TS EAMCET 2023 (Online) 12th May Morning Shift

If a circle passing through $A(1,1)$ touches the $X$-axis, then the locus of the other end of the diameter through $A$ is

If $C(\alpha, \beta)(a<0)$ is the centre of the circle that touches the $Y$-axis at $(0,3)$ and makes an intercept of length 2 units on positive $X$-axis, then $(\alpha, \beta)=$

The image of every point lying on the curve $x^2+y^2=1$ in the line $x+y=1$ satisfies the equation

If $i=\sqrt{-1}$, then $1+i^2+i^4+i^6+\ldots \ldots+i^{2024}=$

If $\frac{1+i \cos \theta}{1-2 i \cos \theta}$ is purely real, then $\cos ^3 \theta+\sin ^2 \theta+\cos \theta+1=$

If $z_1$ and $z_2$ are complex numbers such that $\left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right|$, then the difference in the amplitude of $z_1$ and $z_2$ is

If $\theta=\frac{\pi}{6}$, then the 10 th term of the series $1+(\cos \theta+i \sin \theta)^1+(\cos \theta+i \sin \theta)^2+\ldots$. is

If $\alpha$ and $\beta$ are non-zero integers and $z=(\alpha+i \beta)(2+7 i)$ is a purely imaginary number, then minimum value of $|z|^2$ is

[.] is the greatest integer function, then
\(\int_0^{2 \pi}[|\sin x|+|\cos x|] d x=\)

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

If $f$ is defined on $R$ such that $f(x) f(-x)=9$, then

\(\int_{-23}^{23} \frac{d x}{3+f(x)}=\)

\(\int_{1 / 2}^2\left|\log _{10} x\right| d x=\)

The general solution of $\frac{d y}{d x}+y f^{\prime}(x)-f(x) f^{\prime}(x)=0$, $y \neq f(x)$ is

If $m$ and $n$ are respectively the order and degree of the differential equation of the family of parabolas with origin as its focus and $X$-axis as its axis, then $m n-m+n=$

The derivative of $\frac{1-x^2}{1+x^2}$ with respect to $\frac{2 x}{1+x^2}$ at $x=2$ is

If $\tan y=\cot \left(\frac{\pi}{4}-x\right)$, then $\frac{d y}{d x}=$

If the slope of the tangent drawn to the curve $y=e^{a+b x^2}$ at the point $P(1,1)$ is -2 , then the value of $2 a-3 b$ is

If $x=3 \sqrt{2} \cos ^3 \theta$ and $y=4 \tan ^2 \theta$, then $\left(\frac{d y}{d x}\right)_{\theta=\pi / 4}=$

A particle is travelling in clockwise direction on the ellipse $\frac{x^2}{100}+\frac{y^2}{25}=1$. If the particle leaves the ellipse the point $(-8,3)$ on it and travels along the tangent to the ellipse at that point, then the point where ...

If an ellipse with foci at $(3,3)$ and $(-4,4)$ is passing through the origin, then the eccentricity of that ellipse is

If $f:[2, \infty) \rightarrow R$ is defined by $f(x)=x^2-4 x+5$, then the range of $f$ is

If $t$ is a parameter, $A=(a \sec t, b \tan t)$, $B=(-a \tan t, b \sec t)$ and $O=(0,0)$, then the locus of the centroid of $\triangle O A B$ is

If $f(x)=-|x|$, then $($ fofof $)(x)+($ fofof $)(-x)=$

If the equation of a hyperbola is $9 x^2-16 y^2+72 x-32 y-16=0$, then the equation of conjugate hyperbola is

\(\int \frac{\sin ^8 x-\cos ^8 x}{1-2 \sin ^2 x \cos ^2 x} d x=\)

7. If $\int \sin (101 x)(\sin x)^{99} d x$ $=\frac{\sin (100 x)(\sin x)^\lambda}{\mu}+C$ then, $\frac{\lambda}{\mu}=$

If $n$ is a positive integer greater than 1 and $I_n=\int \frac{\sin n x}{\sin x} d x$, then $I_{n+1}-I_{n-1}=$

If $\frac{x^4}{(x-1)(x-2)(x-3)}=p(x)+\frac{A}{x-1}+\frac{B}{x-2}+\frac{C}{x-3}$, then $p\left(\frac{3}{2}\right)+C=$

If $\int e^x\left(\sin ^2 2 x-8 \cos 4 x\right) d x=e^x f(x)+C$, then $f\left(\frac{\pi}{4}\right)=$

If $\int \frac{x^2\left(\sec ^2 x+\tan x\right)}{(x \tan x+1)^2} d x=\frac{-x^2}{x \tan x+1}+f(x)+C$, then

\(f(x)=\)

If $a, b, c$ and $k$ are non-zero real numbers and $\lim \limits_{x \rightarrow \infty} x\left(a^{1 / x}+b^{1 / x}+c^{1 / x}-3 k^{1 / x}\right)=0$, then $k=$

\(\lim \limits_{x \rightarrow 1} \frac{(2 x-3)(\sqrt{x}-1)}{2 x^2+x-3}=\)

If $\sinh (\log x)=-2$, then $x=$

The range of the function $f(x)=\log _{0.5}\left(x^4-2 x^2+3\right)$ is

If $P$ is a non-singular matrix such that $I+P+P^2+\ldots \ldots+P^n=0(0$ denotes the null matrix $)$, then $P^{-1}=$

If the system of equations
$x+k y+3 z=-2$,
$4 x+3 y+k z=14,$
$2 x+y+2 z=3$ can be solved by matrix inversion method, then

If $A=\left[\begin{array}{ccc}5 & 5 \alpha & \alpha \\ 0 & \alpha & 5 \alpha \\ 0 & 0 & 5\end{array}\right]$ and $\operatorname{det}\left(A^2\right)=25$, then $|\alpha|=$

$P$ is a $3 \times 3$ square matrix and $\operatorname{Tr}(P) \neq 0$. If $\operatorname{Tr}\left(P-P^I\right)+$ $\operatorname{Tr}\left(P+P^T\right)+\frac{\operatorname{Tr}(P)}{\operatorname{Tr}\left(P^T\right)}+\operatorname{Tr}(P) \times...

Two tangents are drawn from the point $(-1,-2)$ to the parabola $y^2=4 x$. If $\theta$ is the angle between these tangents, then $\tan \theta=$

If two circles $x^2+y^2-6 x-6 y+13=0$ and $x^2+y^2-8 y+9=0$ intersect at $A$ and $B$, then the focus of the parabola whose directrix is the line $A B$ and vertex is the point $s(a, b)$ is

The number of ways of arranging all the letters of the word "SUNITHA" so that the vowels always occupy the first, middle and last places is

The number of all four digit numbers that can be formed with the digits $0,1,2,3,4,5$ when the repetition of the digits is not allowed, is

The number of four digit numbers that can be formed using the digits $1,2,3,4,5,6$ and 7 which are divisible by 4 , when the repetition of any digit is not allowed,

A student is given 6 questions in an examination with true or false type of answers. If he writes 4 or more correct answers, he passes in the examination. The probability that he passes in the examination is

In a non-leap year, the probability of getting 53 Sundays or 53 Tuesdays or 53 Thursdays is

If $A$ and $B$ are two events in a random experiment such that $P(A)+P(B)=2 P(A \cap B)$, then

The equation of the parabola with $x+2 y=1$ as directrix and $(1,0)$ as focus is

If $P(X=x)=c\left(\frac{2}{3}\right)^x ; x=1,2,3,4, \ldots$ is a probability distribution function of a random variable $X$, then the value of $c$ is

In a $\triangle A B C$, if $r_1=2 r_2=3 r_3$, then $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=$

In an isosceles right angled triangle, a straight line is drawn from the mid-point of one of the equal sides to the opposite vertex. Then, a pair of possible values of the cotangents of the two angles so formed at that vertex are