List of top Mathematics Questions

If the differential equation of the family of curves given by \[ y=ae^x+b\cos x, \] where \(a\) and \(b\) are arbitrary constants, is \[ y_2(\cos x+\sin x)+y(\cos x-\sin x)=2y_1f(x), \] then \(f(x)=\)

The general solution of the differential equation \[ y(2x+y)\,dx=x(x+y)\,dy \] is

The general solution of the differential equation \[ (4xy^2-2xy+2x^2y^2-x^2y)\,dx=dy \] is

The area (in sq. units) of the region bounded by \[ x=0,\quad x=\frac{\pi}{2}, \] \[ y=0,\quad y=\cos x,\quad \text{and}\quad y=\tan x \] is

Evaluate \[ \int x^2(\log x)^2\,dx. \]

Evaluate \[ \lim_{n\to\infty}\frac{1}{n} \left[ \tan^2\frac{\pi}{4n} +\tan^2\frac{2\pi}{4n} +\cdots+ \tan^2\frac{n\pi}{4n} \right]. \]

Evaluate \[ \int \frac{dx}{(1+\sqrt{x})\sqrt{x-x^2}}. \]

Evaluate \[ \int_{0}^{\pi/2} \frac{x\sin 2x}{1+4\cos^2 2x}\,dx. \]

Evaluate \[ \int_{0}^{\pi/4} \frac{\cos x-\sin x}{9+5\sin 2x}\,dx. \]

If \[ \int \frac{\tan x}{1+\tan x+\tan^2x}\,dx = x-\frac{K}{\sqrt A} \tan^{-1} \left( \frac{K\tan x+1}{\sqrt A} \right) +C, \] then the ordered pair \((K,A)\) is

If \[ \int \frac{x^2-x+1}{x^2+1}\,e^{\cot^{-1}x}\,dx = A(x)e^{\cot^{-1}x}+C, \] then \(A(x)=\)

At any point on the curve \[ by^2=(x+a)^3, \] if the length of the sub-tangent (ST) and the length of the sub-normal (SN) are such that \[ p(\text{SN})=q(\text{ST})^2, \] then \[ \frac{p}{q}= \]

Evaluate \[ \int \frac{1}{(x^4+1)^{5/4}}\,dx. \]

If \[ u=\sqrt{a^2\cos^2\theta+b^2\sin^2\theta} + \sqrt{a^2\sin^2\theta+b^2\cos^2\theta}, \] then the difference between the maximum and minimum values of \(u^2\) is

If the function \[ f(x)=x^3+bx^2+ax \] satisfies the conditions of Rolle's theorem in \([1,3]\) with \[ c=2+\frac{1}{\sqrt3}, \] then \((a,b)=\)

If \[ \sqrt{x}+\sqrt{y}=\sqrt{a}, \] then \[ \left(\frac{d^2y}{dx^2}\right)_{x=a} = \]

The maximum volume (in cu. m) of a right circular cone having slant height \(3\) m is

If \[ \lim_{x\to\infty}\left(1+\frac{a}{x}+\frac{b}{x^2}\right)^{2x}=e^2, \] then \(a=\)

Evaluate \[ \lim_{x\to 0}\frac{x\cdot 2^x-x}{1-\cos x}. \]

If \(f\) is a differentiable function such that \[ f(1)=8 \] and \[ f'(1)=\frac18. \] If \(f\) is invertible and \(g=f^{-1}\), then

If \(f(x)\) is defined by \[ f(x)= \begin{cases} \dfrac{1-\tan x}{4x-\pi}, & x\ne \dfrac{\pi}{4},\; x\in\left[0,\dfrac{\pi}{2}\right] \\[6pt] k, & x=\dfrac{\pi}{4} \end{cases} \] and \(f(x)\) is continuous in \[ \left[0,\frac{\pi}{2}\right], \] then \(k=\)

If \[ f(x)=\cot^{-1}\!\left(\sqrt{\cos 2x}\right), \] then \[ f'\!\left(\frac{\pi}{6}\right)= \]

An angle between the plane \[ x+y+z=5 \] and the line \[ \frac{x-16}{0} = \frac{y-0}{-1} = \frac{z+47}{4} \] is

The direction ratios of a line \(L\) are \((ab,b,b)\) \((b\gt 0)\) and the line \(L\) passes through \[ P(b,b,b). \] If \(Q(x,y,z)\) is a point on the line at a distance of \(b\) units from \(P\), then \[ x+y+z= \]

Area of the quadrilateral formed by joining the extremities of the major axis and minor axis of the ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \] is \(8\sqrt3\). If the distance between its foci is \(4\sqrt2\), then the eccentricity of the ellipse is