AP EAPCET 2023 - 15th May Morning Shift

Let the circle $S \equiv x^2+y^2+2 g x+2 f y+c=0$ cut the circles $x^2+y^2-2 x+2 y-2=0$ and $x^2+y^2+4 x-6 y+9=0$ orthogonally. If the centre of the circle $S=0$ lies on the line $2 x+3 y-2=0$, then $2 g+f=$

The equation of the pair of tangents drawn from the point $(1,1)$ to the circle $x^2+y^2+2 x+2 y+1=0$ is

The locus of a point which is at a distance of 2 units from the line $2 x-3 y+4=0$ and at a distance of $\sqrt{13}$ units from a point $(5,0)$, is

If the chord of contact of the point $P(h, k)$ with respect to the circle $x^2+y^2-4 x-4 y+8=0$ meets the circle in two distinct points and it also makes an angle $45^{\circ}$ with the positive $X$-axis in the positive direction, then $(h, ...

Let $S$ be the circumcircle of the triangle formed by the line $x-2 y-4=0$ with the coordinate axes. If $P(-2,-4)$ is a point in the plane of the circle $S$ and $Q$ is a point on $S$ such that the distance between $P$ and $Q$ is the least, ...

$\left[\sqrt{2}\left(\cos 56^{\circ} 15^{\prime}+i \sin 56^{\circ} 15^{\prime}\right)\right]^8=$

If $z_1=(2,-1)$ and $z_2=(6,3)$, then $\operatorname{amp}\left(\frac{z_1-z_2}{z_1+z_2}\right)=$

The number of all possible solutions of the equation $z^3+\bar{z}=0$ is

$(1+i)^{2024}+(1-i)^{2024}=$

\(\int_0^{50 \pi} \sqrt{1-\cos 2 x} d x=\)

If $f(x)=\frac{x^3+5}{\sqrt{12+x}}$ and $\int_{-5}^5 f(x) d x=\int_0^5(f(x)+g(x)) d x$, then $g(x)=$

If $[x]$ is the greatest integer not exceeding $x$, then \(\int_{-0.5}^{1.5} x^2[x] d x=\)

If $I=\int_{-a}^a\left(x^4-2 x^2\right) d x$, then $I$ is minimum at $a=$

Assertion (A) $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{(\sin x)^{\sqrt{2}} d x}{(\sin x)^{\sqrt{2}}+(\cos x)^{\sqrt{2}}}=\frac{\pi}{12}$
Reason (R) $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{f(x) d x}{f(x)+f\left(\frac{\pi}{2}-x\right)}...

The order and degree of the differential equation $\left(\frac{d^3 y}{d x^3}\right)^{\frac{1}{2}}-2\left(\frac{d y}{d x}\right)^{\frac{1}{4}}+x y=0$ are respectively

The substitution $x=v y$ converts which one of the following differential equation to an equation solvable by variable separable method?

If $\frac{d y}{d x}=f(x, y)$ is a homogeneous differential equation, then the general form of $f(x, y)$ is

If $y=\frac{\log x}{x}$, then the value of $x^2 \frac{d^2 y}{d x^2}+3 x \frac{d y}{d x}+y$ at the point $(\sqrt[3]{e}, \sqrt{e})$ is

Let $x^2+y^2=20$ be the director circle of an ellipse $E$ whose major axis is $X$-axis and minor axis is $Y$-axis. If the length of the latusrectum of $E$ is 2 . Then, the distance between its foci is

If $f(x)=x^3-x$ and $g(x)=\sin 2 x$, then $f\left(g\left(\frac{\pi}{12}\right)\right)=$

For $x \in R$ if $f(x)=\sqrt{\log _{10}\left(\frac{3-x}{x}\right)}$, then the domain of $f$ is

The difference between the focal distances of any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is 6. If $(\sqrt{13}, k)$ is an end point of a latusrectum of this hyperbola, then $k=$

If $\theta$ is the acute angle between the tan
from the point $(2,3)$ to the hyperbola from the point $(2,3)$ to the hyperbola $5 x^2-6 y^2-30=0$, then $\tan \theta=$

If $\int \frac{4 e^x+6 e^{-x}}{9 e^x-4 e^{-x}} d x=A x+B \log \left|\left(9 e^{2 x}-4\right)\right|+C$, then $(A, B)=$

If $\frac{2 x^2+5 x+6}{(x+2)^3}=\frac{a}{x+2}+\frac{b}{(x+2)^2}+\frac{c}{(x+2)^3}$, then $a \cdot b+b \cdot c+c \cdot a=$

If $f(x)=\int \frac{d x}{\left(x^2+2\right)}$ and $f(\sqrt{2})=0$, then $f(0)=$

\(\int e^x\left(\frac{2+\sin 2 x}{1+\cos 2 x}\right) d x=\)

If $f(x)=\left\{\begin{array}{cl}1+6 x-3 x^2, & x \leq 1 \\ x+\log _2\left(b^2+7\right), & x>1\end{array}\right.$ is continuous at all real $x$, then $b=$

If $f: R \rightarrow R$ defined by
$$ f(x)= \begin{cases}\frac{\sin x-\sin \frac{X}{2}}{x}, & x<0 \\ \frac{\sqrt{x^2+x}-\sqrt{x}}{x^{3 / 2}}, & x>0\end{cases} $$
is continuous on $R$, then $f(0)=$

Let $f(x)$ be a real valued function. If $f^{\prime}(x)$ is a constant for all $x \in R, f(0)=2$ and $f^{\prime}(0)=1$, then

If $S=\left[\begin{array}{lll}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{array}\right]$ and $A=\frac{1}{2}\left[\begin{array}{lll}b+c & c-a & b-a \\ c-b & c+a & a-b \\ b-c & a-c & a+b\end{array}\right]$, then $S A S^{-1}=$

Let $A=\left(\begin{array}{l}0 \\ -6 \\ 8\end{array}\right), B=\left(\begin{array}{ccc}3 & 5 & -7 \\ 0 & -1 & 8 \\ 6 & -1 & 0\end{array}\right)$ and $X=\left(\begin{array}{l}x \\ y \\ z\end{array}\right)$. If $D=[\alpha \beta \gamma]^T$ is ...

If $A=\left[\begin{array}{cc}i & 0 \\ 0 & -i\end{array}\right], B=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]$ and $C=\left[\begin{array}{cc}0 & i \\ i & 0\end{array}\right]$, then

Let the equation of the tangent at a point $P$ on the parabola $x^2-4 x-4 y+16=0$ be $2 x-y-5=0$. If the equation of the normal drawn at $P$ to this parabola is $a x+y+c=0$, then $a c=$

The number of natural numbers less than 10000 which are divisible by 5 and that no digit is repeated in the same number, is

A team of 5 students is to be selected from 12 students. If two particular students are to be included in that team, then the number of ways that such team can be selected is

If the number of diagonals of a regular polygon of $n$ sides is 104 , then $n=$

Bag $B_1$ contains 4 white and 2 black balls. Bag $B_2$ contains 3 white and 4 black balls. A bag is chosen at random and a ball is drawn from it at random, then the probability that the ball drawn is white, is

If eight coins are tossed simultaneously, then the probability of getting atleast six heads is

If two subsets $A$ and $B$ are selected at random from a set $S$ containing $n$ elements, then the probability that $A \cap B=\phi$ and $A \cup B=S$, is

If four dice are thrown simultaneously, then the probability that none of the dice shows the number 1 on its face, is

The range of a random variable $X$ is $\{0,1,2\}$. If $P(X=0)=3 C^3, P(X=1)=4 C-10 C^2$ and $P(X=2)=5 C-1$, then the value of $C$ is

In a game, a pair of dice is rolled 24 times. If a person whis the game by not getting 6 on both the dice in any one of the 24 rolls, then the probability that a person wins the game is

In $\triangle A B C$, if $\frac{1}{r_1}, \frac{1}{r_2}$ and $\frac{1}{r_3}$ are in arithmetic progression, then $r_2: r=$

If $P_1, P_2$ and $P_3$ are the lengths of the altitudes drawn from the vertices $A, B$ and $C$ of $\triangle A B C$ respectively, then $\frac{\cos A}{P_1}+\frac{\cos B}{P_2}+\frac{\cos C}{P_3}=$

In $\triangle A B C$, if $\cot \frac{A}{2}: \cot \frac{B}{2}: \cot \frac{C}{2}=3: 7: 9$, then $a: b: c=$

If $(3+2 \sqrt{2})^{x^2-4}+(3-2 \sqrt{2})^{x^2-4}=6$, then $x^4+x^2+5=$

If the equation $x^4+a x^3+b x^2+c x+d=0$ has three equal roots, then that root is

If the values of $k$ for which the euqation $x^2+2(k+2) x+6 k+7=0$ has equal roots are $k_1$ and $k_2$, then $k_1^2+k_2^2=$

If -1 is a twice repeated root of the equation $a\left(x^3+x^2\right)+b x+c=0$, then $a: b: c=$