List of top Mathematics Questions asked in AP EAPCET

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

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

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

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

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

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

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 \(2h(x)-f(x)-g(x)=\)

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.

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

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

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

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 \(k-2=\)

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

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

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 \(p^2+q^2=\)

If the directrix of the parabola \[ x^2+4y-6x+\lambda=0 \] is \(y+1=0\), then which of the following is correct?