AP EAPCET 2024 - 19th May Evening Shift

If Rolle's theorem is applicable for the function $f(x)=x(x+3) e^{-x / 2}$ on $[3,0]$, then the value of $c$ is

If $y=\left(1+\alpha+\alpha^2+\ldots\right) e^{\eta x}$, where $\alpha$ and $n$ are constants, then the relative error in $y$ is

If the equation of tangent at $(2,3)$ on $y^2=a x^3+b$ is $y=4 x-5$, then the value of $a^2+b^2=$

For all $x \in[0,2024]$ assume that $f(x)$ is differentiable, $f(0)=-2$ and $f^{\prime}(x) \geq 5$. Then, the least possible value of $f(2024)$ is

The area (in sq units) of the smaller region lying above the $X$-axis and bounded between the circle $x^2+y^2=2 a x$ and the parabola $y^2=a x$ is

Numerically greatest term in the expansion of $(5+3 x)^6$ When, $x=1$, is

If the circle $S=0$ cuts the circles $x^2+y^2-2 x+6 y=0$, $x^2+y^2-4 x-2 y+6=0$ and $x^2+y^2-12 x+2 y+3=0$ orthogonally, then equation of the tangent at $(0,3)$ on $S=0$ is

If the circles $S \equiv x^2+y^2-14 x+6 y+33=0$ and $S^1 \equiv x^2+y^2-a^2=0(a \in N)$ have 4 common tangents, then possible number of values of $a$ is

The angle subtended by the chord $x+y-1=0$ of the circle $x^2+y^2-2 x+4 y+4=0$ at the origin is

Let $P$ be any point on the circle $x^2+y^2=25$. Let $L$ be the chord of contact of $P$ with respect to the circle $x^2+y^2=9$. The locus of the poles of the lines $L$ with respect to the circle $x^2+y^2=36$ is

If the area of the circum-circle of triangle formed by the line $2 x+5 y+\alpha=0$ and the positive coordinate axes is $\frac{29 \pi}{4} S q$, units, then $|\alpha|=$

The circle $S \equiv x^2+y^2-2 x-4 y+1=0$ cuts the $Y$-axis at $A, B(O A>O B)$. If the radical axis of $S \equiv 0$ and $S' \equiv x^2+y^2-4 x-2 y+4=0$ cuts the $Y$-axis at $C$, then the ratio in which $C$ divides $A B$ is

Real part of $\frac{(\cos a+i \sin a)^6}{(\sin b+i \cos b)^8}$ is

If $m, n$ are respectively the least positive and greatest negative integer value of $k$ such that $\left(\frac{1-i}{1+i}\right)^k=-i$, then $m-n=$

If a complex number $z$ is such that $\frac{z-2 i}{z-2}$ is purely imaginary number and the locus of $z$ is a closed curve, then the area of the region bounded by that closed curve and lying in the first quadrant is $\frac{z-2 i}{z-2}$

\(\int_0^\pi x \sin ^4 x \cos ^6 x d x=\)

If $I_n=\int_0^{\frac{\pi}{4}} \tan ^n x d x$, then $I_{13}+I_{11}=$

If $\lim \limits_{n \rightarrow \infty}\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{4}{n^2}\right)\left(1+\frac{9}{n^2}\right) \ldots\left(1+\frac{n^2}{n^2}\right)\right]^{\frac{1}{n}}=a e^b$, then

\(a+b=\)

The difference of the order and degree of the differential equation $\left(\frac{d^2 y}{d x^2}\right)^{-\frac{7}{2}}\left(\frac{d^3 y}{d x^3}\right)^2-\left(\frac{d^2 y}{d x^2}\right)^{-\frac{5}{2}}\left(\frac{d^4 y}{d x^4}\right)=0$ is

The solution of $x d y-y d x=\sqrt{x^2+y^2} d x$ when $y(\sqrt{3})=1$ is

If $x d y+\left(y+y^2 x\right) d x=0$ and $y=1$ at $x=1$, then

If $y=f(x)$ is a thrice differentiable function and a bijection, then $\frac{d^2 x}{d y^2}\left(\frac{d y}{d x}\right)^3+\frac{d^2 y}{d x^2}=$

If a tangent of slope 2 to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ touches the circle $x^2+y^2=4$, then maximum value of $a b$ is

The domain of the real valued function $f(x)=\frac{1}{\sqrt{\log _{0.5}(2 x-3)}}+\sqrt{4-9 x^2}$ is

If a real valued function $f:[a, \infty) \rightarrow[b, \infty)$ defined by $f(x)=2 x^2-3 x+5$ is a bijection. Then, $3 a+2 b=$

If the product of eccentricities of the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ and the hyperbola $\frac{x^2}{9}-\frac{y^2}{16}=-1$ is 1 , then $b^2=$

The locus of the mid-points of the chords of the hyperbola $x^2-y^2=a^2$ which touch the parabola $y^2=4 a x$ is

\(\int\left(\frac{x}{x \cos x-\sin x}\right)^2 d x=\)

If $\int \frac{\log \left(1+x^4\right)}{x^3} d x=f(x) \log \left(\frac{1}{g(x)}\right)+\tan ^{-1}$
$(h(x))+c$, then $h(x)\left[f(x)+f\left(\frac{1}{x}\right)\right]=$

If $\frac{1}{x^4+1}=\frac{A x+B}{x^2+\sqrt{2} x+1}+\frac{C x+D}{x^2-\sqrt{2} x+1}$, then $B D-A C=$

\(\int \frac{2 \cos 2 x}{(1+\sin 2 x)(1+\cos 2 x)} d x=\)

Let $f(x)=\int \frac{x}{\left(x^2+1\right)\left(x^2+3\right)} d x$. If $f(3)=\frac{1}{4} \log \left(\frac{5}{6}\right)$, then $f(0)=$

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

\(\cosh \left(\sinh ^{-1}(\sqrt{8})+\cosh ^{-1} 5\right)=\)

The function $f(x)=\left\{\begin{array}{ll}\frac{2}{5-x}, & x<3 \\ 5-x, & x \geq 3\end{array}\right.$ is

\(\lim \limits_{x \rightarrow 3} \frac{x^3-27}{x^2-9}=\)

If $f(x)=\left\{\begin{array}{ll}3 a x-2 b, & x>1 \\ a x+b+1, & x<1\end{array}\right.$ and
$\lim \limits_{x \rightarrow 1} f(x)$ exists, then the relation between $a$ and $b$ is

If $f(x)=\left\{\begin{array}{cl}x^\alpha \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x=0\end{array}\right.$
which of the following is true?

Let $f(x)=\min \left\{x, x^2\right\}$ for every real number of $x$, then

The system of equations

\(x+2 y+3 z=6, x+3 y+5 z=9 \text {, }\)

$2 x+5 y+a z=12$ has no solution when $a=$

$$ \left|\begin{array}{ccc} 1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{array}\right|= $$

If $A=\left[\begin{array}{cc}1 & 2 \\ -2 & -5\end{array}\right]$ and $\alpha A^2+\beta A=2 I$ for some $\alpha, \beta \in R$, then $\alpha+\beta=$

The normal drawn at a point $(2,-4)$ on the parabola $y^2 \pm 8 x$ cuts again the same parabola at $(\alpha, \beta)$, then $\alpha+\beta=$

If Set $A$ contains 8 elements, then number of subsets of $A$ which contain at least 6 elements is

The number of different permutations that can be formed by taking 4 letters at a time from the letters of the word 'REPETITION' is

If all the letters of the word MASTER are permuted in all possible ways and words (with or without meaning) thus formed are arranged in dictionary order, then the rank of the word MASTER is

A bag contains 2 white, 3 green and 5 red balls. If three balls are drawn one after the other without replacement, then the probability that the last ball drawn was red is

Out of first 5 consecutive natural numbers, if two different numbers $x$ and $y$ are chosen at random, then the probability that $x^4-y^4$ is divisible by 5 is

In a consignment of 15 articles, it is found that 3 are defective. If a sample of 5 articles is chosen at random from it, then the probability of having 2 defective articles is

There are 2 bags each containing 3 white and 5 black balls and 4 bags each containing 6 white and 4 black balls. If a ball drawn randomly from a bag is found to be black, then the probability that this ball is from the first set of bags is