List of top Mathematics Questions

Let $ e_1 $ and $ e_2 $ be the eccentricities of the ellipse $ \frac{x^2}{b^2} + \frac{y^2}{25} = 1 $ and the hyperbola $ \frac{x^2}{16} - \frac{y^2}{b^2} = 1, $ respectively. If $ b<5 $ and $ e_1 e_2 = 1 $, then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is:
The number of real roots of the equation $ x|x-2| + 3|x-3| + 1 = 0 $ is:

Let $ f(x) = \begin{cases} (1+ax)^{1/x} & , x<0 \\1+b & , x = 0 \\\frac{(x+4)^{1/2} - 2}{(x+c)^{1/3} - 2} & , x>0 \end{cases} $ be continuous at x = 0. Then $ e^a bc $ 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

The sum of all rational numbers in \( (2 + \sqrt{3})^8 \) is:
The number of singular matrices of order 2, whose elements are from the set $ \{2, 3, 6, 9\} $ is:
Let \( \mathbf{a} = 3\hat{i} - \hat{j} + 2\hat{k} \), \( \mathbf{b} = \mathbf{a} \times (\hat{i} - 2\hat{k}) \) and \( \mathbf{c} = \mathbf{b} \times \hat{k} \). Then the projection of \( \mathbf{c} - 2\hat{j} \) on \( \mathbf{a} \) is:
Consider an A.P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is:
If \( \alpha \) and \( \beta \) are the roots of the equation \( 2z^2 - 3z - 2i = 0 \), where \( i = \sqrt{-1} \), then \[ 16 \cdot {Re} \left( \frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}} \right) \cdot {Im} \left( \frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}} \right) \] is equal to:
The area (in sq. units) of the region \((x, y) : 0 \leq y \leq 2|x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3\) is:
Let \( M \) and \( m \) respectively be the maximum and the minimum values of \[ f(x) = \frac{1 + \sin^2 x}{\cos^2 x} + \frac{4 \sin 4x}{\sin^2 x \cos^2 x} \quad {for} \quad x \in \mathbb{R} \] Then \( M^4 - m^4 \) is equal to:
Let the set of all values of $ p \in \mathbb{R} $, for which both the roots of the equation $ x^2 - (p + 2)x + (2p + 9) = 0 $ are negative real numbers, be the interval $ (\alpha, \beta) $. Then $ \beta - 2\alpha $ is equal to:
Let \( \mathbf{a} \) and \( \mathbf{b} \) be two unit vectors such that the angle between them is \( \frac{\pi}{3} \). If \( \lambda \mathbf{a} + 2 \mathbf{b} \) and \( 3 \mathbf{a} - \lambda \mathbf{b} \) are perpendicular to each other, then the number of values of \( \lambda \) in \( [-1, 3] \) is:
The remainder, when \(7^{98}\) is divided by 23, is equal to:
Find the value of the integral \( \int_0^{\frac{\pi}{2}} \sin^2(x) \, dx \).
The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is _____.

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

Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on \(3x + 2y + 2 = 0\). Then the length of the chord of the circle C, whose midpoint is (1, 2), is:
The sum of all values of \( \theta \in [0, 2\pi] \) satisfying \( 2\sin^2\theta = \cos 2\theta \) and \( 2\cos^2\theta = 3\sin\theta \) is:
Let the sum of the focal distances of the point $ P(4, 3) $ on the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be $ 8\sqrt{\frac{5}{3}} $. If for $ H $, the length of the latus rectum is $ \ell $ and the product of the focal distances of the point $ P $ is $ m $, then $ 9\ell^2 + 6m $ is equal to:
The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is:
For $ t>-1 $, let $ \alpha_t $ and $ \beta_t $ be the roots of the equation $ \left( (t + 2)^{\frac{1}{7}} - 1 \right)x^2 + \left( (t + 2)^{\frac{1}{6}} - 1 \right)x + \left( (t + 2)^{\frac{1}{21}} - 1 \right) = 0. $ If $ \lim_{t \to 1^+} \alpha_t = a $ and $ \lim_{t \to 1^+} \beta_t = b $, then $ 72(a + b)^2 $ is equal to:
In an arithmetic progression, if \( S_{40} = 1030 \) and \( S_{12} = 57 \), then \( S_{30} - S_{10} \) is equal to:

Let \( T_r \) be the \( r^{\text{th}} \) term of an A.P. If for some \( m \), \( T_m = \dfrac{1}{25} \), \( T_{25} = \dfrac{1}{20} \), and \( \displaystyle\sum_{r=1}^{25} T_r = 13 \), then \( 5m \displaystyle\sum_{r=m}^{2m} T_r \) is equal to:

Let \( [\cdot] \) denote the greatest integer function. If \[ \int_0^3 \left\lfloor \frac{1}{e^x - 1} \right\rfloor \, dx = \alpha - \log_e 2, \] then \( \alpha^3 \) is equal to: