List of practice Questions

Evaluate the integral \[ \int \frac{\cos 4x + 1}{\cot x - \tan x} \, dx \]

Let \( x \) be the arithmetic mean and \( y, z \) be the two geometric means between any two positive numbers, then \[ \frac{y^3 + z^3}{x y z} = \text{----------} \]

Identify the starting compound from the following data: \[ C_6H_{14}O(X) \text{ on reaction with HI yields a haloalkane (A) and an alcohol (B).} \] Compound (A) on reaction with aqueous NaOH gives an alcohol (C). Compounds (B) and (C) on reaction with CrO\(_3\) in anhydrous medium yields Butanone and Ethanal respectively.

5.8 g of a gas maintained at 95\(^\circ C\) occupies the same volume as 0.368 g of hydrogen gas maintained at a temperature of 17\(^\circ C\) and pressure being the same atmospheric pressure for both the gases. What is the molecular mass of the unknown gas?

A first order reaction proceeds to 90% completion. What will be the approximate time taken for 90% completion in relation to \(t_{1/2}\) of the reaction?

The value of \[ \sin^{-1} \left[ \cos \left( \frac{39\pi}{5} \right) \right] \] is

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