TS EAMCET 2020 (Online) 14th September Morning Shift

If the four points $A, B, C, D$ in the Argand plane represented respectively by the complex numbers $2+i, 4+3 i, 2+5 i, 3 i$ lie on a circle, then the centre of the circle is

The centre and radius of the circumcircle of the triangle formed by the lines $2 x+3 y=10, y=x$ and the $X$-axis are respectively.

$z_1, z_2$ are two fixed points on the Argand plane. If $z$ is a complex number such that $\left|z-z_1\right|+\left|z-z_2\right|=\lambda$, then the locus of $z$ is

The roots of the equation $(x-1)^5=32(x+1)^5$ are

If $\omega$ is a non-real cube root of unity and $x=\omega^2-\omega-3$, then the value of $x^4+6 x^3+10 x^2-12 x-19$ is

Sum of the modulii of the complex roots of the equation $\left(x^2+\frac{1}{x^2}\right)-5\left(x+\frac{1}{x}\right)+6=0$ is

Assertion (A) If $a_1, a_2, \ldots, a_n$ are the $n$ distinct roots of the equation $x^n-2=0$, then $1+\left(1-a_1\right)\left(1-a_2\right) \ldots \left(1-a_{n-1}\right)\left(1-a_n\right)=0$
Reason (R) If $\alpha_1, \alpha_2, \ldots, \alpha...

$$ \begin{array}{r}\mathop {\lim }\limits_{n \to \infty }\left[\frac{n^{3 / 2}}{n^{5 / 2}}-\frac{n^{1 / 2}}{n^{3 / 2}}+\frac{n^{3 / 2}}{(n+2)^{5 / 2}}-\frac{n^{1 / 2}}{(n+3)^{3 / 2}}\right. \\ +\frac{n^{3 / 2}}{(n+4)^{5 / 2}}-\frac{n^{1 / 2...

\(\lim _{n \rightarrow \infty}\left[\frac{n+3}{n^2+1^2}+\frac{n+6}{n^2+2^2}+\frac{n+9}{n^2+3^2}+\ldots+\frac{2}{n}\right]=\)

If the order and degree of the differential equation corresponding to the family of curves $(x-2)^2+(y-a)^2=b^2$, (where $a$ and $b$ are parameters) are $m$ and $n$ respectively, then $m^2+n=$

Consider the differential equation $\frac{d y}{d x}=\frac{1}{a x+4 y+7}$ and the following statements
A. The given differential equation is linear in $x$.
B. The given differential equation is not linear in $y$.
C. The given differential eq...

The solution of the differential equation $x \cos x \frac{d y}{d x}+(x \sin x+\cos x) y=1$ is

If $\operatorname{Lt}_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=e^x(x+1)$ and $f(0)=0$, then $\frac{d}{d x}\left(f(x) e^{-x}\right)+\frac{d}{d x}\left(\frac{f(x)}{x}\right)=$

If $y(x)=\tan ^{-1}\left(\frac{\sqrt{1+a^2 x^2}-1}{a x}\right)$ and $\left(1+a^2 x^2\right) y^{\prime \prime}+g(x) y^{\prime}=0$ then, the sum of the roots of the equation $1+a^2 x^2+g(x)=0$ is

Equation of a common tangent to the circle $x^2+y^2=4$ and to the ellipse $2 x^2+25 y^2=50$ is

The $\theta$ is the angle made by the common tangent to the circle $x^2+y^2=16$ and the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ with positive $X$-axis, then $\cos 2 \theta=$

Given that for any $n \in \mathbf{N}$ there exist an odd integer $q$ and a non-negative integer $r$ such that, $n$ can be written uniquely as $n=q \times 2^r$.
Let $f: \mathbf{N} \rightarrow \mathbf{N} \times \mathbf{N}$ be function defined...

Let $H(x)=3 x^4+6 x^3-2 x^2+1$ and $g(x)$ be a polynomial of degree one. If
$\frac{H(x)}{(x-1)(x+1)(x-2)}=f(x)+\frac{g(x)}{(x-1)(x+1)(x-2)}$ then
$H(-1)+2 H(2)-3 H(1)=$

If $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined by $f(x)=x+2|x+1|+2|x-1|$, then the element in the co-domain, which has unique pre image in the domain is

Suppose the axes $X$ and $Y$ are obtained by rotating the axes $x$ and $y$ an angle $\theta$. If the equation $x^2+2 \sqrt{3} x y-y^2=4 a^2$ is transformed to $X^2-Y^2=2 a^2$ with respect to the $X Y$-axes, then $\theta$ is equal to

For the hyperbola $x^2-y^2-4 x+2 y+c=0$, if the focus is $S(2+2 \sqrt{2}, k)$ and the directrix that is adjacent to $S$ is $x=2+\sqrt{2}$, then $c=$

If $5(f(x))^2=x f(x)+30$ and
$$ \begin{aligned} & \int \frac{\left(3 x^3+\left(1-30 x^2\right) f(x)\right)}{(10 f(x)-x)\left(x^3-f(x)\right)^2} d x \\ & =\frac{A}{B x^3+D f(x)}+C \text { then } A+B+D= \end{aligned} $$

If $\int x[\log (1+x)]^3 d x=\frac{(1+x)^2}{16}(f(x))+(1+x)(g(x))$, then
\(f(x)+g(x)=\)

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

\(\int\left[\frac{x^4-x}{x^{20}}\right]^{1 / 4} d x=\)

If $\sinh \left(2 \tanh ^{-1} x\right)=\frac{11}{60}$, then $x=$

If $f: \mathbf{R} \rightarrow \mathbf{R}$ and $g: \mathbf{R} \rightarrow \mathbf{R}$ be defined by $f(x)=\left\{\begin{array}{cc}x+2, & x>0 \\ 2-x, & x \leq 0\end{array}\right.$ and $g(x)=\left\{\begin{array}{cc}x^2-2 x-2, & 1 \leq x<2 \\ x...

Define $f: R \rightarrow R$ by $f(x)= \begin{cases}(x-a) \frac{e^{\frac{1}{(x-a)}}-1}{\frac{1}{(x-a)}}+1 & \text { for } x \neq a \\ 0, \quad \text { at } x=a\end{cases}$
Then which one of the following is true?

Let $f, g: \mathbf{R} \rightarrow \mathbf{R}$ be functions defined by
$$ f(x)=\left\{\begin{array}{cc} x \sin \left(\frac{1}{x}\right), & \text { for } x \neq 0 \\ 0, & \text { for } x=0 \end{array}\right. $$
and $g(x)=x f(x)$
Consider the ...

For a square matrix $B$ of order 3 , if $B^T=B^{-1}$ and $|B|=1$, then $|B-I|=$

For $\alpha, \beta \in[0,2 \pi]$ and $\gamma \in[0, \pi)$ consider the system of equations
$$ \begin{aligned} & 2 \sin \alpha-\cos \beta+3 \tan \gamma=3 \\ & 4 \sin \alpha+2 \cos \beta-2 \tan \gamma=2 \\ & 6 \sin \alpha-3 \cos \beta+\tan \g...

$$ \text { The rank of } A=\left[\begin{array}{ccc} 1 & x & x+1 \\ 2 x & x^2-x & x^2+x \\ 3 x(x-1) & x\left(x^2-3 x+2\right) & x\left(x^2-1\right) \end{array}\right] \text { is } $$

If $S(a, b)$ is a fixed point and $P(\alpha, \beta)$ is such a variable point that $4\left[(x-a)^2+(y-b)^2\right]=(\alpha x+\beta y+7)^2$ represents a parabola, then the locus of $P(\alpha, \beta)$ is

If $P(-3,2)$ is an end point of the focal chord $P Q$ of the parabola $y^2+4 x+4 y=0$, then the slope of the normal drawn at $Q$ is

Let $S_r=\{x, y, z) / x+y+z=11, x \geq r, y \geq r$, $z \geq r, x, y, z, r$ are integers $\}$ and $n\left(S_r\right)$ represents the number of elements in $S_r$. Then $n\left(S_{2)}+n\left(S_3\right)+n\left(S_4\right)=\right.$

A certain question paper contains three parts $A, B, C$ with four questions in part $A$, five questions in part $B$ and six questions in part $C$. A student is required to answer seven questions choosing at least two questions from each par...

A person tossing a biased coin indefinitely wins the game by getting head for the first time. The probability that he wins the game in odd number of tosses is $3 / 4$. If 5 such coins are tossed at a time then the probability that head appe...

Let $B(\alpha, \beta, \gamma)$ represents that a bag $B$ contains $\alpha$ red balls, $\beta$ green balls and $\gamma$ blue balls. Given $B_1(2,3,2), B_2(3,2,2), B_3(2,2,3)$. A die is rolled. If the die shows up 2 or 3 or 5 , then a ball wi...

If the coefficients $a$ and $b$ of a quadratic expression $x^2+a x+b$ are chosen from the sets $A=\{3,4,5\}$ and $B=\{1,2,3,4\}$ respectively, then the probability that the equation $x^2+a x+b=0$ has real roots is

A random variable $X$ has the following probability distribution
$$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline X=x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline P(X=x) & 0.15 & 0.23 & K & 0.10 & 0.20 & 0.08 & 0.07 & 0.05 \\ \hline \end{array} $$...

If the sides of a triangle are in the ratio $\sqrt{3}: \sqrt{5}: \sqrt{8+\sqrt{15}}$, then the largest angle in that triangle is

In a $\triangle A B C$, if $\tan A: \tan B: \tan C=1: 2: 3$ and $\sin A: \sin B: \sin C=\sqrt{5}: 2 \sqrt{2}: k$, then $k=$

In $\triangle A B C$, if $R=\frac{65}{8}, r r_1=42$ and $r_1-r=6.5$, then $s(s-a)=$

In $\triangle A B C$ if $\angle C=\frac{\pi}{2}$ then
$\tan ^{-1}\left(\frac{a}{b+c}\right)+\tan ^{-1}\left(\frac{b}{c+a}\right)+\tan ^{-1}\left(\frac{c}{a+b}\right)=$

The number of integral values of $x$ satisfying $9 x-2<(x+2)^2<12 x-3$ is

If $2+\sqrt{3}$ is a root of the equation $f(x)=x^4+2 x^3-16 x^2-22 x+7=0$, then which one of the following is not a root of $f(x)=0$ ?

If $f(1)=3$, and $f(n+1)-f(n)=3\left(4^n-1\right)$, then $\forall n \in \mathbf{N}$, $f(n)=$

The mean and standard deviation of 100 observations $x_1, x_2, \ldots, x_{100}$ were calculated as 40 and 5.1 respectively by a student who took by mistake 50 instead of 40 for one observation. Then the correct value of $\sum_{i=1}^{100} x_...

The coefficient of variation of the first 5 prime numbers is

If a variable line is moving such that the intercepts made by it on the coordinate axes are reciprocal to each other, then the points $P(x, y)$ on such lines satisfy