TS EAMCET 2022 (Online) 18th July Morning Shift

A circle passes through the points $(1,2)$, $(3,4)$. If its centre lines on the line $x-y+3=0$, then its radius is equal to

If $1, \omega, \omega^2$ are the cube roots of unity, $n \in N$ and $n>2$ then the least value of $n$ such that $1+\omega$ is a root of $x^n-x=0$ is

\(\sqrt{(-3+4 i)(8+6 i)}=\)

If $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^m=1,2022 < m < 2029$, then $m=$

Let $I=\int_{-\pi / 4}^{\pi / 4} \frac{1}{2-\cos 2 x}\left(\frac{\beta}{\pi}+\log \left(\frac{4+\sin x}{4-\sin x}\right)\right) d x$. Given that $\int \frac{d x}{1+k x^2}=\frac{1}{\sqrt{k}} \tan ^{-1}(\sqrt{k} x)+c, \tan ^{-1}(0)=0$ and $\t...

\(\int_1^4\left(x+\sqrt{x}+\frac{1}{x}\right) d x-\int_1^{2 \log 2} d x=\)

The curve that satisfies the differential equation $x y d y-\left(1+y^2\right) d x=0$ passes through $(1,0)$ and intersects the curve $x^2+3 y^2=3$ at an angle $\theta$. Then, $\frac{2 \theta}{\pi}=$

The degree of the differential equation
\(x\left(\frac{d^2 y}{d x^2}\right)^{1 / 3}+2 x^2\left(\frac{d^2 y}{d x^2}\right)^{5 / 3}+7 \frac{d y}{d x}+y=0\)

The differential equation of the family of circles with fixed radius $r$ units and centre on the line $y=3$, is

If $x=\operatorname{cosec} \theta-\sin \theta, y=\operatorname{cosec}^{2022} \theta-\sin ^{2022} \theta$ and $\left(\frac{d y}{d x}\right)^2=\frac{k\left(y^2+4\right)}{g(x)}$ where $k \in R$, then $10+k-g(2022)=$

If $f(x)=\log _e\left(e^{2 x}\left(\frac{3 x+5}{5-3 x}\right)^{2 / 3}\right), x \neq \frac{-5}{3}, \frac{5}{3}$, then the value of $\frac{d f}{d x}$ at $x=1$, is

When the coordinate axes are rotated about the origin in the positive direction through an angle $\frac{\pi}{4}$, if the equation $49 x^2+25 y^2=1225$ is transformed to $p x^2+q x y+r y^2=t$ and the GCD of $p, q, r, t$ is 1 , then

If the eccentricity and the length of the latusrectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are $\frac{\sqrt{3}}{2}$ and 1 respectively, then the sum of the lengths of major axis and minor axis of the ellipse is

The parametric equations of the ellipse whose focii are $(-3,0),(9,0)$ and eccentricity is $\frac{1}{3}$, are

If $[x]$ represents the greatest integer $\leq x$, then the range of the real valued function $f(x)=\frac{1}{\sqrt{[x]^2+[x]-2}}$ is

The domain of the real valued function $f(x)=\frac{\sqrt{6 x^2+5 x-6}}{\sqrt{4-x}-\sqrt{x+4}}$ is

Let $A\left(\theta_1\right)$ and $B\left(\theta_2\right)$ be two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $S$ be the focus of the hyperbola, If $A, S, B$ are collinear and
a $\cos \left(\frac{\theta_1+\theta_2}{2}\rig...

If $\frac{x^2}{k-\frac{5}{2}}+\frac{y^2}{\frac{7}{3}-k}=1$ ( $k$ is a real number) represents a hyperbola, then the set of all values of $k$ is

If $x>0$ and $x \neq(2 n+1) \frac{\pi}{2}$, then $\int\left(x \sqrt{x}-e^{\log (\sec x \tan x)}+\frac{3 x^2-2 x+1}{x^2}\right) d x=$

Let $f(x)=\int \frac{2 x^3-3 x^2+4 x-5}{x^2} d x$ and $f(1)=1$. Then, $f(5)=$

If $\frac{x^2+3}{\left(x^2+1\right)\left(x^2+2\right)}=\frac{A x+B}{x^2+1}+\frac{C x+D}{x^2+2}$ then $A+B+C+D=$

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

If $\frac{x^2-3 x+2}{(x-4)(x-3)^2}=\frac{A}{x-4}+\frac{B}{x-3}+\frac{C}{(x-3)^2}$ then $A+B+C=$

\(\lim _{x \rightarrow 3^{-}} \frac{x^3-3 x^2-4 x+12}{2 x^3-7 x^2+2 x+3}=\)

\(\lim _{x \rightarrow 0} \frac{2^{2 x}-2^{x+1}+2-\cos 2 x}{x^2}=\)

If $f(x)=\left\{\begin{array}{l}\frac{x^2-16}{x-4} \text { if } x>4 \\ 2 x \quad \text { if } x \leq 4\end{array}\right.$ then $f^{\prime}\left(4^{-}\right)+f^{\prime}\left(4^{+}\right)=$

Let $\alpha, \beta, \gamma$ be real numbers. If $A=\left[\begin{array}{ccc}7 & 3 & \alpha \\ \beta & 1 & -11 \\ -5 & \gamma & 19\end{array}\right]$ is a $3 \times 3$ matrix satisfying $A\left[\begin{array}{c}5 \\ -13 \\ 11\end{array}\right]...

If $[\alpha \beta \gamma]\left[\begin{array}{ccc}1 & 2 & 3 \\ 2 & 3 & -5\end{array}\right]=[352]$, then $\alpha^3+\beta^3+\gamma^3=$

3. Let $A=\left[\begin{array}{ccc}a & 3 & 5 \\ 5 & -1 & 3 \\ 2 & 3 & -4\end{array}\right]$ and $B=\left[\begin{array}{ccc}b & 1 & 4 \\ 4 & c & 1 \\ -3 & 1 & d\end{array}\right]$. If the trace of $A$ is -4 and $A B=\left[\begin{array}{ccc}-1...

$\left|\begin{array}{ccc}1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3\end{array}\right|=$

Let $P Q$ and $R T$ be two focal chords of the parabola $y^2=16 x$. If $P=(4,8)$ are $R=(16,16)$, then $Q T=$

If $y^2=16 x$ is the given parabola, then the point of intersection of the focal chord through the point $(2,2)$ and the double ordinate of length 24 is

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The number of 3-digit odd numbers divisible by 3 that can be formed using the digits $1,2,3,4,5,6$ when repetition is not allowed, is

3 balls are drawn one after the other without replacement from an urn containing 4 red, 5 blue and 6 yellow balls. The probability of getting three different coloured balls is

A pair of dice is thrown twice in succession. The probability of getting prime number on both the dice in first throw and composite numbers on both the dice in second throw is

Two balls are drawn at random from a bag containing 5 black balls and 3 white balls. If the random variable $X$ denotes the number of white balls drawn, then the mean of $X$ is

If the mean and variance of a binomial distribution are 4 and $\frac{4}{3}$ respectively, then $P(X=2)=$

A cube having edge of length 5 cm is painted on all faces and then it is cut into equal cubes of unit volume. A small cube is selected at random and found that a face of it is painted, then the probability that two more faces of it are also...

In a $\triangle A B C$, if $b=7, c=4 \sqrt{3}$ and $A=\frac{\pi}{6}$ then a $\sin B \sin C=$

In a $\triangle A B C$, if $(b+c)^2 \sin ^2 \frac{A}{2}+(b-c)^2 \cos ^2 \frac{A}{2}=K(1-\cos 2 A)$, then $K=$

In $\triangle A B C$, if $B C$ is the hypotenuse, then $r_2+r_3=$

If the extreme value of $3 x-2 x^2+1$ is $k$, then the set of all real values of $x$ for which $k x^2+2 x+1>0$ is

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-5 x^2-2 x+24=0$, then $\frac{\beta \gamma}{\alpha}+\frac{\gamma \alpha}{\beta}+\frac{\alpha \beta}{\gamma}=$

Let $p(x)$ be a quadratic polynomial with real coefficients. If $p(x)=0$ has only purely imaginary roots, then the zeroes of the polynomial $p(p(x))$ are

If $\alpha, \beta, \gamma$ are the roots of the equation $4 x^3+12 x^2-7 x+165=0$ and $\alpha+5, \beta+5, \gamma+5$ are the roots of the equation $a x^3+b x^2+c x+d=0$ then the product of the roots of the second equation is

If $\alpha, \beta, \gamma$ are the roots of the equation $3 x^3-26 x^2+52 x-24=0$ such that $\alpha, \beta, \gamma$ are in geometric progression and $\alpha<\beta<\gamma$, then $3 \alpha+2 \beta+\gamma=$

If $A=\left\{x \in R / \sqrt{x^2-8 x+15} \in R\right\}$ and $B=\left\{x \in R / \frac{x-3}{2 x-5}<\frac{x-6}{2 x-11}\right\}$, then $A \cap B=$

If $\bar{x}$ is the mean of $n$ observations $x_1, x_2, \ldots ., x_n$ then the mean of the absolute deviations of these observations from $\bar{x}$ is

If $\theta$ is the acute angle between the lines $\frac{x}{a}+\frac{y}{b}=1, \frac{x}{b}+\frac{y}{a}=1$, then $\sin \theta=$