List of top Mathematics Questions

The general solution of the differential equation \[ (x+y-1)\,dy=(x-y+1)\,dx \] is

The general solution of the differential equation \[ \frac{dy}{dx}=\frac{2xy-3y^2}{2x^2+3xy} \] is

If \(a\) and \(b\) are arbitrary constants, then the differential equation corresponding to the family of curves \[ ax^2+2hxy=1 \] is

Evaluate: \[ \lim_{n\to\infty}\sum_{r=n}^{2n}\left(\frac{n^3+r^3}{n^4}\right) \]

If \[ \int (x+5)\sqrt{x-5}\,dx = \frac{2(x-5)^{5/2}}{15}f(x)+c, \] then \(f(6)=\)

Evaluate the integral: \[ \int \frac{x}{x^2-5x+4}\,dx \]

Evaluate: \[ \int \frac{e^{2x}-1}{e^{2x}+e^x+1}\,dx \]

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

If the vertical angle of a cone is \(60^\circ\) and the rate of change of its total surface area is \(2\sqrt{3}\,\text{cm}^2/\text{sec}\), then the rate of change of its volume (in \(\text{cm}^3/\text{sec}\)) when its radius is \(5\) cm is:

The surface area of a sphere is \(49\pi\) sq.cm. If it is increased by \(0.016\) sq.cm., then the approximate increase in its volume (in c.c.) is:

The cubic equation \[ 2x^3-3x^2+6x+2=0 \]

Evaluate: \[ \int \frac{dx}{(x^5+1)^{6/5}} \]

If \[ y=\sec^{-1}\left(\frac{1+x^2}{2x}\right) \quad \text{and} \quad x>1, \] then \[ \frac{dy}{dx}= \]

If the normal drawn to the curve \(y^4=16x^3\) at the point of intersection of this curve and the line \(y=2\) meets the \(X\)- and \(Y\)-axes at \(A\) and \(B\) respectively, then \(OA+3OB=\)

If \[ f(x)= \begin{cases} \dfrac{\sqrt{2+\cos x}-1}{(\pi-x)^2}, & x\neq \pi\\ k, & x=\pi \end{cases} \] is continuous at \(x=\pi\), then \(k=\)

The set of all the points at which \[ f(x)=|2-|x|| \] is continuous but not differentiable is:

If \[ f(x)=\sqrt{2^{2x}\log(3x-2)} \] then \(f'(2)=\)

Evaluate: \[ \lim_{x\to \frac{\pi}{3}} \frac{\tan^3x-3\tan x} {\cos\left(x+\frac{\pi}{6}\right)} \]

Evaluate: \[ \lim_{x\to e}\frac{\log x-1}{x-e} \]

If the angle \(\theta\) between the line \[ \frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2} \] and the plane \[ 2x-y+\sqrt{\lambda}z+4=0 \] is such that \(\sin\theta=\frac13\), then the value of \(\lambda\) is:

If \(A(1,1,1)\), \(B(2,3,4)\) and \(C(2,5,7)\) are the vertices of \(\triangle ABC\), then the length of the altitude drawn through the vertex \(A\) is:

A chord is drawn through the focus of the parabola \(y^2=6x\) such that its perpendicular distance from the vertex is \(\frac{\sqrt5}{2}\). Then its slope can be:

A perpendicular is drawn through the vertex \(O\) of the parabola \(y^2=8x\) to any non-vertical tangent meeting the parabola at \(P\). Then \(OP.OQ=\)

An ellipse intersects the hyperbola \(2x^2-2y^2=1\) orthogonally. The eccentricity of the ellipse is reciprocal to that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then the equation of the ellipse is:

If \((K,3,5)\), \((2,-1,2)\) are direction ratios of two lines and the angle between them is \(45^\circ\), then a value of \(K\) is: