List of top Mathematics Questions asked in AP EAPCET

If \[ \int \frac{e^{\sqrt{x}}}{\sqrt{x}}(x+\sqrt{x})\,dx = e^{\sqrt{x}}\left[Ax+B\sqrt{x}+C\right]+K, \] then \(A+B+C=\)

If \[ \int \frac{1+\tan x}{\sin 2x}\,dx=A\log \tan x+B\tan x+C, \] then \(4A-2B=\)

Evaluate \[ \int \frac{1+\tan x\tan(x+a)}{\tan x\tan(x+a)}\,dx \]

If \[ I_n=\int_{0}^{\pi/4}\tan^n x\,dx, \] then \[ \frac{1}{I_2+I_4}+\frac{1}{I_3+I_5}+\frac{1}{I_4+I_6} \] is equal to

Evaluate \[ \int_{0}^{\pi/4} e^{\tan^2\theta}\sin^2\theta\tan\theta\,d\theta \]

Evaluate \[ \int_{\pi/4}^{5\pi/4}\left(|\cos t|\sin t+|\sin t|\cos t\right)\,dt \]

The number of those tangents to the curve \[ y^2-2x^3-4y+8=0 \] which pass through the point \((1,2)\) is

If the straight line \[ x\cos\alpha+y\sin\alpha=p \] touches the curve \[ \left(\frac{x}{a}\right)^n+\left(\frac{y}{b}\right)^n=2 \] at the point \((a,b)\) on it and \[ \frac{1}{a^2}+\frac{1}{b^2}=\frac{k}{p^2}, \] then \(k=\)

If \[ f(x)=\int x^2\cos^2x\left(2x\tan^2x-2x-6\tan x\right)\,dx \] and \(f(0)=\pi\), then \(f(x)=\)

Condition that 2 curves \[ y^2=4ax,\qquad xy=c^2 \] cut orthogonally is

A closed cylinder of given volume will have least surface area when the ratio of its height and base radius is

Two particles \(P\) and \(Q\) located at the points \[ P(t,t^3-16t-3),\qquad Q(t+1,t^3-6t-6) \] are moving in a plane. The minimum distance between the points in their motion is

If \[ f(x)= \begin{cases} \dfrac{x^2\log(\cos x)}{\log(1+x)}, & x\neq 0 \\ 0, & x=0 \end{cases} \] then at \(x=0\), \(f(x)\) is

If \(x\neq0\) and \(f(x)\) satisfies \[ 8f(x)+6f\left(\frac{1}{x}\right)=x+5, \] then \[ \frac{d}{dx}\left(x^2f(x)\right) \] at \(x=1\) is

If \(l,m\;(l\lt m)\) are roots of \(ax^2+bx+c=0\), then \[ \lim_{x\to \alpha}\frac{|ax^2+bx+c|}{ax^2+bx+c}= \]

Let \[ f(x)= \begin{cases} \dfrac{1}{|x|}, & \text{for } |x|\gt 1 \\[0.3cm] ax^2+b, & \text{for } |x|\leq1 \end{cases} \] If \[ \lim_{x\to1}f(x) \quad \text{and} \quad \lim_{x\to-1}f(x) \] exist, then the possible values of \(a\) and \(b\) are

If \((2,3,c)\) are the direction ratios of a ray passing through the point \(C(5,q,1)\) and also the midpoint of the line segment joining the points \(A(p,-4,2)\) and \(B(3,2,-4)\), then \(c(p+7q)=\)

If the equation of the plane which is at a distance of \(\dfrac{1}{3}\) units from the origin and perpendicular to a line whose directional ratios are \((1,2,2)\) is \[ x+py+qz+r=0 \] then \[ \sqrt{p^2+q^2+r^2}= \]

If the vertices and foci of a hyperbola are respectively \((\pm 3,0)\) and \((\pm 4,0)\), then the parametric equations of that hyperbola are

If \(x\)-coordinate of a point \(P\) on the line joining the points \(Q(2,2,1)\) and \(R(5,2,-2)\) is \(4\), then the \(y\)-coordinate of \(P\) is

If \([\,]\) denotes greatest integer function, then \[ \lim_{x\to -\frac{3}{5}}\frac{1}{x}\left[\frac{-1}{x}\right] \] is

The ratio of the largest and shortest distances from the point \((2,-7)\) to the circle \[ x^2+y^2-14x-10y-151=0 \] is

A circle has its centre in the first quadrant and passes through \((2,3)\). If this circle makes intercepts of length \(3\) and \(4\) respectively on \(x=2\) and \(y=3\), its equation is

The image of the point \((3,4)\) with respect to the radical axis of the circles \[ x^2+y^2+8x+2y+10=0 \] and \[ x^2+y^2+7x+3y+10=0 \] is

The focal distances of the point \(\left(\dfrac{4}{\sqrt{5}},\dfrac{3}{\sqrt{5}}\right)\) on the ellipse \(\dfrac{x^2}{4}+\dfrac{y^2}{9}=1\) are