TS EAMCET 2023 (Online) 12th May Morning Shift

If the function $f(x)=\frac{x}{5}+\frac{5}{x},(x \neq 0)$ attains its relative maximum value at $x=\alpha$, then $\sqrt{\alpha^2+2 \alpha-6}=$

The equation of the normal at $t=\frac{\pi}{2}$ to the curve $x=2 \sin t, y=2 \cos t$ is

If the expression $x^3+3 x^2-9 x+\lambda$ is of the form $(x-\alpha)^2(x-\beta)$, then the values of $\lambda$ are

The number of rational terms in the binomial expansion $(\sqrt[4]{5}+\sqrt[5]{4})^{100}$ is

The coefficient of $x^{50}$ in the expansion of $(1+x)^{101}\left(1-x+x^2\right)^{100}$ is

If the term independent of $x$ in the expansion of $\left(\sqrt{x}-\frac{k}{x^2}\right)^{10}$ is 405 , then $k=$

If the inverse of $P(-3,5)$ with respect to a circle is $(1,3)$ then polar of $P$ with respect to that circle is

The equations of the tangents to the circle $x^2+y^2=4$ drawn from the point $(4,0)$ are

If the tangent drawn at the point $P$ on the circle $x^2+y^2+6 x+6 y=2$ meets the straight line $5 x-2 y+6=0$ at a point $Q$ on the $Y$-axis, then the length of $P Q$ is

If a diameter of the circle $x^2+y^2-4 x+6 y-12=0$ is a chord of a circle $S$ whose centre is at $(-3,2)$, then the radius of $S$ is

If a circle passing through $A(1,1)$ touches the $X$-axis, then the locus of the other end of the diameter through $A$ is

If $C(\alpha, \beta)(a<0)$ is the centre of the circle that touches the $Y$-axis at $(0,3)$ and makes an intercept of length 2 units on positive $X$-axis, then $(\alpha, \beta)=$

The image of every point lying on the curve $x^2+y^2=1$ in the line $x+y=1$ satisfies the equation

If $i=\sqrt{-1}$, then $1+i^2+i^4+i^6+\ldots \ldots+i^{2024}=$

If $\frac{1+i \cos \theta}{1-2 i \cos \theta}$ is purely real, then $\cos ^3 \theta+\sin ^2 \theta+\cos \theta+1=$

If $z_1$ and $z_2$ are complex numbers such that $\left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right|$, then the difference in the amplitude of $z_1$ and $z_2$ is

If $\theta=\frac{\pi}{6}$, then the 10 th term of the series $1+(\cos \theta+i \sin \theta)^1+(\cos \theta+i \sin \theta)^2+\ldots$. is

If $\alpha$ and $\beta$ are non-zero integers and $z=(\alpha+i \beta)(2+7 i)$ is a purely imaginary number, then minimum value of $|z|^2$ is

[.] is the greatest integer function, then
\(\int_0^{2 \pi}[|\sin x|+|\cos x|] d x=\)

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

If $f$ is defined on $R$ such that $f(x) f(-x)=9$, then

\(\int_{-23}^{23} \frac{d x}{3+f(x)}=\)

\(\int_{1 / 2}^2\left|\log _{10} x\right| d x=\)

The general solution of $\frac{d y}{d x}+y f^{\prime}(x)-f(x) f^{\prime}(x)=0$, $y \neq f(x)$ is

If $m$ and $n$ are respectively the order and degree of the differential equation of the family of parabolas with origin as its focus and $X$-axis as its axis, then $m n-m+n=$

The derivative of $\frac{1-x^2}{1+x^2}$ with respect to $\frac{2 x}{1+x^2}$ at $x=2$ is

If $\tan y=\cot \left(\frac{\pi}{4}-x\right)$, then $\frac{d y}{d x}=$

If the slope of the tangent drawn to the curve $y=e^{a+b x^2}$ at the point $P(1,1)$ is -2 , then the value of $2 a-3 b$ is

If $x=3 \sqrt{2} \cos ^3 \theta$ and $y=4 \tan ^2 \theta$, then $\left(\frac{d y}{d x}\right)_{\theta=\pi / 4}=$

A particle is travelling in clockwise direction on the ellipse $\frac{x^2}{100}+\frac{y^2}{25}=1$. If the particle leaves the ellipse the point $(-8,3)$ on it and travels along the tangent to the ellipse at that point, then the point where ...

If an ellipse with foci at $(3,3)$ and $(-4,4)$ is passing through the origin, then the eccentricity of that ellipse is

If $f:[2, \infty) \rightarrow R$ is defined by $f(x)=x^2-4 x+5$, then the range of $f$ is

If $t$ is a parameter, $A=(a \sec t, b \tan t)$, $B=(-a \tan t, b \sec t)$ and $O=(0,0)$, then the locus of the centroid of $\triangle O A B$ is

If $f(x)=-|x|$, then $($ fofof $)(x)+($ fofof $)(-x)=$

If the equation of a hyperbola is $9 x^2-16 y^2+72 x-32 y-16=0$, then the equation of conjugate hyperbola is

\(\int \frac{\sin ^8 x-\cos ^8 x}{1-2 \sin ^2 x \cos ^2 x} d x=\)

7. If $\int \sin (101 x)(\sin x)^{99} d x$ $=\frac{\sin (100 x)(\sin x)^\lambda}{\mu}+C$ then, $\frac{\lambda}{\mu}=$

If $n$ is a positive integer greater than 1 and $I_n=\int \frac{\sin n x}{\sin x} d x$, then $I_{n+1}-I_{n-1}=$

If $\frac{x^4}{(x-1)(x-2)(x-3)}=p(x)+\frac{A}{x-1}+\frac{B}{x-2}+\frac{C}{x-3}$, then $p\left(\frac{3}{2}\right)+C=$

If $\int e^x\left(\sin ^2 2 x-8 \cos 4 x\right) d x=e^x f(x)+C$, then $f\left(\frac{\pi}{4}\right)=$

If $\int \frac{x^2\left(\sec ^2 x+\tan x\right)}{(x \tan x+1)^2} d x=\frac{-x^2}{x \tan x+1}+f(x)+C$, then

\(f(x)=\)

If $a, b, c$ and $k$ are non-zero real numbers and $\lim \limits_{x \rightarrow \infty} x\left(a^{1 / x}+b^{1 / x}+c^{1 / x}-3 k^{1 / x}\right)=0$, then $k=$

\(\lim \limits_{x \rightarrow 1} \frac{(2 x-3)(\sqrt{x}-1)}{2 x^2+x-3}=\)

If $\sinh (\log x)=-2$, then $x=$

The range of the function $f(x)=\log _{0.5}\left(x^4-2 x^2+3\right)$ is

If $P$ is a non-singular matrix such that $I+P+P^2+\ldots \ldots+P^n=0(0$ denotes the null matrix $)$, then $P^{-1}=$

If the system of equations
$x+k y+3 z=-2$,
$4 x+3 y+k z=14,$
$2 x+y+2 z=3$ can be solved by matrix inversion method, then

If $A=\left[\begin{array}{ccc}5 & 5 \alpha & \alpha \\ 0 & \alpha & 5 \alpha \\ 0 & 0 & 5\end{array}\right]$ and $\operatorname{det}\left(A^2\right)=25$, then $|\alpha|=$

$P$ is a $3 \times 3$ square matrix and $\operatorname{Tr}(P) \neq 0$. If $\operatorname{Tr}\left(P-P^I\right)+$ $\operatorname{Tr}\left(P+P^T\right)+\frac{\operatorname{Tr}(P)}{\operatorname{Tr}\left(P^T\right)}+\operatorname{Tr}(P) \times...

Two tangents are drawn from the point $(-1,-2)$ to the parabola $y^2=4 x$. If $\theta$ is the angle between these tangents, then $\tan \theta=$

If two circles $x^2+y^2-6 x-6 y+13=0$ and $x^2+y^2-8 y+9=0$ intersect at $A$ and $B$, then the focus of the parabola whose directrix is the line $A B$ and vertex is the point $s(a, b)$ is