List of top Mathematics Questions

The number of real roots of the equation $ x|x-2| + 3|x-3| + 1 = 0 $ is:
Let $ y = y(x) $ be the solution of the differential equation $ (x^2 + 1)y' - 2xy = (x^4 + 2x^2 + 1)\cos x, $ with the initial condition $ y(0) = 1 $. Then $ \int_{-3}^{3} y(x) \, dx \text{ is:} $
Let $ p $ be the number of all triangles that can be formed by joining the vertices of a regular polygon $ P $ of $ n $ sides, and $ q $ be the number of all quadrilaterals that can be formed by joining the vertices of $ P $. If $ p + q = 126 $, then the eccentricity of the ellipse $ \frac{x^2}{16} + \frac{y^2}{n} = 1 $ is:
Let the length of a latus rectum of an ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $ be 10. If its eccentricity is $ e $, and the minimum value of the function $ f(t) = t^2 + t + \frac{11}{12} $, where $ t \in \mathbb{R} $, then $ a^2 + b^2 $ is equal to:
If the locus of $ z \in \mathbb{C} $, such that $ \text{Re} \left( \frac{z - 1}{2z + i} \right) + \text{Re} \left( \frac{ \bar{z} - 1}{2 \bar{z} - i} \right) = 2, $ is a circle of radius $ r $ and center $ (a, b) $, then $ \frac{15ab}{r^2} \text{ is equal to:} $
Let a random variable X take values 0, 1, 2, 3 with $ P(X = 0) = P(X = 1) = p, \, P(X = 2) = P(X = 3), \, \text{and} \, F(X^2) = 2F(X). $ Then the value of $ 8p - 1 $ is:
Let $ a_n $ be the $ n $-th term of an A.P. If $ S_n = a_1 + a_2 + a_3 + \cdots + a_n = 700 $, $ a_6 = 7 $, and $ S_7 = 7 $, then $ a_n $ is equal to:
Let $ \vec{a} $ and $ \vec{b} $ be the vectors of the same magnitude such that $ \frac{| \vec{a} + \vec{b} | + | \vec{a} - \vec{b} |}{| \vec{a} + \vec{b} | - | \vec{a} - \vec{b} |} = \sqrt{2} + 1. \quad \text{Then } \frac{| \vec{a} + \vec{b} |^2}{| \vec{a} |^2} \text{ is:} $

If the four distinct points $ (4, 6) $, $ (-1, 5) $, $ (0, 0) $ and $ (k, 3k) $ lie on a circle of radius $ r $, then $ 10k + r^2 $ is equal to

Consider the sets $A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + y^2 = 25\}$, $B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + 9y^2 = 144\}$, $C = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} : x^2 + y^2 \leq 4\}$, and $D = A \cap B$. The total number of one-one functions from the set $D$ to the set $C$ is:
Let $f, g: (1, \infty) \rightarrow \mathbb{R}$ be defined as $f(x) = \frac{2x + 3}{5x + 2}$ and $g(x) = \frac{2 - 3x}{1 - x}$. If the range of the function $fog: [2, 4] \rightarrow \mathbb{R}$ is $[\alpha, \beta]$, then $\frac{1}{\beta - \alpha}$ is equal to
Evaluate the following limit: $ \lim_{x \to 0^+} \frac{\tan\left(5x^{\frac{1}{3}}\right) \log\left(1 + 3x^2\right)}{\left(\tan^{-1}\left(3\sqrt{x}\right)\right)^2 \left(e^{5x^{\frac{4}{3}}} - 1\right)} $
Solve for \( x \) in the equation \( \frac{2x - 3}{4} = 5 \).

If $ \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p $, then $ 96 \log_e p $ is equal to _______

Let $ \vec{a} = \hat{i} + 2\hat{j} + \hat{k} $, $ \vec{b} = 3\hat{i} - 3\hat{j} + 3\hat{k} $, $ \vec{c} = 2\hat{i} - \hat{j} + 2\hat{k} $ and $ \vec{d} $ be a vector such that $ \vec{b} \times \vec{d} = \vec{c} \times \vec{d} $ and $ \vec{a} \cdot \vec{d} = 4 $. Then $ |\vec{a} \times \vec{d}|^2 $ is equal to _______

If the equation of the hyperbola with foci \( (4, 2) \) and \( (8, 2) \) is \( 3x^2 - y^2 - \alpha x + \beta y + \gamma = 0 \), then \( \alpha + \beta + \gamma \) is equal to _______
Let C be the circle of minimum area enclosing the ellipse E: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) with eccentricity \( \frac{1}{2} \) and foci \( (\pm 2, 0) \). Let PQR be a variable triangle, whose vertex P is on the circle C and the side QR of length 29 is parallel to the major axis and contains the point of intersection of E with the negative y-axis. Then the maximum area of the triangle PQR is:

The shortest distance between the curves $ y^2 = 8x $ and $ x^2 + y^2 + 12y + 35 = 0 $ is:

The distance of the point (7, 10, 11) from the line \( \frac{x - 4}{1} = \frac{y - 4}{0} = \frac{z - 2}{3} \) along the line \( \frac{x - 7}{2} = \frac{y - 10}{3} = \frac{z - 11}{6} \) is

The sum $ 1 + \frac{1 + 3}{2!} + \frac{1 + 3 + 5}{3!} + \frac{1 + 3 + 5 + 7}{4!} + ... $ upto $ \infty $ terms, is equal to

Let I be the identity matrix of order 3 × 3 and for the matrix $ A = \begin{pmatrix} \lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2 \end{pmatrix} $, $ |A| = -1 $. Let B be the inverse of the matrix $ \text{adj}(A \cdot \text{adj}(A^2)) $. Then $ |(\lambda B + I)| $ is equal to _______

Let $ (1 + x + x^2)^{10} = a_0 + a_1 x + a_2 x^2 + ... + a_{20} x^{20} $. If $ (a_1 + a_3 + a_5 + ... + a_{19}) - 11a_2 = 121k $, then k is equal to _______

Let the equation $ x(x+2) * (12-k) = 2 $ have equal roots. The distance of the point $ \left(k, \frac{k}{2}\right) $ from the line $ 3x + 4y + 5 = 0 $ is

The integral $ \int_{0}^{\pi} \frac{8x dx}{4 \cos^2 x + \sin^2 x} $ is equal to

If the domain of the function $ f(x) = \log_7(1 - \log_4(x^2 - 9x + 18)) $ is $ (\alpha, \beta) \cup (\gamma, \delta) $, then $ \alpha + \beta + \gamma + \delta $ is equal to