List of top Mathematics Questions

Show that a function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = \frac{2x{1 + x^2} \) is neither one-one nor onto. Further, find set \( A \) so that the given function \( f : \mathbb{R} \to A \) becomes an onto function.}
The position vectors of vertices of \( \Delta ABC \) are \( A(2\hat{i} - \hat{j} + \hat{k}) \), \( B(\hat{i} - 3\hat{j} - 5\hat{k}) \), and \( C(3\hat{i} - 4\hat{j} - 4\hat{k}) \). Find all the angles of \( \Delta ABC \).
The vertices of triangle \( \triangle ABC \) are \( A(1, 1, 0), B(1, 2, 1) \), and \( C(-2, 2, -1) \). Find the equations of the medians through A and B. Use the equations so obtained to find the coordinates of the centroid.
Find the probability that the airplane will not crash.
Find the general solution of the differential equation: \[ y \, dx = (x + 2y^2) \, dy. \]
(a) If \( y = (\log x)^2 \), prove that \( x^2 y'' + x y' = 2 \).
Find: \[ \int x^2 \log(x^2 - 1) \, dx. \]
Evaluate: \[ \int_{-2}^{2} \sqrt{\frac{2 - x}{2 + x}} \, dx. \]
Find: \[ \int \frac{e^{4x} - 1}{e^{4x} + 1} \, dx. \]
Find: \[ \int \frac{1}{x \left[(\log x)^2 - 3 \log x - 4\right]} \, dx. \]
If \( y = \csc(\cot^{-1} x) \), then prove that \( \sqrt{1 + x^2} \frac{dy}{dx} - x = 0 \).
A pair of dice is thrown simultaneously. If \( X \) denotes the absolute difference of the numbers appearing on top of the dice, then find the probability distribution of \( X \).
Find the particular solution of the differential equation given by: \[ 2xy + y^2 - 2x^2 \frac{dy}{dx} = 0; \quad y = 2, \, \text{when } x = 1. \]
(a) Find the value of \( \sin^{-1}\left( -\frac{1}{2} \right) + \cos^{-1}\left( -\frac{\sqrt{3}}{2} \right) + \cot^{-1}\left( \tan \frac{4\pi}{3} \right) \).
(a) Find the value of \( \sin^{-1}\left( -\frac{1}{2} \right) + \cos^{-1}\left( -\frac{\sqrt{3}}{2} \right) + \cot^{-1}\left( \tan \frac{4\pi}{3} \right) \).
Assertion (A): For two non-zero vectors \( \vec{a} \) and \( \vec{b} \), \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \).
Reason (R): For two non-zero vectors \( \vec{a} \) and \( \vec{b} \), \( \vec{a} \times \vec{b} = -\vec{b} \times \vec{a} \).
Let \( E \) be an event of a sample space \( S \) of an experiment, then \( P(S | E) \) is:
Determine whether the function \( f(x) = x^2 - 6x + 3 \) is increasing or decreasing in \( [4, 6] \).
Assertion (A): For any symmetric matrix \( A \), \( B'A B \) is a skew-symmetric matrix.
Reason (R): A square matrix \( P \) is skew-symmetric if \( P' = -P \).
The common region determined by all the constraints of a linear programming problem is called:
The coordinates of the foot of the perpendicular drawn from the point \( (0, 1, 2) \) on the \( x \)-axis are given by:
The anti-derivative of \( \sqrt{1 + \sin 2x}, \, x \in \left[ 0, \frac{\pi}{4} \right] \) is:
For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true?
The function \( f(x) = x^3 - 3x^2 + 12x - 18 \) is:
The differential equation \( \frac{dy}{dx} = F(x, y) \) will not be a homogeneous differential equation, if \( F(x, y) \) is: