AP EAPCET 2025 - 24th May Morning Shift

$A(a, 0)$ is a fixed point and $\theta$ is a parameter such that $0<\theta<2 \pi$. If $P(a \cos \theta, a \sin \theta)$ is a point on the circle $x^2+y^2=a^2$ and $Q(b \sin \theta,-b \cos \theta)$ is a point on the circle $x^2+y^2=b^2$, the...

$3 x+4 y-43=0$ is a tangent to the circle $S \equiv x^2+y^2-6 x+8 y+k=0$ at a point $P$. If $C$ is the centre of the circle and $Q$ is a point which divides $C P$ in the ratio $-1: 2$, then the power of the point $Q$ with respect to the cir...

If the point $P$ denotes the complex number $z=x+i y$ in the argand plane and $\frac{z-(2-i)}{z+(1+2 i)}$ is purely imaginary number, then the locus of $P$ is

If $(\sqrt{3}-i)^n=2^n, n \in N$, then the least possible value of $n$ is

\((1+\sqrt{5}+i \sqrt{10-2 \sqrt{5}})^5=\)

If $f(t)=\int_0^t \tan ^{(2 n-1)} x d x, n \in N$, then $f(t+\pi)=$

\(\int_0^2 x^8\left(\frac{4}{x^2}-1\right)^{\frac{5}{2}} d x=\)

\(\int_{-2 \pi}^{2 \pi} \sin ^4(2 x) \cos ^6(2 x) d x=\)

The general solution of the differential equation $\frac{d y}{d x}=\frac{2 x+y-3}{2 y-x+3}$

The solution of the differential equation $x^2(y+1) \frac{d y}{d x}+y^2(x+1)^2=0$, when $y(1)=2$, is

If $x \log x \frac{d y}{d x}+y=\log x^2$ and $y(e)=0$, then $y\left(e^2\right)=$

If the slope of the tangent drawn at any point $(x, y)$ on a curve is $(x+y)$, then the equation of that curve is

If $y=\tan ^{-1}\left(\frac{3 x-x^3}{1-3 x^2}\right)+\tan ^{-1}\left(\frac{7 x}{1-12 x^2}\right)$, then at $x=0, \frac{d y}{d x}=$

If $y=\sqrt{\frac{x^4 \sqrt{3 x-5}}{\left(x^2-3\right)(2 x-3)}}$, then $\left(\frac{d y}{d x}\right)_{x=2}=$

If $x^2+y^2+\sin y=4$, then the value of $\frac{d^2 y}{d x^2}$ at $x=-2$ is

The area (in sq. units) of the triangle formed by the tangent and normal to the ellipse $9 x^2+4 y^2=72$ at the point $(2,3)$ with the $X$-axis is

If the normal drawn to the hyperbola $x y=16$ at $(8,2)$ meets the hyperbola again at a point $(\alpha, \beta)$, then $|\beta|+\frac{1}{|\alpha|}=$

If $3 \sqrt{2} x-4 y=12$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{5}{4}$ is its eccentricity, then $a^2-b^2=$

\(\int(\sqrt{\tan x}+\sqrt{\cot x}) d x=\)

If $\int x^{49}\left[\tan ^{-1} x^{50}+\frac{x^{50}}{1+x^{100}}\right] d x=\frac{x^n}{k} f(x)+c$, then
\(f(x)-f\left(\sqrt[k]{x^n}\right)=\)

If $\frac{3 x^3-7 x+1}{(x-2)^5}=\frac{A}{x-2}+\frac{B}{(x-2)^2}+\frac{C}{(x-2)^3}+\frac{D}{(x-2)^4}+\frac{E}{(x-2)^5}, \text { then } A(B+C+D+E)= $

\(\int \frac{x}{\sqrt{x^2-2 x+5}} d x=\)

$\int \frac{\sqrt{x-2}}{2 x+4} d x=$

For $0 < x < 1, \int\left[\tan ^{-1}\left(1-x+x^2\right)+\tan ^{-1}(1-x)\right] d x=$

The domain of the function, $f(x)=\sqrt{\log _e\left(\frac{1}{x^2-4 x+4}\right)}+\sin ^{-1}\left(x^2-2\right)$ is

If $A=\left\{x \in R / \sin ^{-1}\left(\sqrt{x^2+x+1}\right) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\right\}$ and $B=\left\{y \in R / y=\sin ^{-1}\left(\sqrt{x^2+x+1}\right), x \in A\right\}$, then

If $\sinh ^{-1} x=\log 3$ and $\cosh ^{-1} y=\log \frac{3}{2}$, then $\tanh ^{-1}(x-y)=$

If $\cot \left(\cos ^{-1} x\right)=\sec \left\{\tan ^{-1}\left(\frac{a}{\sqrt{b^2-a^2}}\right)\right\}, b>a$ then $x=$

If $f(x)=\left\{\begin{array}{cl}\frac{\left(e^{a x}-1\right) \log (1+x)}{\sin ^2 x}, & \text { if } x>0 \\ 2, & \text { if } x=0 \\ \frac{\cos 4 x-\cos b x}{\tan ^2 x}, & \text { if } x<0\end{array}\right.$ is continuous at $x=0$, then $\s...

$[x]$ represents the greatest integer function. If $\mathop {\lim }\limits_{x \to 0 + } \frac{\cos [x]-\cos (k x-[x])}{x^2}=5$, then $k=$

\(\mathop {\lim }\limits_{x \to 0} \frac{x \tan 2 x-2 x \tan x}{(1-\cos 2 x)^2}=\)

For all $n \in N$, if $n\left(n^2+3\right)$ is divisible by $k$, then the maximum value of $k$ is

If $a$ is the determinant of the adjoint of the matrix $\left[\begin{array}{lll}1 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 3\end{array}\right]$ and $b$ is the determinant of the inverse of the matrix $\left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & -3 & -1...

Consider two systems of 3 linear equations in 3 unknowns $A X=B$ and $C X=D$. If $A X=B$ has unique solution $D$ and $C X=D$ has unique solution $B$, then the solution of $\left(A-C^{-1}\right) X=0$ is

$f(x)$ is an $n$th degree polynomial satisfying $f(x)=\frac{1}{2}\left|\begin{array}{cc}f(x) & f\left(\frac{1}{x}\right)-f(x) \\ 1 & f\left(\frac{1}{x}\right)\end{array}\right|$. If $f(2)=33$, then the value of $f(3)$ is

Tangents are drawn at three points $P\left(t_1\right), Q\left(t_2\right), R\left(t_3\right)$ on the parabola $y^2=x$. Let these tangents intersect each other at the points $L, M, N$. If $t_1=2, t_2=-4, t_3=6$, then the area of the $\triangl...

All letters of the word 'AGAIN' are permuted in all possible ways and the words so formed (with or without meaning) are written as in a dictionary, then the 50th word is

The number of integers between 10 and 10,000 such that in every integer every digit is greater than its immediate preceeding digit, is

The number of ways in which a cricket team of 11 members can be formed out of 6 batsmen, 6 bowlers, 4 all-rounders and 4 wicket keepers by selecting atleast 4 batsmen, atleast 3 bowlers, atleast 2 all-rounders and only one wicket keeper is

An item is tested on a device for its defectiveness. The probability that such an item is defective is 0.3 . The device gives accurate result in 8 out of 10 such tests.
If the device reports that an item tested is not defective, then the pr...

A basket contains 5 apples and 7 oranges and another basket contains 4 apples and 8 oranges. If one fruit is picked out at random from each basket, then the probability of getting one apple and one orange is

Two cards are drawn from a pack of 52 playing cards one after the other without replacement. If the first card drawn is a queen, then the probability of getting a face card from a black suit in the second draw is

If the probability distribution of a random variable $X$ is as follows, then the mean of $X$ is
$$ \begin{array}{ccccc} \hline \boldsymbol{X}=\boldsymbol{x}_{\boldsymbol{i}} & -1 & 0 & 1 & 2 \\ \hline \boldsymbol{P}\left(\boldsymbol{X}=\bol...

In a school there are 3 sections $A, B$ and $C$. Section $A$ contains 20 girls and 30 boys, section $B$ contains 40 girls and 20 boys and section $C$ contains 10 girls and 30 boys. The probabilities of selecting the section $A, B$ and $C$ a...

If $X$ is a binomial variate with mean $\frac{16}{5}$ and variance $\frac{48}{25}$, then $P(X \leq 2)=$

In a $\triangle A B C, A, B$ and $C$ are in arithmetic progression, $r r_3=r_1 r_2$ and $c=10$, then $a^2+b^2+c^2=$

In a $\triangle A B C, \frac{2\left(r_1+r_3\right)}{a c(1+\cos B)}=$

In a $\triangle A B C$, if $a, b, c$ are in arithmetic progression and the angle $A$ is twice the angle $C$, then $\cos A: \cos B: \cos C=$

Two roots of the equation, $a x^4+b x^3+c x^2+d x+e=0$ are positive and equal. If the product of the other two real roots is 1 , then

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $2 x^3+3 x^2-5 x-7=0$, then $\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}=$