TS EAMCET 2022 (Online) 20th July Evening Shift

The absolute maximum value of the function $f(x)=2 x^3-3 x^2-36 x+9$ defined on $[-3,3]$ is

The equation of the tangent to the curve $x^2+y-7=4 x$ at the point $(1,10)$ is

If $\theta$ is the angle between the curves $x^2-y^2=4$ and $y^2=3 x$, then $\tan \theta=$

Numerically greatest term in the expansion of $(2 x-3 y)^{11}$ when $x=\frac{1}{3}$ and $y=\frac{1}{2}$ is

If $L_1, L_2$ and $L_3$ are the chords of contact of the three points $(2,0),(1,-2)$ and $(4,4)$ respectively with respect to the circle $x^2+y^2=3$, then $L_1, L_2$ and $L_3$ are

If $(h, k)$ is the centre of the circle which passes through the origin and cuts the circles $x^2+y^2+4 x+6 y+12=0$ and $x^2+y^2+4 x-6 y+9=0$ orthogonally, then $k-2 h=$

If $(-1,-1)$ is the radical centre of the circles $x^2+y^2+2 g x-4 y+4=0, x^2+y^2+6 x+2 f y+12=0$ and $x^2+y^2+10 y+20=0$, then $g-f=$

The line $4 x+3 y-4=0$ divides the circumference of a circle in the ratio $1: 2$. If $C(5,3)$ is the centre of that circle, then equation of the circle is

The combined equation of the direct common tangents of the circles $x^2+y^2+2 x=0$ and $x^2+y^2-2 y-3=0$

Two sides of a square are along the lines $x=-5$ and $y=4$. The point of intersection of the diagonals is $(3,-4)$. The point of intersection of the tangents drawn to the circumcircle of the square at the two consecutive vertices lying on $...

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 $\frac{R}{r}=$