TS EAMCET 2023 (Online) 14th May Morning Shift

If the function $f(x)=x e^{-x}, x \in R$ attains its maximum value $\beta$ at $x=\alpha$, then $(\alpha, \beta)=$

If a line having slope 2 is a tangent to the curve $y=x^4-6 x^3+13 x^2-12 x+5$ at points $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right), x_1, x_2 \in N$, then $x_1 x_2-y_1 y_2=$

Let $m$ be the slope of the normal $L$ drawn at $(1,2)$ to the curve $x=t^2-7 t+7, y=t^2-4 t-10$ and $a x+b y+c=0$ be the equation of the normal $L$. If GCD of $(a, b, c)$ is 1 , then $m(a+b+c)=$

The area (in sq units) of the region bounded by the curve $y=|\sin 2 x|$ and the $X$-axis in $[0,2 \pi]$ is

If $y=\frac{3}{4}+\frac{3 \cdot 5}{4 \cdot 8}+\frac{3 \cdot 5 \cdot 7}{4 \cdot 8 \cdot 12}+\ldots$ to $\infty$, then

If $n$ is a positive integer and $f(n)$ is the coefficient of $x^n$ in the expansion of $(1+x)(1-x)^n$, then $f(2023)=$

If $(-c, c)$ is the set of all values of $x$ for which the expansion of $(7-5 x)^{\frac{-2}{3}}$ is valid, then $5 c+7=$

A tangent $P T$ is drawn to the circle $x^2+y^2=4$ at the point $P(\sqrt{3}, 1)$. If a straight line $L$ which is perpendicular to $P T$ is a tangent to the circle $(x-3)^2+y^2=1$, then a possible equation of $L$ is

If the parametric equations of the circle passing through the points $(3,4),(3,2)$ and $(1,4)$ is $x=a+r \cos \theta, y=b+r \sin \theta$, then $b^a r^a=$

If a point $P$ moves so that the distance from $(0,2)$ to $P$ is $\frac{1}{\sqrt{2}}$ times the distance of $P$ from $(-1,0)$, then the locus of the point $P$ is

If the angle between the circles $x^2+y^2-4 x-6 y+k=0$ and $x^2+y^2+8 x-4 y+11=0$ is $\frac{\pi}{2}$, then the value of $k$ is

If the radical centre of the given three circles $x^2+y^2=1, x^2+y^2-2 x-3=0$ and $x^2+y^2-2 y-3=0$ is $C(\alpha, \beta)$ and $r$ is the sum of the radii of the given circles, then the circle with $C(\alpha, \beta)$ as centre and $r$ as rad...

If the angle between the pair of tangents drawn to the circle $x^2+y^2-2 x+4 y+3=0$ from the point $(6,-5)$ is $\theta$, then $\cot \theta=$

The radius of a circle touching all the four circles $(x \pm \lambda)^2+(y \pm \lambda)^2=\lambda^2$ is

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

$\alpha, \beta, \gamma$ are the roots of the equation $x^3+2 x^2-x-2=0$, then $\alpha^6+\beta^6+\gamma^6=$

If $i=\sqrt{-1}$, then $\operatorname{Arg}\left[\frac{(1+i)^{2025}}{(1-i)^{2022}}\right]=$

If $x_n=\cos \frac{\pi}{2^n}+i \sin \frac{\pi}{2^n}$, then $\prod_{n=1}^{\infty} x_n=$

If $x=\log \left(y+\sqrt{y^2+1}\right)$, then $y=$

If the roots of the equation $z^2-i=0$ are $\alpha$ and $\beta$, then $|\arg \beta-\arg \alpha|=$

The locus of $z$ such that $\left|\frac{z-i}{z+i}\right|=2$, where $z=x+i y$, is

\(\int_{-1}^1 x|x| d x=\)

If $\int_0^3\left(3 x^2-4 x+2\right) d x=k$, then an integer root of $3 x^2-4 x+2=3 k / 5$ is

\(\int_{-\pi / 2}^{\pi / 2} \sin ^2 x \cos ^2 x(\sin x+\cos x) d x=\)

If the solution of the differential equation $\frac{d y}{d x}=\frac{2 x+3 y}{3 x-2 y}$ is $y=x \tan (f(x))+C$, then $f(x)=$

If the order and degree of the differential equation corresponding to the family of curves $y^2=4 a(x+a)(a$ is parameter) are $m$ and $n$ respectively, then $m+n^2=$

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

If $\sin y=\sin 3 t$ and $x=\sin t$, then $\frac{d y}{d x}=$

If $f(x)=e^x, h(x)=(f \circ f)(x)$, then $\frac{h^{\prime}(x)}{h(x)}=$

On differentiation if we get $f(x, y) d y-g(x, y) d x=0$ from $2 x^2-3 x y+y^2+x+2 y-8=0$, then $\frac{g(2,2)}{f(1,1)}=$

If $x+\sqrt{3} y=3$ is the tangent to the ellipse $2 x^2+3 y^2=k$ at a point $P$, then the equation of the normal to this ellipse at $P$ is

If the line $x \cos \alpha+y \sin \alpha=2 \sqrt{3}$ is a tangent to the ellipse $\frac{x^2}{16}+\frac{y^2}{8}=1$ and $\alpha$ is an acute angle, then $\alpha=$

The period of the function $f(x)=e^{\log (\sin x)}+(\tan x)^3-\operatorname{cosec}(3 x-5)$ is

If ${ }^n C_r$ denotes the number of combinations of $n$ distinct things taken $r$ at a time, then the domain of the function $g(x)={ }^{(16-x)} C_{(2 x-1)}$ is

Let $X=\left\{\left.\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \right\rvert\, a, b, c, d \in R\right\}$. If $f: X \rightarrow R$ is defined by $f(A)=\operatorname{det}(A) . \forall A \in X$, then $f$ is

If the angle between the asymptotes of a hyperbola is $30^{\circ}$, then its eccentricity is

If $\int \frac{x^{49} \tan ^{-1}\left(x^{50}\right)}{\left(1+x^{100}\right)} d x=k\left(\tan ^{-1}\left(x^{50}\right)\right)^2+C$, then $k=$

\(\int \frac{d x}{\left(x^2+1\right)\left(x^2+4\right)}=\)

$\int \frac{d x}{(x-1)^{\frac{3}{4}}(x+2)^{\frac{5}{4}}}=$

If $\int(\log x)^3 x^5 d x=\frac{x^6}{A}\left[B(\log x)^3\right. \left.+C(\log x)^2+D(\log x)-1\right]+k$ and $A, B, C, D$ are integers, then $A-(B+C+D)=$

The quadratic equation whose roots are
$$ l=\lim _{\theta \rightarrow 0}\left(\frac{3 \sin \theta-4 \sin ^3 \theta}{\theta}\right) \text { and } m=\lim _{\theta \rightarrow 0}\left(\frac{2 \tan \theta}{\theta\left(1-\tan ^2 \theta\right)}\r...

\(\lim _{n \rightarrow \infty} \frac{1}{n^3} \sum_{k=1}^n\left(k^2 x\right)=\)

If $A=\left[\begin{array}{ll}0 & 3 \\ 0 & 0\end{array}\right]$ and $f(x)=x+x^2+x^3+\ldots \ldots+x^{2023}$, then $f(A)+I=$

If $A$ is a square matrix of order $3, \operatorname{then}\left|\operatorname{Adj}\left(\operatorname{Adj} A^2\right)\right|=$

If $A$ and $B$ are two square matrices of the same order and $(A B+B A)^T+(A B-B A)^T=2 B A$, then

If $\operatorname{adj}\left[\begin{array}{ccc}1 & 0 & 2 \\ -1 & 1 & -2 \\ 0 & 2 & 1\end{array}\right]=\left[\begin{array}{ccc}5 & m & -2 \\ 1 & 1 & 0 \\ -2 & -2 & n\end{array}\right]$, then $m+n=$

If $x-2 y+k=0$ is a tangent to the parabola $y^2-4 x-4 y+8=0$, then the value of $k$ is

If the points of intersection of the parabolas $y^2=5 x$ and $x^2=5 y$ lie on the line $L$, then the area of the triangle formed by the directrix of one parabola, latus rectum of another parabola and the line $L$ is

There are 10 points in a plane, of which no three points are collinear except 4. Then, the number of distinct triangles that can be formed by joining any three points of these ten points, such that at least one of the vertices of every tria...

A student is asked to answer 10 out of 13 questions in an examination such that he must answer atleast four questions from the first five questions. Then, the total number of possible choices available to him is