List of top Mathematics Questions

If assumed mean of a data is 47.5, \( \sum f_i d_i = 435 \) and \( \sum f_i = 30 \), then mean of that data is:
Show that the function \( f(x) = 7x^2 - 3 \) is an increasing function when \( x>0 \).
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 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 value of the integral \[ \int_0^{\frac{\pi}{2}} \frac{1 + 2 \cos x}{(2 + \cos x)^2} \, dx \] lies in the interval:
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?
If \( B = \sin^2y + \cos^4y \), then for all real \( y \),
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:
Which one of the following is NOT a correct statement?
If a number is increased by 30% and then decreased by 20%, what is the net change?
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
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:
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)
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?
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

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:
 

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?
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?
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:

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