TS EAMCET 2023 (Online) 12th May Evening Shift

The equation of the line perpendicular to the radical axis of two circles $x^2+y^2-5 x+6 y+12=0$, $x^2+y^2+6 x-4 y-14=0$ and passing through $(1,1)$ is

If the angle between the circles
\(x^2+y^2-2 x-4 y+c=0 \text { and } x^2+y^2-4 x-2 y+4=0\)
is $60^{\circ}$, then $c=$

The pole of the straight line $9 x+y-28=0$ with respect to the circle $2 x^2+2 y^2-3 x+5 y-7=0$ is

The equation of the circle whose diameter is the common chord of the circles $x^2+y^2+2 x+3 y+1=0$ and $x^2+y^2+4 x+3 y+2=0$ is

If the imaginary part of $\frac{2 z+1}{i z+1}$ is -2, then the locus of the point representing $z$ in the Argand plane is

\(\text { If } x+i y=\sqrt{\frac{3+i}{1+3 i}}, \text { then }\left(x^2+y^2\right)^2=\)

$\operatorname{Arg}\left(\sin \frac{6 \pi}{5}+i\left(1+\cos \frac{6 \pi}{5}\right)\right)=$

If $i=\sqrt{-1}$, then $(1+i)^{10}+(1-i)^{10}=$

\(\int\limits_2^5 \sqrt{\frac{5-x}{x-2}} d x=\)

\(\int\limits_0^{\frac{\pi}{2}} \sin ^6 x \cos ^4 x d x=\)

\(\lim\limits_{n \rightarrow \infty}\left[\frac{1}{n^2} \sec ^2 \frac{1}{n^2}+\frac{2}{n^2} \sec ^2 \frac{4}{n^2}+\ldots \ldots+\frac{1}{n} \sec ^2 1\right]=\)

\(\int_0^{\pi / 4} \frac{\sec x}{1+2 \sin ^2 x} d x=\)

The general solution of the differential equation $\left(2 x-10 y^3\right) d y+y d x=0, y \neq 0$ is

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

The degree and order of the differential equation of the family of parabolas whose axis is the $X$-axis, are respectively

If $\sec \left(\log _2 y^2\right)=\operatorname{cosec}\left(\log _2 x^2\right)$, then $\frac{d y}{d x}=$

If $e^x=y+\sqrt{y^2-1}$, then $\frac{d y}{d x}=$

If $x=\log p$ and $y=\frac{1}{p}$, then $\frac{d y}{d x}=$

The locus of the mid-points of the intercepted portion of the tangents by the coordinate axes, which are drawn to the ellipse $x^2+2 y^2=2$ is

The product of the lengths of the perpendiculars drawn from the two foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ to the tangent at any point on the ellipse is

Tangents are drawn to the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1$ at all the ends of its latus recta. The area of the quadrilateral, so formed (in sq units) is

The domain of the function $f(x)=\sin ^{-1}\left(\log _2\left(\frac{x^2}{2}\right)\right)$ is

If $f: R \rightarrow R$ is defined by $f(x)=2 x+\sin x, x \in R$, then $f$ is

The range of the function $f(x)=-\sqrt{-x^2-6 x-5}$ is

$P(a \sec \theta, b \tan \theta)$ and $Q(a \sec \phi, b \tan \phi)$ are two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ where, $\phi+\theta=\frac{\pi}{2}$. If $(h, k)$ is the point of intersection of the normals drawn at $P$...

If $\int \frac{x}{(a+x)^5} d x=\frac{1}{k(a+x)^4}(f(x))+C$, then $\frac{f(-a)}{a k}=$

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

\(\text { Match the following items from List I into List II }\)






List-I

List-II





1.

∫







sin

2



⁡

x







cos

4



⁡

x...

If $\int x^4(\log x)^3 d x=x^5\left[A(\log x)^3\right]$ $\left.+B(\log x)^2+C \log x+D\right]+k$, then $A+B+C+5 D=$

If $f(9)=9$ and $f^{\prime}(9)=4$, then $\lim\limits_{x \rightarrow 9} \frac{\sqrt{f(x)}-3}{\sqrt{x}-3}=$

\(\lim\limits_{x \rightarrow 1}(1-x) \tan \left(\frac{\pi}{2} x\right)=\)

\(\sinh (\log (3+\sqrt{8}))=\)

$$ \begin{aligned} &\text { If } \omega \neq 1 \text { is a cube root of unity, then }\\ &\left|\begin{array}{ccc} \omega+\omega^2 & \omega^2+\omega^9 & \omega^9+\omega \\ \omega^{27}+\omega^{31} & \omega^{31}+\omega^{17} & \omega^{17}+\ome...

Let $A$ be a matrix such that $A B$ is a scalar matrix, where $B=\left[\begin{array}{ll}1 & 2 \\ 0 & 3\end{array}\right]$ and $\operatorname{det}(3 A)=27$. Then, $3 A^{-1}+A^2=$

$$ \left|\begin{array}{ccc} \sqrt{3} & 2 \sqrt{5} & \sqrt{5} \\ \sqrt{15} & 5 & \sqrt{10} \\ 3 & \sqrt{15} & 5 \end{array}\right|= $$

If $A$ is a non-singular matrix such that $(A-2 I)$ $(A-3 I)=0$, then $\frac{1}{5} A+\frac{6}{5} A^{-1}=$

If $A$ is a symmetric matrix with real entries, then

The normal at a point on the parabola $y^2=4 x$ passes through $(5,0)$. If there are two more normals to this parabola passing through $(5,0)$, then the equation of one of these normals is

The equations of common tangents to the parabola $y^2=16 x$ and the circle $x^2+y^2=8$ are

The total number of all those 3-digit numbers in which the sum of all the digits in each of them is 10 , is

All the letters of the word 'MOTHER' are written in all possible ways and the strings of letters (with or without meaning), so formed are written as in a dictionary order. Then, the position of the word 'THROEM' is

A student is allowed to select at most $n$ books from a collection of ( $2 n+1$ ) books. If the total number of ways in which he can select at least one book is 255 , then the value of $n$ is

$A, B, C, D$ cut a pack of 52 well shuffled playing cards successively in the same order. If the person who cuts a spade first, wins the game and the game continues until this happens, then the probability that $A$ wins the game is

If $A$ and $B$ are two events of a random experiment such that $P(A \cup B)=P(A \cap B)$, then which one amongst the following four options is not true?

Two bad eggs are mixed accidentally with 10 good ones. If three eggs are drawn at random from this lot in succession without replacement, then the variance of the probability distribution of the number of bad eggs drawn is

If a group of six students including two particular students $A$ and $B$ stand in a row, then the probability of getting an arrangement in which $A$ and $B$ are separated by exactly one student in between them is

$P Q R$ is an isosceles triangle with $P Q=P R$. If the radius of the circumcircle of $\triangle P Q R$ is equal to the length of $P Q$ then $\angle P=$

In $\triangle A B C$, if $\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos \cdot C}{c}$ and side $a=2$, then area of the $\triangle A B C$ (in sq units) is

The set of all values of $x$ which satisfy both the inequations $x^2-1 \leq 0$ and $x^2-x-2 \geq 0$ simultaneously is

The quadratic equations $x^2-6 x+a=0$ and $x^2-c x+6=0$ have one root in common. If the other roots of the first and second equations are integers and are in the ratio $4: 3$, then their common root is