List of top Mathematics Questions

Suppose for a differentiable function $h$, $h(0) = 0$, $h(1) = 1$ and $h'(0) = h'(1) = 2$. If $g(x) = h(e^x) \, e^{h(x)}$, then $g'(0)$ is equal to:

Let $\vec{a} = 6\hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = \hat{i} + \hat{j}$. If $\vec{c}$ is a vector such that \[ |\vec{c}| \geq 6, \quad \vec{a} \cdot \vec{c} = 6 |\vec{c}|, \quad |\vec{c} - \vec{a}| = 2\sqrt{2} \] and the angle between $\vec{a} \times \vec{b}$ and $\vec{c}$ is $60^\circ$, then $|(\vec{a} \times \vec{b}) \times \vec{c}|$ is equal to:

If the function $f(x) = \left(\frac{1}{x}\right)^{2x}; \, x>0$ attains the maximum value at $x = \frac{1}{c}$, then:

A software company sets up m number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of m is equal to :

If $z_1, z_2$ are two distinct complex numbers such that \[ \frac{|z_1 - 2z_2|}{\left| \frac{1}{2} - z_1 \overline{z_2} \right|} = 2, \] then

If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is:

Let ABC be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle ABC and the same process is repeated infinitely many times. If P is the sum of perimeters and Q is be the sum of areas of all the triangles formed in this process, then:

A mixture contains apple juice and water in the ratio 10 : x. When 36 litres of the mixture and 9 litres of water are mixed, the ratio of apple juice and water becomes 5 : 4. The value of x is:

If $ a $ is in the 3rd quadrant, $ \beta $ is in the 2nd quadrant such that $ \tan \alpha = \frac{1}{7}, \sin \beta = \frac{1}{\sqrt{10}} $, then \[ \sin(2\alpha + \beta) = \]

If $ \mathbf{a}, \mathbf{b}, \mathbf{c} $ are the position vectors of the points $ A, B, C $ respectively and \[ 5\mathbf{a} - 3\mathbf{b} - 2\mathbf{c} = \mathbf{0}, \] then find the ratio in which the point $ C $ divides the line segment $ BA $ externally.

Let \[ \vec{a} = 2\hat{i} + \alpha \hat{j} + \hat{k}, \quad \vec{b} = -\hat{i} + \hat{k}, \quad \vec{c} = \beta \hat{j} - \hat{k}, \] where $ \alpha $ and $ \beta $ are integers and $ \alpha \beta = -6 $. Let the values of the ordered pair $ (\alpha, \beta) $ for which the area of the parallelogram of diagonals $ \vec{a} + \vec{b} $ and $ \vec{b} + \vec{c} $ is $ \frac{\sqrt{21}}{2} $, be $ (\alpha_1, \beta_1) $ and $ (\alpha_2, \beta_2) $. Then $ \alpha_1^2 + \beta_1^2 - \alpha_2 \beta_2 $ 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:

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

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:

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

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:

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:

If $\alpha$ satisfies the equation $x^2 + x + 1 = 0$ and $(1 + \alpha)^7 = A + B \alpha + C \alpha^2$, $A, B, C \geq 0$, then $5(3A - 2B - C)$ is equal to ______.

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

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:

Consider the line $L$ passing through the points $(1, 2, 3)$ and $(2, 3, 5)$. The distance of the point $$ \left( \frac{11}{3}, \frac{11}{3}, \frac{19}{3} \right) $$ from the line $L$ along the line $$ \frac{3x - 11}{2} = \frac{3y - 11}{1} = \frac{3z - 19}{2} $$ is equal to:

Let $ f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3) $, $ x \in \mathbb{R} $. Then $ f'(10) $ is equal to ______.