List of top Mathematics Questions

A line $ L $ intersects the lines $ 3x - 2y - 1 = 0 $ and $ x + 2y + 1 = 0 $ at the points $ A $ and $ B $. If the point $ (1,2) $ bisects the line segment $ AB $ and $ \frac{a}{b} x + \frac{b}{a} y = 1 $ is the equation of the line $ L $, then $ a + 2b + 1 = ? $

Let $ \alpha, \beta; \, \alpha > \beta $, be the roots of the equation $$ x^2 - \sqrt{2}x - \sqrt{3} = 0. $$ Let $ P_n = \alpha^n - \beta^n, \, n \in \mathbb{N} $. Then $$ \left( 11\sqrt{3} - 10\sqrt{2} \right) P_{10} + \left( 11\sqrt{2} + 10 \right) P_{11} - 11P_{12} $$ is equal to:

Let $ A $ be a $ 3 \times 3 $ matrix of non-negative real elements such that \[A \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = 3 \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}.\]Then the maximum value of $ \det(A) $ is _____

Evaluate the integral $\int_0^1 \frac{a - bx^2}{(a + bx^2)^2} , dx$:

The eccentricity of the ellipse \[ p x^2 + 5y^2 = 80, \quad \text{where } p > 5, \] is $ \frac{\sqrt{3}}{2} $. Then the value of $ p $ is

Let the length of the focal chord $ PQ $ of the parabola $ y^2 = 12x $ be 15 units. If the distance of $ PQ $ from the origin is $ p $, then $ 10p^2 $ is equal to _____

Consider 10 observations $x_1, x_2, \ldots, x_{10}$ such that \[\sum_{i=1}^{10} (x_i - \alpha) = 2 \quad \text{and} \quad \sum_{i=1}^{10} (x_i - \beta)^2 = 40,\]where $\alpha, \beta$ are positive integers. Let the mean and the variance of the observations be $\frac{6}{5}$ and $\frac{84}{25}$, respectively. The value of $\frac{\beta}{\alpha}$ is equal to:

Let H: $\frac{-x^2}{a^2} + \frac{y^2}{b^2} = 1$ be the hyperbola, whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4\sqrt{3}$. Suppose the point $(\alpha, 6)$, $\alpha>0$ lies on H. If $\beta$ is the product of the focal distances of the point $(\alpha, 6)$, then $\alpha^2 + \beta$ is equal to:

Let $ S $ be the set of positive integral values of $ a $ for which \[ \frac{a x^2 + 2(a + 1)x + 9a + 4}{x^2 - 8x + 32} < 0, \quad \forall x \in \mathbb{R}. \] Then, the number of elements in $ S $ is:

Consider the function $ f : (0, \infty) \rightarrow \mathbb{R} $ defined by $f(x) = e^{-| \log_e x |}.$If $ m $ and $ n $ be respectively the number of points at which $ f $ is not continuous and $ f $ is not differentiable, then $ m + n $ is

Let the median and the mean deviation about the median of 7 observation 170, 125, 230, 190, 210, a, b be 170 and $\frac{205}{7}$respectively. Then the mean deviation about the mean of these 7 observations is :

An urn contains 15 red, 10 white, 60 orange balls, 15 green balls. 2 balls are taken with replacement. Find the probability 1 ball is red and the other ball is white.

If the domain of the function \[ f(x) = \cos^{-1} \left( \frac{2 - |x|}{4} \right) + \left( \log_e (3 - x) \right)^{-1} \] is $[-\alpha, \beta) - \{ \gamma \}$, then $ \alpha + \beta + \gamma $ is equal to:

Let $ f: \mathbb{R} \to \mathbb{R} $ be defined as: $f(x) = \begin{cases} \frac{a - b \cos 2x}{x^2}, & x < 0, \\ x^2 + cx + 2, & 0 \leq x \leq 1, \\ 2x + 1, & x > 1. \end{cases}$
If $ f $ is continuous everywhere in $ \mathbb{R} $ and $ m $ is the number of points where $ f $ is NOT differentiable, then $ m + a + b + c $ equals: