AP EAPCET 2024 - 20th May Morning Shift

If a circle is inscribed in an equilateral triangle of side $a$, then the area of any square (in sq units) inscribed in this circle is

Let $\alpha$ be an integer multiple of 8 . If $S$ is the set of all possible values of $\alpha$ such that the line $6 x+8 y+\alpha=0$ intersects the circle $x^2+y^2-4 x-6 y+9=0$ at two distinct points, then the number of elements in $S$ is

If the circle $x^2+y^2-8 x-8 y+28=0$ and $x^2+y^2-8 x-6 y+25-\alpha^2=0$ have only one common tangent, then $\alpha=$

If the point $P$ represents the complex number $z=x+i y$ in the argand plane and if $\frac{z+i}{z-i}$ is a purely imaginary number, then the locus of $P$ is

$S=\{z \in C /|z+1-i|=1\}$ represents

\(\text { } \int\limits_{-3}^3|2-x| d x=\)

$\int_{-\pi}^\pi \frac{x \sin ^3 x}{4-\cos ^2 x} d x=$

\(\int_{\frac{1}{\sqrt[5]{31}}}^{\frac{1}{\sqrt[5]{242}}} \frac{1}{\sqrt[5]{x^{30}+x^{25}}} d x=\)

$$ \begin{aligned} &\text { The general solution of the differential equation }\\ &(x+y) y d x+(y-x) x d y=0 \text { is } \end{aligned} $$

The sum of the order and degree of the differential equation $\frac{d^4 y}{d x^4}=\left\{c+\left(\frac{d y}{d x}\right)^2\right\}^{3 / 2}$ is

The general solution of the differential equation $\left(y^2+x+1\right) d y=(y+1) d x$ is

For $x<0, \frac{d}{d x}\left[|x|^x\right]=$

If $y=x-x^2$, then the rate of change of $y^2$ with respect to $x^2$ at $x=2$ is

If $y=\tan (\log x)$, then $\frac{d^2 y}{d x^2}=$

Let F and $F^1$ be the foci of the ellipse $\frac{x^2}{4}+\frac{y^2}{b^2}=1(b<2)$ and $B$ is one end of the minor axis. If the area of the triangle $\mathrm{FBF}^1$ is $\sqrt{3}$ sq units, then the eccentricity of the ellipse is

If $P(x)=x^5+a x^4+b x^3+c x^2+d x+e$ is a polynomial such that $P(0)=1, P(1)=2, P(2)=5, P(3)=10$ and $P(4)=17$, then $P(5)=$

Let $f(x)=3+2 x$ and $g_n(x)=(f \circ f \circ f o \ldots$ in times $)(x)$, $\forall n \in N$ if all the lines $y=g_n(x)$ pass through a fixed point $(\alpha, \beta)$, then $\alpha+\beta=$

Let $a > 1$ and $0 < \mathrm{b} < 1$. If $f: R \rightarrow[0,1]$ is defined by $f(x)=\left\{\begin{array}{ll}a^x, & -\infty < x < 0 \\ b^x, & 0 \leq x < \infty\end{array}\right.$, then $f(x)$ is

If a tangent to the hyperbola $x^2-\frac{y^2}{3}=1$ is also a tangent to the parabola $y^2=8 x$, then equation of such tangent with the positive slope is

If a circle of radius 4 cm passes through the foci of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{4}=1$ and concentric with the hyperbola, then the eccentricity of the conjugate hyperbola of that hyperbola is

If $\int \sqrt{\frac{2}{1+\sin x}} d x=2 \log |A(x)-B(x)|+C$ and $0 \leq x \leq \frac{\pi}{2}$, then $B\left(\frac{\pi}{4}\right)=$

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

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

$\int \frac{3 x^9+7 x^8}{\left(x^2+2 x+5 x^8\right)^2} d x=$

$$ \begin{aligned} &\text { If } \int \frac{3}{2 \cos ^3 x \sqrt{2 \sin 2 x}} d x=\frac{3}{2}(\tan x)^B+\frac{3}{10}(\tan x)^A+C \text {, than }\\&A= \end{aligned} $$

$\int \frac{2 x^2-3}{\left(x^2-4\right)\left(x^2+1\right)} d x=A \tan ^{-1} x+B \log (x-2)+C \log (x+2)$, then $6 A+7 B-5 C=$

If $\cos ^4 \frac{\pi}{8}+\cos ^4 \frac{3 \pi}{8}+\cos ^4 \frac{5 \pi}{8}+\cos ^4 \frac{7 \pi}{8}=k$, then $\sin ^{-1}\left(\sqrt{\frac{k}{2}}\right)+\cos ^{-1}\left(\frac{k}{3}\right)=$

\(4 \tan ^{-1} \frac{1}{5}-\tan ^{-1} \frac{1}{70}+\tan ^{-1} \frac{1}{99}=\)

If a function $f(x)=\left\{\begin{array}{cl}\frac{\tan (\alpha+1) x+\tan 2 x}{x} & \text { if } x>0 \\ \beta & \text { at } x=0 \text { is } \\ \frac{\sin 3 x-\tan 3 x}{x^3} & \text { if } x<0\end{array}\right.$
continuous at $x=0$, then $...

$\lim \limits_{x \rightarrow 0} \frac{\sqrt{1+\sqrt{1+x^4}}-\sqrt{2+x^5+x^6}}{x^4}=$

$\lim \limits_{x \rightarrow 1} \frac{\sqrt{x}-1}{\left(\cos ^{-1} x\right)^2}=$

If the solution of the system of simultaneous linear equations $x+y-z=6,3 x+2 y-z=5$ and $2 x-y-2 z+3=0$ is $x=\alpha, y=\beta, z=y$, then $\alpha+\beta=$

Sum of the positive roots of the equation

$$ \left|\begin{array}{ccc} x^2+2 x & x+2 & 1 \\ 2 x+1 & x-1 & 1 \\ x+2 & -1 & 1 \end{array}\right|=0 \text { is } $$

$$ \text { If } A=\left[\begin{array}{lll} 1 & 0 & 2 \\ 2 & 1 & 3 \\ 3 & 2 & 4 \end{array}\right] \text {, then } A^2-5 A+6 I= $$

A common tangent to the circle $x^2+y^2=9$ and parabola $y^2=8 x$ is

The number of positive divisors of 1080 is

If a polygon of $n$ sides has 275 diagonals, then $n$ is

If $a_n=\sum\limits_{r=0}^n \frac{1}{{ }^n C_r}$, then $\sum\limits_{r=0}^n \frac{r}{{ }^n C_r}=$

A, B and C are mutually exclusive and exhaustive events of a random experiment and $E$ is an event that occurs in conjunction with one of the events $\mathrm{A}, \mathrm{B}$ and $C$. The conditional probabilities of $E$ given the happening ...

Two digits are selected at random from the digits 1 through 9. If their sum is even, then the probability that both are odd, is

For the probability distribution of a discrete random variable $X$ as given below, then mean of $X$ is

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S is the sample space and $A, B$ are two events of a random experiment. Match the items of List $A$ with the items of List B

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In a random experiment, two dice are thrown and the sum of the numbers appeared on them is recorded. This experiment is repeated 9 times. If the probability that a sum of 6 appears atleast once is $P_1$ and a sum of 8 appears atleast once i...

$P(A \mid A \cap B)+P(B \mid A \cap B)=$

In $\triangle A B C$, if $r_1=4, r_2=8$ and $r_3=24$, then $a=$

Match the items of List I with those of List II (here, $\Delta$ denotes the area of $\triangle A B C$ )

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If $2 x^2+3 x-2=0$ and $3 x^2+\alpha x-2=0$ have one common root, then the sum of all possible values of $\alpha$ is

If $\alpha$ and $\beta$ are two double roots of $x^2+3(a+3) x-9 a=0$ for different values of $a(\alpha>\beta)$, then the minimum value of $x^2+\alpha x-\beta=0$ is

If the sum of two roots of $x^3+p x^2+q x-5=0$ is equal to its third root, then $p\left(p^2-4 q\right)=$

$ \frac{1}{3 \cdot 7}+\frac{1}{7 \cdot 11}+\frac{1}{11 \cdot 15}+\ldots$ to 50 terms $=$