AP EAPCET 2025 - 22nd May Morning Shift

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

If the probability distribution of a random variable $X$ is as follows, then $P(X \leq 2)=$
$$ \begin{array}{cccccc}\hline x_i & 0 & 1 & 2 & 3 & 4 \\ \hline P\left(X=x_i\right) & 3 k & 5 k & 3 k^2 & 4 k^2+k & 3 k^2 \\ \hline \end{array} $$

The probability that a person $A$ completes a work in a given time is $\frac{2}{3}$ and the probability that another person
$B$ completes the same work in the same time is $\frac{3}{4}$.
If both $A$ and $B$ start doing this work at the same...

If $l, m$ represent any two elements (identical or different) of the set $\{1,2,3,4,5,6,7\}$, then the probability that $l x^2+m x+1>0 \forall x \in R$ is

$A$ and $B$ are playing chess game with each other. The probability that $A$ wins the game is 0.6 . the probability that he loses is 0.3 and the probability its draw is 0.1 . If they played three games, then the probability that $A$ wins at...

$U_1, U_2, U_3$ are three urns. $U_1$ contains 5 red, 3 white, 2 back balls: $U_2$ contains 4 red 4 white, 2 black balls and $U_3$ contains 3 red. 4 white, 3 black balls. If a ball is chosen at random from an urn chosen at random, then the ...

If $X$ follows poisson distribution with variance 2 , then $P(X \geq 3)=$

In $\triangle A B C, \sqrt{\frac{r \cdot r_2}{r_3 r_1}}=$

In a $\triangle A B C, A-B=120^{\circ}, R=8 r$, then $\frac{1+\cos C}{1-\cos C}=$

If $A(0,0,0) B(3,4,0)$ and $C(0,12,5)$ are the vertices of a $\triangle A B C$, then the $x$-coordinate of its incentre is

The equation $6 x^4-5 x^3+13 x^2-5 x+6=0$ will have