AP EAPCET 2025 - 27th May Morning Shift

Circles are drawn through the point $(2,0)$ to cut intercepts of length 5 units on the $X$-axis. If their centre lie in the first quadrant, then their equation is

A circle touches both the coordinate axes and the straight line $L \equiv 4 x+3 y-6=0$ in the first quadrant. If this circle lies below the line $L=0$, then the equation of that circle is

If the lines $3 x-4 y+4=0$ and $6 x-8 y-7=0$ are the tangents to the same circle, then the area of that circle (in sq. units) is

The locus of the third vertex of a right-angled triangle, the ends of whose hypotenuse are $(1,2)$ and $(4,5)$ is

If $z$ and $w$ are two non-zero complex numbers such that $|z w|=1$ and $\arg z-\arg w=\frac{\pi}{2}$, then $\bar{z} w=$

Let $z$ satisfy $|z|=1, z=1-\bar{z}$ and $\operatorname{Im}(z)>0$
Statement $\mathbf{I} z$ is a real number
Statement II Principal argument of $z$ is $\frac{\pi}{3}$.
Then,

If $w_1$ and $w_2$ are two non-zero complex numbers and ${ }a, b$ are non-zero real numbers such that $\left|a w_1+b w_2\right|=\left|a w_1-b w_2\right|$, then $\frac{w_1}{w_2}$ is

If $\sinh ^{-1}(2)+\sinh ^{-1}(3)=\alpha$, then $\sinh \alpha=$

\(\int_0^1 \frac{2 x+5}{x^2+3 x+2} d x=\)

\(\int_0^1 x^{\frac{5}{2}}(1-x)^{\frac{3}{2}} d x=\)

$$ \lim _{n \rightarrow \infty}\left[\begin{array}{c} \frac{1}{n^2} \sec ^2 \frac{1}{n^2}+\frac{2}{n^2} \sec ^2 \frac{4}{n^2}+\frac{3}{n^2} \sec ^2 \\ \frac{9}{n^2}+\ldots+\frac{1}{n^2} \sec ^2 1 \end{array}\right]= $$

The general solution of the differential equation $\left(x \sin \frac{y}{x}\right) d y=\left(y \sin \frac{y}{x}-x\right) d x$ is

The general solution of the differential equation $\cos (x+y) d y=d x$ is

If $A x^3+B x y=4$ ( $A$ and $B$ are arbitrary constants) is the general solution of the differential equation $F(x) \frac{d^2 y}{d x^2}+G(x) \frac{d y}{d x}-2 y=0$, then $F(l)+G(l)=$

If $x=\sqrt{2^{\operatorname{cosec}^{-1} t}}$ and $y=\sqrt{2^{\sec ^{-1} t}},|t| \geq 1$, then $\frac{d y}{d x}=$

If $(a+\sqrt{2} b \cos x)(a-\sqrt{2} b \cos y) =a^2-b^2$, where $a>b>0$, then at $\left(\frac{\pi}{4}, \frac{\pi}{4}\right), \frac{d y}{d x}=$

Assertion (A) The length of the latus rectum of an ellipse is 4 . The focus and its corresponding directrix are respectively $(1,-2)$ and $3 x+4 y-15=0$. Then, its eccentricity is $\frac{1}{2}$.
Reason $(\mathrm{R})$ Length of the perpendic...

If $f: R \rightarrow A$, defined by $f(x)=\cos x+\sqrt{3} \sin x-1$ is an onto function then $A=$

Let $g(x)=1+x-[x]$ and ${ }^{\prime}$
$$ f(x)= \begin{cases}-1, & x<0 \\ 0, & x=0,[x] \text { denotes the greatest integer less } \\ 1, & x>0\end{cases} $$
than or equal to $x$. Then for all $x, f(g(x))=$

If the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passing through the point $(4,6)$ is 2 , then the equation of the tangent to this hyperbola at $(4,6)$ is

A hyperbola passes through the point $P(\sqrt{2}, \sqrt{3})$ and has foci at $( \pm 2,0)$. Then, the point that lies on the tangent drawn to this hyperbola at $P$ is

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

If $\int \frac{5 \tan x}{\tan x-2} d x=a x+b \log |\sin x-2 \cos x|+c$, then $a+b=$

\(\int x \cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right) d x(x>0)=\)

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

\(\int \sin ^{-1}\left(\sqrt{\frac{x}{a+x}}\right) d x=\)

If $\int \frac{x}{x \tan x+1} d x=\log f(x)+k$, then $f\left(\frac{\pi}{4}\right)=$

The equation $\cos ^{-1}(1-x)-2 \cos ^{-1} x=\frac{\pi}{2}$ has

\(\mathop {\lim }\limits_{x \to {\pi \over 4}} \frac{2 \sqrt{2}-(\cos x+\sin x)^3}{1-\sin 2 x}=\)

Let $[x]$ denote the greatest integer less than or equal to $x$. Then,
\(\lim _{x \rightarrow 2^{+}}\left(\frac{[x]^3}{3}-\left[\frac{x}{3}\right]^3\right)=\)

If the function $f$ defined by
$$ f(x)=\left\{\begin{array}{cc} \frac{1-\cos 4 x}{x^2}, & x<0 \\ a, & x=0 \\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & x>0 \end{array}\right. $$
is continuous at $x=0$, then $a=$

The domain of the derivative of the function $f(x)=\frac{x}{1+|x|}$ is

The equation $x^{\frac{3}{4}\left(\log _2 x\right)^2+\log _2 x-\frac{5}{4}}=\sqrt{2}$ has

A value of $\theta$ lying between 0 and $\pi / 2$ and satisfying $\left|\begin{array}{ccc}1+\sin ^2 \theta & \cos ^2 \theta & 4 \sin 4 \theta \\ \sin ^2 \theta & 1+\cos ^2 \theta & 4 \sin 4 \theta \\ \sin ^2 \theta & \cos ^2 \theta & 1+4 \s...

If the system of equations $2 x+p y+6 z=8$, $x+2 y+q z=5$ and $x+y+3 z=4$ has infinitely many solutions, then $p=$

If $x^a y^b=e^m, x^c y^d=e^n, \Delta_1=\left|\begin{array}{ll}m & b \\ n & d\end{array}\right|$, $\Delta_2=\left|\begin{array}{cc}a & m \\ c & n\end{array}\right|, \Delta_3=\left|\begin{array}{cc}a & b \\ c & d\end{array}\right|$, then the ...

A circle is drawn with its centre at the focus of the parabola $y^2=2 p x$ such that it touches the directrix of the parabola. Then, a point of intersection of the circle and the parabola is

If the locus of a point that divides a chord of slope 2 of the parabola $y^2=4 x$ internally in the ratio $1: 2$ is a parabola, then its vertex is

An eight digit number divisible by 9 is to be formed using digits from 0 to 9 without repeating the digits. The number of ways in which this can be done is

A string of letters is to be formed by using 4 letters from all the letters of the word "MATHEMATICS". The number of ways this can be done such that two letters are of same kind and the other two are of different kind is

Two students appeared simultaneously for an entrance exam. If the probability that the first student gets qualified in the exam is $\frac{1}{4}$ and the probability that the second student gets qualified in the same exam is $\frac{2}{5}$, t...

For three events $A, B$ and $C$ of a sample space, $P$ (exactly one of $A$ or $B$ occurs ) $=P$ (exactly one of $B$ or $C$ occurs) $=P($ exactly one of $C$ or $A$ occurs $)=\frac{1}{4}$. If probability of all the three events occurring simu...

On every evening, a student either watches TV or reads a book. The probability of watching TV is $\frac{4}{5}$ If he watches TV, the probability that he will fall asleep is $\frac{3}{4}$ and it is $\frac{1}{4}$ when he reads a book. If the ...

Let $X$ be the random variable taking values $1,2, \ldots n$ for a fixed positive integer $n$. If $P(X=k)=\frac{1}{n}$ for $1 \leq k \leq n$, then the variance of $X$ is

A radar system can detect an enemy plane in one out of ten consecutive scans.
The probability that it can detect an enemy plane atleast twice in four consecutive scans is

$A$ bag $P$ contains 4 red and 5 black balls another bag Q contains 3 red and 6 black balls. If one ball is drawn at random from bag $P$ and two balls are drawn from bag $Q$, then the probability that out of the three balls drawn two are bl...

In $\triangle A B C$, if $A, B, C$ are in arithmetic progression, then
\(\sqrt{a^2-a c+c^2} \cdot \cos \left(\frac{A-C}{2}\right)=\)

In a $\triangle A B C$, if $\sin ^2 B=\sin A$ and $2 \cos ^2 A=3 \cos ^2 B$, then the triangle is

If in $\triangle A B C, B=45^{\circ}, a=2(\sqrt{3}+1)$ and area of $\triangle A B C$ is $6+2 \sqrt{3}$ sq. units, then the side $b=$

If $\alpha$ is the common root of the quadratic equations $x^2-5 x+4 a=0, x^2-2 a x-8=0$, where $a \in R$, then the value $\alpha^4-\alpha^3+68$ is