List of top Mathematics Questions

Find: \[ \int \frac{2x}{(x^2 + 3)(x^2 - 5)} \, dx \]

Let \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) be three vectors such that $\mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c} \text{ and } \mathbf{a} \times \mathbf{b} \neq 0 \text{ Show that } \mathbf{b} = \mathbf{c}$.

Find a vector of magnitude 5 which is perpendicular to both the vectors \( 3\hat{i} - 2\hat{j} + \hat{k} \text{ and } 4\hat{i} + 3\hat{j} - 2\hat{k} \).
If $y = 5 \cos x - 3 \sin x, \text{ prove that } \frac{d^2y}{dx^2} + y = 0$.
Differentiate \(\frac{\sin x}{\sqrt{\cos x}}\) \(\text{ with respect to } x\).

Find the domain of the function \( f(x) = \cos^{-1}(x^2 - 4) \).

If \( |\vec{a}| = 10, |\vec{b}| = 2 \) and \( \vec{a} \cdot \vec{b} = 12 \), then the value of \( |\vec{a} \times \vec{b}| \) is:
Consider the following statements:
Statement-I: The set of all solutions of the linear inequalities 3x + 8 < 17 and 2x + 8 ≥ 12 are x < 3 and x ≥ 2 respectively.
Statement-II: The common set of solutions of the linear inequalities 3x + 8 < 17 and 2x + 8 ≥ 12 is (2,3).
Which of the following is true?
A die has two faces each with number '1', three faces each with number '2' and one face with number '3'. If the die is rolled once, then \(P(1 \text{ or } 3)\) is:
If \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \), \( \vec{b} = \hat{i} - \hat{j} + 4\hat{k} \), and \( \vec{c} = \hat{i} + \hat{j} + \hat{k} \) are such that \( \vec{a} + \lambda \vec{b} \) is perpendicular to \( \vec{c} \), then the value of \( \lambda \) is:
The number of four digit even numbers that can be formed using the digits 0, 1, 2 and 3 without repetition is:
Consider the following statements:
% Statement Statement (I): If either \( |\vec{a}| = 0 \) or \( |\vec{b}| = 0 \), then \( \vec{a} \cdot \vec{b} = 0 \).
% Statement Statement (II): If \( \vec{a} \times \vec{b} = 0 \), then \( \vec{a} \) is perpendicular to \( \vec{b} \).
Which of the following is correct?
The mean deviation about the mean for the data \( 4, 7, 8, 9, 10, 12, 13, 17 \) is:
The equation of the line through the point \( (0, 1, 2) \) and perpendicular to the line \[ \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{-2} \] is:
Consider the following statements. Statement (I): If \( E \) and \( F \) are two independent events, then \( E' \) and \( F' \) are also independent. Statement (II): Two mutually exclusive events with non-zero probabilities of occurrence cannot be independent. Which of the following is correct?
Let the functions \( f \) and \( g \) be \[ f : [0, \frac{\pi}{2}] \to \mathbb{R} \text{ given by } f(x) = \sin x \text{ and } g(x) = \cos x, \text{ where } R \text{ is the set of real numbers}. \] Consider the following statements: Statement (I): \( f \) and \( g \) are one-to-one. Statement (II): \( f + g \) is one-to-one. Which of the following is correct?
Meera visits only one of the two temples A and B in her locality. Probability that she visits temple A is \( \frac{2}{5} \). If she visits temple A, the probability that she meets her friend is \( \frac{1}{3} \). The probability that she meets her friend, whereas it is \( \frac{2}{7} \) if she visits temple B. Meera met her friend at one of the two temples. The probability that she met her friend at temple B is:
Domain of the function \( f(x) = \frac{1{(x-2)(x-5)} \) is:}
The value of the integral \[ \int_0^1 \log(1 - x) \, dx \] is:
The integral \[ \int \frac{dx}{x^2 \left( x^4 + 1 \right)^{3/4}} \] equals:
If \[ y = \frac{\cos x}{1 + \sin x} \] then:

Match the following:
In the following, \( [x] \) denotes the greatest integer less than or equal to \( x \). 

 Choose the correct answer from the options given below:

The function f(x) is given by:

For x < 0:
f(x) = ex + ax

For x ≥ 0:
f(x) = b(x - 1)2


The function is differentiable at x = 0. Then,
General solution of the differential equation \[ \frac{dy}{dx} + y \tan x = \sec x \quad \text{is:} \]
If 'a' and 'b' are the order and degree respectively of the differentiable equation \[ \frac{d^2 y}{dx^2} + \left(\frac{dy}{dx}\right)^3 + x^4 = 0, \quad \text{then} \, a - b = \, \_ \_ \]