TS EAMCET 2023 (Online) 13th May Morning Shift

$x$ and $y$ are two positive integers such that $2 x+3 y=50$. If $x^2 y^3$ is maximum for $x=\alpha$ and $y=\beta$, then $\frac{\alpha}{2}+\frac{\beta}{5}=$

The nearest approximate value of $\sqrt{2023}$ is (let $\Delta x=87$ ).

The slope of the normal drawn at a point $P$ to the curve $y=x^3-10 x^2+31 x-30$ is $-\frac{1}{14}$. If the co-ordinates of $P$ are integers, then the $X$-intercept of the tangent drawn at $P$ to the given curve is

Area of the region bounded by the curve $y=2-x-3 x^2$, the $X$-axis, the $Y$-axis and the line $x=-2$ is

In the expansion of $(x-2 y+3 z)^5$, if the total number of terms is $p$ and the coefficient of $x^2 y z^2$ is $q$, then $\frac{q}{p}=$

If $|x|$ is so small that $x^3$ and higher powers of $x$ can be neglected, then an approximate value of $\frac{1}{\sqrt{4-x}(2+x)^3}$ is

Let $C_0, C_1, C_2, \ldots, C_n$ be the binomial coefficients in the expansion of $(1+x)^n$. If $S_{n+1}=5 \cdot C_0+8 \cdot C_1+11 \cdot C_2+\ldots(n+1)$ terms, then $S_{11}=$

Let $S \equiv x^2+y^2-8 x+10 y+5=0$ be a circle. Let $P(1,1)$ and $Q(1,-1)$ be two points. Then, the point of intersection of the polar of $P$ with respect to $S=0$ and the chord with $Q$ as mid-point to $S=0$ is

Let the line $x-y+1=0$ intersect the circle $x^2+y^2+2 x+2 y+1=0$ in two points $A$ and $B$. If $A B$ is the diameter of the circle $x^2+y^2+2 g x+2 f y+c=0$, then $g+f=$

If $(3,1)$ and $(-2,4)$ are points on a circle $S$ whose centre lies on the line $x-y+1=0$, then the parametric equations of $S$ are

Let 6,8 be the $X$ and $Y$-intercepts made by the circle $S \equiv x^2+y^2+2 g x+2 f y+c=0$, respectively. If $g x+f y+1=0$ is a line passing through the point $(1,-1)$, then the radius of the circle $S=0$ is

If the angle between the circles $x^2+y^2-2 x+2 y+1=0$ and $x^2+y^2+2 x-2 y+k=0$ is $\frac{\pi}{3}$, then

Let a chord $A B$ subtend an angle of $60^{\circ}$ at the centre $C(2,3)$ of a circle $S$. If the equation of $A B$ is $x+y+1=0$, then the equation of the circle $S$ is

If the value of $\sqrt{-5-12 i}+\sqrt{7+24 i}$ is a negative real number $k$, then $k=$

If $i$ is the root of the equation $x^2+1=0$, then
\((1+\sqrt{3} i)^{2023}+(1-\sqrt{3} i)^{2023}=\)

If a point $P$ denotes the complex number $z=x+i y$ in the argand plane and if $\frac{z-(2+i)}{z+(1-2 i)}$ is purely real, then the locus of $P$ is

One of the values of $(\sqrt{3}-i)^{\frac{1}{6}}$ is

If $a x^2-x y-3 y^2-5 x+20 y+c=0$ represents a pair of lines passing through the point $(2,3)$, then $a-c=$

Let $z=x+i y$ be a point in the argand plane. If the amplitude of $\left(\frac{z-3}{z+2 i}\right)$ is $\frac{\pi}{2}$, then the locus of $z$ is

\(\int_{-\frac{\pi}{8092}}^{\frac{\pi}{8092}} \frac{\sec (2023 x)}{1+(2023)^{(2023 x)}} d x=\)

\(\int_0^2 x^{\frac{5}{2}} \sqrt{2-x} d x=\)

\(\int_0^3\left|x^2-3 x+2\right| d x=\)

If $A$ and $B$ are arbitrary constants, then the differential equation having $y=A e^x+B \sin 2 x$ as its general solution is

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

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

If $f(x)=\frac{e^{2 x}-e^{-2 x}}{e^{3 x}+e^{-3 x}}$, then $f^{\prime}(0)=$

If $f(x)=x^{\tan x}+(\tan x)^x$, then $f^{\prime}\left(\frac{\pi}{4}\right)=$

If $f(x)=|x-1|+|x-2|$, then
\(f^{\prime}(-2023)+f^{\prime}\left(\frac{2024}{2023}\right)+f^{\prime}(2023)=\)

If $4 x+2 y+n=0$ is a normal to the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$ then $n=$

Let $A=(2,0)$ and $B=(0,-2)$. Let $P$ be any point such that the sum of the distance of $P$ from $A$ and $B$ is 4 . Then, the equation of the locus of the point $P$ is

Let $P$ be the point to which origin has to be shifted by the translation of axes, so as to remove the first degree terms from the equation $3 x^2+y^2-6 x+4 y+4=0$. If the origin is shifted to $P$ by the translation of axes, then the transf...

Let $S$ and $S^{\prime}$ be the foci of an ellipse $E$ and $B$ be one end of its minor axis. Let $\angle S^{\prime} S B=\pi / 6$ and $(2 \sqrt{3}, 1)$ be a point on $E$. If $X$-axis is the major axis and $Y$-axis is the minor axis of the el...

The domain of the real valued function
\(f(x)=\frac{\sqrt{\log _{10}\left(\frac{x}{x-2}\right)}}{\sqrt{[x]^2-5[x]+6}} \text { is }\)
(Here, $[x]$ denotes the greatest integer function)

If $\frac{6 x^4+13 x^3+2 x^2-x+3}{2 x^2+3 x-2}=f(x)+\frac{A}{a x-1}+\frac{B}{x+b}$, then $f(\mathrm{l})+a \cdot B+b \cdot A=$

If the function $f: R \rightarrow R$ is defined by
$$ f(x)= \begin{cases}2 x-3, & \text { if } x<-2 \\ x^2-1, & \text { if }-2 \leq x \leq 2 \\ 3 x+2, & \text { if } x>2\end{cases} $$
then $f$ is

The range of the real valued function $f(x)=\frac{1}{x-|x|}$ is

If $y=m x+4(m>0)$ is a tangent to the hyperbola $\frac{x^2}{25}-\frac{y^2}{9}=1$, then the point of contact of this tangent is

\(\int \frac{1}{(x-2)\left(x^2+1\right)} d x=\)

\(\int \frac{1}{3 \cos x-4 \sin x+5} d x=\)

\(\int \frac{\sec ^2 x}{(\sec x+\tan x)^2} d x=\)

\(\int \frac{1}{16-7 \sin ^2 x} d x=\)

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

\(\lim _{x \rightarrow 0} \frac{\left(3^{2 x}-\sqrt{x+1}\right) \sin 5 x}{1-\cos 4 x}=\)

$$ \left|\begin{array}{lll} 2 & 3 & 5 \\ 3 & 5 & 2 \\ 5 & 2 & 3 \end{array}\right|+\left|\begin{array}{ccc} 1 & 1 & 1 \\ 7 & 11 & 13 \\ 49 & 121 & 169 \end{array}\right|= $$

If $A=\left[\begin{array}{ccc}k & 5 & 2 \\ 2 & -k & 5 \\ 5 & 2 & -k\end{array}\right]$ and $\operatorname{det} A=190$, then $\operatorname{adj} A=$

If $A=\left[\begin{array}{ccc}1 & 2 & -1 \\ -1 & 0 & 2 \\ 1 & 2 & 0\end{array}\right]$ and $B=\left[\begin{array}{ccc}-3 & -2 & 4 \\ 2 & 2 & -1 \\ -2 & 0 & 3\end{array}\right]$, then $A^2=$

If the unique solution of the simultaneous linear equations $3 x-2 y+z=5 k, 2 x+3 y-2 z=-5 k$, $x+4 y+3 z=k$ is $x=\alpha, y=\beta, z=3$, then $k=$

Normals are drawn from the point $P(8,0)$ to the parabola $y^2=12 x$. If $\theta$ is the acute angle between two non-horizontal normals among them, then $\tan \theta=$

If the focal distance of a point $P\left(2, y_1\right)$ on the parabola $y^2=k x$ is 3 , then the equation of the tangent drawn at $P$ to the given parabola is

If $n, r$ are two positive integers such that $1 \leq r