TS EAMCET 2023 (Online) 13th May Evening Shift

The equation of the circle inscribed in a square formed by the lines $x+y-2=0, x+y-6=0, x-y+1=0$ and $x-y+5=0$ is

If the equation $2 x-3 y+3=0,2 x+y+1=0$ and $6 x+4 y+1=0$ represent the sides of a triangle, then the equation of the circle passing through the vertices of this triangle is

If $\alpha, \beta, \gamma$ and $\delta$ are the roots of the equation $x^4+x^2+1=0$ such that $\alpha+\beta=-1, \gamma+\delta=1, \alpha^2=\beta$ and $\gamma^2=-\delta$, then $\alpha^{2023}+\beta^{2023}+\gamma^{2022}+\delta^{2022}=$

If $\alpha$ is a multiple root of the equation $x^5-6 x^4+11 x^3-2 x^2-12 x+8=0$, then $3 \alpha^2-2 \alpha+1=$

If $i^2=-1$, then $(1+\sqrt{3} i)^{2022}-(\sqrt{3}-i)^{2022}=$

If $\alpha, \beta, \gamma$ are the real roots of the equation $18 x^3-15 x^2-4 x+4=0$ such that $\alpha=\beta$ and $\alpha>\gamma$, then $\alpha+\beta^2+\gamma^3=$

If $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^4+\left(\frac{\sqrt{3}-i}{\sqrt{3}+i}\right)^4=r$ cis $\theta$, then one of the values of $\sqrt{r \operatorname{cis} \theta}$ is

If $z=x+i y$ and the point $P$ in the argand plane represents $z$, then the locus of $z$ satisfying the equation $|z-2|+|z-2 i|=4$ is

When $3^{2023}$ is divided by 16 , the remainder obtained is

One of the values of $(\sqrt{3}-i)^{2 / 5}$ is

If $\alpha, \beta, \gamma$ are the roots of the equation $2 x^3+x^2-13 x+6=0$, then $\alpha^3+\beta^3+\gamma^3=$

\(\int_0^2 x^3(2-x)^4 d x=\)

\(\int_0^\pi \frac{x \cos ^2 x}{1+\sin x} d x=\)

\(\int_0^2 \frac{x}{(2-x)^{\frac{3}{4}}} d x=\)

If $[x]$ represents greatest integer function, then
\(\int_{-2}^2[2-x] d x=\)

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

The general solution of the differential equation $(3 x-4 y)(d x-3 d y)+(6 d x-4 d y)=0$ is

If $f(x)=\sqrt{\log \left(x^2+x+1\right)+\sqrt{\cosh (2 x-3)}}$, then $f^{\prime}(0)=$

If $x=\cos ^3 \theta-\sin ^3 \theta$ and $y=\sqrt[3]{\cos \theta}-\sqrt[3]{\sin \theta}$, then the value of $\frac{d y}{d x}$ at $\theta=\frac{\pi}{4}$ is

If $2 x^2+3 x y-y^2+4 x-5 y+6=0$, then the value of $\frac{d y}{d x}$ at $(x, y)=(1,-2)$ is

If the equations $x=1+2 \cos \theta, y=2+\sin \theta, 0 \leq \theta<2 \pi$ represent an ellipse, then the point of intersection of the normal drawn at $P\left(\frac{\pi}{4}\right)$ to this ellipse and its major axis is

When the origin is shifted to the point $(h, k)$ by translating the coordinates axes, the equation $S \equiv 2 x^2-x y+y^2+2 x+3 y+1=0$ is changed to $S \equiv a x^2+2 h x y+b y^2-3=0$. Again by rotating the coordinate axes about the new or...

In an ellipse, the distance from one of the foci to its corresponding end of the major axis is $4-\sqrt{7}$ and the distance from same focus to one end of the minor axis is 4 . Then, the cosine of the angle subtended by the line segment joi...

The domain of the real valued function $f(x)=\frac{\sqrt{|x|-x}}{\sqrt{x-[x]}}$ is

If $\sinh x=-\frac{4}{3}$, then $\sinh 2 x+\cosh 2 x=$

The range of the function defined by

$$ f(x)=\left\{\begin{array}{lc} 2 x-3, & \text { if } x<-1 \\ 1-x^2, & \text { if }-1 \leq x \leq 1 \text { is } \\ 3 x^2+2, & \text { if } x>1 \end{array}\right. $$

Which one of the following functions is a bijection?

If the equation $x+y+n=0$ represents a normal to the hyperbola $\frac{x^2}{6}-\frac{y^2}{2}=1$, then $n=$

Let $A=(1,2), B=(2,1), C=(-1,-1)$ be three points. If $P$ is a point such that the area of the quadrilateral $P A B C$ is twice the area of the $\triangle P A B$, then the equation of the locus of $P$ is

$\int \frac{1}{\left(x+\frac{2}{x}\right) \sqrt{x^4+4 x^2+3}} d x=$

If $\int \frac{2 \sin 2 x-3 \cos x}{2 \sin ^2 x-3 \sin x+4} d x=f(x)+C$, where $C$ is the constant of integration, then $f\left(\frac{\pi}{2}\right)-f(0)=$

$\int \frac{2 x+3}{\sqrt{3 x^2-2 x+1}} d x=$

If $\frac{3 \pi}{2} < x < \frac{5 \pi}{2}$ and $\int(\sqrt{1-\sin x}+\sqrt{1+\sin x}) d x=f(x)+C$, where $C$ is the constant of integration, then $f\left(\frac{\pi}{3}\right)-f(0)=$

$\mathop {\lim }\limits_{x \to 0} \frac{\tan ^4 x-\sin ^4 x}{x^6}=$

\(\mathop {\lim }\limits_{x \to 2} \frac{\sqrt[3]{6+x}-\sqrt[3]{10-x}}{x-2}=\)

If $A=\left[\begin{array}{lll}b & a & 0 \\ c & 0 & b \\ a & a & b\end{array}\right]$ and $B=\left[\begin{array}{lll}0 & a & b \\ b & 0 & c \\ b & a & a\end{array}\right]$ are two matrices such that $A B=\left[\begin{array}{ccc}2 & 2 & 7 \\...

If $A=\left[\begin{array}{lll}1 & a & 3 \\ b & 2 & c \\ 3 & d & 4\end{array}\right]$ is a symmetric matrix and $B=\left[\begin{array}{ccc}0 & 5 & b \\ -5 & 0 & -7 \\ 6 & c & 0\end{array}\right]$ is a skew-symmetric matrix, then $A B=$

If $x=\alpha, y=\beta, z=\gamma$ is the unique solution of the system of linear equations $2 x-3 y+5 z=12,5 x+2 y+3 z=11$ and $x+2 y-3 z=-3$, then $2 \alpha+5 \beta+3 \gamma=$

If the inverse of the matrix $A=\left[\begin{array}{ccc}-1 & -3 & -2 \\ 0 & 1 & 2 \\ 3 & 4 & 5\end{array}\right]$ is $A^{-1}=\left[\begin{array}{lll}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{array}\right]$, then $a_1+c_2+b_3...

$a x-y+c=0$ is the equation of the common tangent to the parabola $y^2=8 \sqrt{5} x$ and the circle $x^2+y^2=1$. If this tangent makes an acute angle with the positive $X$-axis in the positive direction, then $a^2 c^2=$

If the line $2 x+3 y+n=0$ is a tangent to the parabola $y^2=8 x$, then the equation of the normal drawn at the point $(2 n, 4 \sqrt{n})$ to the parabola $y^2=8 x$ is

All possible 5-digit numbers each having 5 distinct digits are formed using the digits $1,2,3,5,6,8$. Among them, the number of numbers which are divisible by 3 but not by 6 is

All the letters of the word 'INDEED' are taken and permuted in all possible ways to form distinct 6 letter strings (words with or without meaning). If they are listed in dictionary order, then the rank position of the string 'NIDDEE' is

The total number of ways of forming a committee of 5 members out of 7 Indians, 6 Americans, 5 Russians and 4 Australians, so that every committee contains atleast one member from each country is

Two players $A$ and $B$ alternatively toss 3 coins simultaneously. The player who gets 2 heads and 1 tail first, wins the game. If game continues until someone wins and if $A$ begins the game, the probability that B wins the game is

A bag contains 3 red, 5 black and 7 blue balls. If three balls are drawn at random simultaneously from the bag, then the probability of getting at least two blue balls is

In a game, two dice are thrown simultaneously by a person $A$ and two cards are drawn at random simultaneously from a pack of 52 playing cards by a person $B$. They win the game, if $A$ gets a prime score as the sum of the numbers appear on...

In $\triangle A B C$, if $a=7, b=8$ and $c=9$, then $\frac{1}{r_1^2}+\frac{1}{r_2^2}+\frac{1}{r_3^2}=$

In $\triangle A B C$, if $b=6, c=7$ and $\tan \frac{A}{2}=\frac{1}{\sqrt{6}}$, then the inradius of $\triangle A B C$ is

Let the equations $a x^2-7 x+c=0$ and $a x^2+5 x-c=0$ have a common root and $a c \neq 0$. If 3 is a root of $a x^2-7 x+c=0$ other than the common root, then the common root of the given equations is