AP EAPCET 2024 - 21th May Evening Shift

The radical axis of the circle $x^2+y^2+2 g x+2 f y+c=0$ and $2 x^2+2 y^2+3 x+8 y+2 c=0$ touches the circle $x^2+y^2+2 x+2 y+1=0$. Then,

The equation of a circle which touches the straight lines $x+y=2, x-y=2$ and also touches the circle $x^2+y^2=1$ is

The perimeter of the locus of the point $P$ which divides the line segment QA internally in the ratio $1: 2$, where $A=(4,4)$ and $Q$ lies on the circle $x^2+y^2=9$, is

The equation of the circle touching the circle $x^2+y^2-6 x+6 y+17=0$ externally and to which the lines $x^2-3 x y-3 x+9 y=0$ are normal is

If $n$ is an integer and $Z=\cos \theta+i \sin \theta, \theta \neq(2 n+1) \frac{\pi}{2}$, then $\frac{1+Z^{2 n}}{1-Z^{2 n}}=$

Imaginary part of $\frac{(1-i)^3}{(2-i)(3-2 i)}$ is

The square root of $7+24 i$

\(\int\limits_0^{\pi /4} {{{{x^2}} \over {{{(x\,\sin \,x + \cos \,x)}^2}}}dx = }\)

$\int_1^5(|x-3|+|1-x|) d x=$

$\int_0^1 \frac{x}{(1-x)^{\frac{3}{4}}} d x=$

\(\int_{-1}^1\left(\sqrt{1+x+x^2}-\sqrt{1-x+x^2}\right) d x=\)

If $\cos x \frac{d y}{d x}-y \sin x=6 x,\left(0 < x < \frac{\pi}{2}\right)$ and $y\left(\frac{\pi}{3}\right)=0$, then $y\left(\frac{\pi}{6}\right)=$

$\frac{d y}{d x}=\frac{y+x \tan \frac{y}{x}}{x} \Rightarrow \sin \frac{y}{x}=$

The differential equation formed by eliminating arbitrary constants $A, B$ from the equation $y=A \cos 3 x+B \sin 3 x$ is

If $\frac{d}{d x}\left(\frac{1+x^2+x^4}{1+x+x^2}\right)=a x+b$, then $(a, b)=$

The product of perpendiculars from the two foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ on the tangent at any point on the ellipse is

If $3 f(x)-2 f(1 / x)=x$, then $f(2)=$

The real valued function $f: R \rightarrow\left[\frac{5}{2}, \infty\right)$ defined by $f(x)=|2 x+1|+|x-2|$ is

The descending order of magnitude of the eccentricities of the following hyperbolas is
A. A hyperbola whose distance between foci is three times the distance between its directrices.
B. Hyperbola in which the transverse axis is twice the co...

\(\int {\left( {\sum\limits_{r = 0}^\infty {{{{x^r}{3^r}} \over {r!}}} } \right)dx = }\)

$\int \frac{x^4+1}{x^6+1} d x=$

$\int e^x(x+1)^2 d x=$

$\int \frac{\sin ^6 x}{\cos ^8 x} d x=$

$\int \frac{x^5}{x^2+1} d x=$

If $\frac{x+2}{\left(x^2+3\right)\left(x^4+x^2\right)\left(x^2+2\right)}=\frac{A x+B}{x^2+3}+\frac{C x+D}{x^2+2}$ $+\frac{E x^3+F x^2+G x+H}{x^4+x^2}$, then $(E+F)(C+D)(A)=$

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

The real values of $x$ that satisfy the equation $\tan ^{-1} x+\tan ^{-1} 2 x=\frac{\pi}{4}$ is

If $y=\sin ^{-1} x$, then $\left(1-x^2\right) y_2-x y_1=$

$2 \operatorname{coth}^{-1}(4)+\sec h^{-1}\left(\frac{3}{5}\right)=$

\(\mathop {\lim }\limits_{x \to 1} \left( {{{x + {x^2} + {x^3} + ... + {x^n} - n} \over {x - 1}}} \right) =\)

If the function $f(x)=\frac{\sqrt{1+x}-1}{x}$ is continuous at $x=0$, then $f(0)=$

\(\mathop {\lim }\limits_{x \to 0} \left( {{{\sin (\pi {{\cos }^2}x} \over {{x^2}}}} \right) =\)

If $3 A=\left[\begin{array}{ccc}1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b\end{array}\right]$ and $A A^T=I$, then $\frac{a}{b}+\frac{b}{a}=$

$\left|\begin{array}{ccc}a+b+2 c & a & b \\ c & b+c+2 a & b \\ c & a & c+a+2 b\end{array}\right|=$

Assertion (A) : If $B$ is a $3 \times 3$ matrix and $|B|=6$, then $|\operatorname{adj}(B)|=36$
Reason (R) : If $B$ is a square matrix of order $n$, then $|\operatorname{adj}(B)|=|B|^n$

If the ordinates of points $P$ and $Q$ on the parabola $y^2=12 x$ are in the ratio $1: 2$. Then, the locus of the point of intersection of the normals to the parabola at $P$ and $Q$ is

There are 6 different novels and 3 different poetry books on a table. If 4 novels and 1 poetry book are to be selected and arranged in a row on a shelf such that the poetry book is always in the middle, then the number of such possible arr...

If a five-digit number divisible by 3 is to be formed using the numbers $0,1,2,3,4$ and 5 without repetition, then the total number of ways this can be done is

Four-digit numbers with all digits distinct are formed using the digits $1,2,3,4,5,6,7$ in all possible ways.If $p$ is the total number of numbers thus formed and $q$ is the number of numbers greater than 3400 among them, then $p: q=$

Two persons $A$ and $B$ throw a pair of dice alternately until one of them gets the sum of the numbers appeared on the dice as 4 and the person who gets this result first is declared as the winner. If $A$ starts the game, then the probabili...

An urn contains 3 black and 5 red balls. If 3 balls are drawn at random from the urn, the mean of the probability distribution of the number of red balls drawn is

If all the letters of the word 'SENSELESSNESS' are arranged in all possible ways and an arrangement among them is chosen at random, then the probability that all the E's come together in that arrangement is

Bag $A$ contains 3 white and 4 red balls, bag $B$ contains 4 white and 5 red balls and bag $C$ is contains 5 white and 6 red balls. If one ball is drawn at random from each of these three bags, then the probability of getting one white and ...

If $X \sim B(5, p)$ is a binomial variate such that $P(X=3)=P(X=4)$, then $P(|X-3|<2)=$

If two numbers $x$ and $y$ are chosen one after the other at random with replacement from the set of number $\{1,2,3, \ldots \ldots 10\}$. Then, the probability that $\left|x^2-y^2\right|$ is divisible by 6 is

In $\triangle A B C$, if $a=13, b=14$ and $\cos \frac{C}{2}=\frac{3}{\sqrt{13}}$, then $2 r_1=$

In $\triangle A B C$, if $\left(r_2-r_1\right)\left(r_3-r_1\right)=2 r_2 r_3$, then $2(r+R)=$

If 7 and 8 are the length of two sides of a triangle and $a^{\prime}$ is the length of its smallest side. The angles of the triangle are in AP and ' $a$ ' has two values $a_1$ and $a_2$ satisfying this condition. If $a_1 < a_2$, then $2 a_...

If $\alpha$ is a common root of $x^2-5 x+\lambda=0$ and $x^2-8 x-2 \lambda=0(\lambda \neq 0)$ and $\beta, \gamma$ are the other roots of them, then $\alpha+\beta+\gamma+\lambda=$

The equation $x^4-x^3-6 x^2+4 x+8=0$ has two equal roots. If $\alpha, \beta$ are the other two roots of this equation, then $\alpha^2+\beta^2=$