List of top Mathematics Questions

For \(0 < \theta < \frac{pi}{2}\), if the eccentricity of the hyperbola \( x^2 - y^2 \csc^2 \theta = 5 \) is \( \sqrt{7} \) times the eccentricity of the ellipse \( x^2 \csc^2 \theta + y^2 = 5 \), then the value of \( \theta \) is:

If the system of equations
\( 2x + 3y - z = 5 \)  
\( x + \alpha y + 3z = -4 \)  
\( 3x - y + \beta z = 7 \)  
has infinitely many solutions, then \( 13 \alpha \beta \) is equal to:  

Let \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R} \to \mathbb{R} \) be defined as: \(f(x) = \begin{cases} \log_e x, & \text{if } x > 0, \\ e^{-x}, & \text{if } x \leq 0, \end{cases}\) and
\(g(x) = \begin{cases} x, & \text{if } x \geq 0, \\ e^x, & \text{if } x < 0. \end{cases}\) Then \( g \circ f: \mathbb{R} \to \mathbb{R} \) is:

The area enclosed by the curves xy + 4y = 16 and x + y = 6 is equal to :

\(\text{Let } S = \{x \in \mathbb{R} : (\sqrt{3} + \sqrt{2})^x + (\sqrt{3} - \sqrt{2})^x = 10\}\).Then the number of elements in \( S \) is:

Let \(\vec{a} = -5\hat{i} + \hat{j} - 3\hat{k}, \quad \vec{b} = \hat{i} + 2\hat{j} - 4\hat{k},\) and  \(\vec{c} = \left( \left( \left( \vec{a} \times \vec{b} \right) \times \hat{i} \right) \times \hat{i} \right) \times \hat{i}.\) Then  \(\vec{c} \cdot (-\hat{i} + \hat{j} + \hat{k})\) is equal to:  

If \(\tan A = \frac{1}{\sqrt{x(x^2 + x + 1)}}, \quad \tan B = \frac{\sqrt{x}}{\sqrt{x^2 + x + 1}}\) and \(\tan C = \left(x^3 + x^2 + x^{-1}\right)^{\frac{1}{2}}, \quad 0 < A, B, C < \frac{\pi}{2}\),then \( A + B \) is equal to:

If n is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then n is equal to:

The value of the integral \(\int_0^{\frac{\pi}{4}} \frac{x \, dx}{\sin^4(2x) + \cos^4(2x)}\)

If \( A = \begin{bmatrix} \sqrt{2} & 1 \\ -1 & \sqrt{2} \end{bmatrix} \), \( B = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \), \( C = ABA^\top \) and \( X = A^\top C^2 A \), then \( \det (X) \) is equal to:

A bag contains 8 balls, whose colours are either white or black. 4 balls are drawn at random without replacement and it was found that 2 balls are white and other 2 balls are black. The probability that the bag contains equal number of white and black balls is:

If the function \( f(x) = \begin{cases} \frac{72^x - 9^x - 8^x + 1}{\sqrt{2} - \sqrt{1 + \cos x}}, & x \neq 0 \\ a \log_e 2 \log_e 3, & x = 0 \end{cases} \) is continuous at \( x = 0 \), then the value of \( a^2 \) is equal to

Let $I(x) = \int \frac{6}{\sin^2 x (1 - \cot x)^2} dx$. If $I(0) = 3$, then $I\left(\frac{\pi}{12}\right)$ is equal to:

If \[ \int \frac{1}{\sqrt[5]{(x - 1)^4}(x + 3)^6} \, dx = A \left( \frac{\alpha x - 1}{\beta x + 3} \right)^B + C, \] where \(C\) is the constant of integration, then the value of \(\alpha + \beta + 20AB\) is _______.

Let \(a, b, c \in \mathbb{N}\) and \(a<b<c\). Let the mean, the mean deviation about the mean and the variance of the 5 observations \(9, 25, a, b, c\) be \(18, 4\) and \(\frac{136}{5}\), respectively. Then \(2a + b - c\) is equal to _______.

Let a ray of light passing through the point \((3, 10)\) reflects on the line \(2x + y = 6\) and the reflected ray passes through the point \((7, 2)\). If the equation of the incident ray is \(ax + by + 1 = 0\), then \(a^2 + b^2 + 3ab\) is equal to _.

The number of distinct real roots of the equation \[ |x + 1| |x + 3| - 4|x + 2| + 5 = 0, \] is _______.

An arithmetic progression is written in the following way

The sum of all the terms of the 10^th row is ______ .

Let \(P(\alpha, \beta, \gamma)\) be the image of the point \(Q(1, 6, 4)\) in the line \[ \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}. \] Then \(2\alpha + \beta + \gamma\) is equal to _______.

Let \(S\) be the focus of the hyperbola \(\frac{x^2}{3} - \frac{y^2}{5} = 1\), on the positive x-axis. Let \(C\) be the circle with its centre at \(A\left(\sqrt{6}, \sqrt{5}\right)\) and passing through the point \(S\). If \(O\) is the origin and \(SAB\) is a diameter of \(C\), then the square of the area of the triangle \(OSB\) is equal to -

Let \(A\) be the region enclosed by the parabola \(y^2 = 2x\) and the line \(x = 24\). Then the maximum area of the rectangle inscribed in the region \(A\) is ________.

If \[ \alpha = \lim_{x \to 0^+} \left( \frac{e^{\sqrt{\tan x}} - e^{\sqrt{x}}}{\sqrt{\tan x} - \sqrt{x}} \right) \] \[ \beta = \lim_{x \to 0} (1 + \sin x)^{\frac{1}{2\cot x}} \] are the roots of the quadratic equation \(ax^2 + bx - \sqrt{e} = 0\), then \(12 \log_e (a + b)\) is equal to _________.

For $a, b > 0$, let $ f(x) = \begin{cases} \frac{\tan((a+1)x) + b \tan x}{x}, & x < 0, \\ \frac{x}{3}, & x = 0, \\ \frac{\sqrt{ax + b^2x^2} - \sqrt{ax}}{b\sqrt{a x \sqrt{x}}}, & x > 0 \end{cases} $ be a continuous function at $x = 0$. Then $\frac{b}{a}$ is equal to: 

The variance \(\sigma^2\) of the data

Is _______.

 

The area of the region enclosed by the parabola \((y - 2)^2 = x - 1\), the line \(x - 2y + 4 = 0\) and the positive coordinate axes is ______.