List of top Mathematics Questions

If \( 8^{x-1} = \left( \frac{1}{4} \right)^x \), then the value of \[ \frac{1}{\log_{x+14} x - \log_{x+15}} \div \frac{1}{\log_{1-x} 4 - \log_{1-x} 5} \] is
Let \( g: \mathbb{R} \to \mathbb{R} \) and \( h: \mathbb{R} \to \mathbb{R} \) be two functions such that \( h(x) = s \cdot g \cdot n(g(x)) \), where \( \mathbb{R} \) denotes the set of all real numbers and \( s \) and \( n \) stand for the signum and the sine functions respectively. Then, select which of the following is not true?
A sum of money was divided into 2 parts in the ratio 2:5. The first part was invested for 2 years at the annual interest rate of 20% compounded annually. At what rate of simple interest per annum the second part must be invested for 2 years so that the interest earned in both cases is the same?
Let \( \mathbf{a} = e\hat{i} + 3\hat{j} + 4\hat{k}, \mathbf{b} = \hat{i} + 2\hat{j} - \hat{k}, \mathbf{c} = 3\hat{i} + \hat{j} + \lambda \hat{k} \) be the co-terminal edges of a parallelepiped whose volume is 5 units. Then the value of \( \lambda \) is:
In a box, there are 8 red, 7 blue, and 6 green balls. One ball is picked up randomly. What is the probability that it is neither red nor green?

If \( \cos^2(10^\circ) \cos(20^\circ) \cos(40^\circ) \cos(50^\circ) \cos(70^\circ) = \alpha + \frac{\sqrt{3}}{16} \cos(10^\circ) \), then \( 3\alpha^{-1} \) is equal to:
 

Let $A = \{5n - 4n - 1 : n \in \mathbb{N}\}$ and $B = \{16(n - 1): n \in \mathbb{N}\}$ be sets. Then:
 

Consider the matrix \( B = \begin{pmatrix} 0 & 1 & -1 \\ 1 & 0 & -1 \\ 0 & 0 & 1 \end{pmatrix} \). The sum of all the entries of the matrix \( B^{19} \) is
An equilateral triangle is inscribed in the parabola \( y^2 = x \). One vertex of the triangle is at the vertex of the parabola. The centroid of the triangle is:

Let $\vec{a}$ and $\vec{c}$ be unit vectors such that the angle between them is $\cos^{-1} \left( \frac{1}{4} \right)$. If $\vec{b} = 2\vec{c} + \lambda \vec{a}$. Where $\lambda > 0$ and $|\vec{b}| = 4$, then $\lambda$ is equal to: 
 

The maximum value of $\sin(x) + \sin(x + 1)$ is $k \cos^{\frac{1}{2}}$ Then the value of $k$ is:
 

\( \text{A tower subtends angles a, 2a, and 3a respectively at points A, B, and C, which are lying on a horizontal line through the foot of the tower. Then }\) \( \frac{AB}{BC} \) \(\text{ is equal to:}\)

Given the equations \( x + y = 1 \), \( x^2 + y^2 = 2 \), \( x^5 + y^5 = A \). Let N be the number of solution pairs \( (x, y) \) to this system of equations. Then \( N \) is equal to:
A tower subtends an angle of \( 30^\circ \) at a point on the same level as the foot of the tower. At a second point \( h \) meters above the first, the depression of the font of the tower is \( 60^\circ \). What is the horizontal distance of the tower from the point?

If \( x, y, z \) \(\text{ are the three cube roots of 27, then the determinant of the matrix}\) \[ \begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix} \] \(\text{is:}\)

The number of 3-digit integers that are multiple of 6 which can be formed by using the digits 1, 2, 3, 4, 5, 6 without repetition is:
If \( \alpha \) and \( \beta \) are the two roots of the equation \( x^2 + ax + b = 0 \), \( ab \neq 0 \), then the quadratic equation with roots \( \frac{1}{\alpha^3 + \alpha} \) and \( \frac{1}{\beta^3 + \beta} \) is

Let \( f: \mathbb{R} \to \mathbb{R} \) \(\text{ be any function defined as }\) \[ f(x) = \begin{cases} x^\alpha \sin \left( \frac{1}{x^\beta} \right) & \text{for } x \neq 0, \\ 0 & \text{for } x = 0, \end{cases} \] where \( \alpha, \beta \in \mathbb{R} \). Which of the following is true? \( \mathbb{R} \) denotes the set of all real numbers.

There are two coins, say blue and red. For the blue coin, the probability of getting heads is 0.99 and for the red coin, it is 0.01. One coin is chosen randomly and is tossed. The probability of getting heads is:

What is the general solution of the equation \( \cot\theta + \tan\theta = 2 \)? 

A two-digit number is such that the product of its digits is 12. If 36 is added to the number, the digits interchange their places. What is the number?

Let the line $\frac{x}{4} + \frac{y}{2} = 1$ meet the x-axis and y-axis at A and B, respectively. M is the midpoint of side AB, and M' is the image of the point M across the line $x + y = 1$. Let the point P lie on the line $x + y = 1$ such that $\Delta ABP$ is an isosceles triangle with $AP = BP$. Then the distance between M' and P is:
 

The obtuse angle between lines \(2y = x + 1\) and \(y = 3x + 2\) is:

The value of \( \frac{d}{dx} \left( \int 2 \sin x \sin x^2 e^{t^2} \right) \) at \( t = \pi \) is:
The percentage change in area of a rectangle by decreasing its length and increasing its breadth by 5% is: