AP EAPCET 2025 - 21st May Evening Shift

If the curves $y^2=16 x$ and $9 x^2+\alpha y^2=25$ intersect at right angles, then $\alpha=$

If the normal drawn at the point $P$ on the curve $y=x \log x$ is parallel to the line $2 x-2 y=3$, then $P=$

If the function $y=\sin x(1+\cos x)$ is defined in the interval $[-\pi, \pi]$, then $y$ is strictly increasing in the interval

If the velocity of a particle moving on a straight line is proportional to the cube root of its displacement, then its acceleration is

The area of the region (in sq. units) enclosed between the curves $y=|x|, y=[x]$ and the ordinates $x=-1$, $x=0, x=1$ is

If $P_n$ denotes the product of the binomial coefficients in the expansion of $(1+x)^n$, then $\frac{P_{n+1}}{P_n}=$

Coefficient of $x^2$ in the expansion of $\left(x^2+x-2\right)^5$ is

The coefficient of $x^3$ in the expansion of $\frac{x^4+1}{\left(x^2+1\right)(x-1)}$ when it is expressed in terms of positive integral powers of $x$, is

If the circle $S=x^2+y^2+2 g x+4 y+1=0$ bisects the circumference of the circle $x^2+y^2-2 x-3=0$, then the radius of circle $S=0$ is

If $(a, b)$ is the common point for the circles $x^2+y^2-4 x+4 y-1=0$ and $x^2+y^2+2 x-4 y+1=0$, then $a^2+b^2=$

If a circle $S$ passes through the origin and makes an intercept of length 4 units on the line $x=2$, then the equation of the curve on which the centre of $S$ lies is

The angle between the tangents drawn from the point $(2,2)$ to the circle $x^2+y^2+4 x+4 y+c=0$ is $\cos ^{-1}\left(\frac{7}{16}\right)$. If two such circles exist, then sum of the values of $c$ is

A circle touches the line $2 x+y-10=0$ at $(3,4)$ and passes through the point $(1,-2)$. Then, a point that lies on the circle is

If $\left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^{2024}+\left(\frac{1+\cos \theta+i \sin \theta}{1-\cos \theta+i \sin \theta}\right)^{2025}=x+i y$ then the value of $x+y$ at $\theta=\frac{\pi}{2}$ is

If $a \pm i b$ and $b \pm a i$ are the roots of $x^4-10 x^3+50 x^2-130 x+169=0$, then $\frac{a}{b}+\frac{b}{a}=$

If $a=\operatorname{Im}\left(\frac{1+z^2}{2 i z}\right)$ and $z$ is any non-zero complex number such that $|z|=1$, then $a=$

If $(3+4 i)^{2025}=5^{2023}(x+i y)$, then $\sqrt{x^2+y^2}=$

\(\int_0^\pi x \cdot \sin ^5 x \cdot \cos ^6 x d x=\)

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

\(\int_0^{\frac{\pi}{2}} \log |\tan x+\cot x| d x=\)

The general solution of the differential equation $\frac{d y}{d x}+x y=4 x-2 y+8$ is

The general solution of the differential equation $\left(x+2 y^3\right) \frac{d y}{d x}-y=0, y>0$ is

The general solution of the differential equation $\frac{d y}{d x}+\frac{x+y+1}{x-3 y+5}=0$ is

If $x^2+y^2=t-\frac{1}{t}$ and $x^4+y^4=t^2+\frac{1}{t^2}$, then $\frac{d y}{d x}=$

If $y=(a x+b) \cos x$, then
\(y_2+y_1 \sin 2 x+y\left(1+\sin ^2 x\right)=\)

The equation of a chord $A B$ of an ellipse $2 x^2+y^2=1$ is $x-y+1=0$. If $O$ is the origin, then $\sqrt{A O B}=$

The square of the slope of a common tangent drawn to the circle $4 x^2+4 y^2=25$ and the ellipse $4 x^2+9 y^2=36$ is

The set of real values of $x$ such that $f(x)=\sqrt{\frac{[x]-1}{\left.[x]^2-[x]-6\right]}}$ is a real valued function is

Domain of the real valued function $f(x)=\log \left(x^2-1\right)+x \operatorname{coth}^{-1} x$ is

If a function $f: Z \rightarrow Z$ is defined by $f(x)=x-(-1)^x$, then $f(x)$ is

The tangents drawn to the hyperbola $5 x^2-9 y^2=90$ through a variable point $P$ make the angles $\alpha$ and $\beta$ with its transverse axis. If $\alpha, \beta$ are the complementary angles then the locus of $P$ is

If $\theta$ is the acute angle between the asymptotes of a hyperbola $7 x^2-9 y^2=63$, then $\cos \theta=$

If $\int \frac{d x}{(x-1)^{\frac{3}{2}}(x-3)^{\frac{1}{2}}}=\sqrt{f(x)}+C$, then $f(-1)-f(0)=$

If $\int e^{\sin x}(1+\sec x \tan x) d x=e^{\sin x} f(x)+c$, then in $0 \leq x \leq 2 \pi$, then number of solutions of $f(x)=1$ is

If $\int x^2 \cos ^2 x d x=\frac{1}{6} f(x)+g(x) \sin 2 x +h(x) \cos 2 x+c$, then $f(1)+g(2)+h\left(\frac{1}{2}\right)=$

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

$\int\left(\frac{1+x+\sqrt{x+x^2}}{\sqrt{x}+\sqrt{1+x}}\right) d x=$

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

If $x$ is a real number, then the number of solutions of $\tan ^{-1}(\sqrt{x(x+1)})+\sin ^{-1}\left(\sqrt{x^2+x+1}\right)=\frac{\pi}{2}$ is

If $\mathop {\lim }\limits_{x \to {a^ + }} f(x)=p, \mathop {\lim }\limits_{x \to {a^ - }} f(x)=m$ and $f(a)=k$, then which one of the following is true?

\(\mathop {\lim }\limits_{x \to - \infty } \frac{5 x^3-x^2 \sin 5 x}{x \cos 4 x+7|x|^3-4|x|+3}=\)

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

If $A=\left[\begin{array}{ccc}1 & 2 & -2 \\ 2 & -1 & 2 \\ -1 & 1 & -2\end{array}\right]$, then $A+2 A^{-1}=$

If $A=\left[\begin{array}{ccc}a & b & c \\ d & e & f \\ l & m & n\end{array}\right]$ is a matrix such that $|A|>0$ and $\operatorname{adj}(A)=\left[\begin{array}{ccc}0 & 4 & -6 \\ 10 & 8 & 0 \\ 2 & 4 & -4\end{array}\right]$, then $\frac{c d...

In solving a system of linear equations $A X=B$ by Cramer's rule, in the usual notation, if $\Delta_1=\left|\begin{array}{ccc}-11 & 1 & -7 \\ -4 & 1 & -2 \\ 5 & 1 & 1\end{array}\right|$ and $\Delta_3=\left|\begin{array}{ccc}4 & 1 & -11 \\ 1...

The angle between the tangents drawn from the point $(1,4)$ to the parabola $y^2=4 x$ is

The number of ways of distributing 3 dozen fruits (no two fruits are identical) to 9 persons such that each gets the same number of fruits is

If $\binom{p}{q}={ }^p C_q$ and $\sum\limits_{i=0}^m\binom{10}{i}\binom{20}{m-i}$ is maximum, then $m=$

If all the letters of the word COMBINATION are arranged in all possible ways to form 11 letter words (with or without meaning), then the number of words among them in which $C$ and $N$ occupy the end positions and no vowel appears exactly i...

The probability distribution of a random variable $X$ is given below$$ \begin{array}{ccccccc} \hline X & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline P\left(X=x_i\right) & \alpha & \alpha & \alpha & \beta & \beta & 0.3 \\ \hline \end{array} $$
If $\mu$ ...