TS EAMCET 2020 (Online) 10th September Morning Shift

The equation of the circle passing through the points of intersection of the two orthogonal circles $S_1=x^2+y^2+k x-4 y-1=0$, $S_2=3 x^2+3 y^2-14 x+23 y-15=0$ and passing through the point $(-1,-1)$ is

Let $z=x+i y$ be a complex number, $A=\{z /|z| \leq 2\}$ and $B=\{z /(1-i) z+(1+i) \bar{z} \geq 4\}$ Then which one of the following options belongs to $A \cap B$ ?

The solutions of the equation $z^2\left(1-z^2\right)=16, z \in \mathbf{C}$, lie on the curve

\(\left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^8+\left(\frac{1+\cos \theta-i \sin \theta}{1+\cos \theta+i \sin \theta}\right)^{16}=\)

If $z, \bar{z},-z,-\bar{z}$ forms a rectangle of area $2 \sqrt{3}$ square units, then one such $z$ is

\(\mathop {\lim }\limits_{x \to \infty }\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{2^2}{n^2}\right) \ldots \ldots\left(1+\frac{n^2}{n^2}\right)\right]^{1 / n}=\)

\(\int_{\pi / 4}^{\pi / 2} \frac{3 d x}{1+e^{\sqrt{8} \sin \left(x-\frac{3 \pi}{8}\right)}}=\)

If the family of curves $y=a e^{4 x}+b e^{-x}$, where $a, b$ are arbitrary constants represents the general solution of the differential equation
$$ f\left(x, y \frac{d y}{d x}, \frac{d^2 y}{d x^2}\right)=0, \text { then } \frac{d f}{d x}= ...

If the length of the sub tangent at any point $p(x, y)$ on a curve $f(x, y)=0$ is $x+7 y^2$, then $f(x, y)=$

If the general solution of the differential equation $(y-x+1) d y-(y+x+2) d x=0$ is $f(x, y, c)=0$, then the value of $c$ such that $f(1,1, c)=0$ is

let $g(x) \neq 0, g^{\prime}(x) \neq 0, f(x) \neq 0, f^{\prime}(x) \neq 0$. If
$F(x)=f(x) g(x), G(x)=f^{\prime}(x) g^{\prime}(x)$ and
$F^{\prime}(x)=G(x) H(x)=F(x) K(x)$, then $H(x)+K(x)=$

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

Let $f(x)$ and $g(x)$ be twice differentiable functions such that $f(x)=x^2+g^{\prime}(1) x+g^{\prime \prime}(2)$ and $g(x)=f(1) x^2+x f^{\prime}(x)+f^{\prime \prime}(x)$. Then $f(x)-g(x)=$

The eccentricity of an ellipse passing through $(3 \sqrt{2}, \sqrt{10})$ with foci at $(-4,0)$ and $(4,0)$ is

If the product of the lengths of the perpendiculars drawn from the foci to the tangent $y=\frac{-3}{4} x+3 \sqrt{2}$ of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is 9 , then the eccentricity of that ellipse is

Let $f:[0,10] \rightarrow[1,20]$ be a function defined as
$$ f(x)=\left\{\begin{array}{ll} \frac{60-5 x}{3}, & 0 \leq x \leq 6 \\ 10, & 6 \leq x \leq 7 \\ 31-3 x, & 7 \leq x \leq 10 \end{array} \text { then } f\right. \text { is } $$

The domain of the function, $f(x)=\sqrt{\log _{10}\left(\frac{5 x-x^2}{4}\right)}$ is

The equation of the hyperbola, whose eccentricity is $\sqrt{2}$ and whose foci are 16 units apart, is

If $\frac{4 x^2+5 x^4+7}{\left(x^2+1\right)\left(x^4+x^2+1\right)}=\frac{A x+B}{x^2+1} +\frac{C x^3+D x^2+E x+F}{x^4+x^2+1}$, then $B+2(D+F+E)-C \cdot A=$

\(\int \frac{y^2+\sqrt[3]{y^4}+\sqrt[6]{y^2}}{y\left(1+\sqrt[3]{y^2}\right)} d y=\)

For $k \in(1, \infty), \int \frac{1}{1+k \cos x} d x=$

\(\int e^{-3 x}\left(x^2+\sin 4 x\right) d x=\)

If $\int \frac{2 x^{12}+5 x^9}{\left(1+x^3+x^5\right)^3} d x=\frac{x^m}{l\left(1+x^3+x^5\right)^r}+C$, then $\frac{m-l}{r}=$

\(\operatorname{cosec}^{-1}\left[\left(\frac{\tan ^2\left(\frac{\alpha-\pi}{4}\right)-1}{\tan ^2\left(\frac{\alpha-\pi}{4}\right)+1}+\cos \frac{\alpha}{2} \cdot \cot 5 \alpha\right) \sec \frac{11 \alpha}{2}\right]\)

If $\tan ^{-1} \frac{1}{5}+\frac{1}{2} \sec ^{-1} x+\tan ^{-1} \frac{1}{8}=\frac{\pi}{8}$, then $x^2=$

Assertion $(\mathrm{A}) \operatorname{cosech}^{-1}(3)=\log \left(\frac{1+\sqrt{10}}{3}\right)$
Reason (R) $e^{\operatorname{cosech}^{-1} x}$ is a root of the quadratic equation $x p^2-2 p-x=0$
The correct option among the following is

If $\log (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\ldots \ldots \infty$ and $\mathop {\lim }\limits_{x \to 0} \frac{\log (1+x)^{1+x}}{x^2}-\frac{1}{x}=k$, then $12 k=$

If $f(x)=\left\{\begin{array}{ll}k, & \text { for } x=1 \\ \frac{(9 x-1)(\sqrt{x}-1)}{3 x^2+2 x-5}, & \text { for } x \neq 1\end{array}\right.$ is continuous on $[0, \infty)$, then $k=$

In each of the choices given below, a function and an interval are given. The correct choice having a function and the associated interval for which the Lagrange's mean value theorem is not valid is

If $A, B$ are two non singular matrices of order $3,|B|=k$, a positive integer, then match the items of list-I with the items of list-II.


.tg {border-collapse:collapse;border-spacing:0;}

.tg td{border-color:black;border-style:solid;bord...

All the real values of $p, q$ so that the system of equations
\(2 x+p y+6 z=8, x+2 y+q z=5\)
and $\quad x+y+3 z=4$
may have no solution are

If $p$ and $q$ are two distinct real values of $\lambda$ for which the system of equations
$$ \begin{array}{r} (\lambda-1) x+(3 \lambda+1) y+2 \lambda z=0 \\ (\lambda-1) x+(4 \lambda-2) y+(\lambda+3) z=0 \\ 2 x+(3 \lambda+1) y+3(\lambda-1) ...

Consider the parabola $y^2+2 x+2 y-3=0$ and match the items of List-I with those of the List-II.
$$ \begin{array}{llll} \hline & \text { List-I } & & \text { List-II } \\ \hline \text { A. } & 2 x-5=0 & \text { I. } & \text { Vertex } \\ \h...

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 which pass through $(5,0)$, the centroid of the triangle formed by the feet of these three normals is

The total number of three digit and five digit integers which can be formed by using the digits $0,1,2,3,4,5$ but using each digit not more than once in each number is

At an election a voter may vote for any number of candidates not exceeding the number to be elected. If 4 candidates are to be elected out of the 12 contested in the election and voter votes for at least one candidate, then the number of wa...

If $A$ and $B$ are events of a sample space such that $P(A \cup B)=\frac{3}{4}, P(A \cap B)=\frac{1}{4}$ and $P(\bar{A})=\frac{2}{3}$, then $P(\bar{A} \cap B)$ is

Let $X$ and $Y$ be two events of a sample space such that $P(X)=\frac{1}{3}, P(X / Y)=\frac{1}{2}$ and $P(Y / X)=\frac{2}{5}$ then

Let $A$ and $B$ be not mutually exclusive events. If $P(A)=\frac{4}{9}, P(A \cap \bar{B})=\frac{3}{7}$ then $P\left(\frac{B}{A}\right)=$

If $20 \%$ of the bolts produced by a machine are defective then the probability that out of 4 bolts chosen at random, less than 2 bolts will be defective, is

In a book consisting of 600 pages, there are 60 typographical errors. The probability that a randomly chosen page will contain at most two errors, is

In $\triangle A B C, \frac{1+\cos C}{r_1+r_2}+\frac{1+\cos A}{r_2+r_3}+\frac{1+\cos B}{r_1+r_3}=$

In a triangle $A B C$, if $\cos A \cos B+\sin A \sin B \sin C=1$, then $a: b: c=$

In a $\triangle A B C$ if $\angle A=3 \angle B, C A=9$ and $B C=16$, then the length of $A B$ is

Let $S$ be the set of all possible integral values of $\lambda$ in the interval $(-3,7)$ for which the roots of the quadratic equation $\lambda x^2+13 x+7=0$ are all rational numbers. Then the sum of the elements in $S$ is

$\alpha$ is the maximum value of $1-2 x-5 x^2$ and $\beta$ is the minimum value of $x^2-2 x+r$. If $5 \alpha x^2+\beta x+6>0$ for all real values $x$, then the interval in which $r$ lies is

For the equation $x^4+x^3-4 x^2+x-1=0$ the ratio of the sum of the squares of all the roots to the product of the distinct roots is

If $\alpha_1, \beta_1, \gamma_1, \delta_1$ are the roots of the equation $a x^4+b x^3+c x^2+d x+e=0$ and $\alpha_2, \beta_2, \gamma_2, \delta_2$ are the roots of the equation $e x^4+d x^3+c x^2+b x+a=0$ such that $0<\alpha_1<\beta_1<\gamma_...

Let the greatest common divisor of $m, n$ be 1 . If $\frac{1}{1 \cdot 7}+\frac{1}{7 \cdot 13}+\frac{1}{13 \cdot 19}+\ldots \ldots$. upto 20 terms $=\frac{m}{n}$, then $5 m+2 n=$

For the following frequency distribution, the variance is approximately equal to
$$ \begin{array}{cccccc} \hline \begin{array}{c} \text { Class } \\ \text { Interval } \end{array} & 0-5 & 5-10 & 10-15 & 15-20 & 20-25 \\ \hline \text { Frequ...