List of top Mathematics Questions

The value of the integral ∫ sinθ sin2θ (sin⁶θ + sin⁴θ + sin²θ) / √(2sin⁴θ + 3sin²θ + 6) dθ is: (where c is a constant of integration)
A wire is in the form of a circle of radius 38cm. If it is bent into a square what will be the area of the square (in cm2) ?
Let $f:[0, 3] \rightarrow R$ be defined by $f(x) = \min\{x-[x], 1+[x]-x\}$ where $[x]$ is the greatest integer less than or equal to x. Let P denote the set containing all $x \in [0, 3]$ where f is discontinuous, and Q denote the set containing all $x \in [0, 3]$ where f is not differentiable. Then the sum of number of elements in P and Q is equal to _________.
For real numbers $\alpha$ and $\beta$, consider the following system of linear equations:
$x+y-z=2$
$x+2y+\alpha z = 1$
$2x-y+z = \beta$
If the system has infinite solutions, then $\alpha+\beta$ is equal to _________.

Let 

, $x \in [0, \pi]$. Then the maximum value of $f(x)$ is equal to _________. 
 

Let a plane P pass through the point (3, 7, -7) and contain the line, $\frac{x-2}{-3} = \frac{y-3}{2} = \frac{z+2}{1}$. If distance of the plane P from the origin is d, then $d^2$ is equal to _________.

Let $f:(-\frac{\pi}{4}, \frac{\pi}{4}) \rightarrow R$ be defined as 

If f is continuous at $x=0$, then the value of $6a+b^2$ is equal to : 
 

Let $\alpha, \beta$ be two roots of the equation $x^2 + (20)^{1/4}x + (5)^{1/2} = 0$. Then $\alpha^8 + \beta^8$ is equal to :
Let the plane passing through the point (-1, 0, -2) and perpendicular to each of the planes $2x+y-z=2$ and $x-y-z=3$ be $ax+by+cz+8=0$. Then the value of $a+b+c$ is equal to :

Let the domain of the function $f(x) = \log_4(\log_5(\log_3(18x-x^2-77)))$ be $(a, b)$. Then the value of the integral $\int_a^b \frac{\sin^3 x}{\sin^3 x + \sin^3(a+b-x)} dx$ is equal to _________.
If $\log_3 2, \log_3(2^x-5), \log_3(2^x-\frac{7}{2})$ are in an arithmetic progression, then the value of x is equal to _________.
The value of $\lim_{n \to \infty} \frac{1}{n}\sum_{j=1}^{n}\frac{(2j-1)+8n}{(2j-1)+4n}$ is equal to :
The value of the definite integral $\int_{-\pi/4}^{\pi/4} \frac{dx}{(1+e^{x\cos x})(\sin^4 x + \cos^4 x)}$ is equal to :
Let $f:R \rightarrow R$ be a function such that $f(2)=4$ and $f'(2)=1$. Then, the value of $\lim_{x\to 2}\frac{x^2f(2) - 4f(x)}{x-2}$ is equal to :
The compound statement $(P \lor Q) \land (\sim P) \Rightarrow Q$ is equivalent to :
If the mean and variance of the following data : 6, 10, 7, 13, a, 12, b, 12 are 9 and $\frac{37}{4}$ respectively, then $(a-b)^2$ is equal to :
If $\sin\theta + \cos\theta = \frac{1}{2}$, then $16(\sin(2\theta) + \cos(4\theta) + \sin(6\theta))$ is equal to :
Let C be the set of all complex numbers. Let $S_1 = \{z \in C : |z-3-2i|^2 = 8\}$, $S_2 = \{z \in C : \text{Re}(z) \ge 5\}$ and $S_3 = \{z \in C : |z - \bar{z}| \ge 8\}$. Then the number of elements in $S_1 \cap S_2 \cap S_3$ is equal to :
If the area of the bounded region $R = \left\{(x, y) : \max\{0, \log_e x\} \le y \le 2^x, \frac{1}{2} \le x \le 2\right\}$ is, $\alpha(\log_e 2)^{-1} + \beta(\log_e 2) + \gamma$, then the value of $(\alpha + \beta - 2\gamma)^2$ is equal to :
A function \( f \) is defined on \([-3, 3]\) as \( f(x) = \begin{cases} \min\{|x|, 2-x^2\}, & -2 \le x \le 2 \\ |x|, & 2 < |x| \le 3 \end{cases} \), where \( [x] \) denotes the greatest integer \( \le x \). The number of points, where \( f \) is not differentiable in \((-3, 3)\) is ________ .
A plane passes through the points A(1, 2, 3), B(2, 3, 1) and C(2, 4, 2). If O is the origin and P is (2, -1, 1), then the projection of $\vec{OP}$ on this plane is of length :
The following system of linear equations
2x + 3y + 2z = 9
3x + 2y + 2z = 9
x - y + 4z = 8
If for the matrix, A = \( A = \begin{bmatrix} 1 & -\alpha \\ \alpha & \beta \end{bmatrix} \), and \( A A^T = I_2 \), then the value of \( \alpha^4 + \beta^4 \) is :
The number of elements in the set \[ \left\{ A = \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} : a, b, d \in \{-1, 0, 1\} \text{ and } (I - A)^3 = I - A^3 \right\}, \] where \( I \) is the \( 2 \times 2 \) identity matrix, is _________.