AP EAPCET 2024 - 20th May Evening Shift

The circle $x^2+y^2-8 x-12 y+\alpha=0$ lies in the first quadrant without touching the coordinate axes. If $(6,6)$ is an interior point to the circle, then

$A(2,3), B(-1,1)$ are two points. If $P$ is a variable point such that $\angle A P B=90^{\circ}$, then locus of $P$ is

If the locus of the mid-point of the chords of the circle $x^2+y^2=25$, which subtend a right angle at the origin is given by $\frac{x^2}{\alpha^2}+\frac{y^2}{\alpha^2}=1$, then $|\alpha|=$

If $z_1=10+6 i, z_2=4+6 i$ and $z$ is any complex number such that the argument of $\frac{\left(z-z_1\right)}{\left(z-z_2\right)}$ is $\frac{\pi}{4}$,

If $z=x+i y, x^2+y^2=1$ and $z_1=z e^{i \theta}$, then $\frac{z_1^{2 n}-1}{z_1^{2 n}+1}=$

If $\frac{3-2 i \sin \theta}{1+2 i \sin \theta}$ is purely imaginary number, then $\theta=$

$\int_0^{\frac{\pi}{2}} \frac{1}{1+\sqrt{\tan x}} d x=$

$\int\limits_{\frac{-1}{24}}^{\frac{1}{24}} \sec x \log \left(\frac{1-x}{1+x}\right) d x=$

If $[x]$ is the greatest integer function, then $\int_0^5[x] d x=$

$\int_0^\pi \frac{x \sin x}{1+\cos ^2 x} d x=$

Order and degree of the differential equation $\frac{d^3 y}{d x^3}=\left[1+\left(\frac{d y}{d x}\right)^2\right]^{\frac{5}{2}}$, respectively are

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

Integrating factor of the differential equation $\sin x \frac{d y}{d x}-y \cos x=1$ is

If $y=t^2+t^3$ and $x=t-t^4$, then $\frac{d^2 y}{d x^2}$ at $t=1$ is

If the chord of the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ having $(1,1)$ as its middle point is $x+\alpha y=\beta$, then

$f: R \rightarrow R$ is defined by $f(x+y)=f(x)+12 y, \forall x, y \in R$. If $f(1)=6$, then $\sum_{r=1}^n f(r)=$

The domain of the real valued function $f(x)=\sqrt{2+x}+\sqrt{3-x}$ is

If $l_1$ and $l_2$ are the lengths of the perpendiculars drawn from a point on the hyperbola $5 x^2-4 y^2-20=0$ to its asymptotes, then $\frac{l_1{ }^2 l_2{ }^2}{100}=$

If a directrix of a hyperbola centred at the origin and passing through the point $(4,-2 \sqrt{3})$ is $\sqrt{5} x=4$ and e is its eccentricity, then $e^2=$

If $\frac{x^2+3}{x^4+2 x^2+9}=\frac{A x+B}{x^2+a x+b}+\frac{C x+D}{x^2+c x+b}$, then $a A+b B+c C+D=$

$\int \frac{d x}{x\left(x^4+1\right)}=$

$\int \sin ^{-1} \sqrt{\frac{x}{a+x}} d x=$

$\int \frac{d x}{\sqrt{\sin ^3 x \cos (x-a)}}=$

$\int \frac{2-\sin x}{2 \cos x+3} d x=$

$\int \frac{e^{2 x}}{\sqrt[4]{e^x+1}} d x=$

If $\theta=\sec ^{-1}(\cosh u)$, then $u=$

If $\cos ^{-1} 2 x+\cos ^{-1} 3 x=\frac{\pi}{3}$ and $4 x^2=\frac{a}{b}$, then $a+b$ is equal to

If $f(x)=\left\{\begin{array}{cc}2 x+3, & x \leq 1 \\ a x^2+b x, & x>1\end{array}\right.$
is differentiable, $\forall x \in R$, then $f^{\prime}(2)=$

The values of $a$ and $b$ for which the function$ f(x)=\left\{\begin{array}{cl}1+|\sin x|^{\frac{a}{\sin x \mid}} & \frac{-\pi}{6} < x < 0 \\ b, & x=0 \quad \text { is continuous at } x=0 \\ e^{\frac{\tan 2 x}{\tan 3 x},} & 0 < x < \frac{\p...

If $\lim \limits_{x \rightarrow 0} \frac{e^x-a-\log (1+x)}{\sin x}=0$, then $a=$

$\lim \limits_{x \rightarrow \frac{\pi}{4}} \frac{4 \sqrt{2}-(\cos x+\sin x)^5}{1-\sin 2 x}=$

In the interval $[0,3]$ The function $f(x)=|x-1|+|x-2|$ is

$\left|\begin{array}{ccc}a & b & c \\ a^2 & b^2 & c^2 \\ 1 & 1 & 1\end{array}\right|$ is not equal to

Let $A, B, C, D$ and $E$ be $n \times n$ matrices each with non-zero determinant. If $A B C D E=I$, then $C^{-1}=$

If $A=\left[a_{i j}\right], 1 \leq i, j \leq n$ with $n \geq 2$ and $a_{i j}=i+j$ is a matrix, then the rank of $A$ is

The number of ways in which 17 apples can be distributed among four guests such that each guest gets at least 3 apples is .

The number of numbers lying between 1000 and 10000 such that every number contains the digit 3 and 7 only once without repetition is

A test containing 3 objective type of questions is conducted in a class. Each question has 4 options and only one option is the correct answer. No two students of the class have answered identically and no student has written all correct an...

For a binomial variate $X \sim B(n, p)$ the difference between the mean and variance is 1 and the difference between their square is 11 . If the probability of $P(x=2)=m\left(\frac{5}{6}\right)^n$ and $n=36$, then $m: n$

A box $P$ contains one white ball, three red ball and two black balls. Another box $Q$ contains two white balls, three red balls and four black balls. If one ball is drawn at random from each one of the two boxes, then the probability that ...

A person is known to speak false once out of 4 times, If that person picks a card at random from a pack of 52 cards and reports that it is a king, then the probability that it is actually a king is

The probability that a man failing to hit a target is $\frac{1}{3}$. If he fires 4 times, then the probability that he hits the target at least thrice is

If five-digit numbers are formed from the digits $0,1,2,3,4$ using every digit exactly only once. Then, the probability that a randomly chosen number from those numbers is divisible by 4 is

Two natural numbers are chosen at random from 1 to 100 and are multiplied. If $A$ is the event that the product is an even number and $B$ is the event that the product is divisible by 4 , then $P(A \cap \bar{B})=$

In $\triangle A B C, b c-r_2 r_3=$

In $\triangle A B C$, if $4 r_1=5 r_2=6 r_3$, then $\sin ^2 \frac{A}{2}+\sin ^2 \frac{B}{2}+\sin ^2 \frac{C}{2}=$

In $\triangle A B C, r_1 \cot \frac{A}{2}+r_2 \cot \frac{B}{2}+m_3 \cot \frac{C}{2}=$

If $O(0,0,0), A(3,0,0)$ and $B(0,4,0)$ form a triangle, then the incentre of $\triangle O A B$ is

Let $[r]$ denote the largest integer not exceeditio $r$ and the roots of the equation $3 x^2+6 x+5+\alpha\left(x^2+2 x+2\right)=0$ are complex number when ever $\alpha>L$ and $\alpha

For any real value of $x$. If $\frac{11 x^2+12 x+6}{x^2+4 x+2} \notin(a, b)$, then the value $x$ for which $\frac{11 x^2+12 x+6}{x^2+4 x+2}=b-a+3$ is