TG EAPCET 2025 (Online) 4th May Morning Shift

If the curves $y^2=12 x-3$ and $y^2=12-k x$ cut each other orthogonally, then the length of the sub-tangent at $(1, b)$ on the curve $y^2=12-k x$ is

If the extreme values of the function $f(x)=(2 \sqrt{6}+1) \cos x+(2 \sqrt{2}-\sqrt{3}) \sin x-6$ are $m$ and $M$ then $\sqrt{\left|M^2-m^2\right|}=$

If $x=2 \sqrt{2} \sqrt{\cos 2 \theta}$ and $y=2 \sqrt{2} \sqrt{\sin 2 \theta}, 0<\theta<\frac{\pi}{4}$, then the value of $\frac{d y}{d x}$ at $\theta=22 \frac{1}{2}^{\circ}$ is

If $x$ and $y$ are two positive real numbers such that $x y=4$, then the minimum value of $\left(\sqrt{x}+\frac{y^2}{2}\right)$ is

The real valued function $f(x)=\frac{x^2}{2}-\log \left(x^2+x+1\right)$ is

A rod of length 41 m with an end $A$ on the floor and another end $B$ on the wall perpendicular to the floor is sliding away horizontally from the wall at the rate of $3 \mathrm{fit} / \mathrm{min}$. When the end $B$ is at the height of 9 f...

There is a possible error of 0.02 cm in measuring the base diameter of a right circular cone as 14 cm . If the semi-vertical angle of the cone is $45^{\circ}$, then the approximate error in its volume is (in $\mathrm{cu} . \mathrm{cm}$ )

If $C_0, C_1, C_2, \ldots, C_{10}$ represent the binomial coefficients in the expansion of $(1+x)^{10}$, then
\(C_0 C_6+C_1 C_7+C_2 C_8+C_3 C_9+C_4 C_{10}=\)

When $|x|<\frac{1}{2}$ the coefficient of $x^6$ in the expansion of $\left(\frac{2-x}{1+2 x}\right)^2$ is

The power of a point $(2,-1)$ with respect to a circle $C$ of radius 4 is 9 . The centre of the circle $C$ lies on the lines $x+y=0$ and in the 2nd quadrant. If ( $\alpha, \beta$ ) is the centre of the circle $C$ then $\beta-\alpha=$

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