TS EAMCET 2020 (Online) 14th September Evening Shift

If the radical axis of the circles $x^2+y^2+2 \alpha x+2 \beta y+c=0$ and $x^2+y^2+\frac{3}{2} x+4 y+c=0$ touches the circle $x^2+y^2+2 x+2 y+1=0$, then $4 \alpha \beta-8 \alpha-3 \beta+10=$

If the parametric equations of the circle passing through the points $(3,4),(3,2)$ and $(1,4)$ is $x=a+r \cos \theta$, $y=b+r \sin \theta$, then $b^a r^a=$

From a point $P$ on the circle $x^2+y^2-4 x-6 y+9=0$, a pair of tangents $P Q$ and $P R$ are drawn touching the circle $x^2+y^2-4 x-6 y+12=0$ at $Q$ and $R$. If $C$ is the centre of the concentric circles, then the area of the $\triangle C ...

Let $z$ be a complex number such that $|z|-z=2+i$, where $i=\sqrt{-1}$. Then, $|z|=$

If the amplitude of $z-2-3 i$ is $\pi / 4$, then the locus of $z=x+i y$ is

For $n>1$ and $n \in \mathbf{N}$, if $z_1, z_2, \ldots, z_n$ are the roots of the equation $(z+1)^n=z^n$, then $\sum_{i=1}^n \frac{\cot ^{-1}\left(2\left|\operatorname{Im} z_i\right|\right)-1}{2 \operatorname{Re} z_i}=$

If $f(x)=\frac{1}{x^3} \int_5^x\left(2 u^2-u f^{\prime}(u) d u\right.$, then $f^{\prime}(5)=$

Assertion (A) $\int\limits_{-a}^a f(x) d x=\int_0^a(f(x)+f(-x)) d x$
Reason (R) $\int\limits_a^b f(x) d x=\int_{g(a)}^{g(b)} f(g(u)) g^{\prime}(u) d u$
The correct option among the following is

If $\cos x+\cos 2 x+\ldots+\cos n x=\frac{A(x)}{2 \sin x / 2}$, then $\int\limits_0^\pi A(x) d x=$

The order and degree of the differential equation $\frac{d^2 y}{d x^2}+y+\left(\frac{d y}{d x}-\frac{d^3 y}{d x^3}\right)^{3 / 2}=0$, are respectively.

The general solution of the differential equation $\frac{d y}{d x}=\frac{2 x-3 y+4}{3 x+2 y-7}$ is

The general solution of $\frac{d y}{d x}=\frac{x+y+1}{y-x+1}$ is

\(\frac{d}{d x}\left[\operatorname{cosech}^{-1}(\tan 2 x)\right]=\)

Let $f: R \rightarrow R$ be defined by $f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}$ for all $x$ and $y$. If $f^{\prime}(0)$ exists and equals -1 and $f(0)=1$, then $f(2)=$

If $a \alpha^2+b \beta^2+c \alpha \beta+d=0$ is the transformed equation of $4 x^2+\sqrt{3} x y+5 y^2-4=0$ obtained by using $\alpha=\frac{\sqrt{3}}{2} x+\frac{y}{2}$ and $\beta=-\frac{x}{2}+\frac{\sqrt{3}}{2} y$, then $c(a+b+d)=$

If tangents are drawn to the ellipse $x^2+2 y^2=2$, then the locus of the mid-points of the intercepts made by those tangents between the coordinate axes is

The area (in sq. units) of the quadrilateral formed by the tangents drawn at the end points of the latus rectum to the ellipse $S \equiv \frac{x^2}{16}+\frac{y^2}{12}=1$ is

If $f: Z \rightarrow N$ is defined by
$$ f(n)=\left\{\begin{array}{cll} 2 n, & \text { if } & n>0 \\ 1, & \text { if } & n=0, \text { then } f \text { is } \\ -2 n-1, & \text { if } & n<0 \end{array}\right. $$

If $\frac{x^5-5}{x^3+x^2}=f(x)+\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x+1}$, then the larger value of $K$ for which $f(K)+A+B+C=1$, is

If $p, q$ are the eccentricities of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and its conjugate hyperbola respectively, then the area of the square (in sq. units) formed by the points of intersection of the ellipse $\frac{x^2}{p^2}+...

If $\int \frac{(x-1) d x}{(x+1) \sqrt{x^3+x^2+x}}=A \cdot \tan ^{-1} \sqrt{f(x)}+$ constant, then the ordered pair $(A, f(-1))=$

If $f\left(\frac{2 x+3}{3 x+5}\right)=x+4, x \neq \frac{-5}{3}, \frac{2}{3}$ and $\int f(x) d x=A x+B \ln |3 x-2|+C$, then $3 B-A=$

If $\int e^x\left(\frac{x^2-8 x+19}{(x-1)^5}\right) d x=\frac{e^x(l x+m)}{(x-1)^4}+C$, then $4 l+m=$

\(\int \frac{d x}{(x-2) \sqrt{x^2-3 x+5}}=\)

Domain of $\cos ^{-1}\left[\log _5\left(x^2+7 x+15\right)\right]$ is

If $\sum\limits_{n=1}^k \tan ^{-1}\left(\frac{1}{n^2+3 n+3}\right)=\tan ^{-1} \alpha$, then $\alpha=$

The set of values of $x$ such that $\tan ^{-1}\left(\frac{x}{x-2}\right)-\tan ^{-1}\left(\frac{x}{2 x-1}\right)=\tan ^{-1}\left(\frac{2}{3}\right)$ is

Let $[x]$ denote the greatest integer less than or equal to $x$ and $k \geq 2$ be an integer. Then
\(\mathop {Lt}\limits_{x \to k} \frac{\sin \left(2 \pi\left([x]-\left[\frac{x}{k}\right]\right)-x\right)+\sin k}{x-k}=\)

Define $f(x)=\left\{\begin{array}{ll}1+x, & 0 \leq x \leq 2 \\ 3-x, & 2 < x \leq 3\end{array}\right.$.
If $f \circ f(x)$ is discontinuous at $a$ and $b$ in $[0,3]$ and $a

If $a$ and $b$ are any two real numbers, then
$$ \left|\begin{array}{ccc} 2 a-2 b-4 & 4 a & 4 a \\ 4 & 2-b-a & 4 \\ 2 b & 2 b & b-a-2 \end{array}\right|= $$

Let $A=\left[\begin{array}{ccc}2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & x\end{array}\right]$ and $A^2=A$. If $r$ is the rank of $A$, then $r+x=$

Let $a, b, c, d \in \mathbf{R}$ be such that $a d-b c \neq 0$ and $e$ be a positive number other than 1 .
If $x^a y^b=e^m, x^c y^d=e^n, \Delta_1=\left|\begin{array}{ll}m & b \\ n & d\end{array}\right|, \Delta_2=\left|\begin{array}{cc}a & m ...

If $P Q$ is a focal chord of the parabola $y^2=4 x$ with focus $S$ and $P=(4,4)$, then $S Q=$

If the parabola $x^2=4 a y,(a>0)$ makes an intercept of length $\sqrt{40}$ units on the line $y=1+2 x$ then $4 a=$

Consider the following statements:
I. The number of positive integral solutions of $x_1+x_2+x_3+x_4=10$ is 286 .
II. If $25!=10^n \times k,(k \in \mathbf{N})$, then $n=6$
Which one of the following options is true?

A student is allowed to select at least $(n+1)$ books but not all books from a collection of ( $2 n+1$ ) books. If the total number of ways in which he can select these books is 255 , then the number of books in that collection is

If a man throws a die until he gets a number bigger than 3 , then the probability that he gets a 5 in his last throw is

A diagnostic test has the probability 0.95 of giving a positive result when applied to a person suffering from a certain disease and a probability 0.10 of giving a positive result when given to a non-sufferer. It is estimated that $0.5 \%$ ...

Consider the following statements
Assertion (A) If $P_1, P_2, P_3$ are probability of happening of three independent events, then probability of happening of atleast one of them is $1-\left[\left(1-P_1\right)\left(1-P_2\right)\left(1-P_3\ri...

If probability function of a discrete random variable $X$ is $P(X=r)=r / k, r=1,2,3,4,5$, then $P\left(X=2\right.$ or $\left.X=\frac{k}{3}\right)$, is

If the probability that an individual will suffer a reaction from an injection of a drug is 0.001 , then the probability that out of 2000 individuals having that injection, more than 2 individuals will suffer a reaction, is

If the sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one, then the area (in sq. units) of that triangle is

In $\triangle A B C, A D$ and $B E$ are medians drawn from $A$ and $B$. If $A D=\frac{7}{2}, \angle D A B=\frac{\pi}{8}$ and $\angle A B E=\frac{\pi}{4}$, then the area (in sq. units) of $\triangle A B C$ is

If the radius of the incircle of a triangle with sides $5 k, 6 k$ and $5 k$ is 6 , then the largest angle of that triangle is

If $x$ is real, then the maximum and minimum values of $\frac{x^2+14 x+9}{x^2+2 x+3}$ are respectively

When $\mathbf{R}$ is the set of all real numbers,
\(\left\{x \in \mathbf{R}: \frac{\sqrt{12-x-x^2}}{x+10} \leq \frac{\sqrt{12-x-x^2}}{2 x+9}\right\}=\)

If $\alpha$ and $\beta$ are two complex roots of the equation $6 x^6-25 x^5+31 x^4-31 x^2+25 x-6=0$, then $\alpha+\beta=$

Let $f(n)=A(-2)^n+B(-3)^n \forall A, B \in \mathbf{R}$ and $n \in \mathbf{N}-\{1,2\}$. If $f(n)+a f(n-1)+b f(n-2)=0$, then $(a+b)(b-a)=$

If $1+\frac{\cos \theta}{2}+\frac{\cos 2 \theta}{4}+\frac{\cos 3 \theta}{8}+\ldots \ldots=\frac{a-2 \cos \theta}{5+b \cos \theta}$ for some $a, b \in \mathbf{R}$, then $(a-b)^2=$

In a discrete data $\frac{1 \text { th }}{4}$ of the observations are equal to $a$, another $\frac{1 \text { th }}{4}$ of the observations are equal to $-a$. Out of the remaining, half of them are equal to $b$ and the rest are equal to $-b$...