AP EAPCET 2025 - 22nd May Morning Shift

If the tangent drawn at the point $(\alpha, \beta)$ on the curve $x^{\frac{2}{3}}+y^{\frac{2}{3}}=4$ is parallel to the line $\sqrt{3 x}+y=1$, then $\alpha^2+\beta^2=$

The displacement $S$ of a particle measured from a fixed point $O$ on a line is given by $S=t^3-16 t^2+64 t-16$. Then, the time at which displacement of the particle is maximum is

If the extreme value of the function $f(x)=\frac{4}{\sin x}+\frac{1}{1-\sin x}$ in $\left[0, \frac{\pi}{2}\right]$ is $m$ and it exists at $x=k$, then $\cos k=$

If $y=|\cos x-\sin x|+|\tan x-\cot x|$, then

\(\left(\frac{d y}{d x}\right)_{x=\frac{\pi}{3}}+\left(\frac{d y}{d x}\right)_{x=\frac{\pi}{6}}=\)

The area of the region lying between the curves $y=\sqrt{4-x^2}, y^2=3 x$ and the $Y$-axis is

The number of all possible values of $k$ for which the expansion $(\sqrt{x}+\sqrt[k]{y})^{10}$ will have exactly nine irrational terms is

If $k$ is a positive integer and $10^k$ is a divisor of the number $9^{11}+11^9$, then the greatest value of $k$ is

If the circle passing through $(3,5),(5,5)$ and $(3,-3)$ cuts the circle $x^2+y^2+2 x+2 f y=0$ orthogonally, then $f=$

A circle $S \equiv x^2+y^2-16=0$ intersects another circle $S^{\prime}=0$ of radius 5 units such that their common chord is of maximum length. If the slope of that chord is $\frac{3}{4}$, then the centre of such a circle $S^{\prime}=0$ is

Let $\theta$ be the angle between the circles $S \equiv x^2+y^2+2 x-2 y+c=0$ and $S^{\prime} \equiv x^2+y^2-6 x-8 y+9=0$. If $c$ is an integer and $\cos \theta=\frac{5}{16}$, then the radius of the circle $S=0$ is

If $Q$ is the inverse point of $P(-1,1)$ with respect to the circle $x^2+y^2-2 x+2 y=0$, then the line containing $Q$ is

Length of the common chord of two circles of same radius is $2 \sqrt{17}$. If one of the two circles is $x^2+y^2+6 x+4 y-12=0$, then acute angle between the two circles is

If $z=x+i y$ and if the point $P$ in the argand diagram represents $z$, then the locus of the point $P$ satisfying the equation $2|z-2-3 i|=3|z+i-2|$ is a circle with centre

If $z$ is a non-real root of $x^7=1$, then $1+3 z+5 z^2+7 z^3+9 z^4+11 z^5+13 z^6=$

The minimum value of $|z-1|+|z-5|$ is

If $\cosh 2 x=199$, then $\cot h x=$

\(\mathop {\lim }\limits_{n \to \infty }\left(\frac{1}{1^2+n^2}+\frac{2}{2^2+n^2}+\frac{3}{3^2+n^2}+\ldots+\frac{n}{n^2+n^2}\right)=\)

$\int_{\frac{-\pi}{4}}^{\frac{\pi}{3}}\left|\tan \left(x-\frac{\pi}{6}\right)\right| d x=$

\(\int_0^\pi \frac{x \sin x}{\sin ^2 x+2 \cos ^2 x} d x=\)

The general solution of the differential equation $\frac{d y}{d x}=\frac{2 x^2-x y-y^2}{x^2-y^2}$ is

The general solution of the differential equation $y+\cos x\left(\frac{d y}{d x}\right)-\cos ^2 x=0$ is

If the degree of the differential equation corresponding to the family of curves $y=a x+\frac{1}{a}$ (where $a \neq 0$ is an arbitary constant) is $r$ and it's order is $m$. Then, the solution of $\frac{d y}{d x}=\frac{y}{2 x}, y(\mathrm{l}...

Assertion (A) If $y=f(x)=(|x|-|x-1|)^2$, then $\left(\frac{d y}{d x}\right)_{x=1}=1$
Reason (R) $\mathop {\lim }\limits_{x \to a} \frac{f(x)-f(a)}{x-a}$ exist, then it is called derivative of $f(x)$ at $x=a$.

If $x=2 \cos ^3 \theta$ and $y=3 \sin ^2 \theta$, then $\frac{d y}{d x}=$

The angle between the tangents drawn from a point $(-3,2)$ to the ellipse $4 x^2+9 y^2-36=0$ is

Let $f: N \rightarrow N$ be a function such that $f(x+y)=f(x)+f(y)+x y$ for every $x, y \in N$. If $f(\mathbb{l})=2$, then $\sum_{k=0}^{10} f(k)=$

If a real valued function $f:[-1,2] \rightarrow B$ defined by
$$ f(x)= \begin{cases}1-x, & \text { when }-1 \leq x \leq 1 \\ x-1, & \text { when } 1 < x \leq 2\end{cases} $$
is a surjection, then $B=$

The sum of the least positive integer and the greatest negative integer in the range of the function $f(x)=\frac{x^2-5 x+7}{x^2-5 x-7}$ is

The interval in which the curve represented by $f(x)=2 x+\log \left(\frac{x}{2+x}\right)$ is

By rotating the axes about the origin in anti-clockwise direction with certain angle, if the equation $x^2+4 x y+y^2=1$ is transformed to $\frac{x^2}{a^2}-\frac{y^2}{b^2}=l$, then $\sqrt{\frac{a^2+b^2}{a^2}}=$

If a tangent to the hyperbola $x y=-1$ is also a tangent to the parabola $y^2=8 x$, then the equation of that tangent is

The distance between the tangents of the hyperbola $2 x^2-3 y^2=6$ which are perpendicular to the line $x-2 y+5=0$ is

If $\frac{x+1}{(x-1)^2\left(x^2+1\right)}=\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C x+D}{x^2+1}$, then $\sqrt{3 A^2+4 D^2+5 C^2+B^2}=$

If $\int \frac{1}{\cot \frac{x}{2} \cot \frac{x}{3} \cot \frac{x}{6}} d x=A \log \left|\cos \frac{x}{2}\right| +B \log \left|\cos \frac{x}{3}\right|+C \log \left|\cos \frac{x}{6}\right|+k$, then $A+B+C=$

\(\int \frac{1}{9 \cos ^2 x-24 \sin x \cos x+16 \sin ^2 x} d x=\)

\(\int \frac{\sin x+\cos x}{\sin x-\cos x} d x=\)

\(\int \frac{x^4-1}{x^2 \sqrt{x^4+x^2+1}} d x=\)

\(\int \frac{(3 x-2) \tan \left(\sqrt{9 x^2-12 x+1}\right)}{\sqrt{9 x^2-12 x+1}} d x=\)

The range of the real valued function $f(x)=\cos ^{-1}(-x)+\sin ^{-1}(-x)+\operatorname{cosec}^{-1}(x)$ is

The horizontal distance between a tower and a building is $10 \sqrt{3}$ units. If the angle of depression of the foot of the building from the top of the tower is $60^{\circ}$ and the angle of elevation of the top of the building from the f...

If $\mathop {\lim }\limits_{x \to 0} \frac{\cos 2 x-\cos 4 x}{1-\cos 2 x}=k$, then $\lim\limits_{x \rightarrow k} \frac{x^k-27}{x^{k+1}-81}=$

If the function $f(x)=\left\{\begin{array}{l}1+\cos x, x \leq 0 \\ a-x, 02\end{array}\right.$ everywhere, then $a^2+b^2=$

\(\mathop {\lim }\limits_{y \to 0} \frac{\sqrt{1+\sqrt{1+y^4}}-\sqrt{2}}{y^4}=\)

Consider the matrices $A=\left[\begin{array}{ccc}x & y & 0 \\ -3 & 1 & 2 \\ 1 & -2 & z\end{array}\right]$ and $B=\left[\begin{array}{ccc}1 & -2 & -2 \\ 2 & 0 & 1 \\ 2 & 1 & 0\end{array}\right]$
If the cofactors of the elements $z, 1$ in 3rd...

The value of $p$ and $q$ is that system of equations $2 x+p y+6 z=8, x+2 y+q z=5$ and $x+y+3 z=4$ may have no solution are

$A$ is the set of all matrices of order 3 with entries 0 or 1 only. $B$ is the subset of $A$ consisting of all matrices with determinant value 1 . If $C$ is the subset of $A$ consisting of all matrices with determinant value -1 , then

$P Q$ is a focal chord of the parabola $y^2=4 x$ with focus $S$. If $P=(4,4)$, then $S Q=$

The number of divisors of 7 ! is

All the letters of the word LETTER are arranged in all possible ways and the words (with or without meaning) thus formed are arranged in dictionary order.
Then, the rank of the word TETLER is

5-digit numbers are formed by using the digits $0,1,2$, $3,5,7$ without repetetion and all of them are arranged in ascending order. Then, the rank of the number 70513 is