AP EAPCET 2025 - 23rd May Morning Shift

The radius of the circle passing through the points of intersection of the circles $x^2+y^2+2 x+4 y+1=0$, $x^2+y^2-2 x-4 y-4=0$ and intersecting the circle $x^2+y^2=6$ orthogonally is

If the power of the point $(1,6)$ with respect to the circle $x^2+y^2+4 x-6 y-a=0$ is -16 , then $a=$

From a point $P(-4,0)$, two tangents are drawn to the circle $x^2+y^2-4 x-6 y-12=0$ touching the circle at $A$ and $B$. If the equation of the circle passing through $P, A$ and $B$ is $x^2+y^2+2 g x+2 f y+c=0$, then $(g, f)=$

If $\theta$ is the angle between the tangents drawn from the point $(-1,-1)$ to the circle $x^2+y^2-4 x-6 y+c=0$ and $\cos \theta=-\frac{7}{25}$, then the radius of the circle is

If $1, \omega, \omega^2$ are the cube roots of unity, then
$$ 1\left(2+\frac{1}{\omega}\right)\left(2+\frac{1}{\omega^2}\right)+2\left(3+\frac{1}{\omega}\right)\left(3+\frac{1}{\omega^2}\right) +3\left(4+\frac{1}{\omega}\right)\left(4+\frac...

\((1+\sqrt{3} i)^6-(\sqrt{3}+i)^6=\)

For any two non-zero complex numbers $z_1$ and $z_2$, if $\left|z_1+z_2\right|^2=\left|z_1\right|^2+\left|z_2\right|^2$, then

\(\int_0^x \frac{t^2}{\sqrt{a^2+t^2}} d t=\)

\(\int\limits_0^{\frac{3 \pi}{2}} \frac{\cos ^3 x}{\cos ^3 x+\sin ^3 x} d x=\)

\(\int_{\frac{5}{6}}^\pi \cos ^{-4} x d x=\)

The differential equation for which $y^2=4 a(x+a)$ ( $a$ is the parameter) is the general solution is

The general solution of the differential equation $\frac{d y}{d x}=\frac{2 x y-4 x+y-2}{2 x y+x-4 y-2}$ is

The general solution of the differential equation $\sec (x-y+1) d y=d x$ is

If $x=\sqrt{2} e^t(\sin t-\cos t)$ and $y=\sqrt{2} e^t(\sin t+\cos t)$, then $\left(\frac{d^2 y}{d x^2}\right)_{t=\frac{\pi}{4}}=$

If the tangents drawn from a point $P$ to the ellipse $4 x^2+9 y^2-16 x+54 y+61=0$ are perpendicular, then the locus of $P$ is

\(\text { Consider the following statements. }\)
$$ \begin{array}{cl} \hline \text { Statement I } & \begin{array}{l} \text { A function } f: A \rightarrow B \text { is said to be one-one if and } \\ \text { only if } f(x) \neq f(y) \Righ...

$x+y+3=0,2 x-y+1=0$ are the equations of the asymptotes of a hyperbola.
If $(1,-2)$ is a point on this hyperbola, then the equation of its conjugate hyperbola is

If $\theta$ is the acute angle between the tangents drawn from the point $(1,1)$ to the hyperbola $4 x^2-5 y^2-20=0$, then $\tan \theta=$

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

\(\int \frac{1}{\cos x}\left[\frac{1}{\sin x}-\frac{1}{\sin x+3 \cos x}\right] d x=\)

\(\int \frac{x^4-16 x^2+2 x+8}{x^3-4 x^2+2} d x=\)

\(\int \frac{\sec ^2 x}{(\sec x+\tan x)^{\frac{5}{2}}} d x=\)

\(\int \frac{x+1}{x^3-1} d x=\)

If the equation $2 \cot ^{-1}\left(x^2+2 x+k\right)=\pi-3 \tan ^{-1} \left(x^2+2 x+k\right)$ has two distinct real solutions, then all the values of $k$ lie in the interval

The range of the real valued function $f(x)=\cos ^{-1}\left(\frac{3}{\sqrt{9 x^2-12 x+22}}\right)$ is

If $y=\sin ^{-1} \frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}$ and $\frac{-3 \pi}{2}

\(\sec h^{-1}(\sin \alpha)=\)

If $y=\log \left(\sec \left(\tan ^{-1} x\right)\right)(x>0)$, then $\frac{d y}{d x}$ at $x=1$ is

If a function,
$$ f(x)=\left\{\begin{array}{cc} \frac{\sqrt[3]{1+a x^2+b x^3}-\sqrt[3]{1-a x^2-b x^3}}{x^2}, & x<0 \\ 5, & x=0 \\ \frac{\tan 3 x-\sin 3 x}{b x^3}, & x>0 \end{array}\right. $$
is continuous at $x=0$, then the geometric mean o...

The quadratic equation whose roots are $l=\lim\limits_{\theta \rightarrow 0}\left(\frac{3 \sin \theta-4 \sin ^3 \theta}{\theta}\right)$ and $m=\lim\limits_{\theta \rightarrow 0}\left(\frac{2 \tan \theta}{\theta\left(1-\tan ^2 \theta\right)}...

\(\mathop {\lim }\limits_{x \to \infty } \frac{3 x+4 \cos ^2 x}{\sqrt{x^2-5 \sin ^2 x}}=\)

$A=\left[\begin{array}{ccc}0 & k & k \\ k & -4 & -6 \\ k & -3 & -5\end{array}\right]$ is a singular matrix for

$$ \left|\begin{array}{ll} 2 & 1 \\ 3 & 1 \end{array}\right|+\left|\begin{array}{cc} 1 & \frac{1}{3} \\ 3 & 1 \end{array}\right|+\left|\begin{array}{cc} \frac{1}{2} & \frac{1}{9} \\ 3 & 1 \end{array}\right|+\left|\begin{array}{cc} \frac{1}{...

If $A=\left[\begin{array}{ccc}1 & 2 & x \\ 4 & -1 & 7 \\ 2 & 4 & -6\end{array}\right]$ and the rank of $A$ is 2 , then the value of $x$ is equal to

The lengths of the two focal chords of the parabola $y^2=16 x$ is 25 units each. If these two chords cut the parabola at $A, B, C$ and $D$, then the area (in sq. units) of the quadrilateral formed by $A, B, C$ and $D$ is

If 3 sisters and 8 brothers are together playing a game, then the number of ways in which all the sisters and brothers are to be seated around a circle such that all the three sisters are not seated together is

Out of 8 students in a classroom, 4 of them are chosen and they are arranged around a table.
If the remaining 4 are arranged in a row, then the total number of arrangements that can be made with those 8 students is

Three letters are chosen at random from the letters of the word VARIABLE and all possible three letter words (with or without meaning) are formed with them.
Then, the probability of getting a three letter word having a consonent as its midd...

A rational number is selected at random from the distinct rational numbers of the form $p / q$ formed with $p$ and $q$ belonging to the set $\{1,2,3,4,5,6\}$. The probability that the rational number selected is a proper fraction, is

In a shoe rack there are 4 pairs of shoes and 4 shoes. are drawn one after the other at random without replacement. Then, the probability of getting atleast one correct pair of shoes among the four shoes drawn is

The probability distribution of a discrete random variable $X$ is given below
$$ \begin{array}{lllll} \hline X=x & -1 & 0 & 1 & 2 \\ \hline P(X=x) & \frac{1}{3} & \frac{1}{6} & \frac{1}{6} & \frac{1}{3} \\ \hline \end{array} $$
Then, the va...

If the average number of accidents occurring at a particular junction on a highway in a week is 5 , then the probability that atmost one accident occurs in a particular week is

If 3 squares are chosen at random from the 64 squares of a chess board, then the probability that all of them lie along the same diagonal line is

In $\triangle A B C$, if $a: b: c=4: 5: 6$, then $\frac{\cos A+3 \cos C}{\cos B}=$

In $\triangle A B C$, if $a=6, b=8$ and $c=10$, then $\frac{2 r_2 r_3}{r r_1}=$

In $\triangle A B C$ if $\cos A \cos B+\sin A \sin B \sin C=1$, then $\sin A+\sin B+\sin C=$

If $\alpha, \beta$ are the roots of the equation $x^2+b x+c=0$ satisfying the conditions $\alpha+\beta=5$ and $\alpha^3+\beta^3=60$, then $3 c+2=$

If $\alpha, \beta, \gamma$ are the roots of the equation,
$$ \begin{aligned} & x^3+a x^2+b x+c=0, \text { then }(\alpha+\beta-2 \gamma) \\ & (\beta+\gamma-2 \alpha)(\gamma+\alpha-2 \beta)= \end{aligned} $$

If the sum of two roots of the equation $x^4+2 x^3-7 x^2-8 x+12=0$ is zero, then the sum of the squares of the other two roots is

The sum of all integers between 1 and 100 (both inclusive) which are divisible by 5 or 13 is