TG EAPCET 2025 (Online) 4th May Morning Shift

If the line $x+y=2$ cuts the circle $x^2+y^2+2 x-4 y+4=0$ at two points $A$ and $B$, then the radius of the circle passing through $A, B$ and orthogonal to $x^2+y^2-2 x-4 y-4=0$ is

The tangents drawn from a point $(2,-1)$ touch the circle $x^2+y^2+4 x-2 y+1=0$ at the points $A$ and $B$. If $C$ is the centre of the circle, then the area (in sq. units) of the $\triangle A B C$ is

If the length of the chord $2 x+3 y+k=0$ of the circle $x^2+y^2-2 x+4 y-11=0$ is $2 \sqrt{3}$, then the sum of all possible values of $k$ is

The angle between the tangents drawn from the point $P(k, 6 k)$ to the circle $x^2+y^2+6 x-6 y+2=0$ is $2 \tan ^{-1}\left(\frac{4}{3}\right)$. If the coordinates of $P$ are integers, then $k=$

If $\theta$ is the angle between the circles $x^2+y^2-4 x+2 y-4=0$ and $x^2+y^2-2 x+4 y-11=0$ then $\sin \theta=$

One of the roots of the equation $(x+1)^4+81=0$ is

\(\left(\frac{1+i}{1-i}\right)^{228}=\)

\((1-i \sqrt{3})^{2025}=\)

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

\(\int_0^{\frac{\pi}{2}} \sqrt{\tan x d x}=\)

\(\int_{-4}^5 \frac{1}{\sqrt{20+x-x^2}} d x=\)

\(\int_0^{\frac{\pi}{2}} \frac{d x}{\cos x-\sqrt{3} \sin x}=\)

If $y=f(x)$ is the solution of the differential equation $\left(1+\cos ^2 x\right) f^{\prime}(x)-4 \sin 2 x-f(x) \sin 2 x=0$ when $f(0)=0$, then $f\left(\frac{\pi}{3}\right)=$

The differential equation corresponding to the family of ellipses $\frac{x^2}{a^2}+\frac{y^2}{4}=1$, where ' $a$ ' is an arbitrary constant is

If $y=\tan ^2\left(\cos ^{-1} \sqrt{\frac{1+x^2}{2}}\right)$, then $\frac{d y}{d x}=$

If $y=x^{\log x}+(\log x)^x, x>1$, then $\left(\frac{d y}{d x}\right)_{x=e}=$

Let $e$ be the eccentricity of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.
If $a=5, b=4$ and the equation of the normal drawn at one end of the latus rectum that lies in the first quadrant is $l x+m y=27$ then $l+m=$

If $P$ is any point on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ and $S, S^{\prime}$ are its foci, then the maximum area (in sq. units) of $\triangle S P S^{\prime}=$

The domain of the real valued function $f(x)=\log _{\sqrt{2}}\left(\sqrt{x^2+x}+\sqrt{x^2-x}\right)$ is

If $\frac{x+1}{x^3(x-1)}=\frac{a}{x}+\frac{b}{x^2}+\frac{c}{x^3}+\frac{d}{x-1}$, then

If $A=(0,1), B=(1,2), C=(-2,1)$, then the equation of the locus of a point $P$ such that area of $\triangle P A B=$ area of $\triangle P A C$ is

If the latus rectum through one of the foci of a hyperbola $\frac{x^2}{9}-\frac{y^2}{b^2}=1$ subtends a right angle at the farther vertex of the hyperbola, then $b^2=$

If $\int x^3 \sin 3 x d x=\frac{1}{27}[f(x) \cos 3 x+g(x) \sin 3 x]+C$, then $f(\mathrm{l})+g(\mathrm{l})=$

If $I_1=\int \sin ^6 x d x$ and $I_2=\int \cos ^6 x d x$, then $I_1+I_2=$

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

If $\int \frac{1}{(x+2) \sqrt{x^2+x+2}} d x=$

The domain of the derivative of the function $f(x)=\cos ^{-1}(2 x-5)-\sin ^{-1}(x-2)$ is

\(\sin ^{-1}(-\cos 2)+\cos ^{-1}(\sin 3)+\tan ^{-1}(\cot 5)=\)

If $[x]$ is the greatest integer function and
$$ f(x)=\left\{\begin{array}{cc} 2[x]-\frac{x}{|x|}, & x \neq 0 \\ 1, & x=0 \end{array}\right. $$
is a real valued function, then $f$ is

If $\mathop {\lim }\limits_{x \to 0} \frac{3^{x^3}-\left(1-x^3\right)^{\frac{2}{3}}}{x^2 \sin x}=p+\log q$, then $p q=$

If $A=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 1 \\ 1 & 2 & 1\end{array}\right]$ then $|\operatorname{adj}|\left(A^2\right) \mid=$

If $x=\alpha, y=\beta, z=\gamma$ is the solution of the system of equations $2 x+3 y+z=-1,3 x+y+z=4$, $x-3 y-2 z=1$, then the value of $\beta$ is

The positive value of ' $a$ ' for which the system of linear homogeneous equations $x+a y+z=0, a x+2 y-z=0$, $2 x+3 y+z=0$ has non-trivial solution is

If $L(p, q), q>3$ is one end of the latus rectum of the parabola $(y-2)^2=3(x-1)$, then the equation of the tangent at $L$ to this parabola is

A normal chord $P Q$ drawn at a point $P$ on the parabola $y^2=5 x$ subtends a right angle at the vertex. If $P$ lies in the first quadrant, then the other end $Q$ of the normal chord is

\({ }^{20} P_5-{ }^{19} P_5=\)

All possible words (with or without meaning) the contain the word 'GENTLE' are formed using all the letters of the word 'INTELLIGENCE'. Then, the number of words in which the word 'GENTLE' appears among the first nine positions only is

5 boys and 5 girls have to sit around a table. The number of ways in which all of them can sit so that no two boys and no two girls are together is

The range of a discrete random variable $X$ is $\{1,2,3\}$ and the probabilities of its elements are given by $P(X=1)=3 k^3, P(X=2)=2 k^2$ and $P(X=3)=7-19 \mathrm{k}$. Then, $P(X=3)=$

Among every 8 units of a product, one is likely to be defective. If a consumer has order 5 units of that product, then the probability that atmost one unit is defective among them is

If two smallest squares are chosen at random on a chess board, then the probability of getting these squares such that they do not have a side in common is

An urn contains 7 red, 5 white and 3 black balls. Three balls are drawn randomly one after the other without replacement. If it is known that first ball drawn is red and the second ball drawn is white, then the probability that the third ba...

Let $A$ and $B$ be two events in a random experiment . If $P(A \cap \bar{B})=0.1, P(\bar{A} \cap B)=0.2$ and $P(B)=0.5$, then $P(\bar{A} \cap \bar{B})=$

If $a=3, b=5, c=7$ are the sides of a $\triangle A B C$, then $\cot A+\cot B+\cot C=$

Let $p_1, p_2$ and $p_3$ be the altitudes of a $\triangle A B C$ drawn through the vertices $A, B$ and $C$ respectively. If $r_1=4$, $r_2=6, r_3=12$ are the ex-radii of $\triangle A B C$, then $\frac{1}{p_1^2}+\frac{1}{p_2^2}+\frac{1}{p_3^2...

If $\alpha, \beta, \gamma, \delta$ and $\varepsilon$ are the roots of the equation $x^5+x^4-13 x^3-13 x^2+36 x+36=0$ and $\alpha<\beta<\gamma<\delta<\varepsilon$ then $\frac{\varepsilon}{\alpha}+\frac{\delta}{\beta}+\frac{1}{\gamma}=$

If one of the roots of the equation $6 x^3-25 x^2+2 x+8=0$ is an integer and $\alpha>0, \beta<0$ are the other two roots, then $\frac{4}{\alpha}+\frac{1}{\beta}=$

If $\alpha, \beta$ are the roots of the equation $x^2+3 x+k=0$ and $\alpha+\frac{1}{\alpha}, \beta+\frac{1}{\beta}$ are the roots of the equation $4 x^2+p x+18=0$, then $k$ satisfies the equation

If $f(x)=x^2+b x+c$ and $f(1+k)=f(1-k) \forall k \in R$, for two real numbers $b$ and $c$ then

If $f(x)$ is a second degree polynomial such that $f(x) \geq 0 \forall x \in R, f(-3)=0$ and $f(0)=18$, then $f(3)=$