AP EAPCET 2025 - 21st May Morning Shift

If the intercept made by a variable circle on the X -axis and $Y$-axis are 8 and 6 units respectively, then the locus of the centre of the circle is

If $z_1$ and $z_2$ are two of the $n$th roots of unity such that the line segment joining them subtends at a right angle at the origin, then for a positive integer $k, n$ takes the form

\((\sqrt{\sqrt{2}+1}+i \sqrt{\sqrt{2}-1})^8=\)

If $i=\sqrt{-1}$, then $\sum\limits_{n=2}^{30} i^n+\sum\limits_{n=30}^{65} i^{n+3}=$

\(\int_0^{\pi / 2} \frac{\sin x}{1+\cos x+\sin x} d x=\)

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

Let $H(x)=3 x^4+6 x^3-2 x^2+1$ and $g(x)$ be a linear polynomial. If $\frac{H(x)}{(x-1)(x+1)(x-2)}=f(x) +\frac{g(x)}{(x-1)(x+1)(x-2)}$, then
$H(-1)+2 H(2)-3 H(1)=$

\(\int_{\pi / 4}^{\pi / 3} \frac{\cos x-\sin x}{\sin 2 x} d x=\)

The general solution of the equation $\frac{d y}{d x}+\frac{1}{x} y=\frac{1}{x} e^x$

The general solution of the differential equation
\(\left(x \sin \frac{y}{x}\right) d y=\left(y \sin \frac{y}{x}-x\right) d x\)

The differential equation corresponding to the family of parabolas whose axis is along $x=1$ is

If $5 f(x)+3 f\left(\frac{1}{x}\right)=x+2$ and $y=x f(x)$, then $\frac{d y}{d x}$ at $x=1$ is equal to

If the tangents are drawn to the ellipse $x^2+2 y^2=2$, then the locus of the mid-points of the intercepts made by the tangents between the coordinate axes is

If $f: R \rightarrow R$ and $g: R \rightarrow R$ are two functions defined by $f(x)=2 x-3$ and $g(x)=5 x^2-2$, then the least value of the function $(g \circ f)(x)$ is

The domain and range of a real valued function $f(x)=\cos x-3$ are respectively

One of the latus recta of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ subtends an angle $2 \tan ^{-1}\left(\frac{3}{2}\right)$ at the centre of the hyperbola. If $b^2=36$ and $e$ is the eccentricity of the given hyperbola, then $\sqrt...

If the equation of the hyperbola having $(8,3),(0,3)$ as foci and $\frac{4}{3}$ as eccentricity is $\frac{(x-\alpha)^2}{p}-\frac{(y-\beta)^2}{q}=1$, then $p+q=$

\(\int \frac{\sin 2 x}{\sin ^2 x+3 \cos x-3} d x\)

If $\int \frac{d x}{\sin ^3 x+\cos ^3 x}=A \log \left|\frac{\sqrt{2}+t}{\sqrt{2}-t}\right|+B \tan ^{-1}(t)+C$, then $\left(\frac{B}{A}, t\right)=$

\(\int \frac{e^{\sin x}(\sin 2 x-8 \cos x)}{2(\sin x-3)^2} d x=\)

\(\int(\log x)^3 x^4 d x=\)

If $\int\left(3 t^2 \sin \frac{1}{t}-t \cos \frac{1}{t}\right) d t=f(t) \sin \left(\frac{1}{t}\right)+C$ then $f(2)=$

\(\tan ^{-1} \frac{\sqrt{8-2 \sqrt{15}}}{\sqrt{15}+1}+\tan ^{-1} \frac{1}{\sqrt{5}}=\)

The derivative of $\sec ^{-1}\left(\frac{1}{2 x^2-1}\right)$ with respect to $\sqrt{1-x^2}$ at $x=\frac{1}{2}$ is

If a real valued function
$$ f(x)=\left\{\begin{array}{cl} \frac{x^2+(a+3) x+(a+1)}{x+3} & , \text { when } x \neq-3 \\ -\frac{5}{2} & , \text { when } x=-3 \end{array}\right. $$
is continuous at $x=-3$, then $\lim _{x \rightarrow a}\left(x...

\(\mathop {\lim }\limits_{x \to 0} \frac{x \tan 2 x-2 x \tan x}{(1-\cos 3 x)(\operatorname{cosec} x-\cot x)^2}=\)

Consider the following functions
I. $f(x)= \begin{cases}\frac{1}{2}-x & , x<\frac{1}{2} \\ \left(\frac{1}{2}-x\right)^2 & , x \geq \frac{1}{2}\end{cases}$
II. $f(x)=|3 x-1|$
III. $f(x)=x|x|$
IV. $f(x)=|x|$
Then, on $[0,1]$ Lagrange's mean v...

\(\mathop {\lim }\limits_{x \to \infty }\left[\frac{n+1}{n^2+1^2}+\frac{n+2}{n^2+2^2}+\frac{n+3}{n^2+3^2}+\ldots+\frac{n+2 n}{n^2+4 n^2}\right]=\)

\(\mathop {\lim }\limits_{x \to \infty } \frac{(\sqrt{2})-\sqrt{1+\cos x}}{\sqrt{15+\cos 2 x-4}}=\)

Match the functions in Column I with their properties in Column II. In the following [ $x$ ] denotes the greatest integer less than or equal to $x$.
.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:so...

If $A=\left[\begin{array}{ccc}x & 2 & 1 \\ -2 & y & 0 \\ 2 & 0 & -1\end{array}\right], x$ and $y$ are non-zero numbers, trace of $A=0$ and determinant of $A=-6$, then the minor of the elements 1 of $A$ is

If $A$ and $B$ are both $3 \times 3$ matrices, then which of the following statements are true?
(i) $A B=0 \Rightarrow A=0$ or $B=0$
(ii) $A B=I_3 \Rightarrow A^{-1}=B$
(iii) $(A-B)^2=A^2-2 A B+B^2$

$A=\left[\begin{array}{ccc}1 & -1 & 2 \\ -2 & 3 & -3\end{array}\right]$ is the given matrix and $A^T$ represents the transpose of $A$, then $A A^T-A-A^T=$

If $L$ is the normal drawn to the parabola $y^2=8 x$ at the point $t=\frac{1}{\sqrt{2}}$, then the foot of the perpendicular drawn from the focus of the parabola on to the normal $L$ is

The number of all five letter words (with or without meaning) having atleast one repeated letter than can be formed by using the letters of the word INCONVENIENCE is

The number of ways of arranging all the letters of the word PERFECTION such that there must be exactly two consonants between any two vowels is

The number of all possible positive integrals solutions of the equation $x y z=30$ is

If $A$ and $B$ are events of a random experiment such that $P(A \cup B)=\frac{3}{4}, P(A \cap B)=\frac{1}{4}, P(\overline{\mathrm{~A}})=\frac{2}{3}$, then $P(\overline{\mathrm{~A}} \cap \mathrm{~B})=$

$70 \%$ of the total employees of a factory are men. Among the employees of that factory 30\% of men and $15 \%$ of women are technical assistants. If an employee chosen at random is found to be a technical assistant, then the probability t...

A random variable $X$ follows a binomial distribution in which the difference between its mean and variance is 1. if $2 P(x=2)=3 P(x=1)$, then $n^2 P(x>1)=$

If a discrete random variable $X$ has the probability distribution $P(X=x)=k \frac{2^{2 x+1}}{(2 x+1)!}, x=0,1,2 \ldots \infty$, then $k=$

Two cards are drawn at random from a pack of 52 playing cards. If both the cards drawn are found to be black in colour, then the probability that atleast one of them is face card is

A person is known to speak the truth in 3 out of 4 occasions. If he throws a die and reports that it is six, then the probability that it actually six is

In a $\triangle A B C$ if $r_1=2 r_2=3 r_3$, then $a: b$ is

If the area of a $\triangle A B C$ is $4 \sqrt{5}$ sq units. Length of the side $C A$ is 6 units and $\tan \frac{B}{2}=\frac{\sqrt{5}}{4}$, then its smallest side is of length

In $\triangle A B C$, the sum of the lengths of two sides is $x$ and the product of those lengths is $y$. If $c$ is the length of its third side and $x^2-c^2=y$, then the circumradius of that triangle is

If the harmonic mean of the roots of the equation $\sqrt{2} x^2-b x+(8-2 \sqrt{5})=0$ is

All the values of $k$ such that the quadratic expression $2 k x^2-(4 k+1) x+2$ is negative for exactly three integrals values of $x$, lie in the interval

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3-13 x^2+k x+189=0$ such that $\beta-\gamma=2$, then $\beta+\gamma: k+\alpha=$

For all $n \in N$, if $1^3+2^3+3^3+\ldots n^3>x$, then a value of $x$ among the following is