List of top Mathematics Questions

The shortest distance between the lines \[ \frac{x + 1}{1} = \frac{y - 3}{-1} = \frac{z - 1}{-1} \quad \text{and} \quad \frac{x}{3} = \frac{y - 1}{2} = \frac{z + 1}{-1} \] is:
The value of \[ \tan^{-1} \left( \frac{\sin 2 - 1}{\cos 2} \right) \] is:

If l, m, n are the pth, qth and rth terms of a G.P. respectively and l, m, n > 0, then

\[ \begin{vmatrix} \log_l p & 1 \\ \log_m q & 1 \\ \log_n r & 1 \end{vmatrix} \]

The equation of the image of the line $ 2y - x = 1 $ obtained by the reflection on the line $ 4y - 2x = 5 $ is:
If $ A $ is the A.M. of the roots of the equation $ x^2 - 2ax + b = 0 $ and $ G $ is the G.M. of the roots of the equation $ x^2 - 2bx + a^2 = 0 $, $ a>0 $, then:
The function $ f(x) = x^3 - 3x $ is:
The differential equation of all circles of radius $ a $ is:
The mapping $ f: \mathbb{R} \to \mathbb{R} $ such that $ f(x) = |x - 1| $, $ x \in \mathbb{R} $ is:
The area bounded by the curves $ y = \log_e x $ and $ y = (\log_e x)^2 $ is:
Evaluate the integral: \[ \int_0^2 |x^2 + x - 2| \, dx \]
The period of \( 2 \sin 4x \cos 4x \) is:
If the slope of the tangent drawn at any point \((x, y)\) on the curve \(y = f(x)\) is \(y = 6x^2 + 10x - 9\) and \(f(2) = 0\), then \(f(-2) = \)?
If the period of the function \[ f(x) = 2\cos(3x+4) - 3\tan(2x-3) + 5\sin(5x) - 7 \] is \( k \), then \( f(x) \) is periodic with fundamental period \( k \). Find \( k \).

If \( \sqrt{5} - i\sqrt{15} = r(\cos\theta + i\sin\theta), -\pi < \theta < \pi, \) then

\[ r^2(\sec\theta + 3\csc^2\theta) = \]

Let \( z \) be a non-zero complex number such that \[ z = \frac{16}{\bar{z}}. \] Then the locus of \( z \) is:
(c) The value of the integral \( \int \sqrt{1 + \sin 2x} \,dx \) is:
If \[ f(x) = 3x^{15} - 5x^{10} + 7x^5 + 50\cos(x - 1), \] then \[ \lim_{h \to 0} \frac{f(1 - h) - f(1)}{h^2 + 3h} \] is:
Let \( a, b, c \) be positive numbers. If \( a + b + c \geq K \left[ (a + b)(b + c)(c + a) \right]^{1/3} \), then the maximum value of \( K \) is:

Evaluate the integral: \[ \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{dx}{\sec^2 x + (\tan^{2022} x - 1)(\sec^2 x - 1)} \]

Let \( z = x + iy \), where \( y>0 \). If \( z + \overline{z} = 6 \) and \( |z| + | \overline{z} | = 10 \), then \( z = \)

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
Evaluate $$ \cot \left( \sum_{n=1}^{50} \tan^{-1} \left( \frac{1}{1 + n + n^2} \right) \right).= $$
If $y = \tan^{-1} \frac{x}{1+2x^2} + \tan^{-1} \frac{x}{1+6x^2} + \tan^{-1} \frac{x}{1+12x^2}$, then $\left(\frac{dy}{dx}\right)_{x=\frac{1}{2}} =$

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: