AP EAPCET 2025 - 27th May Morning Shift

Consider the quadratic equation $a x^2+b x+c=0$, where $2 a+3 b+6 c=0$ and let $g(x)=\frac{a x^3}{3}+\frac{b x^2}{2}+c x$
Statement I The given quadratic equation $a x^2+b x+c=0$ has atleast one root in $(0,1)$.
Statement II Rolle's theorem...

The difference between the absolute maximum and absolute minimum values of the function $f(x)=2 x^3-15 x^2+36 x-30$ on $[-1,4]$ is

The angle between the curves $y^2=x$ and $x^2=y$ at the point $(1,1)$ is

If $f(x)=x e^{x(1-x)}, x \in R$, then $f(x)$ is

The area (in sq units) of the region given by $R=\left\{(x, y) ; \frac{y^2}{2} \leq x \leq y+4\right\}$ is

The remainder obtained when $(2 m+1)^{2 n}(m, n \in N)$ is divided by 8 is

\(\sum_{r=1}^{15} r^2\left(\frac{{ }^{15} C_r}{{ }^{15} C_{r-1}}\right)=\)

\(\frac{1}{81^n}-{ }^{2 n} C_1 \frac{10}{81^n}+{ }^{2 n} C_2 \frac{10^2}{81^n}-\ldots+\frac{10^{2 n}}{81^n}=\)

If $x$ is positive real number and the first negative term in the expansion of $(1+x)^{\frac{27}{5}}$ is $t_k$, then $k=$

If the smallest circle through the points of intersection of $x^2+y^2=a^2$ and $x \cos \alpha+y \sin \alpha=p, 0

Circles are drawn through the point $(2,0)$ to cut intercepts of length 5 units on the $X$-axis. If their centre lie in the first quadrant, then their equation is

A circle touches both the coordinate axes and the straight line $L \equiv 4 x+3 y-6=0$ in the first quadrant. If this circle lies below the line $L=0$, then the equation of that circle is

If the lines $3 x-4 y+4=0$ and $6 x-8 y-7=0$ are the tangents to the same circle, then the area of that circle (in sq. units) is

The locus of the third vertex of a right-angled triangle, the ends of whose hypotenuse are $(1,2)$ and $(4,5)$ is

If $z$ and $w$ are two non-zero complex numbers such that $|z w|=1$ and $\arg z-\arg w=\frac{\pi}{2}$, then $\bar{z} w=$

Let $z$ satisfy $|z|=1, z=1-\bar{z}$ and $\operatorname{Im}(z)>0$
Statement $\mathbf{I} z$ is a real number
Statement II Principal argument of $z$ is $\frac{\pi}{3}$.
Then,

If $w_1$ and $w_2$ are two non-zero complex numbers and ${ }a, b$ are non-zero real numbers such that $\left|a w_1+b w_2\right|=\left|a w_1-b w_2\right|$, then $\frac{w_1}{w_2}$ is

If $\sinh ^{-1}(2)+\sinh ^{-1}(3)=\alpha$, then $\sinh \alpha=$

\(\int_0^1 \frac{2 x+5}{x^2+3 x+2} d x=\)

\(\int_0^1 x^{\frac{5}{2}}(1-x)^{\frac{3}{2}} d x=\)

$$ \lim _{n \rightarrow \infty}\left[\begin{array}{c} \frac{1}{n^2} \sec ^2 \frac{1}{n^2}+\frac{2}{n^2} \sec ^2 \frac{4}{n^2}+\frac{3}{n^2} \sec ^2 \\ \frac{9}{n^2}+\ldots+\frac{1}{n^2} \sec ^2 1 \end{array}\right]= $$

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

The general solution of the differential equation $\cos (x+y) d y=d x$ is

If $A x^3+B x y=4$ ( $A$ and $B$ are arbitrary constants) is the general solution of the differential equation $F(x) \frac{d^2 y}{d x^2}+G(x) \frac{d y}{d x}-2 y=0$, then $F(l)+G(l)=$

If $x=\sqrt{2^{\operatorname{cosec}^{-1} t}}$ and $y=\sqrt{2^{\sec ^{-1} t}},|t| \geq 1$, then $\frac{d y}{d x}=$

If $(a+\sqrt{2} b \cos x)(a-\sqrt{2} b \cos y) =a^2-b^2$, where $a>b>0$, then at $\left(\frac{\pi}{4}, \frac{\pi}{4}\right), \frac{d y}{d x}=$

Assertion (A) The length of the latus rectum of an ellipse is 4 . The focus and its corresponding directrix are respectively $(1,-2)$ and $3 x+4 y-15=0$. Then, its eccentricity is $\frac{1}{2}$.
Reason $(\mathrm{R})$ Length of the perpendic...

If $f: R \rightarrow A$, defined by $f(x)=\cos x+\sqrt{3} \sin x-1$ is an onto function then $A=$

Let $g(x)=1+x-[x]$ and ${ }^{\prime}$
$$ f(x)= \begin{cases}-1, & x<0 \\ 0, & x=0,[x] \text { denotes the greatest integer less } \\ 1, & x>0\end{cases} $$
than or equal to $x$. Then for all $x, f(g(x))=$

If the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passing through the point $(4,6)$ is 2 , then the equation of the tangent to this hyperbola at $(4,6)$ is

A hyperbola passes through the point $P(\sqrt{2}, \sqrt{3})$ and has foci at $( \pm 2,0)$. Then, the point that lies on the tangent drawn to this hyperbola at $P$ is

If $\frac{x^2}{\left(x^2+2\right)\left(x^4-1\right)}=\frac{A}{x^2-1}+\frac{B}{x^2+1}+\frac{C}{x^2+2}$, then $A+B-C=$

If $\int \frac{5 \tan x}{\tan x-2} d x=a x+b \log |\sin x-2 \cos x|+c$, then $a+b=$

\(\int x \cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right) d x(x>0)=\)

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

\(\int \sin ^{-1}\left(\sqrt{\frac{x}{a+x}}\right) d x=\)

If $\int \frac{x}{x \tan x+1} d x=\log f(x)+k$, then $f\left(\frac{\pi}{4}\right)=$

The equation $\cos ^{-1}(1-x)-2 \cos ^{-1} x=\frac{\pi}{2}$ has

\(\mathop {\lim }\limits_{x \to {\pi \over 4}} \frac{2 \sqrt{2}-(\cos x+\sin x)^3}{1-\sin 2 x}=\)

Let $[x]$ denote the greatest integer less than or equal to $x$. Then,
\(\lim _{x \rightarrow 2^{+}}\left(\frac{[x]^3}{3}-\left[\frac{x}{3}\right]^3\right)=\)

If the function $f$ defined by
$$ f(x)=\left\{\begin{array}{cc} \frac{1-\cos 4 x}{x^2}, & x<0 \\ a, & x=0 \\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & x>0 \end{array}\right. $$
is continuous at $x=0$, then $a=$

The domain of the derivative of the function $f(x)=\frac{x}{1+|x|}$ is

The equation $x^{\frac{3}{4}\left(\log _2 x\right)^2+\log _2 x-\frac{5}{4}}=\sqrt{2}$ has

A value of $\theta$ lying between 0 and $\pi / 2$ and satisfying $\left|\begin{array}{ccc}1+\sin ^2 \theta & \cos ^2 \theta & 4 \sin 4 \theta \\ \sin ^2 \theta & 1+\cos ^2 \theta & 4 \sin 4 \theta \\ \sin ^2 \theta & \cos ^2 \theta & 1+4 \s...

If the system of equations $2 x+p y+6 z=8$, $x+2 y+q z=5$ and $x+y+3 z=4$ has infinitely many solutions, then $p=$

If $x^a y^b=e^m, x^c y^d=e^n, \Delta_1=\left|\begin{array}{ll}m & b \\ n & d\end{array}\right|$, $\Delta_2=\left|\begin{array}{cc}a & m \\ c & n\end{array}\right|, \Delta_3=\left|\begin{array}{cc}a & b \\ c & d\end{array}\right|$, then the ...

A circle is drawn with its centre at the focus of the parabola $y^2=2 p x$ such that it touches the directrix of the parabola. Then, a point of intersection of the circle and the parabola is

If the locus of a point that divides a chord of slope 2 of the parabola $y^2=4 x$ internally in the ratio $1: 2$ is a parabola, then its vertex is

An eight digit number divisible by 9 is to be formed using digits from 0 to 9 without repeating the digits. The number of ways in which this can be done is

A string of letters is to be formed by using 4 letters from all the letters of the word "MATHEMATICS". The number of ways this can be done such that two letters are of same kind and the other two are of different kind is