List of top Mathematics Questions

If \[ \int \frac{\sin\left(x-\frac{\pi}{4}\right)}{2+\sin 2x}\,dx = -\frac{1}{\sqrt2}\tan^{-1}(f(x))+C, \] then \(f(x)=\)

For \(n\in N\), if \[ I_n=\int \frac{\sin nx}{\sin x}\,dx = \frac{2}{n-1}\sin(n-1)x+I_{n-2} \] and \[ \int_0^\pi \frac{\sin nx}{\sin x}\,dx=\frac{k\pi}{2}, \] then \(k=\)

If \[ f(x)=\sin(\tan^{-1}x), \] then \[ \int_0^1 xf''(x)\,dx= \]

If \(a\) and \(b\) are respectively the order and degree of the differential equation \[ y^2(y'')^2+3x(y')^{1/3}+x^2y^2=\sin x, \] then

Evaluate \[ \int \frac{dx}{(1+\sqrt{x})^{2022}} \]

If an open cylinder of given surface area has maximum volume, then its radius is

If \[ x+y=k,\quad x\gt 0,\quad y\gt 0, \] then \(x^2+y^2\) is minimum, if

If \[ \int \frac{x^8+4}{x^4-2x^2+2}\,dx=Ax^5+Bx^3+Cx+K, \] then \(5A+3B+C=\)

Evaluate \[ \int \sqrt{x+\sqrt{x^2+2}}\,dx \]

Equation of tangent to the curve \[ y=x+\frac{4}{x^2} \] which is parallel to \(x\)-axis is

The normal to the curve \(y=f(x)\) at the point \((3,4)\) makes an angle \(\frac{3\pi}{4}\) with positive \(x\)-axis, then \(f'(3)=\)

The minimum distance of a point on the curve \[ y=x^2-4 \] from the origin is

Assertion (A): \[ f(x)=|x| \] is differentiable at \(x=a\neq 0\) and continuous but not differentiable at \(x=0\). Reason (R): If a function is differentiable at a point, then it is continuous at that point. But the converse is not true.

If \[ \frac{d}{dx}\left(\frac{x^2+1}{(x^2+5)(x^2+9)}\right) = \frac{2x(x^2+1)}{(x^2+5)(x^2+9)} \left[ \frac{1}{f(x)}-\frac{1}{g(x)}-\frac{1}{h(x)} \right], \] then \(2h(x)-f(x)-g(x)=\)

If \[ \lim_{x\to -a}\frac{x^7+a^7}{x+a}=7, \] then \(a=\)

Evaluate \[ \lim_{x\to \infty}x^3\left\{\sqrt{x^2+\sqrt{x^4+1}}-\sqrt2x\right\} \]

Let \[ f(x)= \begin{cases} 4, & -\infty\lt x\lt -\sqrt5 \\ x^2-1, & -\sqrt5\leq x\leq \sqrt5 \\ 4, & \sqrt5\lt x\lt \infty \end{cases} \] If \(k\) is the number of points where \(f(x)\) is not differentiable, then \(k-2=\)

Evaluate \[ \lim_{x\to \frac{\pi}{2}} \frac{1-\tan \frac{x}{2}}{1+\tan \frac{x}{2}} \cdot \frac{1-\sin x}{(\pi-2x)^3} \]

A ray makes angles \(\frac{\pi}{3}\) and \(\frac{\pi}{4}\) with \(Y\)-axis and \(Z\)-axis respectively. Then the value of the sine of the angle made by the ray with \(X\)-axis is

A plane meets the \(X, Y, Z\)-axes in \(A, B, C\) respectively. If the centroid of the triangle \(ABC\) is \((2,-3,5)\), then the perpendicular distance from origin to the given plane is

The foci of the ellipse \[ 9x^2+25y^2=225 \] are

If \((1,2)\) is the focus, \(x+2y=0\) is the directrix and \(\sqrt{2}\) is the eccentricity of a hyperbola, then the equation of the hyperbola is

The locus of the point of intersection of the lines \[ \sqrt3x-y-4\sqrt3k=0 \] and \[ \sqrt3kx+ky-4\sqrt3=0 \] for different real values of \(k\) is a hyperbola \(H\). If \(e\) is the eccentricity of \(H\), then \(4e^2=\)

If the centroid of a triangle with vertices \[ (4,p,-3),\quad (-1,-1,2),\quad (3,5,-8) \] is given by the midpoint of \[ (1,4,-2) \] and \[ (q,2,-4), \] then \(p^2+q^2=\)

Suppose \(A(2,3)\) and \(B\) are the points of intersections of two circles. The points \(P\) lying on one circle and \(Q\) lying on the other circle are such that \(BP\) and \(BQ\) constitute the diameters of the circles respectively. If the slopes of the radical axis and \(PQ\) are \(\frac{3}{4}\) and \(\frac{a}{b}\) respectively, then the value of \(3a+4b\) is