TS EAMCET 2020 (Online) 11th September Evening Shift

$L_1$ and $L_2$ are two common tangents to two circles. If $L_1$ touches the two circles at $A(1,1)$ and $B(0,1)$ and $L_2$ touches the two circles at $C\left(\frac{3}{5}, \frac{4}{5}\right), D\left(\frac{-1}{5}, \frac{7}{5}\right)$, then t...

The centre of the smallest circle which cuts the circles $x^2+y^2-2 x-4 y-4=0$ and $x^2+y^2-10 x+12 y+52=0$ orthogonally is

The point/points of intersection of the common tangents of the two circles $x^2+y^2-8 x-6 y+21=0$ and $x^2+y^2-2 y-15=0$ is/are

If $P A$ and $P B$ are the tangents drawn from the point $P(1,1)$ to the circle $x^2+y^2+g x+g y-2=0$ with $C$ as the centre, then the area (in sq. units) of the quadrilateral $P A C B$ is

The number of points $z$ on the Argand plane which satisfy the conditions $\operatorname{Re}\left(\frac{z-2}{z-4 i}\right)=0$ and $\lim \left(\frac{z-2}{z-4 i}\right)=1$ simultaneously is

If $(\sqrt{3}+i)^{10}=a+b i, a, b \in \mathbf{R}$, then the values of $a$ and $b$ are respectively

If $z$ is a complex number such that $z^2+z+1=0$, then $\left(z+\frac{1}{z}\right)^3+\left(z^2+\frac{1}{z^2}\right)^3+\left(z^3+\frac{1}{z^3}\right)^3+\ldots . .+\left(z^{2020}+\frac{1}{z^{2020}}\right)^3=$

Let the roots of the equation $E_1 \equiv x^3+x^2+l x+n=0$ be $x_i,(i=1,2,3)$ and the roots of $E_2 \equiv x^3+a x^2+b x+c=0$ be $\frac{x_i-1}{2}$. If the equation $E_2=0$ is a equation of class one, then the roots of these two equations ex...

If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $x^4+x^2+1=0$, then $\frac{\alpha^3+\beta^3+\gamma^3+\delta^3}{\alpha^6+\beta^6+\gamma^6+\delta^6}=$

If
$$ f(x)=\left|\begin{array}{ccc} 1+\sin x+\sin 2 x+\sin 3 x & \frac{3+\sin 2 x}{2} & \frac{-2+\sin 3 x}{3} \\ 3+4 \sin x & \frac{3}{2} & \frac{4}{3} \sin x \\ 1+\sin x & \frac{1}{2} \sin x & \frac{1}{3} \end{array}\right| $$
then $\int_0...

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

If $\alpha$ and $\beta$ are respectively the order and degree of the differential equation for which $a x^2+b y^2=1$ is the general solution, then the eccentricity of the ellipse $\alpha x^2+\beta y^2=1$ is

The solution of the differential equation $x d y-y d x=\sqrt{x^2+y^2} d x$, given that $y=1$ when $x=\sqrt{3}$, is

If the solution $y(x)$ of the differential equation $\sin x \frac{d y}{d x}+y \cos x=e^{2 x}, x \in(0, \pi)$ satisfies $y\left(\frac{\pi}{2}\right)=0$, then $y\left(\frac{\pi}{6}\right)=$

Match the functions of List-I with derivates given in List-II
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If $f(x)=\frac{x-1}{e^x}$, then $f^{\prime}(0)+f^{\prime \prime}(0)=$

$$ \begin{aligned} & \text { If }\left(\frac{d y}{d x}\right)=\frac{1}{\left(\frac{d x}{d y}\right)} \text { and } \frac{d^2 x}{d y^2}\left(\frac{d y}{d x}\right)^3+\frac{d^2 y}{d x^2}=k \text {, then } \\ & e^{k f(x)}-k f(x)= \end{aligned}...

If $\pi / 3, \theta$ are the eccentric angles of the ends of a focal chord of the ellipse $\frac{x^2}{16}+\frac{y^2}{12}=1$, then $\tan \theta=$

If $x+2 y+k=0, k>0$ is a tangent to the ellipse $2 x^2+y^2=2$, then the equation of the normal to the given ellipse at $\left(\frac{1}{\sqrt{2}}, \frac{k}{3}\right)$, is

The number of bijective functions $f: \mathbf{Z} \rightarrow \mathbf{Z}$ such that $f(x+y)=f(x)+f(y) \forall x, y \in \mathbf{Z}$, is

For each $n \in \mathbf{N}$, let $A_n=\{(n+1) k / k \in \mathbf{N}\}$ and $X=\bigcup_{n \in \mathbf{N}} A_n \cdot A$ mapping $f: X \rightarrow N$ defined by $f(x)=x$, $\forall x \in \mathbf{X}$, is

If $(8,2)$ is a point on the hyperbola whose length of the transverse axis is 12 and conjugate axis is $x=0$, then the eccentricity of that hyperbola is

If the partial fractions decomposition of $\frac{x^4+24 x^2+28}{\left(x^2+1\right)^3}$ is $\frac{A}{x^2+1}+\frac{B}{\left(x^2+1\right)^2}+\frac{C}{\left(x^2+1\right)^3}$ then $B-2 A+C=$

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

\(\int \frac{d x}{x \ln (x) \ln ^2(x) \ln ^3(x) \ldots \ln ^m(x)}=\frac{(\ln (x))^K}{K}+C \Rightarrow 2 K=\)

If $I_m=\int x^m \cos n x d x=g(x)-\frac{m(m-1)}{n^2} I_{m-2}$, then $g(x)=$

Let $I_n=\int \sec ^n x d x$. If $5 I_6-4 I_4=f(x)$, then $f\left(\frac{\pi}{4}\right)$ is equal to

For the least possible value of $n \in \mathbf{Z}$ the solution $(x, y)$ of the equations $\cos ^{-1} x+\left(\sin ^{-1} y\right)^2=\frac{n \pi^2}{4}$ and $\cos ^{-1} x\left(\sin ^{-1} y\right)^2=\frac{\pi^4}{16}$, is

If $x=\left(\tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{8}\right)$, then $\frac{\sin x+\cos x}{\tan x}=$

If for $|x|>1, \tanh ^{-1}\left(\frac{1}{x}\right)+\operatorname{coth}^{-1}(x)=\log _e(f(x))$, then $f(-5)=$

\(\mathop {\lim }\limits_{x \to 0} \frac{1-\cos (1-\cos x)}{\sin ^4 x}=\)

At $x=0, f(x)=\left\{\begin{array}{l}\frac{x}{|x|+2 x^2}, x \neq 0 \\ k, \quad x=0\end{array}\right.$ is

Let $I$ be a unit matrix of order 6 . Let $A=\left(a_{i j}\right)$ be a square matrix of order 6 such that $a_{i j}=\left\{\begin{array}{l}1, \text { if } i+j=7 \\ 0, \text { if } i+j \neq 7\end{array}\right.$ then $\left(A(\operatorname{ad...

Let $a, b, c \notin\{0,1\}$. If the system of equations
$$ \begin{aligned} & \Pi_1 \equiv x+a y+a z=0 \\ & \Pi_2 \equiv b x+y+b z=0 \\ & \Pi_3 \equiv c x+c y+z=0 \end{aligned} $$
has a non-trivial solution, then the system of equations $\Pi...

$A$ is a singular matrix of order five. $B$ is another matrix having the rank $\rho(B)$ equal to the $\operatorname{rank} \rho(A)$ and $B$ has a non-zero minor of order 3. Then which one of the following is true?

If $m x-y+c=0$ is a normal at a point $P$ on the parabola $y^2=16 x$ and the focal distance of $P$ is 40 units, then $|c|=$

If all the vertices of an equilateral triangle lie on the parabola $y^2=16 x$ and one of them coincides with the vertex of that parabola, then the length of the side of that triangle is

For $n=1,2,3, \ldots .50$, let
$$ A=\left\{a_n / a_n=\left\{\begin{array}{ll} (-1)^{\frac{n}{2}}\left(\frac{n}{2}\right), & \text { if } n \text { is even } \\ (-1)^{\frac{n-1}{2}}\left(\frac{n-1}{2}\right), & \text { if } n \text { is odd ...

If $\alpha$ represents the number of arrangements of $p$ men and $q$ women in a row such that all men are together and $\beta$ represents the number of circular arrangements of the same people with the same condition, then $\alpha: \beta$ i...

4-digit numbers are formed using the digits 4, 5, 6, 7, 8, 9 allowing repetition of the given digits. If a number is chosen at random from those numbers thus formed, then the probability that it is exactly divisible by 3 is

If $E_1, E_2 \ldots, E_n$ are an independent events such that $P\left(E_r\right)=\frac{1}{1+r},(r=1,2, \ldots, n)$, then the probability that atleast one of $E_1, E_2, \ldots, E_n$ happens is

An urn contains five balls. Two balls are drawn at random and they are found to be white. The probability that all the balls in the urn are white, is

If the probability function of a random variable $X$ is given by $P(X=n)=\frac{k(n+1)}{3 n}$ for $n \in \mathbf{N} \cup\{0\}$ where $k$ is a constant, then $P(X<2)=$

An observer counts 240 vehicles per hour at a specific location on a highway. Assuming that the arrival of vehicles at the location follows Poisson distribution, the probability that more than two vehicles arrive over a 30 sec time interval...

In a triangle $A B C$, if $a

In a triangle $A B C$, if $c=9, s=10$ and $\Delta=10 \sqrt{2}$ then $b\left[1+\sqrt{2} \tan \left(\frac{A-B}{2}\right)\right]=$

In a $\triangle A B C, \cot A+\cot B+\cot C=$

For $n>2$ and $n \in \mathbf{N}$, the product of the roots of $(x-n)\left(\left(x^2-2 n x\right)^2+\left(2 n^2-5\right)\left(x^2-2 n x\right)\right. \left.+\left(n^4-5 n^2+4\right)\right)=0$ is divisible by

If $\alpha, \beta$ are the roots of $a x^2+b x+c=0$ then $\left(\frac{\alpha}{a \beta+b}\right)^3-\left(\frac{\beta}{a \alpha+b}\right)^3=$

The maximum value of $\left\{x \in \mathbf{R} / \sqrt{x+2}>\sqrt{8-x^2}\right\}=$