List of top Mathematics Questions

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 _.

An arithmetic progression is written in the following way

The sum of all the terms of the 10^th row 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 _________.

If the term independent of \(x\) in the expansion of \[ \left( \sqrt{ax^2} + \frac{1}{2x^3} \right)^{10} \] is 105, then \(a^2\) is equal to:

The area of the region in the first quadrant inside the circle \(x^2 + y^2 = 8\) and outside the parabola \(y^2 = 2x\) is equal to:

If the shortest distance between the lines \[ \frac{x - \lambda}{2} = \frac{y - 4}{3} = \frac{z - 3}{4} \] and \[ \frac{x - 2}{4} = \frac{y - 4}{6} = \frac{z - 7}{8} \] is \(\frac{13}{\sqrt{29}}\), then a value of \(\lambda\) is:

The sum of all possible values of \(\theta \in [-\pi, 2\pi]\), for which \[ \frac{1 + i \cos\theta}{1 - 2i \cos\theta} \] is purely imaginary, is equal to:

Let $\alpha = \sum_{r=0}^n (4r^2 + 2r + 1) \binom{n}{r}$ and $\beta = \left( \sum_{r=0}^n \frac{\binom{n}{r}}{r+1} \right) + \frac{1}{n+1}$. If $140 < \frac{2\alpha}{\beta} < 281$, then the value of $n$ is _____.

Let $z$ be a complex number such that $|z + 2| = 1$ and $\text{Im}\left(\frac{z+1}{z+2}\right) = \frac{1}{5}$. Then the value of $|\text{Re}(z+2)|$ is:

Let $\text{P}(x, y, z)$ be a point in the first octant, whose projection in the xy-plane is the point $\text{Q}$. Let $\text{OP} = \gamma$; the angle between $\text{OQ}$ and the positive x-axis be $\theta$; and the angle between $\text{OP}$ and the positive z-axis be $\phi$, where $\text{O}$ is the origin. Then the distance of $\text{P}$ from the x-axis is:

If the solution $y(x)$ of the given differential equation \[(e^y + 1) \cos x \, dx + e^y \sin x \, dy = 0\]passes through the point $\left(\frac{\pi}{2}, 0\right)$, then the value of $e^{y\left(\frac{\pi}{6}\right)}$ is equal to ________.

If the system of equations \[2x + 7y + \lambda z = 3,\]\[3x + 2y + 5z = 4,\]\[x + \mu y + 32z = -1\]has infinitely many solutions, then $(\lambda - \mu)$ is equal to ________.

Let $\alpha, \beta$ be roots of $x^2 + \sqrt{2}x - 8 = 0$. If $U_n = \alpha^n + \beta^n$, then \[ \frac{U_{10} + \sqrt{12} U_9}{2 U_8} \] is equal to ________.

If the shortest distance between the lines \[ \frac{x - \lambda}{3} = \frac{y - 2}{-1} = \frac{z - 1}{1} \] and \[ \frac{x + 2}{-3} = \frac{y + 5}{2} = \frac{z - 4}{4} \] is \[ \frac{44}{\sqrt{30}}, \] then the largest possible value of $|\lambda|$ is equal to ________.

In a triangle $ABC$, $BC = 7$, $AC = 8$, $AB = \alpha \in \mathbb{N}$ and $\cos A = \frac{2}{3}$. If \[ 49 \cos(3C) + 42 = \frac{m}{n}, \] where $\gcd(m, n) = 1$, then $m + n$ is equal to ________.

Let $[t]$ denote the largest integer less than or equal to $t$. If \[ \int_0^1 \left(\left[x^2\right] + \left\lfloor \frac{x^2}{2} \right\rfloor\right) dx = a + b\sqrt{2} - \sqrt{3} - \sqrt{5} + c\sqrt{6} - \sqrt{7}, \] where $a, b, c \in \mathbb{Z}$, then $a + b + c$ is equal to ________.

If \[ S(x) = (1 + x) + 2(1 + x)^2 + 3(1 + x)^3 + \ldots + 60(1 + x)^{60}, \, x \neq 0, \] and \[ (60)^2 S(60) = a(b)^b + b, \] where $a, b \in \mathbb{N}$, then $(a + b)$ is equal to ________.

The length of the latus rectum and directrices of a hyperbola with eccentricity $e$ are 9 and $x = \pm \frac{4}{\sqrt{3}}$, respectively. Let the line $y - \sqrt{3}x + \sqrt{3} = 0$ touch this hyperbola at $(x_0, y_0)$. If $m$ is the product of the focal distances of the point $(x_0, y_0)$, then $4e^2 + m$ is equal to ________.

Let $[t]$ denote the greatest integer less than or equal to $t$. Let $f: [0, \infty) \to \mathbb{R}$ be a function defined by \[ f(x) = \left[\frac{x}{2} + 3\right] - \left[\sqrt{x}\right]. \] Let $S$ be the set of all points in the interval $[0, 8]$ at which $f$ is not continuous. Then \[ \sum_{a \in S} a \] is equal to ________.

If $A$ is a square matrix of order 3 such that \[ \det(A) = 3 \] and \[ \det(\text{adj}(-4 \, \text{adj}(-3 \, \text{adj}(3 \, \text{adj}((2A)^{-1}))))) = 2^{m^3 n}, \] then $m + 2n$ is equal to:

If \[\int \frac{1}{a^2 \sin^2 x + b^2 \cos^2 x} \, dx = \frac{1}{12} \tan^{-1}(3 \tan x) + \text{constant},\]then the maximum value of $a \sin x + b \cos x$ is:

If the area of the region \[ \left\{(x, y) : \frac{a}{x^2} \leq y \leq \frac{1}{x}, \, 1 \leq x \leq 2, \, 0<a<1 \right\} \] is \[ (\log 2) - \frac{1}{7}, \] then the value of $7a - 3$ is equal to:

Let $\vec{a} = 2\hat{i} + \hat{j} - \hat{k}$, $\vec{b} = \left((\vec{a} \times (\hat{i} + \hat{j})) \times \hat{i}\right) \times \hat{i}$. Then the square of the projection of $\vec{a}$ on $\vec{b}$ is:

Let $f(x) = \frac{1}{7 - \sin 5x}$ be a function defined on $\mathbb{R}$. Then the range of the function $f(x)$ is equal to:

If P(6, 1) be the orthocentre of the triangle whose vertices are A(5, –2), B(8, 3) and C(h, k), then the point C lies on the circle.