List of top Mathematics Questions

The direction cosines of the vector \(\hat{i} + \hat{j} - 2\hat{k}\) are
If the orders of the matrices A and B are \(m \times n\) and \(n \times p\) respectively, then the order of AB will be
\[ \left( \sqrt{2} + 1 + i \sqrt{2} - 1 \right)^8 = ? \]
For all $n \in \mathbb{N}$, if $n(n^2+3)$ is divisible by $k$, then the maximum value of $k$ is
If the matrix A = \(\begin{bmatrix} 0 & 2y & z
x & y & -z
x & -y & z \end{bmatrix}\) satisfies the equation AA' = I, then find the values of x, y and z.
Given any two events A and B are such that \( P(A) = \frac{1}{2}, P(B) = \frac{1}{4} \) and \( P(A \cap B) = \frac{1}{8} \), then find \( P(\text{not A and not B}) \).
If a function \( f: \mathbb{R} \to \{ x \in \mathbb{R} : x \in (-1, 1) \} \) is defined as \( f(x) = \frac{x}{1+|x|}, \ x \in \mathbb{R} \), then prove that f is one-one and onto.
If \( x\sqrt{1+y} + y\sqrt{1+x} = 0\), \(-1 < x < 1\), then prove that \(\frac{dy}{dx} = -\frac{1}{(1+x)^2} \).
If \( A = \begin{bmatrix} 3 & 3 & 1 \\ 3 & 4 & 1 \\ 4 & 3 & 1 \end{bmatrix} \), then verify that \( A \, (\text{adj } A) = |A| I \) and find \( A^{-1} \).
Prove that: \(\int_0^{\pi/4} \log_e(1 + \tan \theta) \, d\theta = \frac{\pi}{8} \log_e 2\).
Solve the integral \(\int \frac{\sec^2 2\theta}{(\cot \theta - \tan \theta)^2} \, d\theta\).
Show that \( y = c_1 e^{ax} \cos(bx) + c_2 e^{ax} \sin(bx) \), where \( c_1, c_2 \) are constants, is a solution of the differential equation \( \frac{d^2y}{dx^2} - 2a\frac{dy}{dx} + (a^2 + b^2)y = 0 \).
If \( A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix} \), then verify that \( A \cdot \text{adj}(A) = |A| I \) and find \( A^{-1} \).
If \( x\sqrt{1+y} + y\sqrt{1+x} = 0 \) for \( -1<x<1 \), then prove that \( \frac{dy}{dx} = -\frac{1}{(1+x)^2} \).
Express the matrix \( A = \begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix} \) as the sum of a symmetric and a skew-symmetric matrix.
Does the relation defined by \( R = \{(x, y) : y \text{ is divisible by } x\} \) on the set \( A = \{1, 2, 3, 4, 5, 6\} \), an equivalence relation?
Does the function \( f(x) = \begin{cases} x+5 & \text{if } x \le 1 \\ x-5 & \text{if } x > 1 \end{cases} \) continuous at \( x = 1 \)?
If \( \begin{bmatrix} x+y & 2 \\ 5+z & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix} \), then find the values of \(x, y, z\).
A relation \(R = \{(a, b) : a = b - 1,\; b \geq 3\}\) is defined on set \(\mathbb{N}\), then
Find the area bounded by the ellipse \(\frac{x^2}{16} + \frac{y^2}{25} = 1\).
If \(y = 600 e^{-7x} + 500 e^{7x}\), then show that \(\frac{d^2y}{dx^2} = 49y\).
There are 10 white and 5 black balls in a bag. Two balls are drawn one by one. First ball is not placed back before the second is taken out. Assume that the taking out of each ball from the bag is equally likely. What is the probability that both balls taken out are white?
Prove that the radius of the right circular cylinder of maximum curved surface inscribed in a cone is half of the radius of the cone.
Prove that \(|\vec{a} \cdot \vec{b}| \le |\vec{a}| |\vec{b}|\) is always true for any two vectors \(\vec{a}\) and \(\vec{b}\).
Solve: \(\int \frac{3x - 5}{x^3 - x^2 - x + 1} dx\)