List of top Mathematics Questions

Which of the following rational number has terminating decimal?

The number of real solutions of the equation x |x + 5| + 2|x + 7| – 2 = 0 is _____.

Let a line perpendicular to the line \( 2x - y = 10 \) touch the parabola \( y^2 = 4(x - 9) \) at the point \( P \). The distance of the point \( P \) from the centre of the circle \[ x^2 + y^2 - 14x - 8y + 56 = 0 \] is _____.

Let \( a > 0 \) be a root of the equation \( 2x^2 + x - 2 = 0 \). If \[ \lim_{x \to \frac{1}{a}} \frac{16 \left( 1 - \cos(2 + x - 2x^2) \right)}{1 - ax^2} = \alpha + \beta \sqrt{17}, \] where \( \alpha, \beta \in \mathbb{Z} \), then \( \alpha + \beta \) is equal to _____.

Let the point \((-1, \alpha, \beta)\) lie on the line of the shortest distance between the lines \[\frac{x + 2}{-3} = \frac{y - 2}{4} = \frac{z - 5}{2} \quad \text{and} \quad \frac{x + 2}{-1} = \frac{y + 6}{2} = \frac{z - 1}{0}.\] Then \((\alpha - \beta)^2\) is equal to ______.

The number of solutions of \[\sin^2 x + (2 + 2x - x^2) \sin x - 3(x - 1)^2 = 0, \quad \text{where } -\pi \leq x \leq \pi,\] is

Let \( y = y(x) \) be the solution of the differential equation\[\frac{dy}{dx} + \frac{2x}{\left( 1 + x^2 \right)^2} y = x e^{\frac{1}{1+x^2}}, \quad y(0) = 0. \] Then the area enclosed by the curve \[ f(x) = y(x) e^{\frac{1}{1+x^2}} \]and the line \( y - x = 4 \) is _______.

Let the mean and the standard deviation of the probability distribution be be $\mu$ and $\sigma$, respectively. If $\sigma - \mu = 2$, then $\sigma + \mu$ is equal to ________.

If $y(\theta) = \frac{2\cos\theta + \cos2\theta}{\cos3\theta + 4\cos2\theta + 5\cos\theta + 2}$, then at $\theta = \frac{\pi}{2}, y'' + y' + y$ is equal to:

Let $\alpha \beta \neq 0$ and $A = \begin{bmatrix} \beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2\alpha \end{bmatrix}$. If $B = \begin{bmatrix} 3\alpha & -9 & 3\alpha \\ -\alpha & 7 & -2\alpha \\ -2\alpha & 5 & -2\beta \end{bmatrix}$ is the matrix of cofactors of the elements of A, then det(AB) is equal to:

Let $\beta(m, n) = \int_{0}^{1}x^{m-1}(1-x)^{n-1}dx$, $m, n > 0$. If $\int_{0}^{1}(1-x^{10})^{20}dx = a\beta(b,c)$, then $100(a+b+x)$ equals

Let ABCD and AEFG be squares of side 4 and 2 units, respectively. The point E is on the line segment AB and the point F is on the diagonal AC. Then the radius r of the circle passing through the point F and touching the line segments BC and CD satisfies :

The coefficients a, b, c in the quadratic equation ax² + bx + c = 0 are from the set {1, 2, 3, 4, 5, 6}. If the probability of this equation having one real root bigger than the other is p, then 216p equals :

Let the set $S = \{2, 4, 8, 16, ..., 512\}$ be partitioned into 3 sets $A, B, C$ with equal number of elements such that $A \cup B \cup C = S$ and $A \cap B = B \cap C = A \cap C = \phi$. The maximum number of such possible partitions of $S$ is equal to:

Let the circle $C_{1}: x^{2}+y^{2}-2(x+y)+1=0$ and $C_{2}$ be a circle having centre at $(-1, 0)$ and radius 2. If the line of the common chord of $C_{1}$ and $C_{2}$ intersects the y-axis at the point P, then the square of the distance of P from the centre of $C_{1}$ is:

Let $f, g: \mathbb{R} \rightarrow \mathbb{R}$ be defined as: $f(x) = |x - 1|$ and $g(x) = \begin{cases} e^x, & x \geq 0 \\ x + 1, & x \leq 0 \end{cases}$ Then the function $f(g(x))$ is

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 ________.

By taking out a hemisphere from both the ends of a wooden solid cylinder, an item is formed. If the height of the cylinder is 10 cm and radius of the base is 3.5 cm, find the total surface area of the item.

From the top of a tower of height 50 m, the angles of depression of the top and bottom of a pillar are $45^\circ$ and $60^\circ$ respectively. Find the height of the pillar.

A flagstaff stands on a tower. At a distance 10 m from the tower, the angles of elevation of the top of the tower and flagstaff are $45^\circ$ and $60^\circ$ respectively. Find the length of the flagstaff.

Find the sum of 51 terms of an A.P. whose second and third terms are 14 and 18 respectively.

Solve the following equations: \[ \dfrac{3}{2}x - \dfrac{5}{3}y = -2, \quad \dfrac{x}{3} + \dfrac{y}{2} = \dfrac{13}{6} \]

If the perimeter of a garden of area 800 m$^2$ is 120 m and its length is twice the breadth, then find its length and breadth.