List of top Mathematics Questions

Let the foci of a hyperbola $ H $ coincide with the foci of the ellipse $ E : \frac{(x - 1)^2}{100} + \frac{(y - 1)^2}{75} = 1 $ and the eccentricity of the hyperbola $ H $ be the reciprocal of the eccentricity of the ellipse $ E $. If the length of the transverse axis of $ H $ is $ \alpha $ and the length of its conjugate axis is $ \beta $, then $ 3\alpha^2 + 2\beta^2 $ is equal to:

The vectors $\overrightarrow{AB} = 3\hat{i} + 4\hat{k}$ and $\overrightarrow{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}$ are the sides of a $\triangle ABC$. The length of the median through $A$ is:

When \( |x| < 2 \), the coefficient of \( x^2 \) in the power series expansion of

\[ \frac{x}{(x-2)(x-3)} \]

is:

If \( z = 1 + i \), then the maximum value of \( |z + 12 + 9i| \) is
By shifting the origin to the point \( (h,5) \) by the translation of coordinate axes, if the equation \[ y = x^2 - 9x^2 + cx - d \] transforms to \( Y = X^2 \), then \( \left( \frac{d - c}{h} \right) \) is: \
The eccentricity of the ellipse \[ p x^2 + 5y^2 = 80, \quad \text{where } p > 5, \] is \( \frac{\sqrt{3}}{2} \). Then the value of \( p \) is
Evaluate $$ \cot \left( \sum_{n=1}^{50} \tan^{-1} \left( \frac{1}{1 + n + n^2} \right) \right).= $$
Evaluate the integral: $\int_{-\pi}^{\pi} x^2 \sin(x) \, dx$
If the mirror image of the point $P(3, 4, 9)$ in the line $\frac{x-1}{3} = \frac{y+1}{2} = \frac{z-2}{1}$ is $(\alpha, \beta, \gamma)$, then $14(\alpha + \beta + \gamma)$ is:
If $y = \tan^{-1} \frac{x}{1+2x^2} + \tan^{-1} \frac{x}{1+6x^2} + \tan^{-1} \frac{x}{1+12x^2}$, then $\left(\frac{dy}{dx}\right)_{x=\frac{1}{2}} =$
If lines $\frac{x - 1}{-3} = \frac{y - 2}{2k} = \frac{z - 3}{2}$ and $\frac{x - 1}{3k} = \frac{y - 5}{1} = \frac{z - 6}{-5}$ are mutually perpendicular, then $k$ is equal to:

Let $E_1$ and $E_2$ be two independent events of a random experiment such that
$P(E_1) = \frac{1}{2}, \quad P(E_1 \cup E_2) = \frac{2}{3}$.
Then match the items of List-I with the items of List-II:

two independent events

The correct match is:

Given that $ A^{-1} = \frac{1}{7} \begin{bmatrix} 2 & 1 \\ -3 & 2 \end{bmatrix} $, matrix $ A $ is:

(b) Order of the differential equation: $ 5x^3 \frac{d^3y}{dx^3} - 3\left(\frac{dy}{dx}\right)^2 + \left(\frac{d^2y}{dx^2}\right)^4 + y = 0 $

Consider the following Linear Programming Problem $ P $: Minimize $ x_1 + 2x_2 $, subject to 
$ 2x_1 + x_2 \leq 2 $, 
$ x_1 + x_2 = 1 $, 
$ x_1, x_2 \geq 0 $. 
The optimal value of the problem $ P $ is equal to:

Which of the following is a null set?
Which of the following statement is true?
If $a_i, b_i, c_i \in \mathbb{R}$ ($i = 1, 2, 3$) and $x \in \mathbb{R}$, and
$\begin{vmatrix} a_1 + b_1x & a_1x + b_1 & c_1 \\ a_2 + b_2x & a_2x + b_2 & c_2 \\ a_3 + b_3x & a_3x + b_3 & c_3 \end{vmatrix} = 0$,
then:

Let $D = \{(x, y) \in \mathbb{R}^2 : x > 0 \text{ and } y > 0\}$. If the following second-order linear partial differential equation 
$y^2 \frac{\partial^2 u}{\partial x^2} - x^2 \frac{\partial^2 u}{\partial y^2} + y \frac{\partial u}{\partial y} = 0$ on $D$ 
is transformed to 
$\left( \frac{\partial^2 u}{\partial \eta^2} - \frac{\partial^2 u}{\partial \xi^2} \right) + \left( \frac{\partial u}{\partial \eta} + \frac{\partial u}{\partial \xi} \right) \frac{1}{2\eta} + \left( \frac{\partial u}{\partial \eta} - \frac{\partial u}{\partial \xi} \right) \frac{1}{2\xi} = 0$ on $D$, 
for some $a, b \in \mathbb{R}$, via the coordinate transform $\eta = \frac{x^2}{2}$ and $\xi = \frac{y^2}{2}$, then which one of the following is correct?

If $a^2 x^4 + b^2 y^4 = c^6$, then the maximum value of $xy$ is:
Let $\alpha, \beta$ be the roots of the equation $x^2 + 2\sqrt{2}x - 1 = 0$. The quadratic equation, whose roots are $\alpha^4 + \beta^4$ and $\frac{1}{10} \left( \alpha^6 + \beta^6 \right)$, is:

Consider the following limit: $ \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{0}^{\infty} e^{-x / \epsilon} \left( \cos(3x) + x^2 + \sqrt{x + 4} \right) dx. $ 
Which one of the following is correct?

Given that: $M = \int_{0}^{\infty} \frac{\log t}{1 + t^3} \, dt$, and $N = \int_{-\infty}^{\infty} \frac{e^{2t} t}{1 + e^{3t}} \, dt$. Then, the relation between $M$ and $N$ is:

$ \lim_{x \to -\frac{3}{2}} \frac{(4x^2 - 6x)(4x^2 + 6x + 9)}{\sqrt{2x - \sqrt{3}}} $

The value of $\frac{1 \times 2^2 + 2 \times 3^2 + \dots + 100 \times (101)^2}{1^2 \times 2 + 2^2 \times 3 + \dots + 100^2 \times 101}$ is: