AP EAPCET 2024 - 22th May Evening Shift

Parametric equations of the circle $2 x^2+2 y^2=9$ are

The normal drawn at $(1,1)$ to the circle $x^2+y^2-4 x+6 y-4=0$ is

The triangle $P Q R$ is inscribed in the circle $x^2+y^2=25$. If $Q=(3,4)$ and $R=(-4,3)$, then $\angle Q P R$ is equal to

If $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ and $\alpha_5$ are the roots of $x^5-5 x^4+9 x^3-9 x^2+5 x-1=0$, then $\frac{1}{\alpha_1^2}+\frac{1}{\alpha_2^2}+\frac{1}{\alpha_3^2}+\frac{1}{\alpha_4^2}+\frac{1}{\alpha_5^2}$ is equal to

If $P(x, y)$ represents the complex number $z=x+iy$ in the argand plane and $\arg \left(\frac{z-3 i}{z+4}\right)=\frac{\pi}{2}$, then the equation of the locus of $P$ is

$\arg \left[\frac{(1+i \sqrt{3})(-\sqrt{3}-i)}{(1-i)(-i)}\right]$ is equal to

If $M=\int\limits_0^{\infty} \frac{\log t}{1+t^3} d t$ and $N=\int\limits_{-\infty}^{\infty} \frac{t e^{2 t}}{1+e^{3 t}} d t$, then

$\int\limits_{-2}^2\left(4-x^2\right)^{\frac{5}{2}} d x$ is equal to

$\int\limits_{-5 \pi}^{5 \pi}(1-\cos 2 x)^{\frac{5}{2}} d x$ is equal to

$\int_0^1 \sqrt{\frac{2+x}{2-x}} d x$ is equal to

\(\mathop {\lim }\limits_{x \to \infty }\left[\left(1+\frac{1}{n^3}\right)^{\frac{1}{n^3}}\left(1+\frac{8}{n^3}\right)^{\frac{4}{n^3}}\left(1+\frac{27}{n^3}\right)^{\frac{9}{n^3}} \ldots . .(2)^{\frac{1}{n}}\right] \text { is equaln }\)

The differential equation of the family of hyperbols having their centres at origin and their axes along coordinates axes is

The general solution of the differential equation $\left(x y+y^2\right) d x-\left(x^2-2 x y\right) d y=0$ is

If $f(x)=5 \cos ^3 x-3 \sin ^2 x$ and $g(x)=4 \sin ^3 x+\cos ^2 x$, then the derivative of $f(x)$ with respect to $g(x)$ is

If $y=\tan ^{-1} \frac{x}{1+2 x^2}+\tan ^{-1} \frac{x}{1+6 x^2}+\tan ^{-1} \frac{x}{1+12 x^2}$, then $\left(\frac{d y}{d x}\right)_{x=\frac{1}{2}}$ is equal to

If $y=1+x+x^2+x^3+\ldots \ldots \infty$ and $|x|<1$, then $y^{\prime \prime}$ is equal to

The length of the latusrectum of $16 x^2+25 y^2=400$ is

The range of the real valued function $f(x)=\frac{15}{3 \sin x+4 \cos x+10}$ is

Define the function, $f, g$ and $h$ from $R$ to $R$ such that $f(x)=x^2-1, g(x)=\sqrt{x^2+1}$ and $h(x)= \begin{cases}0, \text { if } & x \leq 0 \\ x, \text { if } & x \geq 0\end{cases}$ consider the following statements
(i) fog is invertib...

If the equation $\frac{x^2}{7-k}+\frac{y^2}{5-k}=1$ represents a hyperbola, then

The line $21 x+5 y=k$ touches the hyperbola $7 x^2-5 y^2=232$, then $k$ is equal to

$\int e^{4 x^2+8 x-4}(x+1) \cos \left(3 x^2+6 x-4\right) d x$ is equal to

$\int\left[(\log 2 x)^2+2 \log 2 x\right] d x$ is equal to

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

\(\text { If } \frac{13 x+43}{2 x^2+17 x+30}=\frac{A}{2 x+5}+\frac{B}{x+6} \text {, then } A+B \text { is equal to }\)

$\int \sin ^4 x \cos ^4 x d x$ is equal to

If $\int \log \left(6 \sin ^2 x+17 \sin x+12\right) \cos x d x=f(x)+c$, then $f\left(\frac{\pi}{2}\right)$ is equal to

$\cot \left(\sum\limits_{n=1}^{50} \tan ^{-1}\left(\frac{1}{1+n+n^2}\right)\right)$ is equal to

\(\mathop {\lim }\limits_{x \to \infty } \frac{[2 x-3]}{x} \text { is equal to }\)

$\mathop {\lim }\limits_{x \to 0}\frac{\cos 2 x-\cos 3 x}{4 x-\cos 5 x}$ is equal to $\cos 4 x-\cos 5 x$

If a real valued function $f(x)=\left\{\begin{array}{cl}\frac{2 x^2+(k+2) x+9}{3 x^2-7 x-6}, & \text { for } x \neq 3 \\ 1, & \text { for } x=3\end{array}\right.$ is continuous at $x=3$ and $l$ is a finite value, then $l-k$ is equal to

If $\alpha, \beta$ and $\gamma(\alpha<\beta<\gamma)$ are the values of $x$ such that $\left[\begin{array}{ccc}x-2 & 0 & 1 \\ 1 & x+3 & 2 \\ 2 & 0 & 2 x-1\end{array}\right]$ is a singular matrix, then $2 \alpha+3 \beta+4 \gamma$ is equal to

$A=\left[\begin{array}{lll}0 & 1 & 2 \\ 2 & 3 & 0 \\ 4 & 0 & 3\end{array}\right]$ and $B$ is a matrix such that $A B=B A$.If $A B$ is not an identity matrix, then the matrix that can be taken as $B$ is

The system of linear equations $x+2 y+z=-3$, $3 x+3 y-2 z=-1$ and $2 x+7 y+7 z=-4$ has

Equation of the line touching both parabolas $y^2=4 x$ and $x^2=-32 y$ is

If there are 6 alike fruits, 7 alike vegetables and 8 alike biscuits, then the number of ways of selecting any number of things out of them such that at least one from each category is selected, is

There were two women participating with some men in a chess tournament. Each participant played two games with the other. The number of games that the men played between themselves is 66 more than that of the men played with the women. Then...

The number of ways of arranging 9 men and 5 women around circular table, so that no two women come together are

A radar system can detect an enemy plane in one out of 10 consecutive scans. The probability that it cannot detect an enemy plane at least two times in four consecutive scans, is

If each of the coefficients $a, b$ and $c$ in the equation $a x^2+b x+c=0$ is determined by throwing a die, then the probability that the equation will have equal roots, is

$A$ and $B$ throw a pair of dice alternately and they note the sum of the numbers appearing on the dice. $A$ wins if he throws 6 before $B$ throws 7 and $B$ wins if he throws 7 before $A$ throws 6 . If $A$ begins then, the probability of hi...

If a random variable $X$ has the following probability distribution, then its variance is nearly
$$ \begin{array}{clllllll} \hline X=x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline P(X=x) & 0.05 & 0.1 & 2 K & 0 & 0.3 & K & 0.1 \\ \hline \end{arr...

$E_1$ and $E_2$ are two independent events of a random experiment such that $P\left(E_1\right)=\frac{1}{2}$ and $P\left(E_1 \cup E_2\right)=\frac{2}{3}$. Then, match the items of List I with the items of List II.
$$ \begin{array}{lll} \hlin...

A bag contains 4 red and 5 black balls. Another bag contains 3 red and 6 black balls. If one ball is drawn from first bag and two balls from the second bag at random. The probability that out of the three, two are black and one is red, is

In $a \triangle A B C$ if $r: R: r_2=1: 3: 7$, then $\sin (A+C)+\sin B$ is equal to

In a $\triangle A B C$, if $a=13, b=14$ and $c=15$, then $r_1=$

In $\triangle A B C,\left(r_1+r_2\right) \operatorname{cosec}^2 \frac{C}{2}$ is equal to

If ' $a$ ' is a rational number, then the roots of the equation $x^2-3 a x+a^2-2 a-4=0$ are

The set of all real values ' $a$ ' for which $-1<\frac{2 x^2+a x+2}{x^2+x+1}<3$ holds for all real values of $x$ is

The quotient, when $3 x^5-4 x^4+5 x^3-3 x^2+6 x-8$ is divided by $x^2+x-3$ is