TS EAMCET 2023 (Online) 14th May Evening Shift

The number of common tangents of the circles $x^2+y^2-4=0$ and $x^2+y^2-6 x-8 y-24=0$ is

If the equation of the circle whose radius is $\sqrt{10}$ and which touches the circle $x^2+y^2+2 x+8 y-23=0$ externally at the point $(1,2)$ is $x^2+y^2+a x+b y+c=0$, then $|a+b+c|=$

The equation of a circle passing through $(-6,3)$ and touching both the coordinates axes is

If the circumcenter of the triangle formed by the points $A(a, 3), B(b, 5)$ and $C(a, b)$ is $(1,1)$, then out of all the possible coordinates of $C$ the sum of the absolute values of the distinct coordinates of $C$ is

The area (in sq units) of the triangle formed by the $x$-axis, the tangent and the normal drawn to the circle $x^2+y^2=10 x$ at the point $(9,3)$ is

If $\alpha$ and $\beta$ are the roots of the equation $x^2+x+1=0$, then $(\alpha+\beta)^2+\left(\alpha^2+\beta^2\right)^2+\left(\alpha^3+\beta^3\right)^2+\ldots+\left(\alpha^{12}+\beta^{12}\right)^2=$

If a polynomial $P(x)$ given by
$P(x)=2 x^4+a x^3+b x^2+c x+d$ is such that $P(1)=4$,
$P(2)=7, P(3)=12$ and $P(4)=19$, then $P(5)=$

If $z=x+i y$ is a complex number such that $z \bar{z}^3+\bar{z} z^3=350$ and $x, y$ are integers, then $|z|=$

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+x^2+x+1=0$, then match the items of List I with those of List II


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If $x=a+b, y=a \alpha+b \beta, z=a \beta+b \alpha$ and $\alpha, \beta$ are the complex cube roots of unity, then $x^3+y^3+z^3=$

The least positive integral value of $n$ such that $\left[\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right]^n=1$ is

If $z=\frac{3+2 i \cos \theta}{1-2 i \sin \theta}$ is a purely imaginary number, then
\(\sin ^2 \theta+\cos ^2 3 \theta=\)

\(\int_0^{\pi / 2} \frac{x \tan x \sec ^2 x}{\tan ^4 x+1} d x=\)

\(\int_3^6 \frac{\sqrt{x}}{\sqrt{9-x}+\sqrt{x}} d x=\)

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

If $a$ and $b$ are the arbitrary constants, then the differential equation corresponding to the family of curves given by $y=x[a \cos (\log x)+b \sin (\log x)]$ is

The general solution of the differential equation $\frac{d y}{d x}+\frac{y}{x}=x^2$ is

If the solution for the differential equation $y^2 d x+\left(x^2-x y-y^2\right) d y=0$ at $(2,1)$ is $x+y=k\left(x y^2-y^3\right)$, then $k=$

Match the values of $\frac{d y}{d x}$ at $x=\frac{\pi}{3}$ for the following system of curves in parametric form given in List-I with those of the items in List-II


List-I
List-II


(i)

x
=
a
(
θ
−
sin
⁡
θ
)
,
y
...

If $y=x \sin x$ and $\frac{\frac{d y}{d x}-\frac{y}{x}}{x \frac{d y}{d x}-y}$ at $x=\alpha$ is 1 , then $\alpha=$

If $f(x)=\sqrt{x}(x \geq 0)$ and $g(x)=1+x^2$, then $(f \circ g)^{\prime}(1)=$

If an ellipse with its axes as coordinate axes, $2 a$ and $2 b$ as the lengths of its major and minor axes respectively passes through the points $(2,2)$ and $(3,1)$, then $3 a^2+5 b^2=$

The values of $c$ such that the line $y=4 x+c$ touches the ellipse $\frac{x^2}{4}+\frac{y^2}{1}=1$ is

Let $f: R \rightarrow R$ be a function defined by
$$ f(x)=\left\{\begin{array}{cc} x^2-4 x+3, & \text { if } x<2 \\ x-3, & \text { if } x \geq 2 \end{array}\right. $$
Then, the number of real numbers $x$ for which $f(x)=8$ is

If $f(x)$ and $g(x)$ are two real valued functions such that $f(x)=3 x-2$ and $g(x)=x^2+2$, then $[(g \circ f)+(f \circ g)](x)=$

If $f(x)$ is a real valued function defined by $f(x)=\frac{a x^{10}+b x^8+c x^6+d x^4+e x^2+12 x+15}{x}(x \neq 0)$ and $f(4)=-4$, then $f(-4)=$

If the line $2 x+\sqrt{6} y=2$ touches the hyperbola $x^2-2 y^2=4$, then the coordinates of the point of contact are

\(\int \frac{\tan x}{\sec ^2 x\left(1+\sec ^6 x\right)^{\frac{2}{3}}} d x=\)

$\int \frac{1+\sqrt{3} \cot x}{1-\sqrt{3} \cot x} d x=$

\(\int \frac{1}{(x-1)^{\frac{5}{7}}(x+1)^{\frac{9}{7}}} d x=\)

$$ \begin{aligned} & \text { If } \int \frac{1}{\operatorname{cosec} x+\cos x} d x=\frac{1}{2 \sqrt{3}} \log |f(x)| \\ & -\int \frac{\cos x-\sin x}{2+\sin 2 x} d x+c, \text { then at } x=\frac{\pi}{3},|f(x)|= \end{aligned} $$

If a function $f$ is defined by $f(x)=\frac{\cot ^3 x-\tan x}{\cos (x+\pi / 4)},(x \neq \pi / 4)$, then $\lim _{x \rightarrow \pi / 4} f(x)=$

$$ \begin{array}{r} \lim _{x \rightarrow 0} \frac{2 \tan x+\cos x-1+x}{\sqrt{4 \sin ^2 x+2 \tan x+1}}= \\ -\sqrt{3 \tan ^2 x+\sin x+1} \end{array} $$

If $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 5\end{array}\right]$ and $\alpha, \beta \in R$ are such that $\alpha A^2-\beta A=2 I$, then $\alpha^2+\beta=$

If the system of equations $x+y+z=5, x+2 y+2 z=6$ and $x+3 y+\lambda z=\mu(\lambda, \mu \in R)$ is solvable by Matrix Inversion Method, then

If $X_{4 \times 3}, Y_{4 \times 3}$ and $P_{2 \times 3}$ are the matrices, then the order of the matrix $\left[P\left(X^T Y\right)^{-1} P^T\right]^T$ is

If $\left|\begin{array}{ccc}(1+\alpha)^2 & (1+2 \alpha)^2 & (1+3 \alpha)^2 \\ (2+\alpha)^2 & (2+2 \alpha)^2 & (2+3 \alpha)^2 \\ (3+\alpha)^2 & (3+2 \alpha)^2 & (3+3 \alpha)^2\end{array}\right|=k$ and $\alpha=-2$, then $k=$

If $\mathbf{A B}$ is the focal chord of the parabola $y^2=16 x$ and $A=(1,-4)$, then the equation of the normal to the parabola at the point $B$ is

If one of the vertices of an equilateral triangle inscribed in the parabola $y^2=12 x$ coincides with the vertex of the parabola, then the area (in sq units) of that triangle is

The number of ways in which 6 men and 4 women can be seated around a table, so that a particular man and a particular woman never sit adjacent to each other is

The number of odd numbers greater than 600000 that can be formed by using the digits $3,6,7,8,9,0$ without repetition is

There are three sections in a question paper, each section containing 4 questions. If a candidate has to answer only 5 questions from this paper without leaving any section, then the number of ways in which a candidate can make the choice o...

If a matrix is chosen at random from the set of all $3 \times 3$ non-zero matrices whose entries are the elements of the set $\{-1,0,1\}$, then the probability that the matrix is skew-symmetric is

\(\text { A random variable } X \text { has the following distribution, }\)
$$ \begin{array}{lllllll} \hline X=x_i & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline P\left(X=x_i\right) & 0.1 & k & 0.2 & 2 k & 3 k & k \\ \hline \end{array} $$
Then, the ...

A boy throws an unbiased die. Whenever he gets 1 on the die he has a further chance to throw it once again immediately. The probability that the boy gets a score of 7 in this process is

There are 10 coins in a box out of which 8 are normal and the remaining are with heads on both sides. A coin is chosen at random from the box and tossed 6 times. If it shows heads each time, then the probability that the selected coin has h...

Assertion (A) In $\triangle A B C$, if $r=6, r_2=36, R=15$, then $c^2+a^2=b^2$.
Reason (R) In $\triangle A B C$, if $r: R: r_2=1: 2.5: 6$, then $B=90^{\circ}$. The correct option among the following is

In $\triangle A B C$, if $a, b, c$ are in arithmetic progression and $A=2 C$, then $b: c=$

If $\alpha, \beta$ are the roots of $x^2+a x+2=0$ and $1 / \alpha, 1 / \beta$ are the roots of $x^2-b x+c=0$, then
$$ \left(\alpha+\frac{1}{\beta}\right)\left(\beta+\frac{1}{\alpha}\right)\left(\alpha-\frac{1}{\alpha}\right)\left(\beta-\fra...

The sum of all the real values of $x$ satisfying the equation $\left(x^2-7 x+11\right)^{x^2-6 x-7}=1$ is