TS EAMCET 2020 (Online) 10th September Evening Shift

If the circle $S_1: x^2+y^2=16$ intersects another circle $S_2$ of radius 5 units such that the common chord is of maximum length and slope $\frac{3}{4}$, then the centre of the circle $S_2$ is

Assertion (A) If $z$ is a complex number such that $|z| \geq 3$, then the least value of $\left|z+\frac{3}{z}\right|$ is 1 .
Reason (R) $\left|z_1-z_2\right| \leq\left|z_1\right|+\left|z_2\right|$, for any two complex numbers $z_1, z_2$
The...

\(\text { If }\left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^{2020}+\left(\frac{1+\cos \theta+i \sin \theta}{1-\cos \theta+i \sin \theta}\right)^{2021}=x+i y,\)
then the value of $x+y$ at $\theta=\frac{\pi}{2}$ i...

If $\omega$ is a complex cube root of unity, then $\sum_{x=1}^{10}\left((\omega x+2)\left(\omega^2 x+2\right)-3\right)$

If $z_1=x_1+i y_1, z_2=x_2+i y_2, z_3=x_1+\frac{i x_2}{2}, z_4=2 y_1+i y_2$ are complex numbers such that $\left|z_1\right|=1,\left|z_2\right|=2$ and $\operatorname{Re} \left(\begin{array}{ll}z_1 & z_2\end{array}\right)=0$, then

\(\int_0^{\pi / 2} \frac{d x}{4+5 \sin x}\)

\(\mathop {\lim }\limits_{x \to \infty } \frac{\pi}{2 n}\left[\sin \frac{\pi}{2 n}+\sin \frac{2 \pi}{2 n}+\ldots+\sin \frac{\pi}{2}\right]=\)

The differential equation for which $y=a x^2+b x+c$ is the general solution is

The general solution of the differential equation
$(3 y-7 x+7) d x+(7 y-3 x+3) d y=0$ is

If $y=e^{a x}(\cos b x+\sin b x)$ satisfies the equation $\frac{d^2 y}{d x^2}-K \frac{d y}{d x}+L y=0$, then $L+b K=$

The general solution of the differential equation $(3 y-7 x+7) d x+(7 y-3 x+3) d y=0$ is

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

Let $f:[2,5] \rightarrow \mathbf{R}$ be a differentiatiable function and $\frac{f(5)}{f(2)}=1$. If there is a $c \in(2,5)$ such that $c f^{\prime}(c)=2 f(c)-2 c^3$, then $f(x)=$

Let $f:[2,5] \rightarrow \mathbf{R}$ be a differentiatiable function and $\frac{f(5)}{f(2)}=1$. If there is a $c \in(2,5)$ such that $c f^{\prime}(c)=2 f(c)-2 c^3$, then $f(x)=$

If $\alpha$ is such a minimum value for which the inverse of $f(x)=x^2+3 x-3$ exists in $[\alpha, \infty)$ and $g$ is the inverse of the $f$, then at $x=\alpha+\frac{5}{2}, \frac{d g}{d x}$

$$ \begin{aligned} & \text { If } f(x)=\tan ^{-1}\left(\frac{1}{\sin ^2 x+\sin x+1}\right) \\ & \quad+\tan ^{-1}\left(\frac{1}{\sin ^2 x+3 \sin x+3}\right)+\tan ^{-1} \end{aligned} $$
$\left(\frac{1}{\sin ^2 x+5 \sin x+7}\right)+\ldots+$ up...

The ellipse having its foci $(0, \pm 1)$ and major axis of length $\sqrt{5}$ is

An ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $\frac{2 \sqrt{2}}{3}$ is inscribed in a circle $x^2+y^2=18$ such that the length of its major axis is equal to the diameter of this circle. The locus of the poles of all the ...

If $f(x)=x-\frac{1}{x}, x \neq 0$, then $3 f(x)=$

Let $[\cdot]$ denote greatest integer function. If $f(x)=[x]$ and $g(x)=3\left[\frac{x}{3}\right]$, then the set of all real $x$ such that $f(x)=g(x)$ is

A function $f: \mathbf{R} \rightarrow \mathbf{R}$ is such that $f(\mathrm{l})=2$ and $f(x+y)=f(x) \cdot f(y) \forall x, y$. The area (in square units) enclosed by the lines $2|x|+5|y| \leq 4$ expressed interms of $f(1), f(2)$ and $f(4)$ is

If the circle $x^2+y^2=a^2$ intersects the hyperbola $x y=b^2$ at four points $\left(x_1, y_1\right),\left(x_2, y_2\right),\left(x_3, y_3\right),\left(x_4, y_4\right)$, then $y_1 \quad y_2 \quad y_3 y_4=$

For $x \in\left(\frac{3 \pi}{4}, \pi\right), \int(\sqrt{1+\sin 2 x}+\sqrt{1-\sin 2 x}) d x=$

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

$$ \text { If } \begin{aligned} & \int x^3(\log x)^2 d x=x^4\left[A(\log x)^2+B(\log x)\right. \\ &+C \log e]+K, \text { then } A+B+C \end{aligned} $$

$$ \begin{aligned} & \text { If } \int \frac{9 x+15}{x^3-6 x-9} d x=A \log |g(x)| \\ & \quad+B \log |f(x)|+C, \text { then } \frac{(A-B) g(4)}{f(-1)}= \end{aligned} $$

$$ \begin{aligned} & \text { If } \int \frac{x^2\left(x \sec ^2 x+\tan x\right)}{(x \tan x+1)^2} d x=A \log (|x \sin x+\cos x|) \\ & +B \frac{f(x)}{(x \tan x+1)}+C \text {, then } f(A+B)= \end{aligned} $$

If $\sin ^{-1}\left(\frac{12}{x}\right)+\sin ^{-1}\left(\frac{5}{x}\right)=\frac{\pi}{2}$, then $x=$

The value of ' $a$ ' for which the function
$f(x)=\left\{\begin{array}{cl}\frac{1-\cos 4 x}{x^2}, & x<0 \\ \frac{a}{\sqrt{x}}, & x=0 \text { is continuous at } x=0, \text { is } \\ \frac{\sqrt{16+\sqrt{x}}-4}{\sqrt{16+}} & \end{array}\right...

\(\mathop {\lim }\limits_{x \to 0} \frac{1-\cos \left(x^2+\pi(x+2)\right)}{x^2}=\)

\(\log (9+3 \sqrt{2}(2+\sqrt{5})+4 \sqrt{5})=\)

If $C$ and $D$ are two $n \times n$ non-singular matrices over the set of real number $\mathbf{R}$ such that $C D=-D C$, then $n$ is

Let $A=\left[\begin{array}{ccc}1 & 4 & 2 \\ 2 & -1 & 4 \\ -3 & 7 & -6\end{array}\right]$ and $B=\left[b_{i j}\right]_{3 \times 3}$ with $b_{11}=2$, $b_{13}=-2, b_{12}=0$ is such that $A B=\left[\begin{array}{ccc}2 & 14 & -4 \\ 4 & 1 & -8 \\...

A is a $m \times n$ matrix of rank 4 . If A contains an $m$ th order non singular sub matrix and $A^T A$ is a $7 \times 7$ matrix, then the number of rows of $A$ is

For the parabola $y=\frac{h^3}{3} x^2+\frac{h^2}{2} x-h+\frac{3}{4 h^3}$, if the equation of directrix is $y=k$, then $k: h$

The equation of the common tangent of the parabolas $x^2=108 y$ and $y^2=32 x$ is

Numbers between 1 and 10,000 are formed using the digits 2 and 3 only once and the digit 4 twice. If the numbers thus formed are arranged in increasing order and $x, y$ represent the ranks of 4324 and 324 respectively then $x-y=$

If $x$ and $y$ represent the number of arrangements of the letters of word ATRAPATRAM such that (i) all A's are together and (ii) no two A's are together respectively, then $x+y$

A die is thrown thrice. If getting 1 or 6 in a single throw is considered as success, then the variance of the number of successes is

In a hospital, on an average if there are 35 births in a weak, then the probability that there will be less than 3 births in a day, is

Four boxes $A, B, C$ and $D$ contain 5000, 3000, 2000 and 1000 fuses respectively. The percentages of defective fuses in these boxes are $3 \%, 2 \%, 1 \%$ and $0.5 \%$ respectively. If a fuse selected at random from one of the boxes is fou...

In an examination there are four Yes/No type of questions. The probability that the answer by the student to a question without guess to be correct is $2 / 3$. The probability that a student guesses a correct answer is $1 / 2$. A student wr...

If $A_1, A_2, \ldots, A_{15}$ are the events of a random experiment, then which one of the following is true?

In a $\triangle A B C,\left(b^2-c^2\right) \cot A+\left(c^2-a^2\right) \cot B=$

In a $\triangle A B C, \frac{\Delta^2}{a^2+b^2+c^2}\left(\frac{1}{r_1^2}+\frac{1}{r_2^2}+\frac{1}{r_3^2}+\frac{1}{r^2}\right)=$

If $R: r_1: r=5: 12: 2$, then $r+r_3+r_2-r_1=$

If $\alpha$ and $\beta$ are the real roots of the equation $\sqrt{\frac{5 x}{x-2}}+\sqrt{\frac{x-2}{5 x}}=\frac{29}{10}$ and $\alpha>\beta$, then $\sqrt{\alpha^2-11^4 \beta^2}=$

The minimum value of $\frac{9 \cdot 3^{2 x}+6 \cdot 3^x+4}{9 \cdot 3^{2 x}-6 \cdot 3^x+4}$ is

$p$ is non-zero real number. If the equation whose roots are the squares of the roots of the equation $x^3-p x^2+p x-1=0$ is identical with the given equation, then $p=$

If $S_n$ is the sum of the first $n$ terms of the series $1^2+2 \times 2^2+3^2+2 \times 4^2+5^2+2 \times 6^2+\ldots \infty$, then, when $n$ is even $S_n=$