AP EAPCET 2023 - 15th May Evening Shift

If the points of contact of the tangents drawn from $(0,0)$ to the curve $y=x^2+3 x+4$ are $(\alpha, \beta)$ and $(\gamma, \delta)$, then $\beta+\delta=$

If $(2, a)$ and $(b, 19)$ are two stationary points of the curve $y=2 x^3-15 x^2+36 x+c$, then $a+b+c=$

If the angle between the curves $y=e^{2(1+x)-4}$ and $x^2 y=1$ at the point $(1,1)$ is $\theta$, then $|\sin \theta|+|\cos \theta|=$

The area (in sq units) of the region bounded by the curves $y=4|\cos x|$ and $y=-|\cos x|$ from $x=-\frac{\pi}{2}$ to $\frac{\pi}{2}$ is

The coefficient of the highest power of $x$ in the expansion of $\left(x+\sqrt{x^2-1}\right)^8+\left(x-\sqrt{x^2-1}\right)^8$ is

If $C_j={ }^n C_j$, then $C_0 C_r+C_1 C_{r+1}+C_2 C_{r+2}+\ldots+C_{n-r} C_n=$

Let $A(1,2)$ be the centre and 3 be the radius of a circle $S$. Let $B(-1,-1)$ be the centre and $r$ be the radius of another circle $S^{\prime}$. If $\frac{\pi}{3}$ is the angle between the circles $S$ and $S^{\prime}$, then the number of ...

The distance between the centres of similitude of the circles $x^2+y^2+6 x-8 y+16=0$ and $x^2+y^2-2 x-2 y+1=0$ is

If $P\left(\frac{\pi}{3}\right)$ and $Q\left(\frac{2 \pi}{3}\right)$ represent two points on the circle $x^2+y^2-4 x+6 y-12=0$ in parametric form, then the length of the chord $P Q$ is

Let $A(2,3), B(3,-1)$ and $C(-3,2)$ be three points. If the centre of the circle passing through $A, B$ and $C$ is $(h, k)$, then $2 k-4 h=$

Let $P$ and $Q$ be the inverse points with respect to the circle $S \equiv x^2+y^2-4 x-6 y+k=0$ and $C$ be the centre of the circle $S=0$ such that $C P \cdot C Q=4$. If $P=(1,2)$ and $Q=(a, b)$, then $2 a=$

\(S=\{z \in C /|z-1+i|=1\} \text { represents }\)

If $\left|z-\frac{2}{z}\right|=2$, then the greatest value of $|z|$ is

One of the 15 th roots of -1 is

The product of the four values of $(1+i \sqrt{3})^{3 / 4}$ is

If $f(x)=\int_0^x\left[(a+1)(t+1)^2-(a-1)\left(t^2+t+1\right)\right] d t$, then a possible positive value of ' $a$ ', for which $f^{\prime}(x)=0$ has equal roots, is

\(\int_0^1(\sqrt{10})^{2 x} d x=\)

\(\int_0^{\frac{\pi}{4}} \frac{\cos ^2 x}{\cos ^2 x+4 \sin ^2 x} d x=\)

\(\int_0^{\frac{\pi}{2}}\left(\frac{\sin \left(\frac{\pi}{4}+x\right)+\sin \left(\frac{3 \pi}{4}+x\right)}{\cos x+\sin x}\right) d x=\)

If $\alpha$ and $\beta$ are respectively, the order and degree of the differential equation $y=e^{\left(\frac{d y}{d x}+\frac{d^2 y}{d x^2}\right)}$, then the value of $\alpha+\alpha^\beta+\alpha^{2 \beta}+\ldots+\alpha^{2023 \beta}=$

The general soluiton of the differential equation $\tan x \tan y d x+\cos ^2 x \operatorname{cosec}^2 y d y=0$ is

If $x^\alpha \frac{d y}{d x}=y^\beta(\gamma \log x+\delta \log y+1)$ is a homogeneous differential equation, then

If $f(x+a y)+g(x-a y)=0$, then $a \frac{d y}{d x}=$

If $x^x y^y=e^e$, then $\left(\frac{d^2 y}{d x^2}\right)_{(e, e)}=$

If $f(x)=|x-5|+|x+5|+|x-4|+|x+4|$, then $\frac{f^{\prime}(1)-f^{\prime}(-6)}{f^{\prime}(-1)+f^{\prime}(6)}=$

Let the length of the latusrectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ be equal to the length of its semi-major axis. If the radius of its director circle is $\sqrt{3}$ and $e$ is its eccentricity, then the length of its latusr...

If $f(a)=\log \left|\frac{1-a}{1+a}\right|$ for $a \neq\{-1,1\}$, then the set of values of all ' $a$ ', for which $f\left(\frac{2 a}{1+a^2}\right)>0$ is

If a real valued function $f$ is defined by $f(x)=\frac{a x+\sqrt{a^2-x^2}}{b x}$, then $f$ is

Let $(1,2)$ be the focus and $x+y+1=0$ be the directrix of a hyperbola $H$. If $\sqrt{3}$ is the eccentricity of $H$, then its equation is

If $A(4,0)$ and $B(-4,0)$ are two points, then the locus of a point $P$ such that $P A-P B=4$ is

If $S \equiv \frac{x^2}{k-7}+\frac{y^2}{11-k}-1=0, k \in R-\{7,11\}$, then which one of the following statements is incorrect?

If $\frac{x+2}{x^2-3}$ is one of the partial fractions of $\frac{3 x^3-x^2-2 x+17}{x^4+x^2-12}$, then the other partial fraction of it is

If $f(x)=\int \frac{5 x^8+7 x^6}{\left(x^2+2 x^7+1\right)^2} d x(x \geq 0)$ and $f(0)=0$, then the value of $f(1)=$

\(\int x^3(\log x)^2 d x=\)

$\int \frac{d x}{4+5 \cos x}=$

If $\int \frac{1}{x\left[(\log x)^2+4 \log x-1\right]} d x=A \log \left[\frac{\log x+B}{\log x+C}\right]+K$,
where $K$ is the constant of integration, then

Let $[x]$ represents the greatest integer not more than $x$. The discontinuous points of the function $f(x)=\frac{5+[x]}{\sqrt{11+[x]-6 \sqrt{2+[x]}}}$ lies in the interval

If $f(x)=\left\{\begin{array}{cc}\frac{x-[x]}{x-2}, & x > 2 \\ b, & x=2 \\ \frac{\left|x^2-x-2\right|}{a\left(2+x-x^2\right)}, & -1 < x \leq 2 \\ 2 a-b, & x \leq-1\end{array}\right.$
is continuous on $R$, then $\lim _{x \rightarrow 0} \frac...

The function $f(x)=\sqrt{\frac{3 x^2-5 x-2}{2 x^2-7 x+5}}$ has discontinuous points at $x=$

If the co-factors of the elements 3, 7 and 6 of the matrix $\left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & -1 & 7 \\ 2 & 4 & 6\end{array}\right]$ are $a, b$ and $c$ respectively, then $\left[\begin{array}{lll}a & b & c\end{array}\right]\left[\be...

Let $B=\left(\begin{array}{ccc}2 & 6 & 4 \\ 1 & 0 & 1 \\ -1 & 1 & -1\end{array}\right)$ and $C=\left(\begin{array}{ccc}-1 & 0 & 1 \\ 1 & 1 & 3 \\ 2 & 0 & 2\end{array}\right)$. If a matrix $A$ is such that $B A C=I$, then $A^{-1}=$

If $\operatorname{det}(A B)=(\operatorname{det} A)(\operatorname{det} B)$ and $A$ is a non-singular matrix of order $3 \times 3$, then $\operatorname{det}(\operatorname{adj} A)=$

If $A=\left\{\left(\begin{array}{ll}a & b \\ c & d\end{array}\right): a, b, c, d \in\{-1,1\}\right\}$, then the number of singular matrices in $A$ is

The perpendicular distance from the origin to the focal chord drawn through the point $(4,5)$ to the parabola $y^2-4 y-3 x+7=0$ is

The number of six digit natural numbers that can be formed with the digits $2,3,4,0,5,6,7,8$ is

The number of non-negative integral solutions of $x_1+x_2+x_3+x_4=10$ is

Three screws are drawn at random from a lot of 50 screws containing 5 defective ones. Then, the probability of the event that all 3 screws drawn are non-defective, assuming that the drawing is (a) with replacement (b) without replacement re...

From a collection of eight cards numbered 1 to 8 , if two cards are drawn at random, one after the other with replacement, then the probability that the product of numbers that appear on the cards is a perfect square is

If $P(X=x)=k\left(\frac{3}{8}\right)^x, x=1,2,3 \ldots$ is the probability distribution function of a discrete random variable $X$, then $k=$

A coin is tossed three times. Let $A$ be the event of "getting three heads" and $B$ be the event of "getting a head on the first toss". Then, $A$ and $B$ are