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List of top Mathematics Questions

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:

The value of the integral \[ \int_{-1}^{2} \log_e \left( x + \sqrt{x^2 + 1} \right) \, dx \] is:

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $ i $-th roll than the number obtained in the $ (i-1) $-th roll, $ i = 2, 3 $, is equal to:

Let $ a, ar, ar^2, \dots $ be an infinite G.P. If $$ \sum_{n=0}^\infty ar^n = 57 \quad \text{and} \quad \sum_{n=0}^\infty a^3 r^{3n} = 9747, $$ then $ a + 18r $ is equal to:

The integral \[ \int_{1/4}^{3/4} \cos\left( 2 \cot^{-1} \sqrt{\frac{1 - x}{1 + x}} \right) \, dx \] is equal to:

If $ \log_e y = 3 \sin^{-1}x $, then $ (1 - x)^2 y'' - xy' $ at $ x = \frac{1}{2} $ is equal to:

Let $$ B = \begin{bmatrix} 1 & 3 \\ 1 & 5 \end{bmatrix} $$ and $A$ be a $2 \times 2$ matrix such that $$ AB^{-1} = A^{-1}. $$ If $BCB^{-1} = A$ and $$ C^4 + \alpha C^2 + \beta I = O, $$ then $2\beta - \alpha$ is equal to:

The sum of the coefficients of $ x^{2/3} $ and $ x^{-2/5} $ in the binomial expansion of $$ \left( x^{2/3} + \frac{1}{2} x^{-2/5} \right)^9 $$ is:

\[ \lim_{x \to \frac{\pi}{2}} \frac{\int_{x^3}^{\left(\frac{\pi}{2}\right)^3} \left( \sin\left(2t^{1/3}\right) + \cos\left(t^{1/3}\right) \right) \, dt}{\left( x - \frac{\pi}{2} \right)^2} \] is equal to:

Between the following two statements: {Statement-I:} Let \[ \vec{a} = \hat{i} + 2\hat{j} - 3\hat{k} \quad \text{and} \quad \vec{b} = 2\hat{i} + \hat{j} - \hat{k}. \] Then the vector $ \vec{r} $ satisfying \[ \vec{a} \times \vec{r} = \vec{a} \times \vec{b} \quad \text{and} \quad \vec{a} \cdot \vec{r} = 0 \] is of magnitude $ \sqrt{10} $. {Statement-II:} In a triangle $ \triangle ABC $, \[ \cos 2A + \cos 2B + \cos 2C \geq -\frac{3}{2}. \]

Let the range of the function \[ f(x) = \frac{1}{2 + \sin 3x + \cos 3x}, \, x \in \mathbb{R} \, \text{be } [a, b]. \] If $ \alpha $ and $ \beta $ are respectively the arithmetic mean (A.M.) and the geometric mean (G.M.) of $ a $ and $ b $, then $ \frac{\alpha}{\beta} $ is equal to:

If the variance of the frequency distribution is 160, then the value of $ c \in \mathbb{N} $ is: \[ \begin{array}{|c|c|c|c|c|c|c|} \hline x & c & 2c & 3c & 4c & 5c & 6c \\ \hline f & 2 & 1 & 1 & 1 & 1 & 1 \\ \hline \end{array} \]

Two vertices of a triangle $ \triangle ABC $ are $ A(3, -1) $ and $ B(-2, 3) $, and its orthocentre is $ P(1, 1) $. If the coordinates of the point $ C $ are $ (\alpha, \beta) $ and the centre of the circle circumscribing the triangle $ \triangle PAB $ is $ (h, k) $, then the value of \[ (\alpha + \beta) + 2(h + k) \] equals:

Let the foci of a hyperbola $ H $ coincide with the foci of the ellipse $ E : \frac{(x - 1)^2}{100} + \frac{(y - 1)^2}{75} = 1 $ and the eccentricity of the hyperbola $ H $ be the reciprocal of the eccentricity of the ellipse $ E $. If the length of the transverse axis of $ H $ is $ \alpha $ and the length of its conjugate axis is $ \beta $, then $ 3\alpha^2 + 2\beta^2 $ is equal to:

The area (in square units) of the region enclosed by the ellipse $$ x^2 + 3y^2 = 18 $$ in the first quadrant below the line $ y = x $ is:

Let $ z $ be a complex number such that the real part of \[ \frac{z - 2i}{z + 2i} \] is zero. Then, the maximum value of $ |z - (6 + 8i)| $ is equal to:

Let \[ \int_{0}^{x} \sqrt{1 - (y'(t))^2} \, dt = \int_{0}^{x} y(t) \, dt, \quad 0 \leq x \leq 3, \, y \geq 0, \, y(0) = 0. \] Then, at $ x = 2 $, $ y'' + y + 1 $ is equal to:

$\lim_{x \to 0} \frac{e - (1 + 2x)^{\frac{1}{2x}}}{x} \quad \text{is equal to:}$

In the expansion of \[ (1 + x)(1 - x^2) \left( 1 + \frac{3}{x} + \frac{3}{x^2} + \frac{1}{x^3} \right)^5, \quad x \neq 0, \]the sum of the coefficients of $ x^3 $ and $ x^{-13} $ is equal to ____

Using integration, evaluate the area of the region bounded by the curve $ y = x^2 $, the lines $ y = 1 $ and $ y = 3 $, and the y-axis.

It is known that 20% of the students in a school have above 90% attendance and 80% of the students are irregular. Past year results show that 80% of students who have above 90% attendance and 20% of irregular students get “A” grade in their annual examination. At the end of a year, a student is chosen at random from the school and is found to have an “A” grade. What is the probability that the student is irregular?

Find the general solution of the differential equation $ y \, dx - x \, dy + (x \log x) \, dx = 0 $.

Given that $ f(x) = \frac{\log x}{x} $, find the point of local maximum of $ f(x) $.

The number of arbitrary constants in the general solution of the differential equation dy/dx + y = 0 is:

The area (in sq. units) of the region bounded by the curve $ y = x $, x-axis, $ x = 0 $ and $ x = 2 $ is: