TS EAMCET 2022 (Online) 20th July Evening Shift

If $z=\frac{-1-i \sqrt{3}}{2}$, then $\sum_{k=1}^{2022}\left(z^k+\frac{1}{z^k}\right)^2=$

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2 x+2=0$, then $\alpha^{2020}+\beta^{2020}=$

If $\int_{-a}^a f(x) d x=\int_0^a f(x) d x+\int_0^a g(x) d x$, then $g(x)=$

\(\int_0^4| | x-2|-x| d x=\)

The general solution of the differential equation $d x=(2 x+3 y-4) d y$ is

$f\left(x, y, c_1, c_2\right)=0$ is an equation containing two arbitrary constants $c_1$ and $c_2$. If the differential equation having $f\left(x, y, c_1, c_2\right)=0$ as its general solution is of $k$ th order, then the differential equat...

If $l$ and $m$ are respectively the order and the degree of the differential equation $f(x) y^{\prime \prime}+g(x) y^{\prime}=\frac{4 y}{x}$ whose general solution is $y=a x^2+b x^2 \log x$, then $f(m)+g(m)=$

If $x^2+y^2=t-\frac{1}{t}, x^4+y^4=t^2+\frac{1}{t^2}$, then $\frac{d y}{d x}=$

If $f(x)=\sum_{p=1}^7 p^2 \sin ^{-1}\left(\frac{4}{5} \sin (p x)-\frac{3}{5} \cos (p x)\right)$, then the value of $\frac{d f}{d x}$ at $x=1$ is [given that $\sin ^{-1}(\sin x)=x$ ])

If $y=\frac{a x+b}{c x+d}$, then $\frac{d x}{d y}=$

If $m$ is the length of the latusrectum and $n$ is the length of the major-axis of the ellipse $25 x^2+16 y^2-150 x-64 y-111=0$, then the ordered pair $(m, n)=$

If $P(\theta)$ and $Q\left(\frac{\pi}{2}+\theta\right)$ are two points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the locus of mid-point of $P Q$ is $\frac{x^2}{\alpha^2}+\frac{y^2}{\beta^2}=1$, then $\frac{a+b}{\alpha+\beta}=$

If $[x]$ represents the greatest integer function, then the set of all real values of $x$ for which $f(x)=\sqrt{\frac{[x]-x}{x-[x]}}$ is real is

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

Assertion (A) $\operatorname{coth} x=\frac{1-k}{1+k}(0 < k < 2)$.
Reason (R) The graph of $y=\tanh x$ always lies between the lines $y=-1$ and $y=1$
The correct option among the following is

If $P\left(\frac{\pi}{6}\right)$ is a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1, S, S$ are its foci and $S P+S P=2 | S P-S P$|, then $e=$

Let $S$ be the focus of the hyperbola $x^2-2 y^2=1$ lying on the positive $X$-axis. Let $P(-1,1)$ be a given point. Then, the area of the triangle formed by the line $P S$ with the coordinate axes is (in sq. units)

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

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

If $x \neq(2 n+1) \frac{\pi}{2}, n \in Z$ and $\cos x \neq \frac{-1}{2}$, then
\(\int\left(\frac{\sin x+\sin 2 x}{1+\cos x+\cos 2 x}\right)^2 d x=\)

If $f(x)=\int \frac{2-3 \sin ^2 x}{1+\cos 2 x} d x$ and $f\left(\frac{\pi}{4}\right)=1$, then $f(0)=$

Given that $\int \frac{1}{x^2+a^2} d x=\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right)+c$.
$$ \begin{aligned} & \text { If } \int \frac{1}{x^4+3 x^2+1} d x=a \tan ^{-1}\left(\frac{b\left(x^2-1\right)}{x}\right) \\ & +c \tan ^{-1}\left(\frac{...

\(\lim _{x \rightarrow 0} \frac{4[\sin (2022 x)-\sin (2020 x)]}{x[\cos (2022 x)+2 \cos (2021 x)+\cos (2020 x)]}=\)

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

If $\left[\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right]$ is a skew-symmetric matrix and $b, c$ and $f$ are non-zero real numbers, then $\frac{b}{c}=$

In the matrix $\left[\begin{array}{ccc}-1 & x & 3 \\ -4 & -5 & -6 \\ -7 & y & 9\end{array}\right]$, if the cofactors of -6 and -7 are respectively 22 and 27 , then $5 x+y=$

Consider the simultaneous linear equations $\beta x+\alpha y-z=-1,3 x-\beta y+\alpha z=0 \alpha x+\beta y+z=1$, In the usual notation used in Crammer's rule, given that $\frac{\Delta_1}{\Delta}=-1, \frac{\Delta_2}{\Delta}=1, \frac{\Delta_3}...

If $\left|\begin{array}{cc}2+3 i & i \\ 1-2 i & -i\end{array}\right|=x+i y$, then $x+y=$

If $A$ is a $2 \times 2$ matrix such that $\operatorname{det} A=-21$ and trace of $A^3$ is 2024 , then the trace of $A$ is

The equation of the given curve is $x^2-4 x+4 y-8=0$. Match the following
$$ \begin{array}{lll} \hline & \text { List I } & \text { List II } \\ \hline \text { (A) } & \text { Focus } & \text { (I) }(4,2) \\ \hline \text { (B) } & \text { V...

If one end of a focal chord of the parabola $y^2=\frac{8}{a} \times(a>0)$ is at $(1,4)$, then the length of this focal chord is

The total number of ways of selecting 4 letters from all the letters of the word TSEAMCET is

If ${ }^m P_r-{ }^{(m-1)} p_r=a \cdot{ }^{(m-1)} P_s$, then $a-s=$

A random variable $X$ has the following probability distribution
$$ \begin{array}{llllllllll} \hline X=\mathbf{x}_i & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline P\left(X=\mathbf{x}_i\right) & 10 k & 9 k & 8 k & 8 k & 6 k & 5 k & 4 k & 3 k ...

Two cards are drawn from a pack of 52 playing cards one after the other. If $p_1$ is the probability of getting a queen in the first draw and a diamond card in the second draw when the first card drawn is replaced and $p_2$ is the probabili...

The probability of getting a king and a spade card when two cards are drawn simultaneously from a pack of 52 playing card is

If $X$ is a Poisson variate such that $\frac{5}{3} k=P(X=2) =P(X=3)$, then $P(X=5)=$

Bag $A$ contains 4 white and 2 black balls, bag $B$ contains 3 white and 3 black balls and bag $C$ contains 2 white and 4 black balls. If a bag is chosen at random and a ball is chosen at random from it, then the probability that the ball d...

In a $\triangle A B C$, if $\frac{a}{\tan A}=\frac{b}{\tan B}=\frac{c}{\tan C}$, then $\cos ^2 A+\cos ^2 B+\cos ^2 C=$

In $\triangle A B C$, if $a=7, b=10$ and $c=11$, then $\frac{R}{r}=$

In $\triangle A B C$, if $a=7, b=8, \tan C=\frac{3 \sqrt{5}}{2}$ and $C$ is an acute angle, then $c=$

Let $a$ be a common root of the equations $x^3-2 x-25 \lambda=0,3 x^3-8 x-\frac{175}{3} \lambda=0$ and $\lambda>0$. Then, $\lambda=$

If the sum of two roots of the equation $x^3-7 p x^2+5 q x-6 r=0$ is zero, then

Statement I The set of solutions of $|x|^2-4|x|+3<0$ is the interval $(-3,3)$
Statement II If $x<3$ or $x>5$, then $x^2-8 x+15>0$
Which of the above statements is (are) true?

The roots of a cubic equation $f(x)=0$ are diminished by $\frac{-3}{2}$ so, as to remove the term containing $x^2$ and the transformed equation is $8 x^3-54 x-78=0$. Then, the equation $f(x)=0$ is

If $\alpha$ and $\beta$ are the irrational roots of the equation $3 p^2 x^3+p x^2+q x+3=0$ when $p=1$ and $q=-7$, then $|\alpha-\beta|=$

If $6 x-x^2+12$ attains its extreme value $\beta$ at $x=\alpha$, then $\beta=$

Statement I The range of the ungrouped data does not change even if certain intermediate observations are removed
Statement II The value of the mean deviation of an ungrouped data about the median is always less than or equal to the value o...

By rotating the axes through an angle of $30^{\circ}$ in the anti-clockwise direction about the origin, the equation $4 x^2+12 x y+9 y^2+6 x+9 y+2=0$ becomes $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ becomes, then

If $\theta$ is the acute angle between the pair of lines $12 x^2+2 h x y+7 y^2=0$ and $\tan \theta=\frac{8}{19}$, then $h=$