List of top Mathematics Questions

Let \( f: \mathbf{R} \rightarrow \mathbf{R} \) be a thrice differentiable odd function satisfying \( f'(x) \ge 0 \), \( f''(x)=f(x) \), \( f(0)=0 \), \( f'(0)=3 \). Then \( 9f(\ln 3) \) equals:

If \[ y(x)=\det\begin{pmatrix} \sin x & \cos x & \sin x+\cos x+1\\ 27 & 28 & 27\\ 1 & 1 & 1 \end{pmatrix} \] for \(x\in\mathbb{R}\), then \(\frac{d^2y}{dx^2}+y\) equals:

Let \(y=f(x)\) satisfy \[ \frac{dy}{dx}+\frac{xy}{x^2-1}=\frac{x^6+4x}{\sqrt{1-x^2}},\ -1<x<1 \] with \(f(0)=0\). If \[ 6\int_{-1/2}^{1/2} f(x)\,dx = 2\pi-\alpha, \] then \(\alpha^2\) equals:

A bag contains 5 red balls, 4 blue balls and 3 green balls. Two balls are drawn at random without replacement. The probability that both balls are red is:

Find the shortest distance between the skew lines \[ \vec r=(1,0,0)+\lambda(1,1,0) \] and \[ \vec r=(0,1,1)+\mu(0,1,1) \]

Find the equation of the tangent to \[ y=x^3-3x+1 \] at the point where \(x=2\).

If \[ \int \frac{2x+1}{x^2+x+1}\,dx \] is equal to:

The differential equation of the family of curves \[ y=ce^{2x} \] is:

If \[ \int_{0}^{1}(3x^2+2x+1)\,dx=k, \] then the value of \(k\) is:

If a random variable \(X\) has probability distribution \[ P(X=x)=kx, \] \[ x=1,2,3,4, \] then the value of \(k\) is:

The number of 5-digit numbers that can be formed using the digits \(1,2,3,4,5,6\) without repetition and divisible by \(5\) is:

If \[ z=\frac{1+i}{1-i}, \] then \(z^8\) equals:

Find the distance from the point \[ (1,1,1) \] to the plane \[ x+y+z=3. \]

Find the rate of change of the area of a circle with respect to its radius \(r\) when \[ r=3\text{ cm}. \]

Find \[ \frac{dy}{dx} \] if \[ y=\log(\sin x). \]

Find the value of \[ \cos\left(\sin^{-1}\left(\frac{1}{2}\right)\right). \]

Calculate \[ \int e^x\,dx. \]

If \[ P(A)=0.5, \qquad P(B)=0.3, \qquad P(A\cap B)=0.1, \] find \[ P(A|B). \]

Is \[ f(x)=x^2 \] continuous at \(x=0\)?

Find the angle between the vectors \[ \vec{a}=\hat{i}+\hat{j}-\hat{k} \] and \[ \vec{b}=\hat{i}-\hat{j}+\hat{k}. \]

Minimize \[ Z=2x+3y \] subject to \[ x+y\ge5, \qquad x,y\ge0. \]

Find the order and degree of the differential equation: \[ \sqrt{1+\left(\frac{dy}{dx}\right)^2}=\frac{d^2y}{dx^2} \]

The differential equation among the following, whose general solution is \( y=A e^{5x}+B e^{-4x} \) is

The substitution required to reduce the differential equation \( \frac{dy}{dx}+\sin y \cos y \sin x = \sin 2x \cos^{2}y \) to a linear differential equation in z is

If \( \int_{0}^{3} x\left(\sqrt[5]{9-x^{2}}\right) \, dx = k \cdot 3^{1/k} \), then \( k = \)