List of practice Questions

Evaluate the integral: \[ I = \int_{-5\pi}^{5\pi} \left(1 - \cos 2x \right)^{\frac{5}{2}} dx. \]

The system $ x + 2y + 3z = 4, \, 4x + 5y + 3z = 5, \, 3x + 4y + 3z = \lambda $ is consistent and $ 3\lambda = n + 100 $, then $ n = ? $

If $ \alpha, \beta $ are the roots of the equation $ x^2 - 6x - 2 = 0 $, $ \alpha > \beta $, and $ a_n = \alpha^n - \beta^n, n \geq 1 $, then the value of $ \frac{a_{10} - 2 a_8}{2 a_9} $ is:

S is the sample space and A, B are two events of a random experiment. Match the items of List A with the items of List B.

Then the correct match is:

If $ \omega $ is a complex cube root of unity and if $ Z $ is a complex number satisfying $ |Z - 1| \leq 2 $ and \[ |\omega^2 Z - 1 - \omega| = a, \] then the set of possible values of $ a $ is:

The independent term in the expansion of $ (1 + x + 2x^2) \left( \frac{3x^2}{2} - \frac{1}{3x} \right)^9 $ is:

A line $ L $ passes through $ (1,2,-3) $ and $ (3,3,-1) $, and a plane $ \pi $ passes through $ (2,1,-2), (-2,-3,6), (0,2,-1) $. If $ \theta $ is the angle between $ L $ and $ \pi $, then $ 27 \cos^2 \theta = $ ?

The solution of the differential equation \[ x dy - y dx = \sqrt{x^2 + y^2} dx \] when $ y(\sqrt{3}) = 1 $ is:

A test containing 3 objective type of questions is conducted in a class. Each question has 4 options and only one option is the correct answer. No two students of the class have answered identically and no student has written all correct answers. If every student has attempted all the questions, then the maximum possible number of students who have written the test is:

The angle between the planes $ \vec{r} \cdot (12\hat{i} + 4\hat{j} - 3\hat{k}) = 5 $ and $ \vec{r} \cdot (5\hat{i} + 3\hat{j} + 4\hat{k}) = 7 $ is:

Evaluate the integral \[ \int \frac{x^4 + 1}{x^6 + 1} dx. \]

If \[ \cos x \frac{dy}{dx} - y \sin x = 6x, \quad (0 < x < \frac{\pi}{2}) \quad \text{and} \quad y(\frac{\pi}{3}) = 0, \quad \text{then} \quad y(\frac{\pi}{6}) = \]

If $$ y = 1 + x + x^2 + x^3 + \dots \quad \text{and} \quad |x| < 1, \text{ then } y'' = $$

The algebraic equation of degree 4 whose roots are the translates of the roots of the equation $ x^4 + 5x^3 + 6x^2 + 7x + 9 = 0 $ by $ -1 $ is:

If \[ \frac{1}{(3x+1)(x-2)} = \frac{A}{3x+1} + \frac{B}{x-2} \quad {and} \quad \frac{x+1}{(3x+1)(x-2)} = \frac{C}{3x+1} + \frac{D}{x-2}, \] then \[ \frac{1}{(3x+1)(x-2)} = \frac{A}{3x+1} + \frac{B}{x-2}, { find } A + 3B = 0, A:C = 1:3, B:D = 2:3. \]

In a class consisting of 40 boys and 30 girls, 30% of the boys and 40% of the girls are good at Mathematics. If a student selected at random from that class is found to be a girl, then the probability that she is not good at Mathematics is:

If $$ \lim\limits_{x \to \infty} \frac{\left(\sqrt{2x+1} + \sqrt{2x-1}\right) + \left(\sqrt{2x+1} - \sqrt{2x-1}\right) P x^4 - 16} {(x+\sqrt{x^2 - 2}) + (x - \sqrt{x^2 - 2})} = 1, $$ then P = ?

If $ n \geq 2 $ is a natural number and $ 0<\theta<\frac{\pi}{2} $, then \[ \int \frac{(\cos^n \theta - \cos \theta)^{1/n}}{\cos^{n+1} \theta} \sin \theta \, d\theta = \]

If 5 letters are to be placed in 5-addressed envelopes, then the probability that at least one letter is placed in the wrongly addressed envelope is:

If a variable straight line passing through the point of intersection of the lines x - 2y + 3 = 0 and 2x - y - 1 = 0 intersects the X and Y axes at A and B respectively, then the equation of the locus of a point which divides the segment AB in the ratio -2 : 3 is:

$\int\frac{2x^{2}\cos(x^{2})-\sin(x^{2})}{x^{2}}dx=$

If five-digit numbers are formed from the digits 0, 1, 2, 3, 4 using every digit exactly only once, then the probability that a randomly chosen number from those numbers is divisible by 4 is

If the function \[ f(x) = \frac{\sqrt{1+x} - 1}{x} \] is continuous at $ x = 0 $, then $ f(0) $ is:

The differential equation formed by eliminating arbitrary constants $ A $ and $ B $ from the equation \[ y = A \cos 3x + B \sin 3x \] is:

In a triangle ABC, if $ (r_1 + r_2) \csc^2 \frac{C}{2} = $