TS EAMCET 2023 (Online) 12th May Evening Shift

$f: R \rightarrow R$ is a function defined by $f(x)=\frac{1}{e^x+2 e^{-x}}$
Assertion (A) : $f(c)=\frac{1}{3}$ for some values of $c \in R$
Reason (R) : $0 < f(x) \leq \frac{1}{2 \sqrt{2}}$ for all $x \in R$
Then, which of the following opt...

Electric current $(I)$ is measured by galvanometer, the current being proportional to the tangent of the angle ( $\theta$ ) of deflection. If the deflection is read as $45^{\circ}$ and an error of $1 \%$ is made in reading it, the percentag...

If the equation of a tangent drawn to the curve $y=\cos (x+y),-1 \leq x \leq 1+\pi$ is $x+2 y=k$, then $k=$

For all real values of $x$, the minimum value of $\frac{1-x+\lambda^2}{1+x+x^2}$ is

The area (in sq units) bounded by the curve $y=2 x-x^2$ and the line $y=-x$ is

If $(1+x)^n=c_0+c_1 x+c_2 x^2+\ldots \ldots+c_n x^n$ for $n \in N$, then $c_0+\frac{c_1}{2}+\frac{c_2}{3}+\ldots \ldots+\frac{c_n}{n+1}=$

The number of integral terms in the expansion of $(\sqrt{3}+\sqrt[8]{5})^{256}$ is

The expansion of $\left(1+x+x^2\right)^{-3 / 2}$ in powers of $x$ is valid, if

If a circle passing through $(1,-2)$ has $x-y=2$ and $2 x+3 y=14$ as its diameters, then the radius of the circle is

The number of common tangents to the circles $x^2+y^2-2 x-6 y+9=0$ and $x^2+y^2+6 x-2 y+1=0$ is

The equation of the line perpendicular to the radical axis of two circles $x^2+y^2-5 x+6 y+12=0$, $x^2+y^2+6 x-4 y-14=0$ and passing through $(1,1)$ is

If the angle between the circles
\(x^2+y^2-2 x-4 y+c=0 \text { and } x^2+y^2-4 x-2 y+4=0\)
is $60^{\circ}$, then $c=$

The pole of the straight line $9 x+y-28=0$ with respect to the circle $2 x^2+2 y^2-3 x+5 y-7=0$ is

The equation of the circle whose diameter is the common chord of the circles $x^2+y^2+2 x+3 y+1=0$ and $x^2+y^2+4 x+3 y+2=0$ is

If the imaginary part of $\frac{2 z+1}{i z+1}$ is -2, then the locus of the point representing $z$ in the Argand plane is

\(\text { If } x+i y=\sqrt{\frac{3+i}{1+3 i}}, \text { then }\left(x^2+y^2\right)^2=\)

$\operatorname{Arg}\left(\sin \frac{6 \pi}{5}+i\left(1+\cos \frac{6 \pi}{5}\right)\right)=$

If $i=\sqrt{-1}$, then $(1+i)^{10}+(1-i)^{10}=$

\(\int\limits_2^5 \sqrt{\frac{5-x}{x-2}} d x=\)

\(\int\limits_0^{\frac{\pi}{2}} \sin ^6 x \cos ^4 x d x=\)

\(\lim\limits_{n \rightarrow \infty}\left[\frac{1}{n^2} \sec ^2 \frac{1}{n^2}+\frac{2}{n^2} \sec ^2 \frac{4}{n^2}+\ldots \ldots+\frac{1}{n} \sec ^2 1\right]=\)

\(\int_0^{\pi / 4} \frac{\sec x}{1+2 \sin ^2 x} d x=\)

The general solution of the differential equation $\left(2 x-10 y^3\right) d y+y d x=0, y \neq 0$ is

The general solution of the differential equation $\left(x \sin \frac{y}{x}\right) d y=\left(y \sin \frac{y}{x}-x\right) d x$ is

The degree and order of the differential equation of the family of parabolas whose axis is the $X$-axis, are respectively

If $\sec \left(\log _2 y^2\right)=\operatorname{cosec}\left(\log _2 x^2\right)$, then $\frac{d y}{d x}=$

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

If $x=\log p$ and $y=\frac{1}{p}$, then $\frac{d y}{d x}=$

The locus of the mid-points of the intercepted portion of the tangents by the coordinate axes, which are drawn to the ellipse $x^2+2 y^2=2$ is

The product of the lengths of the perpendiculars drawn from the two foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ to the tangent at any point on the ellipse is

Tangents are drawn to the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1$ at all the ends of its latus recta. The area of the quadrilateral, so formed (in sq units) is

The domain of the function $f(x)=\sin ^{-1}\left(\log _2\left(\frac{x^2}{2}\right)\right)$ is

If $f: R \rightarrow R$ is defined by $f(x)=2 x+\sin x, x \in R$, then $f$ is

The range of the function $f(x)=-\sqrt{-x^2-6 x-5}$ is

$P(a \sec \theta, b \tan \theta)$ and $Q(a \sec \phi, b \tan \phi)$ are two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ where, $\phi+\theta=\frac{\pi}{2}$. If $(h, k)$ is the point of intersection of the normals drawn at $P$...

If $\int \frac{x}{(a+x)^5} d x=\frac{1}{k(a+x)^4}(f(x))+C$, then $\frac{f(-a)}{a k}=$

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

\(\text { Match the following items from List I into List II }\)






List-I

List-II





1.

∫







sin

2



⁡

x







cos

4



⁡

x...

If $\int x^4(\log x)^3 d x=x^5\left[A(\log x)^3\right]$ $\left.+B(\log x)^2+C \log x+D\right]+k$, then $A+B+C+5 D=$

If $f(9)=9$ and $f^{\prime}(9)=4$, then $\lim\limits_{x \rightarrow 9} \frac{\sqrt{f(x)}-3}{\sqrt{x}-3}=$

\(\lim\limits_{x \rightarrow 1}(1-x) \tan \left(\frac{\pi}{2} x\right)=\)

\(\sinh (\log (3+\sqrt{8}))=\)

$$ \begin{aligned} &\text { If } \omega \neq 1 \text { is a cube root of unity, then }\\ &\left|\begin{array}{ccc} \omega+\omega^2 & \omega^2+\omega^9 & \omega^9+\omega \\ \omega^{27}+\omega^{31} & \omega^{31}+\omega^{17} & \omega^{17}+\ome...

Let $A$ be a matrix such that $A B$ is a scalar matrix, where $B=\left[\begin{array}{ll}1 & 2 \\ 0 & 3\end{array}\right]$ and $\operatorname{det}(3 A)=27$. Then, $3 A^{-1}+A^2=$

$$ \left|\begin{array}{ccc} \sqrt{3} & 2 \sqrt{5} & \sqrt{5} \\ \sqrt{15} & 5 & \sqrt{10} \\ 3 & \sqrt{15} & 5 \end{array}\right|= $$

If $A$ is a non-singular matrix such that $(A-2 I)$ $(A-3 I)=0$, then $\frac{1}{5} A+\frac{6}{5} A^{-1}=$

If $A$ is a symmetric matrix with real entries, then

The normal at a point on the parabola $y^2=4 x$ passes through $(5,0)$. If there are two more normals to this parabola passing through $(5,0)$, then the equation of one of these normals is

The equations of common tangents to the parabola $y^2=16 x$ and the circle $x^2+y^2=8$ are

The total number of all those 3-digit numbers in which the sum of all the digits in each of them is 10 , is