List of top Mathematics Questions

Let $ a $ be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle $ \alpha $ with the positive $ x $-axis and the equations of its diagonals are $\left( \sqrt{3} + 1 \right) x + \left( \sqrt{3} - 1 \right) y = 0$ and $\left( \sqrt{3} - 1 \right) x - \left( \sqrt{3} + 1 \right) y + 8\sqrt{3} = 0.$ Then $ a^2 $ is equal to

If $ \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + ... \infty = \frac{\pi^4}{90}, $ $ \frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + ... \infty = \alpha, $ $ \frac{1}{2^4} + \frac{1}{4^4} + \frac{1}{6^4} + ... \infty = \beta, $ then $ \frac{\alpha}{\beta} $ is equal to:

Let the area of the triangle formed by the lines $ \frac{x + 2}{-3} = \frac{y - 3}{3} = \frac{z - 2}{1} $, $ \frac{x - 3}{5} = \frac{y}{-1} = \frac{z - 1}{1} $ be $ A $. Then $ A^2 $ is equal to:

Let $ f(x) = x - 1 $ and $ g(x) = e^x $ for $ x \in \mathbb{R} $. If $\frac{dy}{dx} = \left( e^{-2\sqrt{x}} g\left(f\left(f(x)\right)\right) - \frac{y}{\sqrt{x}} \right), \, y(0) = 0,$ then $ y(1) $ is

The number of integral terms in the expansion of $ \left( 5^{\frac{1}{2}} + 7^{\frac{1}{8}} \right)^{1016} $ 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 orthocentre of the triangle formed by the lines $ y = x + 1 $, $ y = 4x - 8 $, and $ y = mx + c $ is at $ (3, -1) $, then $ m - c $ is:

Let $ A = \{(\alpha, \beta) \in \mathbb{R} \times \mathbb{R} : |\alpha - 1| \leq 4 \text{ and } |\beta - 5| \leq 6\} $ and $ B = \{(\alpha, \beta) \in \mathbb{R} \times \mathbb{R} : 16(\alpha - 2)^2 + 9(\beta - 6)^2 \leq 144\}. $ Then:

The number of solutions of the equation $ \cos 2\theta \cos \left( \frac{\theta}{2} \right) + \cos \left( \frac{5\theta}{2} \right) = 2 \cos^3 \left( \frac{5\theta}{2} \right) $ in the interval $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right ]$ is:

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:} $

Consider the lines $ L_1: x - 1 = y - 2 = z $ and $ L_2: x - 2 = y = z - 1 $. Let the feet of the perpendiculars from the point $ P(5, 1, -3) $ on the lines $ L_1 $ and $ L_2 $ be $ Q $ and $ R $ respectively. If the area of the triangle $ PQR $ is $ A $, then $ 4A^2 $ is equal to:

Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be $ 2a $ and $ 2b $, respectively, and one focus and the corresponding directrix of this hyperbola be $ (-5, 0) $ and $ 5x + 9 = 0 $, respectively. If the product of the focal distances of a point $ (\alpha, 2\sqrt{5}) $ on the hyperbola is $ p $, then $ 4p $ is equal to:

If the range of the function $ f(x) = \frac{5 - x}{x^2 - 3x + 2}, \quad x \neq 1, 2 $ is $ (-\infty, \alpha] \cup [\beta, \infty) $, then $ \alpha^2 + \beta^2 $ is equal to:

The sum of the series $ 2 \times 1 \times 20C_4 - 3 \times 2 \times 20C_5 + 4 \times 3 \times 20C_6 - 5 \times 4 \times 20C_7 + ... + 18 \times 17 \times 20C_{20}, \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:

If the area of the region $ \{(x, y) : 1 + x^2 \leq y \leq \min(x + 7, 11 - 3x)\} $ is $ A $, then $ 3A $ is equal to:

Let $ f : \mathbb{R} \to \mathbb{R} $ be a polynomial function of degree four having extreme values at $ x = 4 $ and $ x = 5 $. If $ \lim_{x \to 0} \frac{f(x)}{x^2} = 5, \text{ then } f(2) \text{ is equal to:} $

If $ \int \left( \frac{1}{x} + \frac{1}{x^3} \right) \left( \sqrt[23]{3x^{-24}} + x^{-26} \right) \, dx $ is equal to $ -\frac{\alpha}{3(\alpha + 1)} \left( 3x^\beta + x^\gamma \right)^{\alpha + 1} + C, \quad x>0, $ where $ \alpha, \beta, \gamma \in \mathbb{Z} $ and $ C $ is the constant of integration, then $ \alpha + \beta + \gamma $ is equal to _______.

If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms 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 $ 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:} $