List of top Mathematics Questions

Let $f(x) = \begin{cases} \dfrac{1}{|x|}, & |x|>1 \\ ax^2 + b, & |x| \leq 1 \end{cases}$. If $\lim_{x \to 1} f(x)$ and $\lim_{x \to -1} f(x)$ exist, then the possible values for $a$ and $b$ are
For a Poisson distribution:
- Mean $ = \ell $
- Variance $ = m $
- $ \ell + m = 8 $
Then evaluate: $$ e^4 \cdot \left[1 - P(X>2)\right] $$
$\sum_{k=0}^{440} i^k = x + iy \Rightarrow x^{100} + x^{99} y + x^{342} y^2 + x^{97} y^3 =$
The perimeter of a sector is constant. If its area is to be maximum, the sectorial angle should be:
Let \( \vec{OA} = -4\hat{i} + 3\hat{k} \), \( \vec{OB} = 14\hat{i} + 2\hat{j} - 5\hat{k} \), and vector \( \vec{OD} \) bisects \( \angle AOB \), with \( |\vec{OD}| = \sqrt{6} \). Then: \[ \vec{OD} = ? \]
If $\vec{a} + \vec{b} + \vec{c} = 0$, and $|\vec{a}| = 7$, $|\vec{b}| = 5$, $|\vec{c}| = 3$, then the angle between $\vec{b}$ and $\vec{c}$ is
The set of all \( x \) for which \( \sin x \leq x \) is:
Let $$ I_n = \int (\cos^n x + \sin^n x) \, dx,\quad \text{and} \quad \frac{n - 1}{n} I_{n - 2} = \frac{\sin x \cos x}{n} f(x) $$ Then, what is $ f(x) $?

The mean deviation about the mean for the given data:  

Marks Obtained0–2020–4040–6060–8080–100
Number of Students10812911
Out of 205 students:
- 105 pass in English
- 70 pass in Mathematics
- 30 pass in both
How many students fail in both subjects?
Condition that two curves \( y^2 = 4ax \) and \( xy = c^2 \) cut orthogonally is:
If the curves \( y = x^3 - 3x^2 - 8x - 4 \) and \( y = 3x^2 + 7x + 4 \) touch each other at a point \( P \), then the equation of the common tangent at \( P \) is
Evaluate the integral \( \int \frac{x^2 - 2}{x^3 \sqrt{x^2 - 1}} \, dx \)
Suppose a parabola with focus at $(0,0)$ has $x - y + 1 = 0$ as its tangent at the vertex. Then the equation of its directrix is
If the point $(a, 8, -2)$ divides the line segment joining the points $(1, 4, 6)$ and $(5, 2, 10)$ in the ratio $m:n$, then $\dfrac{2m}{n} - \dfrac{a}{3} =$
In a triangle $ABC$, $\dfrac{a}{b} = 2 + \sqrt{3}$ and $\angle C = 60^\circ$. Then the measure of $\angle A$ is
Evaluate: \[ \lim_{x \to 0} \frac{\sqrt{11 + |x|} - 6\sqrt{2 + |x|}}{6 - 2\sqrt{2 + |x|}} \]
If the point \( (2, \lambda) \) lies inside the circles \( x^2 + y^2 = 13 \) and \( x^2 + y^2 + x - 2y = 14 \), then \( \lambda \) lies in the set:
If \( \int \frac{1 + \sqrt{\tan x}}{\sin 2x} dx = A \log \tan x + B \tan x + C \), then \( 4A - 2B = \):
If $f(x) = ax^2 + bx + c$ for some $a, b, c \in \mathbb{R}$ with $a + b + c = 3$ and \[ f(x+y) = f(x) + f(y) + xy,\quad \forall x, y \in \mathbb{R} \] Then $\sum_{n=1}^{10} f(n) = $ ?
The least value of $n$ so that ${}^{n-1}C_3 + {}^{n-1}C_4>{}^nC_3$ is
The length of the intercept on the line $4x - 3y - 10 = 0$ by the circle $x^2 + y^2 - 2x + 4y - 26 = 0$ is
If the equation of the tangent to the circle \(x^2+y^2=a^2\) at \(P(x_1, y_1)\) is \(x x_1 + y y_1 = K\), then K =
By eliminating the arbitrary constants from \[ y = (a + b)\sin(x + c) - de^{x + te^t} \] the differential equation obtained is of order:
If $(a, b, c)$ are the direction ratios of a line joining the points $(4, 3, -5)$ and $(-2, 1, -8)$, then the point $P = (a, 3b, 2c)$ lies on the plane