List of top Mathematics Questions

If \[ \int \frac{x^3}{\sqrt{1 + x^2}} \, dx = A(1 + x^2)^{\frac{3}{2}} + B(1 + x^2)^{\frac{1}{2}} + C, \text{ then } A + B = \]
Evaluate the integral: \[ \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{1 + \sqrt{\cot x}} \, dx = \]
The area bounded by the curves \( y = x^2 \) and \( y - 6 = -|x| \) is:
If \( m \in \mathbb{Z}^+, n = 2m \) and \[ \int_0^{\frac{\pi}{2}} \sin^n x \cos^m x \, dx = K(m) \int_0^{\frac{\pi}{2}} \sin^m x \, dx, \text{ then } \frac{2^{n-1}(m - 1)!}{(2m - 1)!} \cdot K(m) = \]
If \( 5f(x) + 3f\left( \frac{1}{x} \right) = 2 - \frac{1}{x}, x \ne 0 \), then \( \int_1^2 f\left( \frac{1}{x} \right) dx = \)
\(\int \frac{8^{1+x} + 4^{1+x}}{2^{2x}} dx =\):
If $I = \int_{1}^{3} \sqrt{3 + x + x^2} dx$, then $I$ lies in the interval
\( \lim_{n \to \infty} \frac{1}{n} \left( \frac{1}{e^{1/n}} + \frac{1}{e^{2/n}} + \frac{1}{e^{3/n}} + \dots + \frac{1}{e^{n/n}} \right) = \)
\( \lim_{x \to 0} \left( \frac{(1 + y)^{1/x} - 1}{y} \right) = \)
The number of positive even divisors of 6300 is:
If a seven digit number formed with distinct digits 4, 6, 9, 5, 3, \( x \) and \( y \) is divisible by 3, then the number of such ordered pairs \( (x, y) \) is
The number of natural numbers less than 500 in which no two digits are repeated is:
If matrix \( D_1 = \text{diag}(a, b, c) \), matrix \( D_2 = \text{diag}(3, 3, 3) \) and \( A \) is a skew-symmetric matrix of order 3, then
\[ \text{Tr}(D_1 D_2 A + D_2 D_1 + D_1 A + D_2 A) - \text{Tr}(D_1 + D_2) = ? \]
If \( A + 2B = \begin{bmatrix} 1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1 \end{bmatrix} \) and \( 2A - B = \begin{bmatrix} 2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2 \end{bmatrix} \), then \( \text{Tr}[A] - \text{Tr}[B] \) equals:
The sum of the distinct values of \( x \) for which the matrix \( A = \begin{bmatrix} 1 & 1 & x \\ 1 & x & 1 \\ x & 1 & 1 \end{bmatrix} \) has no inverse, is:
The rank of the matrix \( A = \begin{bmatrix} 1 & 1 & 3 \\ 2 & 2 & 1 \\ 1 & 1 & 1 \end{bmatrix} \) is:
The number of natural numbers less than 10000 which are divisible by 5 and that no digit is repeated in the same number, is
A team of 5 students is to be selected from 12 students. If two particular students are to be included in that team, then the number of ways that such team can be selected is:
The least value of \( n \) such that \( {}^{n-1}C_6 + {}^{n-1}C_7<{}^nC_8 \) is
Find the value of \( (3 + \sqrt{8})^5 + (3 - \sqrt{8})^5 \):
The coefficient of the highest power of \( x \) in the expansion of \[ \left( x + \sqrt{x^2 - 1} \right)^3 + \left( x - \sqrt{x^2 - 1} \right)^8 \text{ is:} \]
The greatest term in the expansion of $ (1+x)^{15} $, when $ x = \frac{1}{2} $ is
The coefficient of \( x^r \) in the expansion of \( \frac{1}{\sqrt{(1 - 2x)^3}} \) is
If the numerically greatest term in the expansion of \( (2 - 3x)^9 \) when \( x = 1 \) is \( P_1^q P_2^r P_3^s P_4^t \) (where \( P_1 < P_2 < P_3 < P_4 \) are the first four prime numbers), then \( \alpha + \beta + \gamma + \delta = \):
If \( (2 - 5x)^{\frac{-1}{5}} = a_0 + a_1 x + a_2 x^2 + ... \), then \( \frac{a_1}{a_2} = \)