List of top Mathematics Questions

The matrix \( A = \begin{bmatrix} a & -1 & 4 \\ -3 & 0 & 1 \\ -1 & 1 & 2 \end{bmatrix} \) is not invertible only if \( a = \)

Evaluate the integral \[ \int_{-5}^{5} \log \left( \frac{7 - x}{7 + x} \right) \, dx \]

The direction cosines of the line which bisects the angle between the positive direction of Y and Z axes are

Out of 100 people selected at random, 10 have common cold. If five persons are selected at random from the group, then the probability that at most one person will have common cold is

Evaluate \[ \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{\tan x}{\tan x + \cot x} \, dx \]

The equation of normal to the curve \( y = \sin \left( \frac{\pi x}{4} \right) \) at the point (2, 5) is

The rate at which the metal cools in moving air is proportional to the difference of temperatures between the metal and air. If the air temperature is 290K and the metal temperature drops from 370K to 330K in 10 minutes, then the time required to drop the temperature up to 295K is

For \( f(x) = [x] \), where \( [x] \) is the greatest integer function, which of the following is true, for every \( x \in \mathbb{R} \)?

Let \( G \) be the centroid of a triangle ABC and \( O \) be any other point in that plane, then \[ OA + OB + OC + OG = \]

\( \cos(36^\circ - A) \cos(36^\circ + A) + \cos(54^\circ + A) \cos(54^\circ - A) = \)

The points A \( (-a, -b) \), B \( (0, 0) \), C \( (a, b) \) and D \( (a^2, ab) \) are

The cosine of the angle included between the lines \[ \vec{r_1} = (2i + j - 2k) + \lambda (i - 2j - 2k) \quad \text{and} \quad \vec{r_2} = (i + j + 3k) + \mu (3i + 2j - 6k) \] where \( \lambda, \mu \in \mathbb{R} \) is

The maximum value of \( z = 3x + 5y \), subject to \( 3x + 2y \leq 18 \), \( x \leq 4 \), \( y \leq 6 \), \( x, y \geq 0 \) is

The micro-organisms double themselves in 3 hours. Assuming that the quantity increases at a rate proportional to itself, then the number of times it multiplies itself in 18 years is

The value of \[ \tan^{-1} \left( \frac{1}{3} \right) + \tan^{-1} \left( \frac{1}{5} \right) + \tan^{-1} \left( \frac{1}{7} \right) + \tan^{-1} \left( \frac{1}{8} \right) \] is

The area included between the parabolas \( y^2 = 5x \) and \( x^2 = 5y \) is

For any non-zero vectors \( \vec{a} \) and \( \vec{b} \), \( \left| \vec{b} \times (\vec{a} \times \vec{b}) \right| = \)

If \( \sin(x + y) + \cos(x + y) = \sin \left[ \cos^{-1} \left( \frac{1}{3} \right) \right] \), then \[ \frac{dy}{dx} = \]

Evaluate the integral \[ \int \frac{1 + 2e^{-x}}{1 - 2e^{-x}} \, dx \]

The integral \[ \int_0^{\frac{\pi}{2}} \log \left( \frac{\sqrt{1 - \cos 2x}}{\sqrt{1 + \cos 2x}} \right) dx \] is

The Cartesian coordinates of the point on the parabola \[ y^2 = x \quad \text{whose parameter is} \quad \frac{-4}{3} \] are

If \[ A = \{x, y, z\}, \quad B = \{1, 2\}, \text{ then the total number of relations from set } A \text{ to set } B \text{ are} \]

If \[ p \rightarrow \left( \sim p \vee q \right) \text{ is false, then the truth values of } p \text{ and } q \text{ are respectively} \]

The approximate value of the function \[ f(x) = x^3 - 3x + 5 \quad \text{at} \quad x = 1.99 \] is

The radius of the circle passing through the points \[ (5, 7), \quad (2, -2), \quad (-2, 0) \] is