List of top Mathematics Questions

If \(y=(\tan^{-1} x)^2\), show that \((x^2+1)^2 \frac{d^2y}{dx^2} + 2x(x^2+1)\frac{dy}{dx} = 2\).
Prove that \(\int_0^{\pi/2} \log(\cos x) \, dx = -\frac{\pi}{2} \log 2\).
If \(A = \begin{bmatrix} 2 & -3 & 5
3 & 2 & -4
1 & 1 & -2 \end{bmatrix}\), find \(A^{-1}\). Using \(A^{-1}\) solve the following system of equations:
2x - 3y + 5z = 11
3x + 2y - 4z = -5
x + y - 2z = -3.
Consider the vectors $$ \vec{x} = \hat{i} + 2\hat{j} + 3\hat{k},\quad \vec{y} = 2\hat{i} + 3\hat{j} + \hat{k},\quad \vec{z} = 3\hat{i} + \hat{j} + 2\hat{k}. $$ For two distinct positive real numbers $ \alpha $ and $ \beta $, define $$ \vec{X} = \alpha \vec{x} + \beta \vec{y} - \vec{z},\quad \vec{Y} = \alpha \vec{y} + \beta \vec{z} - \vec{x},\quad \vec{Z} = \alpha \vec{z} + \beta \vec{x} - \vec{y}. $$ If the vectors $ \vec{X}, \vec{Y}, \vec{Z} $ lie in a plane, then the value of $ \alpha + \beta - 3 $ is ________.
If the incentre of the triangle formed by lines $$ x - 2 = 0, \quad x + y - 1 = 0, \quad x - y + 3 = 0 $$ is $ (\alpha, \beta) $, then find $ \beta $.
If \( l, m \) represent any two elements (identical or different) of the set \( \{1, 2, 3, 4, 5, 6, 7\} \), then the probability that \( lx^2 + mx + 1>0 \,\, \forall x \in \mathbb{R} \) is
The value of the integral \[ \int_0^{\frac{\pi}{2}} \frac{1 + 2 \cos x}{(2 + \cos x)^2} \, dx \] lies in the interval:
The number of accidents per week in a town follows a Poisson distribution with mean 2. If the probability that there are three accidents in two weeks time is \( k e^{-6} \), then the value of \( k \) is:
If \( B = \sin^2y + \cos^4y \), then for all real \( y \),
Consider the sample space \( \Omega = \{(x, y): x, y \in \{1, 2, 3, 4\} \} \) where each outcome is equally likely. Let \( A = \{ x \geq 2 \} \) and \( B = \{ y > x \} \) be two events. Then which of the following is NOT true?
Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships as 30° and 45° respectively. If the lighthouse is 100 m high, the distance between the two ships is
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
Which one of the following is NOT a correct statement?
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?
Let \( A \) and \( B \) be two square matrices of the same order satisfying \( A^2 + 5A + 5I = 0 \) and \( B^2 + 3B + I = 0 \) respectively, where \( I \) is the identity matrix. Then the inverse of the matrix \( C = BA + 2B - 2A + 4I \) is:

Let \( A = (1, 2, 3, \dots, 20) \). Let \( R \subseteq A \times A \) such that \( R = \{(x, y) : y = 2x - 7 \} \). Then the number of elements in \( R \) is equal to:
 

Ten years ago, Ram was twice as old as Shyam. Ten years from now, the ratio of their ages will be 3:2. What is Ram's present age?
The slope of the normal line to the curve \( x = t^2 + 3t - 8 \) and \( y = t^2 - 2t - 5 \) at the point \( (2, -1) \) is
The circle \( x^2 + y^2 + ax + by + \gamma = 0 \) is the image of the circle \( x^2 + y^2 - 6x - 10y + 30 = 0 \) across the line \( 3x + y = 2 \). The value of \( [\alpha + \beta + \gamma] \) is (where \( [.] \) represents the floor function)
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?
If a number is increased by 30% and then decreased by 20%, what is the net change?
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:
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?

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

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: