AP EAPCET 2024 - 20th May Evening Shift

The length of the subnormal at any point on the curve $y=\left(\frac{x}{2024}\right)^k$ is constant, if the value of $k$ is

The acute angle between the curves $x^2+y^2=x+y$ and $x^2+y^2=2 y$ is

$p_1$ and $p_2$ are the perpendicular distances from the origin to the tangent and normal drawn at any point on the curve $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ respectively. If $k_1 p_1^2+k_2 p_2^2=a^2$, then $k_1+k_2=$

Equation of a tagent line of the parabola $y^2=8 x$, which passes through the point $(1,3)$ is

A' value of $C$ according to the Lagrange's mean value theorem for $f(x)=(x-1)(x-2)(x-3)$ in $[0,4]$ is

If the coefficients of $x^5$ and $x^6$ are equal in the expansion of $\left(a+\frac{x}{5}\right)^{65}$, then the coefficient of $x^2$ in the expansion of $\left(a+\frac{x}{5}\right)^4$ is.

If $|x|<\frac{2}{3}$, then the 4th term in the expansion of $(3 x-2)^{\frac{2}{3}}$ is :

The radical centre of the circles $x^2+y^2+2 x+3 y+1=0$, $x^2+y^2+x-y+3=0, x^2+y^2-3 x+2 y+5=0$

The largest among the distances from the point $P(15,9)$ to the points on the circle $x^2+y^2-6 x-8 y-11=0$ is

The equation of the circle whose diameter is the common chord of the circles $x^2+y^2-6 x-7=0$ and $x^2+y^2-10 x+16=0$ is

The circle $x^2+y^2-8 x-12 y+\alpha=0$ lies in the first quadrant without touching the coordinate axes. If $(6,6)$ is an interior point to the circle, then

$A(2,3), B(-1,1)$ are two points. If $P$ is a variable point such that $\angle A P B=90^{\circ}$, then locus of $P$ is

If the locus of the mid-point of the chords of the circle $x^2+y^2=25$, which subtend a right angle at the origin is given by $\frac{x^2}{\alpha^2}+\frac{y^2}{\alpha^2}=1$, then $|\alpha|=$

If $z_1=10+6 i, z_2=4+6 i$ and $z$ is any complex number such that the argument of $\frac{\left(z-z_1\right)}{\left(z-z_2\right)}$ is $\frac{\pi}{4}$,

If $z=x+i y, x^2+y^2=1$ and $z_1=z e^{i \theta}$, then $\frac{z_1^{2 n}-1}{z_1^{2 n}+1}=$

If $\frac{3-2 i \sin \theta}{1+2 i \sin \theta}$ is purely imaginary number, then $\theta=$

$\int_0^{\frac{\pi}{2}} \frac{1}{1+\sqrt{\tan x}} d x=$

$\int\limits_{\frac{-1}{24}}^{\frac{1}{24}} \sec x \log \left(\frac{1-x}{1+x}\right) d x=$

If $[x]$ is the greatest integer function, then $\int_0^5[x] d x=$

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

Order and degree of the differential equation $\frac{d^3 y}{d x^3}=\left[1+\left(\frac{d y}{d x}\right)^2\right]^{\frac{5}{2}}$, respectively are

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

Integrating factor of the differential equation $\sin x \frac{d y}{d x}-y \cos x=1$ is

If $y=t^2+t^3$ and $x=t-t^4$, then $\frac{d^2 y}{d x^2}$ at $t=1$ is

If the chord of the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ having $(1,1)$ as its middle point is $x+\alpha y=\beta$, then

$f: R \rightarrow R$ is defined by $f(x+y)=f(x)+12 y, \forall x, y \in R$. If $f(1)=6$, then $\sum_{r=1}^n f(r)=$

The domain of the real valued function $f(x)=\sqrt{2+x}+\sqrt{3-x}$ is

If $l_1$ and $l_2$ are the lengths of the perpendiculars drawn from a point on the hyperbola $5 x^2-4 y^2-20=0$ to its asymptotes, then $\frac{l_1{ }^2 l_2{ }^2}{100}=$

If a directrix of a hyperbola centred at the origin and passing through the point $(4,-2 \sqrt{3})$ is $\sqrt{5} x=4$ and e is its eccentricity, then $e^2=$

If $\frac{x^2+3}{x^4+2 x^2+9}=\frac{A x+B}{x^2+a x+b}+\frac{C x+D}{x^2+c x+b}$, then $a A+b B+c C+D=$

$\int \frac{d x}{x\left(x^4+1\right)}=$

$\int \sin ^{-1} \sqrt{\frac{x}{a+x}} d x=$

$\int \frac{d x}{\sqrt{\sin ^3 x \cos (x-a)}}=$

$\int \frac{2-\sin x}{2 \cos x+3} d x=$

$\int \frac{e^{2 x}}{\sqrt[4]{e^x+1}} d x=$

If $\theta=\sec ^{-1}(\cosh u)$, then $u=$

If $\cos ^{-1} 2 x+\cos ^{-1} 3 x=\frac{\pi}{3}$ and $4 x^2=\frac{a}{b}$, then $a+b$ is equal to

If $f(x)=\left\{\begin{array}{cc}2 x+3, & x \leq 1 \\ a x^2+b x, & x>1\end{array}\right.$
is differentiable, $\forall x \in R$, then $f^{\prime}(2)=$

The values of $a$ and $b$ for which the function$ f(x)=\left\{\begin{array}{cl}1+|\sin x|^{\frac{a}{\sin x \mid}} & \frac{-\pi}{6} < x < 0 \\ b, & x=0 \quad \text { is continuous at } x=0 \\ e^{\frac{\tan 2 x}{\tan 3 x},} & 0 < x < \frac{\p...

If $\lim \limits_{x \rightarrow 0} \frac{e^x-a-\log (1+x)}{\sin x}=0$, then $a=$

$\lim \limits_{x \rightarrow \frac{\pi}{4}} \frac{4 \sqrt{2}-(\cos x+\sin x)^5}{1-\sin 2 x}=$

In the interval $[0,3]$ The function $f(x)=|x-1|+|x-2|$ is

$\left|\begin{array}{ccc}a & b & c \\ a^2 & b^2 & c^2 \\ 1 & 1 & 1\end{array}\right|$ is not equal to

Let $A, B, C, D$ and $E$ be $n \times n$ matrices each with non-zero determinant. If $A B C D E=I$, then $C^{-1}=$

If $A=\left[a_{i j}\right], 1 \leq i, j \leq n$ with $n \geq 2$ and $a_{i j}=i+j$ is a matrix, then the rank of $A$ is

The number of ways in which 17 apples can be distributed among four guests such that each guest gets at least 3 apples is .

The number of numbers lying between 1000 and 10000 such that every number contains the digit 3 and 7 only once without repetition is

A test containing 3 objective type of questions is conducted in a class. Each question has 4 options and only one option is the correct answer. No two students of the class have answered identically and no student has written all correct an...

For a binomial variate $X \sim B(n, p)$ the difference between the mean and variance is 1 and the difference between their square is 11 . If the probability of $P(x=2)=m\left(\frac{5}{6}\right)^n$ and $n=36$, then $m: n$

A box $P$ contains one white ball, three red ball and two black balls. Another box $Q$ contains two white balls, three red balls and four black balls. If one ball is drawn at random from each one of the two boxes, then the probability that ...