TS EAMCET 2022 (Online) 18th July Evening Shift

One of the values of $(-32 i)^{\frac{2}{5}}$ is

If $\cos \alpha+\cos \beta+\cos \gamma=0$ and $\sin \alpha+\sin \beta+\sin \gamma=0$ then $\cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma=$

If the point $(x, y)$ satisfies the equation $\frac{x+i(x-2)}{3+i}-i =\frac{2 y+i(1-3 y)}{i-3}$, then $x+y=$

\(\int_1^2 x \sqrt{4-x^2} d x=\)

If $[x]$ denotes the greatest integer function of $x$ and
\(\int_{-3 / 2}^{3 / 2}[2 x-3] d x=k, \text { then }\left|k+\frac{1}{2}\right|=\)

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

The differential equation corresponding to the family of curves given by $a x^2+b y^2=1$, where $a$ and $b$ are arbitrary constants is

For the differential equation
\(\sqrt{\frac{d^2 y}{d x^2}}=\sqrt[3]{\left[y \frac{d y}{d x}+x \sin \left(\frac{d y}{d x}\right)\right]^2}\)

If $f(x)=\sin \left(\cosh \left(\frac{x^2+1}{x^2+2}\right)\right)$, then $f^{\prime}(1)=$

If $a f(x)+b f\left(\frac{1}{x}\right)=x+1$, and $\frac{d}{d x}\left(x^2 f(x)\right)=2 x^2+2 x+\frac{1}{3}$, then $a-b$

Let $S \equiv \frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0, S \equiv \frac{x^2}{\alpha^2}+\frac{y^2}{\beta^2}-1=0$ be two intersecting ellipses. If $P(a \cos \theta, b \sin \theta)$ and $Q\left(a \cos \left(\frac{\pi}{2}+\theta\right), b \sin \left(...

$P\left(\theta_1\right)$ and $Q\left(\theta_2\right)$ are two points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $e$. If $P S Q$ is a focal chord and $\tan \left(\frac{\theta_1}{2}\right) \tan \left(\frac{\theta_2}{...

Let $R$ be the set of all real number
Statement I The function $f:\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \rightarrow R$ defined by $f(x)=\sec x+\tan x$ is one-one function.
Statement II The function $f:[0, \infty) \rightarrow R$ defined...

Let $R$ be the set of all real numbers. Let $f: R \rightarrow R$ be a function defined by
$$ f(x)=\left\{\begin{array}{rcc} 2 x-5, & \text { if } & x<-3 \\ x+2, & \text { if } & -3 \leq x<5 \\ 3 x+1, & \text { if } & x \geq 5 \end{array}\ri...

Let $S$ be the focus of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ lying on the positive $X$ - axis and $P\left(5, y_1\right)$ be point on the hyperbola. Then $S P=$

If $P(\theta)=\left(x_1, \frac{3 \sqrt{5}}{2}\right), 0<\theta<\frac{\pi}{2}$ is a point on the hyperbola $\frac{x^2}{25}-\frac{y^2}{9}=1$, where $\theta$ is the parameter in its parametric form, then $2 x_1+9 \sin ^2 \theta=$

If the perimeter of a triangle is 20 and two of its vertices are $(-5,0)$ and $(6,0)$, then the locus of the third vertex is

If $\int e^x \cos x d x=\frac{e^x}{2}(\cos x+\sin x)$ and
\(\int \frac{\cos \left(\log \left(\frac{2 x+3}{3-2 x}\right)\right)}{(3-2 x)^2} d x=\frac{f(x)}{24}[\cos (g(x))+\sin (g(x))]+c\)
then $g(1)=$

If $\int \frac{1+\cos 8 x}{\tan 2 x-\cot 2 x} d x=f(x) \cdot \cos (g(x))+c$, then $f\left(\frac{1}{4}\right)+g\left(\frac{1}{4}\right)=$

Let $x \neq \frac{-3}{5}, \frac{2}{5}$, if $f\left(\frac{2 x+1}{5 x+3}\right)=x+2$, then $\int f(x) d x=$

$$ \begin{aligned} & \text { If } \int \frac{(x+3)}{(x-1)^2(2 x-1)} d x \\ & =\frac{A}{x-1}+B \log (2 x-1)+C \log (x-1)+k, \text { then } A+B+C= \end{aligned} $$

Let $A=\left(a_{i j}\right)$ be an $n \times n$ matrix defined by $a_{i j}=\left\{\begin{array}{cc}k^i, & \forall i=j \\ 0, & \text { otherwise }\end{array}\right.$. If $m=$ trace of $A$ and $\lim _{k \rightarrow 1} \frac{n-m}{1-k}=171$, th...

\(\mathop {\lim }\limits_{x \to \infty } {x^3}\left[\sqrt{x^2+\sqrt{x^4+1}}-\sqrt{2 x}\right]=\)

Let $f(x)=\left\{\begin{array}{ccc}3-x & \text { if } & x<-3 \\ 6 & \text { if } & -3 \leq x \leq 3 . \text { Let } \alpha \text { be the number } \\ 3+x & \text { if } & x>3\end{array}\right.$ of points of discontinuity of $f$ and $\beta$ ...

If $\frac{x^2+7}{\left(x^2+1\right)(x-2)}=\frac{A}{x-2}+\frac{B x+C}{x^2+1}$, then the determinant of the matrix $\left[\begin{array}{ll}A & B \\ C & \frac{2}{5}\end{array}\right]$ is

If $A, P, B$ are $3 \times 3$ matrices. If $|-B|=5,\left|B A^T\right|=15$, $\left|P^T A P\right|=-27$, then one of the values of $|P|$ is

Let $x=\alpha, y=\beta, z=\gamma$ be the unique solution of the system of simultaneous linear equations $2 x+3 y-2 z+4=0,3 x-4 y+3 z+5=0$, $k x-2 y+z+3=0$. If $\alpha=-2$, then $k=$

If $A+B=\left[\begin{array}{lll}2 & 1 & 2 \\ 1 & 2 & 0 \\ 0 & 2 & 2\end{array}\right], A B=\left[\begin{array}{lll}1 & 2 & 2 \\ 1 & 1 & 0 \\ 1 & 2 & 1\end{array}\right]$, then $A^2+B(A+B)=$

If $A$ is a $3 \times 3$ matrix and $|A|=\frac{1}{2}$, then $\left|A^{-1}(\operatorname{Adj}(\operatorname{Adj} A))\right|^{-1}=$

The vertex and the focus of the parabola $2 x^2+5 y-6 x+1=0$ respectively, are

The axis of a parabola is along the line $y=x$ and the distance of its vertex $A$ from $(0,0)$ is $\sqrt{2}$ and that of its focus $S$ from $(0,0)$ is $2 \sqrt{2}$. If $A$ and $S$ lie in first quadrant, then the equation of the parabola in ...

$a, b, c$ are three particular speakers among the 10 speakers of a meeting. The number of ways of arranging all 10 speakers on the dias in a row so that all the three speakers $a, b, c$ do not sit together is

The exponent of 6 in 72 ! is

The variance of a Poisson variate $X$ is 2 . Then, $P(X \geq 3)=$

Two dice are thrown and the sum of the numbers appearing on the dice is observed to be a multiple of 4 . If $p$ is the conditional probability that number 4 has appeared atleast once, then $3 p+2=$

In a random experiment of throwing 5 coins, the number of heads is defined as a random variable. The mean of the random variable is

When two dice are thrown, the probability of getting a prime number on die and a composite number on the other is

Let $A, B, C$ be three pairwise independent events of a random experiment. If $P(\bar{B} \cup \bar{C})=\frac{1}{2}, P(A)>0, P(B)=b$ and $P(C)=c, P((\bar{B} \cap \bar{C} \mid A)=$

If $\Delta$ denotes the area of $\triangle A B C$, then $(b \sin C+c \sin B)(b \cos C+c \cos B)=$

In $\triangle A B C$, if $A$ is acute, $C$ is obtuse, $\sin A=\frac{3 \sqrt{3}}{14}, a=3$ and $b=5$, then $c=$

Let $A$ be the area of in-circle and $A_1, A_2, A_3$ be the area of ex-circles of a triangle. If $A_1=4, A_2=9, A_3=16$, then $A=$

Let the transformed equation of $2 x^4-8 x^3+3 x^2-1=0$ so that the term containing the cubic power of $x$ is absent be $2 x^4+b x^2+c x+d=0$. Then, $b=$

If the quadratic equations $x^2-7 x+3 c=0$ and $x^2+x-5 c=0$ have a common root, then for non-zero real value of $c$ the sign of the expression $x^2-3 x+c$ is

II. Let $f(x)=\frac{6 x^2-18 x+21}{6 x^2-18 x+17}$. If $m$ is the maximum value of $f(x)$ and $f(x)>n \forall x \in R$. Then, $14 m-7 n=$

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+x^2+x+r=0$ and $\alpha^3+\beta^3+\gamma^3=5$, then $r=$

If $\tan 15^{\circ}$ and $\tan 30^{\circ}$ are the roots of equation $x^2+p x+q=0$, then $p q=$

If $1-\sqrt{2}$ and $2+i$ are the roots of the equation $x^4+b x^3+c x^2+d x+e=0$ where $b, c, d, e$ are rational numbers, then the roots of the equation $b x^2+c x+d=0$ are

If $\frac{5}{2}$ is the sum of two roots of the equation $6 x^6-25 x^5+31 x^4-31 x^2+25 x-6=0$ then the sum of all non-real roots of the equation is

The mean deviation from the mean of the discrete data $1,3,4,7,11,18,29,47,78$ is

The transformed equation of $3 X^2+4 X Y+Y^2-8 X-4 Y-4=0$ is $f(X, Y)=a X^2+2 h X Y+b Y^2+c=0$ when the origin is shifted to a new point by the translation of axes. Then, $f(1,1)=$