AP EAPCET 2025 - 21st May Morning Shift

The value of the Rolle's theorem for the function $f(x)=2 \sin x+\sin 2 x$ in the interval $[0, \pi]$ is

If $\alpha$ and $\beta(\alpha>\beta)$ are the multiple roots of the equation $4 x^4+4 x^3-23 x^2-12 x+36=0$, then $2 \alpha-\beta=$

If the function $y=g(x)$ representing the slopes of the tangents drawn to the curve $y=3 x^4-5 x^3-12 x^2+18 x+3$ is strictly increasing, then the domain of $g(x)$ is

The area (in square units) of the triangle formed by the $X$-axis, the tangent and the normal drawn at $(1,1)$ to the curve $x^3+y^3=2 x y$ is

If $(1+x)^n=\sum_{r=0}^n C, x^r$, then the value of $C_0+\left(C_0+C_1\right)+\left(C_0+C_1+C_2\right)+\ldots+ \left(C_0+C_1+C_2+\ldots+C_n\right)$ is

If $x$ is so large that terms containing $x^{-3}, x^{-4}, x^{-5}, \ldots$ can be neglected, then the approximate value of $\left(\frac{3 x-5}{4 x^2+3}\right)^{-1 / 5}$ is

The slope of the non-vertical tangent drawn from the point $(3,4)$ to the circle $x^2+y^2=9$ is

From a point $P$ on the circle $x^2+y^2=4$, two tangents are drawn to the circle $x^2+y^2-6 x-6 y+14=0$. If $A$ and $B$ are the points of contact of those lines, then the locus of the centre of the circle passing through the points $P$, $A$...

If the product of the lengths of the perpendicular drawn from the ends of a diameter of the circle $x^2+y^2=4$ on the line $x+y+1=0$ is maximum, then the two ends of that diameter are

If the acute angle between the circles $S \equiv x^2+y^2+2 k x+4 y-3=0$ and $S^{\prime} \equiv x^2+y^2-4 x+2 k y+9=0$ is $\cos ^{-1}\left(\frac{3}{8}\right)$ and the centre of $S^{\prime}=0$ lies in the first quadrant, then the radical axis...

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 $n^2 P(x>1)=$