AP EAPCET 2025 - 23rd May Evening Shift

The slope of one of the direct common tangents drawn to the circles $x^2+y^2-2 x+4 y+1=0$ and $x^2+y^2-4 x-2 y+4=0$ is

If $P(\alpha, \beta)$ is the radical centre of the circles $S \equiv x^2+y^2+4 x+7=0, S^{\prime}=2 x^2+2 y^2+3 x+5 y+9=0$ and $S^{\prime \prime} \equiv x^2+y^2+y=0$, then the length of the tangent drawn from $P$ to $S^{\prime}=0$ is

After the coordinate axes are rotated through an angle $\frac{\pi}{4}$ in the anti-clockwise direction without shifting the origin, if the equation $x^2+y^2-2 x-4 y-20=0$ transforms to $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ in the new coordi...

Sum of the squares of the imaginary roots of the equation $z^8-20 z^4+64=0$ is

If $z$ is a complex number such that $\frac{z-1}{z-i}$ is purely imaginary and locus of $z$ represents a circle with centre $(\alpha, \beta)$ and radius $r$, then $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=$

If the least positive integer $n$ satisfying the equation $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^n=-1$ is $p$ and the least positive integer $m$ satisfying the equation $\left(\frac{1-\sqrt{3 i}}{1+\sqrt{3} i}\right)^m=\operatorname{ci...

\(\int_0^{400 \pi} \sqrt{1-\cos 2 x} d x=\)

\(\int_{-\pi / 2}^{\pi / 2} \sin ^2 x \cos ^2 x(\sin x+\cos x) d x=\)

\(\int_{1 / 5}^{1 / 2} \frac{\sqrt{x-x^2}}{x^3} d x=\)

The equation of the curve passing through the point $(0, \pi)$ and satisfying the differential equation $y d x=\left(x+y^3 \cos y\right) d y$ is

The general solution of the differential equation $(x-(x+y) \log (x+y)) d x+x d y=0$ is

If the order and degree of the differential equation $x \frac{d^2 y}{d x^2}=\left(1+\left(\frac{d^2 y}{d x^2}\right)^2\right)^{-1 / 2}$ are $k$ and $l$ respectively, then $k, l$ are the roots of

If $y=(\log x)^{1 / x}+x^{\log x}$, at $x=e, \frac{d y}{d x}=$

If $y=\sqrt{\cosh x+\sqrt{\cosh x}}$, then $\frac{d y}{d x}=$

If $y=\tan ^{-1} \sqrt{x^2-1}+\sinh ^{-1} \sqrt{x^2-1}, x>1$, then $\frac{d y}{d x}=$

The equation of the normal drawn at the point $(\sqrt{2}+1,-1)$ to the ellipse $x^2+2 y^2-2 x+8 y+5=0$ is

If $f(x)=(x+1)^2-1, x \geq-1$, then $\left\{x \mid f(x)=f^{-1}(x)\right\}$ is

$[t]$ denotes the greatest integer function and $[t-m]=[t]-m$ when $m \in Z$.
If $k=2[2 x-1]-1$ and $3[2 x-2]+1=2[2 x-1]-1$, then the range of $f(x)=[k+5 x]$ is

If a hyperbola has asymptotes $3 x-4 y-1=0$ and $4 x-3 y-6=0$, then the transverse and conjugate axes of that hyperbola are

If $3 x+2 \sqrt{2} y+k=0$ is a normal to the hyperbola $4 x^2-9 y^2-36=0$ making positive intercepts on both the axes, then $k=$

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

\(\int \frac{\sec ^2 x}{\sin ^7 x} d x-\int \frac{7}{\sin ^7 x} d x=\)

If $\int\left(x^6+x^4+x^2\right) \sqrt{2 x^4+3 x^2+6} d x=f(x)+c$, then $f(3)=$

If $\int \frac{1}{\left((x+4)^3(x+1)^5\right)^{1 / 4}} d x=A \cdot\left(\frac{x+4}{x+1}\right)^n+C$

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

If $\int \frac{d x}{2 \cos x+3 \sin x+4}=\frac{2}{\sqrt{3}} f(x)+C$, then $f\left(\frac{2 \pi}{3}\right)=$

The number of solution of $\tan ^{-1} 1+\frac{1}{2} \cos ^{-1} x^2-\tan ^{-1} \left(\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right)=0$ is

The interval in which the function $f(x)=\tan ^{-1}(\sin x+\cos x)$ is an increasing function is

\(\tanh ^{-1}(\sin \theta)=\)

\(\mathop {\lim }\limits_{x \to 0} \frac{(\operatorname{cosec} x-\cot x)\left(e^x-e^{-x}\right)}{\sqrt{3}-\sqrt{2+\cos x}}=\)

If a real valued function
$$ f(x)=\left\{\begin{array}{cc} (1+\sin x)^{\cos x}, & -\pi / 2 < x < 0 \\ a, & x=0 \\ \frac{e^{2 / x}+e^{3 / x}}{a e^{2 / x}+b e^{3 / x}}, & 0 < x < \pi / 2 \end{array}\right. $$
is continuous at $x=0$, then $a b...

\(\mathop {\lim }\limits_{x \to 0} \frac{x^2 \sin ^2(3 x)+\sin ^4(6 x)}{(1-\cos 3 x)^2}=\)

If $\alpha$ is a real root of the equation $x^3+6 x^2+5 x-42=0$, then the determinant of the matrix
$\left[\begin{array}{lll}\alpha-1 & \alpha+1 & \alpha+2 \\ \alpha-2 & \alpha+3 & \alpha-3 \\ \alpha+4 & \alpha-4 & \alpha+5\end{array}\right...

The rank of the matrix $\left[\begin{array}{cccc}2 & -3 & 4 & 0 \\ 5 & -4 & 2 & 1 \\ 1 & -3 & 5 & -4\end{array}\right]$ is

If $P=\left[\begin{array}{lll}1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4\end{array}\right]$ is the adjoint of a matrix $A$ and det $A=4$, then the value of $\alpha$ is

If the tangents of the parabola $y^2=8 x$ passing through the point $P(1,3)$ touches the parabola at $A$ and $B$, then the area (in sq. units) of $\triangle P A B$ is

If all possible 4 -digit numbers are formed by choosing 4 different digits from the given digits $1,2,3,5,8$ then the sum of all such 4 -digit numbers is

The number of ways of forming the ordered pairs $(p, q)$ such that $p>q$ by choosing $p$ and $q$ from the first 50 natural numbers is

The number of ways in which a committee of 7 members can be formed from 6 teachers, 5 fathers and 4 students in such a way that at least one from each group is included and teachers form the majority among them, is

A die is thrown twice. Let A be the event of getting a prime number when the die is thrown first time and $B$ be the event of getting an even number when the die is thrown second time. Then, $P(A / \bar{B})=$

There are 8 boys and 7 girls in a class room. If the names of all those children are written on paper slips and 3 slips are drawn at random from them, then the probability of getting the names of one boy and two girls or one girl and two bo...

If $X \sim B(9, p)$ is a binomial variate satisfying the equation $P(X=3)=P(X=6)$, then $P(X<3)=$

A four member committee is to be formed from a group containing 9 men and 5 women. If a committee is formed randomly, then the probability that it contains atleast one woman is

A bag contains 5 balls of unknown colours. There are equal chances that out of these five balls, there may be 0 or 12 or or 3 or 4 or 5 red balls, A ball is taken out from the bag at random and is found to be red. The probability that it is...

In a $\triangle A B C$, if $\left(r_1-r_3\right)\left(r_1-r_2\right)-2 r_2 r_3=0$, then $a^2-b^2=$

If the median $A D$ of the $\triangle A B C$ is bisected at $E$ and $B E$ meets $A C$ in $E$, then $A F: A C=$

In $\triangle A B C$, if $a=8, b=10, c=12$, then $\frac{r}{R}=$

In $\triangle A B C$, if $a=13, b=8, c=7$, then $\cos (B+C)=$

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+p x^2+q x+r=0$, then $\alpha^3+\beta^3+\gamma^3=$

If the roots of the equation $x^2+2 a x+b=0$ are real, distinct and differ atmost by 2 m , then $b$ lies in the interval