List of top Mathematics Questions

If variance of \(X\) is \(4\), then the variance of \(Y=4X+2\) is

Correlation coefficient is independent of

A random variable \(X\) has the probability density function \[ f(x)=Kx,\qquad 0<x<2, \] then the ratio between \(K\) and the mean of \(X\) is

If \[ A= \begin{bmatrix} 2026 & 2025 & 2024 2025 & 2024 & 2023 2024 & 2023 & 2022 \end{bmatrix}, \] then \(\left|A^{2026}-A^{2025}\right|=\)

If \({}^nC_r={}^{\,n}C_{r-1}\) and \({}^{\,n}P_{r-1}=9\,{}^{\,r}P_r\), then \((n,r)=\)

If \(\alpha,\beta\) are the roots of \(ax^2+bx+c=0\), then \[ \left(\frac{\alpha}{\beta}-\frac{\beta}{\alpha}\right)^2 = \]

If the sum of the three numbers which are in A.P. is \(27\) and the product of the first and last is \(77\), then the numbers are

If the general solution of the differential equation \[ \frac{dy}{dx} + \frac{2x+2y-1}{x+y-5} = 0 \] is \[ ax+by-9\log(x+y+p)=c, \] and \(b,a,p\) are in G.P., then the common ratio of this G.P. is

If \[ f(x)=\frac{x^3}{1-3x+3x^2}, \] then \[ \lim_{n\to\infty} \frac{2}{n} \sum_{r=1}^{n} f\!\left(\frac{r}{n}\right) = \]

The integrating factor of the linear differential equation in \(x\) given by \[ \frac{dy}{dx}=\frac{1}{3x+y+2} \] is

The area of the region bounded by the curves \[ y=x^2+1,\qquad x=y^2+1, \] \(X\)-axis, \(Y\)-axis and \(x=2\) (in sq. units) is

Evaluate \[ \int \frac{x^7-1} {x^8\sqrt{\,1-2x^7+7x^{14}\,}} \,dx. \]

Evaluate \[ \int \frac{\sin^2x\tan x} {\cos^6x+\sin^4x\cos^2x+\cos^4x\sin^2x+\sin^6x}\,dx. \]

Let \(f:\mathbb{R}\rightarrow\mathbb{R}\) be an odd function and \[ \int_{-1}^{1}x^3f''(x)\,dx=\frac85. \] If \[ g(x)=xf(x), \] \[ g'(1)=\frac83 \] and \[ \int_0^1g(x)\,dx=\frac{2}{15}, \] then \(f(1)=\)

If the line \[ 2x+y=7 \] touches the curve \[ y=f(x) \] at \[ x=\frac12, \] then the sum of the lengths of the normal and subnormal is

If \(P\) is any point on the curve \[ y^2=4ax, \] other than the origin, then the length of the subtangent at \(P\), \(y\)-coordinate of \(P\) and the length of the subnormal at \(P\) are in

If \[ \int \frac{e^{\cos x}\sin x}{e^{\cos x}+e^{-\cos x}}\,dx = -\frac12\left[f(x)+\log\!\left(e^{\cos x}+e^{-\cos x}\right)\right]+c \] and \[ f\!\left(\frac{\pi}{3}\right)=\frac12, \] then the number of points at which \(f(x)\) attains the maximum value in \[ [-2\pi,2\pi] \] is

If \[ \int \frac{\cos\left(\frac{7x}{2}\right)}{\cos\left(\frac{x}{2}\right)}\,dx = a\sin x+b\sin2x+c\sin^3x-dx+k, \] then \(a+b+c+d=\)

If the local maximum value of the real valued function \[ f(x)=2(x^3-1)-3ax(x+4a),\qquad a>0 \] is \(54\), then \(a=\)

\(f(x)\) is a cubic polynomial and the curve represented by \(y=f(x)\) passes through the origin. The function \(y=f(x)\) is an increasing function in \[ (-\infty,0)\cup(1,\infty) \] and a decreasing function in \[ (0,1). \] If \[ f'(2)=6, \] then the \(c\) of Lagrange's Mean Value Theorem on the interval \([0,2]\) satisfies the equation

If \[ y=\frac{x^4+x^2+1}{x^2+x+1}, \] \[ \frac{dy}{dx}=y_1 \] and \[ \frac{d^2y}{dx^2}=y_2, \] then the value of \[ \left| \frac{(1+y_1^2)^{3/2}}{y_2} \right| \] at \[ x=\left(\frac12+\sqrt2\right) \] is

If \(f(x)\) is a real valued bijective function and twice differentiable function. If \(g(x)\) is inverse of \(f(x)\) and \(f(0)=a\), then \(g''(a)=\)

The edges of a cuboid are measured with a scale having an error of \(0.01\) cm per cm. If the dimensions of the cuboid are \[ (l,b,h)=(12,5,7), \] then the percentage error in the volume of the cuboid is

If a function \[ f:[1,7]\rightarrow\mathbb{R} \] is defined as \[ f(x)= \begin{cases} 2x^2-1, & x\le\sqrt7,[2mm] \sqrt{7x}+6, & \sqrt7[2mm] \sin\pi x, & 5\le x\le6,[2mm] x-[x], & 6<x\le7, \end{cases} \] then the number of points of discontinuity in \([1,7]\) is

Evaluate \[ \frac{d}{dx}\left(\sin^{-1}(\tan x)+\tan^{-1}(\sin x)\right). \]