TS EAMCET 2023 (Online) 13th May Morning Shift

Let 6,8 be the $X$ and $Y$-intercepts made by the circle $S \equiv x^2+y^2+2 g x+2 f y+c=0$, respectively. If $g x+f y+1=0$ is a line passing through the point $(1,-1)$, then the radius of the circle $S=0$ is

If the angle between the circles $x^2+y^2-2 x+2 y+1=0$ and $x^2+y^2+2 x-2 y+k=0$ is $\frac{\pi}{3}$, then

Let a chord $A B$ subtend an angle of $60^{\circ}$ at the centre $C(2,3)$ of a circle $S$. If the equation of $A B$ is $x+y+1=0$, then the equation of the circle $S$ is

If the value of $\sqrt{-5-12 i}+\sqrt{7+24 i}$ is a negative real number $k$, then $k=$

If $i$ is the root of the equation $x^2+1=0$, then
\((1+\sqrt{3} i)^{2023}+(1-\sqrt{3} i)^{2023}=\)

If a point $P$ denotes the complex number $z=x+i y$ in the argand plane and if $\frac{z-(2+i)}{z+(1-2 i)}$ is purely real, then the locus of $P$ is

One of the values of $(\sqrt{3}-i)^{\frac{1}{6}}$ is

If $a x^2-x y-3 y^2-5 x+20 y+c=0$ represents a pair of lines passing through the point $(2,3)$, then $a-c=$

Let $z=x+i y$ be a point in the argand plane. If the amplitude of $\left(\frac{z-3}{z+2 i}\right)$ is $\frac{\pi}{2}$, then the locus of $z$ is

\(\int_{-\frac{\pi}{8092}}^{\frac{\pi}{8092}} \frac{\sec (2023 x)}{1+(2023)^{(2023 x)}} d x=\)

\(\int_0^2 x^{\frac{5}{2}} \sqrt{2-x} d x=\)

\(\int_0^3\left|x^2-3 x+2\right| d x=\)

If $A$ and $B$ are arbitrary constants, then the differential equation having $y=A e^x+B \sin 2 x$ as its general solution is

The general solution of the differential equation $\frac{d y}{d x}=\sin (x-y)+\cos (x-y)$ is

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

If $f(x)=\frac{e^{2 x}-e^{-2 x}}{e^{3 x}+e^{-3 x}}$, then $f^{\prime}(0)=$

If $f(x)=x^{\tan x}+(\tan x)^x$, then $f^{\prime}\left(\frac{\pi}{4}\right)=$

If $f(x)=|x-1|+|x-2|$, then
\(f^{\prime}(-2023)+f^{\prime}\left(\frac{2024}{2023}\right)+f^{\prime}(2023)=\)

If $4 x+2 y+n=0$ is a normal to the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$ then $n=$

Let $A=(2,0)$ and $B=(0,-2)$. Let $P$ be any point such that the sum of the distance of $P$ from $A$ and $B$ is 4 . Then, the equation of the locus of the point $P$ is

Let $P$ be the point to which origin has to be shifted by the translation of axes, so as to remove the first degree terms from the equation $3 x^2+y^2-6 x+4 y+4=0$. If the origin is shifted to $P$ by the translation of axes, then the transf...

Let $S$ and $S^{\prime}$ be the foci of an ellipse $E$ and $B$ be one end of its minor axis. Let $\angle S^{\prime} S B=\pi / 6$ and $(2 \sqrt{3}, 1)$ be a point on $E$. If $X$-axis is the major axis and $Y$-axis is the minor axis of the el...

The domain of the real valued function
\(f(x)=\frac{\sqrt{\log _{10}\left(\frac{x}{x-2}\right)}}{\sqrt{[x]^2-5[x]+6}} \text { is }\)
(Here, $[x]$ denotes the greatest integer function)

If $\frac{6 x^4+13 x^3+2 x^2-x+3}{2 x^2+3 x-2}=f(x)+\frac{A}{a x-1}+\frac{B}{x+b}$, then $f(\mathrm{l})+a \cdot B+b \cdot A=$

If the function $f: R \rightarrow R$ is defined by
$$ f(x)= \begin{cases}2 x-3, & \text { if } x<-2 \\ x^2-1, & \text { if }-2 \leq x \leq 2 \\ 3 x+2, & \text { if } x>2\end{cases} $$
then $f$ is

The range of the real valued function $f(x)=\frac{1}{x-|x|}$ is

If $y=m x+4(m>0)$ is a tangent to the hyperbola $\frac{x^2}{25}-\frac{y^2}{9}=1$, then the point of contact of this tangent is

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

\(\int \frac{1}{3 \cos x-4 \sin x+5} d x=\)

\(\int \frac{\sec ^2 x}{(\sec x+\tan x)^2} d x=\)

\(\int \frac{1}{16-7 \sin ^2 x} d x=\)

\(\mathop {\lim }\limits_{x \to 2 + }\left([x]^2-[x]-2\right)+\mathop {\lim }\limits_{x \to- 3 - }\left([x]^2-4[x]+3\right)=\)

\(\lim _{x \rightarrow 0} \frac{\left(3^{2 x}-\sqrt{x+1}\right) \sin 5 x}{1-\cos 4 x}=\)

$$ \left|\begin{array}{lll} 2 & 3 & 5 \\ 3 & 5 & 2 \\ 5 & 2 & 3 \end{array}\right|+\left|\begin{array}{ccc} 1 & 1 & 1 \\ 7 & 11 & 13 \\ 49 & 121 & 169 \end{array}\right|= $$

If $A=\left[\begin{array}{ccc}k & 5 & 2 \\ 2 & -k & 5 \\ 5 & 2 & -k\end{array}\right]$ and $\operatorname{det} A=190$, then $\operatorname{adj} A=$

If $A=\left[\begin{array}{ccc}1 & 2 & -1 \\ -1 & 0 & 2 \\ 1 & 2 & 0\end{array}\right]$ and $B=\left[\begin{array}{ccc}-3 & -2 & 4 \\ 2 & 2 & -1 \\ -2 & 0 & 3\end{array}\right]$, then $A^2=$

If the unique solution of the simultaneous linear equations $3 x-2 y+z=5 k, 2 x+3 y-2 z=-5 k$, $x+4 y+3 z=k$ is $x=\alpha, y=\beta, z=3$, then $k=$

Normals are drawn from the point $P(8,0)$ to the parabola $y^2=12 x$. If $\theta$ is the acute angle between two non-horizontal normals among them, then $\tan \theta=$

If the focal distance of a point $P\left(2, y_1\right)$ on the parabola $y^2=k x$ is 3 , then the equation of the tangent drawn at $P$ to the given parabola is

If $n, r$ are two positive integers such that $1 \leq r

Among the positive divisors of the number 12600 , if $n_1$ is the number of divisors which are multiples of 3 and $n_2$ is the number of divisors which are multiples of 14 , then $n_1+n_2=$

The number of ways in which $n$ boys and $n$ girls can be arranged in a row such that all the boys are together and all the girls are also together is equal to

Three persons $A, B$ and $C$ attended a recruitment test, The ratio of the chances of $A, B, C$ in getting through the test is $1: 2: 3$ and their probabilities to face the interview successfully are $0.8,0.7,0.6$, respectively. If one of t...

Two cards are drawn at random one after the other with replacement from a pack of 52 playing cards. Then, the variance of the random variable of the number of spade cards among the drawn cards is

If two cards are drawn at random simultaneously from a pack of 52 playing cards, then the probability of getting a face card and a spade card other than the face card is

If three unbiased dice are rolled simultaneously, then the probability that all the three dice show distinct numbers is

In $\triangle A B C$, if $A$ is an acute angle, $b=6, c=9$ and $\sin A=\frac{2 \sqrt{14}}{9}$, then $3 a(\cos B+\cos C)=$

If the roots of the equation $x^3-11 x^2+36 x-36=0$ are the ex-radii of a $\triangle A B C$, then the perimeter of the $\triangle A B C$ is

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+4 x^2-9 x-36=0$ and $\alpha<\beta<\gamma$, then $\alpha+2 \beta+3 \gamma=$

The values of $\frac{x^2-2 x+1}{x^2+x-1}$ do not lie in the interval