, 3, 4) \) from the line \[ 1 - x = \frac{y}{2} = \frac{1}{3}(1 + z) \] is

If \[ \begin{bmatrix} 2 + x & 3 & 4 \\ 1 & -1 & 2 \\ x & 1 & -5 \end{bmatrix} \] is a singular matrix, then \( x \) is

The sum of the degree and order of the following differential equation \[ \left( 1 - \left( \frac{dy}{dx} \right)^2 \right)^{\frac{3}{2}} = kx \frac{d^2y}{dx^2} \] is

Consider the first 10 natural numbers. If we multiply each number by \( -1 \) and add 1 to each number, the variance of the numbers so obtained is

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length \( x \). The maximum area enclosed by the park is

Let \[ f(x) = \cos^{-1}(3x - 1) \] then the domain of \( f(x) \) is equal to}

The solution of the differential equation \[ \frac{dy}{dx} + y \cos x = \frac{1}{2} \sin 2x \] is

The area bounded by the curve \[ y^2 = 4a^2(x - 1) \] and the lines \[ x = 1, \quad y = 4a \] is

If \[ f(x) = \sin^{-1} \left( \frac{2x+1}{1+4x^2} \right) \] then \( f'(0) \) is equal to

P is a point on the line segment joining the points \( (3, 2, -1) \) and \( (6, 2, -1) \).

The area of the upper half of the circle whose equation is \( (x - 1)^2 + y^2 = 1 \) is given by

In the set \( W \) of whole numbers, an equivalence relation \( R \) is defined as follows: \( a R b \) iff both \( a \) and \( b \) leave the same remainder when divided by 5. The equivalence class of 1 is given by

The function defined by \[ f(x) = \begin{cases} \frac{\sin x}{x} + \cos x & \text{if } x>0 \\ -5k & \text{if } x = 0 \\ \frac{4(1 - \sqrt{1 - x})}{x} & \text{if } x<0 \end{cases} \] is continuous at

If \[ f(x) = 2x - \tan^{-1} x - \log(x + \sqrt{x^2 + 1}) \] is monotonically increasing, then