AP EAPCET 2025 - 22nd May Evening Shift

If the circles $x^2+y^2-2 \lambda x-2 y-7=0$ and $3\left(x^2+y^2\right)-8 x+29 y=0$ are orthogonal, then $\lambda=$

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

If $x^6=(\sqrt{3}-i)^5$, then the product of all of its roots is

If $k \in N$, then $\lim\limits_{n \rightarrow \infty}\left[\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+\ldots .+\frac{1}{k n}\right]=$

\(\int_{-1}^4 \sqrt{\frac{4-x}{x+1}} d x=\)

\(\int_0^{\pi / 4} \frac{\cos ^2 x}{\cos ^2 x+4 \sin ^2 x} d x=\)

\(\int_{5 \pi}^{25 \pi}|\sin 2 x+\cos 2 x| d x=\)

The differential equation of the family of circles passing through the origin and having centre on $X$-axis is

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

The general solution of the differential equation $\frac{d y}{d x}+\frac{\sec x}{\cos x+\sin x} y=\frac{\cos x}{1+\tan x}$ is

If $g$ is the inverse of the function $f(x)$ and $g(x)=x+\tan x$, then $f^{\prime}(x)=$

If $\sqrt{x-x y}+\sqrt{y-x y}=1$, then $\frac{d y}{d x}=$

Let $A_1$ be the area of the given ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
Let $A_2$ be the area of the region bounded by the curve which is the locus of mid-point of the line segment joining the focus of the ellipse and a point $P$ on ...

The set of all real values of $x$ for which $f(x)=\sqrt{\frac{|x|-2}{|x|-3}}$ is a well defined function is

If the equation of the tangent of the hyperbola $5 x^2-9 y^2-20 x-18 y-34=0$ which makes an angle $45^{\circ}$ with the positive $X$-axis in positive direction is $x+b y+c=0$, then $b^2+c^2=$

If the distance between the foci of a hyperbola $H$ is 26 and distance between its directrices is $\frac{50}{13}$, then the eccentricity of the conjugate hyperbola of the hyperbola $H$ is

If $\frac{a x+5}{\left(x^2+b\right)(x+3)}=\frac{x+21}{12\left(x^2+b\right)}+\frac{c}{12(x+3)}$, then $b^2=$

\(\int\left(\sum_{r=0}^{\infty} \frac{x^r 2^r}{r!}\right) d x=\)

\(\int \frac{d x}{12 \cos x+5 \sin x}=\)

If $\int \frac{\cos ^3 x}{\sin ^2 x+\sin ^4 x} d x=c-\operatorname{cosec} x-f(x)$, then $f\left(\frac{\pi}{2}\right)=$

\(\int \frac{13 \cos 2 x-9 \sin 2 x}{3 \cos 2 x-4 \sin 2 x} d x=\)

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

If $\frac{1}{2} \sin ^{-1}\left(\frac{3 \sin 2 \theta}{5+4 \cos 2 \theta}\right)=\tan ^{-1} x$, then $x=$

If $\operatorname{sech}^{-1} x=\log 2$ and $\operatorname{cosech}^{-1} y=-\log 3$, then $(x+y)=$

If $y=\tan ^{-1}\left(\frac{x}{1+2 x^2}\right)+\tan ^{-1}\left(\frac{x}{1+6 x^2}\right)$, then $\frac{d y}{d x}=$

$[x]$ denotes the greater integer less than or equal to $x$. If $\{x\}=x-[x]$ and $\lim\limits_{x \rightarrow 0}-\frac{\sin ^{-1}(x+[x])}{2-\{x\}}=\theta$, then $\sin \theta+\cos \theta=$

\(\mathop {\lim }\limits_{n \to \infty } \frac{1}{n^3} \sum\limits_{k=1}^n k^2 x=\)

Let $f: R \rightarrow R$ be defined by
$$ f(x)=\left\{\begin{array}{cc} a-\frac{\sin [x-1]}{x-1}, & \text { if } x>1 \\ 1, & \text { if } x=1 \\ b-\left[\frac{\sin [x-1]-[x-1]}{([x-1])^3},\right. & \text { if } x<1 \end{array}\right. $$
whe...

If $A=\left[\begin{array}{lll}1 & 2 & 3 \\ 1 & 3 & 5 \\ 2 & 1 & 6\end{array}\right]$ and $|\operatorname{adj}(\operatorname{adj} A)|(\operatorname{adj} A)^{-1}=k A$, then $k=$

If the values $x=\alpha, y=\beta, z=\gamma$ satisfy all the 3 equations $x+2 y+3 z=4,3 x+y+z=3$ and $x+3 y+3 z=2$, then $3 \alpha+\gamma=$

The number of solutions of the system of equations $2 x+y-z=7, x-3 y+2 z=1, x+4 y-3 z=5$ is

If the perpendicular distance from the focus of a parabola $y^2=4 a x$ to its directrix is $\frac{3}{2}$, then the equation of the normal drawn at $(4 a,-4 a)$ is

If ${ }^{27} P_{r+7}=7722{ }^{25} P_{(r+4)}$, then $r=$

If the number of diagonals of a regular polygon is 35 , then the number of sides of the polygon is

If four letters are chosen from the letters of the word ASSIGNMENT and are arranged in all possible ways to form 4 letter words (with or without meaning), then total number of such words that can be formed is

An unbiased coin is tossed 8 times. The probability that head appears consecutively at least 5 times is

A box contains twelve balls of which 4 are red, 5 are green and 3 are white. If three balls are drawn at random simultaneously from the box, then the probability that exactly 2 balls have the same colour is

There are three families $F_1, F_2, F_3 . F_1$ has 2 boys and 1 girl; $F_2$ has 1 boy and 2 girls; $F_3$ has 1 boy and 1 girl. A family is randomly chosen and a child is chosen from that family randomly. If it is known that the child thus s...

An urn $A$ contains 4 white and 1 black ball; urn $B$ contains 3 white and 2 black balls and urn $C$ contains 2 white and 3 black balls. One ball is transferred randomly from $A$ to $B$; later one ball is transferred randomly from $B$ to $C...

If the probability distribution of a discrete random variable $X$ is given by
$P(X=k)=\frac{2^{-k}(3 k+1)}{2^c}, k=0,1,2, \ldots \ldots \infty$, then $P(X \leq c)=$

In a binomial distribution, if $n=4$ and $P(X=0)=\frac{16}{81}$, then $P(X=4)=$

If the sides $a, b, c$ of the $\triangle A B C$ are in harmonic progression, then $\operatorname{cosec}^2 A / 2, \operatorname{cosec}^2 B / 2, \operatorname{cosec}^2 C / 2$ are in

In $\triangle A B C$, if $r=3$ and $R=5$, then $\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c a}=$

$f(x)$ is a quadratic polynomial satisfying the condition $f(x)+f\left(\frac{1}{x}\right)=f(x) f\left(\frac{1}{x}\right)$. If $f(-1)=0$, then the range of $f$ is

If $\alpha \neq 0$ and zero are the roots of the equation $x^2-5 k x+\left(6 k^2-2 k\right)=0$, then $\alpha=$

When the roots of $x^3+\alpha x^2+\beta x+6=0$ are increased by 1 , if one of the resultant values is the least root of $x^4-6 x^3+11 x^2-6 x=0$, then

Let ' $a$ ' be a non-zero real number. If the equation whose roots are the squares of the roots of the cubic equation $x^3-a x^2+a x-1=0$ is identical with this cubic equation, then ' $a$ ' =

The set of all real values of $x$ satisfying the inequation $\frac{8 x^2-14 x-9}{3 x^2-7 x-6}>2$ is

\(\sum\limits_{k=1}^n k(k+1)(k+2) \ldots(k+r-1)=\)

Variance of the following discrete frequency distribution is
$$ \begin{array}{llllll} \hline \text { Class Interval } & 0-2 & 2-4 & 4-6 & 6-8 & 8-10 \\ \hline \text { Frequency } & 2 & 3 & 5 & 3 & 2 \\ \hline \end{array} $$