TS EAMCET 2022 (Online) 19th July Evening Shift

Let $f(x)=\left\{\begin{array}{cc}1+6 x-3 x^2 & x \leq 1 \\ x+\log _2\left(b^2+7\right) & x>1\end{array}\right.$. Then, the set of all possible values of $b$ such that $f(1)$ is the maximum value of $f(x)$ is

The equation of the normal to the curve $\sin y=\sqrt{3} x \sin \left(\frac{\pi}{6}+y\right)$ at $x=0$, is

Assertion (A) The curves $y^2=4 x$ and $x^2=-2 y$ intersect at $(1,2)$ orthogonally.
Reason (R) If the product of the slopes of the tangents drawn to two curves at their point of intersection is -1 , then the curves are said to cut each oth...

The expansion of $(a+x)^n$ contains 15 terms. When $x=1$ the ratio of the neighbouring terms to the middle term in this expansion is 16 . Then, the positive integral value of ' $a$ ' is

If $A(1,1), B(-1,1)$ and $C(-1,-1)$ are three points and a point $P$ moves such that $(P A)^2=(P B)^2+(P C)^2$, then the equation of the locus of $P$ is

If $A=(0,-2)$ and $B$ is any point on the circle $x^2+y^2-2 x-2 y+1=0$, then the maximum value of $(\mathbf{A B})^2$ is

If the circle $x^2+y^2-6 x-12 y+1=0$ cuts another circle $C$ orthogonally and the centre of the circle $C$ is $(-4,2)$, then its radius of

If $(\alpha, \beta)$ is the pole of the line $3 x-5 y+6=0$ with respect to the circle $x^2+y^2-10 x+14 y+46=0$, then $\alpha+\beta=$

If the circles $x^2+y^2-16 x-20 y+164=r^2(r>0)$ and $x^2+y^2-8 x-14 y+29=0$ intersect in two distinct points, then the maximum possible integral value of $r$ is

The radius of the circle passing through the points $(-1,1),(2,-1)$ and $(1,0)$ is

$O(0,0)$ and $A(1,0)$ are centres of two units circles $C_1$ and $C_2$, respectively. $C_3$ is also a unit circle having its centre above $X$ - axis and passing through $O$ and $A$. The equation of the common tangent to $C_1$ and $C_3$ whic...

If $-i$ and $\alpha$ are the roots of the equation $i z^2-2(i+1) z+(2-i)=0, \tan \theta=\frac{-1}{2}$ and $\theta \in 4$ th quadrant, then $5^3 \cos 6 \theta=$

If $e^{i t}=\cos t+i \sin t$ and $e^{-i t}=\cos t-i \sin t$, then $\cosh (x+i y)-\cosh (x-i y)=$

If $z=\alpha+i \beta$ satisfies the equation $|z|-z=1+2 i$ and $|z|=\sqrt{\alpha^2+\beta^2}$, then $z \bar{z}=$

If $1, \alpha_1, \alpha_2, \alpha_3, \ldots \alpha_{n-1}$ are $n$th roots of unity then $\sum\limits_{1 \le i < f \le n - 1}^{} {} {a_i}{a_j} = $

If $(2-i)$ is one of the roots of the equation $x^4-9 x^3+31 x^2-49 x+30=0$ and $\alpha, \beta(\alpha<\beta)$ are its real roots, then $2 \alpha-\beta=$

\(\int_0^{\pi / 2} \sin ^4 \theta \cos ^3 \theta d \theta=\)

$\int_0^3\left(\sin \left(\frac{\pi}{3} x\right)-\cos \left(\frac{\pi}{3} x\right)\right) d x=$

Statement I The differential equation corresponding to the family of circles having their centres on $Y$-axis and fixed radius $k$ is $\left(x^2-k^2\right)\left(\frac{d y}{d x}\right)^2+x^2=0$
Statement II The differential equation correspo...

If $m$ and $n$ are respectively the order and the degree of the differential equation representing the family of curves $y^2-5 a x-5 a^{3 / 2}=0(a>0$ is a parameter), then the value of $m-n$ is

The general solution of $\left(\left(1+x^2\right) y \sin x-2 x y\right) d x-\log y^{1+x^2} d y=0$ is

The value of $\frac{d}{d x}\left[\log \left(\sin \sqrt{\frac{x^2+1}{x^2+2}}\right)\right]$ when $x=\sqrt{2}$, is

If $f(x)=\frac{1+\sec x}{2(\sec x-1)}$ for $0

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

\(\frac{d}{d x}\left[\left(x^{\frac{5}{2}}-x^{\frac{3}{2}}+1\right)\left(x^2-3 x+5\right)\right]=\)

If $S$ is the focus of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ lying on the positive $X$ - axis and $P(\theta)$ is a point on the ellipse such that $S P=1$, then $\cos \theta=$

Statement I The equation of the directrix of the ellipse $4 x^2+y^2-8 x-4 y+4=0$ is $3 y=6-4 \sqrt{3}$
Statement II The equation of the latusrectum of the ellipse $x^2+4 y^2-4 x-8 y+4=0$ is $y=2+\sqrt{3}$
Which of the above statement(s) is ...

Let $f: A \rightarrow B$ be defined as $f(x)=\frac{1}{2}-\tan \left(\frac{\pi x}{2}\right)$ and $g: B \rightarrow C$ be defined as $g(x)=\sqrt{3+4 x-4 x^2}$. If $A, B$ and $C$ are subsets of $R$ and $f$ is an onto function, then the range o...

If $D$ is the domain and $G$ is the range of the real valued function $f(x)=\sqrt{\frac{1-x^2}{1+x^2}}$, then $D \cap G=$

A hyperbola having its centre at the origin is passing through the point $(5,2)$ and has transverse axis of length 8 along the $X$-axis. Then, the eccentricity of its conjugate hyperbola is

If $e_1$ is the eccentricity of the hyperbola $x=\sec \theta$, $y=\sqrt{2} \tan \theta$ and $e_2$ is the eccentricity of the hyperbola $x=\sqrt{2} \sec \theta$ and $y=\tan \theta$, then $\frac{e_2^2}{e_1^2}=$

Let $g(x)$ be the anti-derivative of $f(x)$. Then, the function for which $\log _e\left(1+(g(x))^2\right)+c$ is an anti-derivative is

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

$\int \sqrt{4 \cos ^2 x-5 \sin ^2 x} \cos x d x=$

If $f(x)=\int\left[\tan ^2 x+\cot ^2 x+\frac{4\left(\sin ^3 x+\cos ^3 x\right)}{\sin ^2 2 x}\right] d x$ and $f\left(\frac{\pi}{4}\right)=0$, then $3\left[f\left(\frac{\pi}{6}\right)+2\right]=$

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

If $x=\log _e\left(\cot \left(\frac{\pi}{4}+\theta\right)\right)$, then $\lim _{\theta \rightarrow 0} \frac{\theta}{(\sinh x)(\cosh x)}=$

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

Let $\alpha, \beta$ and $\gamma$ be real numbers.
If $\left[\begin{array}{ccc}7 & 5 & \alpha \\ \beta & 2 & 11 \\ 3 & \gamma & 1\end{array}\right]\left[\begin{array}{l}1 \\ 3 \\ 2\end{array}\right]=\left[\begin{array}{c}\alpha+\beta \\ -2 \...

Let $A$ and $B$ be two $3 \times 3$ matrices and $C$ be a $3 \times 3$ unit matrix such that $A B-C$ is a non-singular matrix. Let $D=(A B-C)^{-1}$. Then, consider the following statements.
Statement I $\operatorname{det}(B A)=\operatorname...

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

Let $A=\left[\begin{array}{ll}0 & 1 \\ 1 & k\end{array}\right], k \in R$ and $A^3=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$. If $d=228$, then $b+c=$

If $x^2=8 a y$ is the transformed equation of $x^2-4 y+6 x+15=0$ when the origin is shifted to the point $(\alpha, \beta)$ by translation of axes, then $2 \alpha+8 \beta^2=$

Let $L L^{\prime}$ be the latusrectum and $P Q$ be the focal chord of the parabola $y^2=16 x$. If $P=(1,4)$ and $P, L$ lie in the same quadrant, then $L Q=$

If $P\left(\frac{1}{2}, 4\right)$ and $Q$ are the ends of a focal chord of the parabola $y^2=32 x$ and $S$ is the focus of the parabola, then $S Q=$

The number of ways of arranging the letters of the word LINEAR so that the letters N and R do not come together and E and A come together is

15 lines are concurrent at a point $P$. A line $L$ is not passing through $P$ intersects all the 15 lines and forms triangles with them. Then, the number of triangles having $L$ as one of its side is

$A$ and $B$ are two independent events $P(A)=\frac{2}{5}, P(B)=\frac{1}{3}$.
Match the following :


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In a university campus, the probability that a person chosen at random is an engineering student is $\frac{1}{5}$. The probability of having atmost two engineering students in a sample of 8 people is

A bag contains 3 white and 6 red balls. Four balls are drawn at a time randomly. Then, the probability of getting at least two red balls is