AP EAPCET 2025 - 23rd May Morning Shift

If the lengths of the tangent, subtangent, normal and subnormal for the curve $y=x^2+x-1$ at the point $(1,1)$ are $a, b, c$ and $d$ respectively, then their increasing order is

If the volume of a sphere is increasing at the rate of 12 c.c. $/ \mathrm{sec}$, then the rate (in $\mathrm{sq} . \mathrm{cm} / \mathrm{sec}$ ) at which its surface area is increasing, when the diameter of the sphere is 12 cm is

If $\frac{1}{2} \leq \frac{x^2+x+a}{x^2-x+a} \leq 2 \forall x \in R$, then $a=$

$P$ and $Q$ are the ends of a diameter of the circle $x^2+y^2=a^2\left(a>\frac{1}{\sqrt{2}}\right) . s$ and $t$ are the lengths of the perpendiculars drawn from $P$ and $Q$ onto the line $x+y=1$ respectively. When the product st is maximum,...

Let $P(x)=x^4+a x^3+b x^2+c x+d$ be such that $x=0$ is the only real root of $P^1(x)=0$. If $P(-1)

The area (in sq. units) of the region bounded by the lines $x=0, x=\frac{\pi}{2}$ and $f(x)=\sin x, g(x)=\cos x$ is

If $-\frac{2}{3} < x < \frac{2}{3}$, then the value of the 5 th term in the expansion of $\frac{1}{\sqrt[3]{2-3 x}}$ when $x=\frac{1}{2}$ is

If the coefficients of $x^{10}$ and $x^{11}$ in the expansion of $\left(1+\alpha x+\beta x^2\right)(1+x)^{11}$ are 396 and 144 respectively, then $\alpha^2+\beta^2=$

If the equation of the polar of the point $(\alpha,-1)$ with respect to the circle $x^2+y^2-4 x-6 y-12=0$ is $y=\beta$, then $4(\alpha+\beta)=$

When the axes are rotated through an angle $\theta$ about origin in anti-clockwise direction and then translated to the new origin $(2,-2)$, if the transformed equation the equation of $x^2+y^2=4$ is $X^2+Y^2+a X+b Y+c=0$ then $a+b+c=$

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