TG EAPCET 2025 (Online) 2nd May Morning Shift

If $A, B$ are the points of contact of the tangents drawn from the point $(-3,1)$ to the circle $x^2+y^2-4 x+2 y-4=0$, then the equation of the circumcircle of the $\triangle P A B$ is

If the angle between the circles $x^2+y^2-2 x+k y+1=0$ and $x^2+y^2-k x-2 y+1=0$ is $\cos ^{-1}\left(\frac{1}{4}\right)$ and $k<0$, then the point which lies on the radical axis of the given circle is

A circle $C$ passing through the point $(1,1)$ bisects the circumference of the circle $x^2+y^2-2 x=0$. If $C$ is orthogonal to the circle $x^2+y^2+2 y-3=0$, then the centre of the circle $C$ is

If $\frac{2+3 i}{i-2}-\frac{4 i-3}{3+4 i}=x+i y$, then $3 x+y=$

Let $z=x+i y$ and $P(x, y)$ be a point on the argand plane. If $z$ satisfies the condition $\arg \left(\frac{z-3 i}{z+2 i}\right)=\frac{\pi}{4}$, then the locus of $P$ is

If $\omega$ is a complex cube root of unity and $x=\omega^2-\omega+2$, then

The product of all the values of $(\sqrt{3}-i)^{\frac{3}{7}}$ is

\(\int_0^2 \sqrt{(x+3)(2-x)} d x=\)

\(\int_0^{\pi / 4} x^2 \sin 2 x d x\)

\(\int_{-2 \pi}^{2 \pi} \sin ^4 x \cos ^6 x d x=\)

If $\cos x \frac{d y}{d x}=y \sin x-1, x \neq(2 n+1) \frac{\pi}{2}, n \in Z$ is the differential equation corresponding to the curve $y=f(x)$ and $f(0)=1$, then $f(x)$

The general solution of the differential equation $2 d x+d y=(6 x y+4 x-3 y) d x$ is

If $y=f(\cosh x)$ and $f^{\prime}(x)=\log \left(x+\sqrt{x^2-1}\right)$, then $\frac{d^2 y}{d x^2}=$

If $\left(x^2-3 x+2\right)^{\frac{y}{x^{2-1}}}=x+2$, then $\left(\frac{d y}{d x}\right)_{x=0}=$

If $x=\frac{t^2}{1+t^5}, y=\frac{2 t^3}{1+t^5}$ and $t \neq-1$ is a perimeter, then $\frac{d y}{d x}=$

If $S$ and $S^{\prime}$ are the foci of an ellipse $\frac{x^2}{169}+\frac{y^2}{144}=1$ and the point $B$ lying on positive $Y$-axis is one end of its minor axis, then the incentre of the $\triangle S B S^{\prime}$ is

One of the foci of an ellipse is $(2,-3)$ and its corresponding directrix is $2 x+y=5$. If the eccentricity of the ellipse is $\frac{\sqrt{5}}{3}$, then the coordinates of the other focus are

If $f(x)=\tan \left(\frac{\pi}{\sqrt{x+1}+4}\right)$ is a real valued function, then the range of $f$ is

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

If the product of the perpendicular distances from any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ to its asymptotes is $\frac{36}{13}$ and its eccentricity is $\frac{\sqrt{13}}{3}$, then $a-b=$

\(\int \frac{2 \sin x-3 \cos x}{4 \cos x-3 \sin x} d x=\)

\(\int e^{4 x}(\sin 3 x-\cos 3 x) d x=\)

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

If $\int(x+2) \sqrt{x^2-x+2} d x=\frac{1}{3} f(x)+\frac{5}{8} g(x)+\frac{35}{16} h(x)+C$ then $f(-1)+g(-1)+h\left(\frac{1}{2}\right)=$

The range of the real value function $f(x)=\sin ^{-1}\left(\sqrt{x^2+x+1}\right)$ is

\(\tan ^{-1} \frac{3}{5}+\tan ^{-1} \frac{6}{41}+\tan ^{-1} \frac{9}{191}=\)

If $2 \tanh ^{-1} x=\sinh ^{-1}\left(\frac{4}{3}\right)$, then $\cosh ^{-1}\left(\frac{1}{x}\right)=$

If $f(x)=\sqrt{\cos ^{-1} \sqrt{1-x^2}}$, then $f^{\prime}\left(\frac{1}{2}\right)=$

If $[x]$ is the greatest integer function, then
\(\mathop {\lim }\limits_{x \to 3} \frac{(3-|x|+\sin |3-x|) \cos [9-3 x]}{|3-x|[3 x-9]}\)

Let ' $a$ ' be a positive real number. If a real valued function
$f(x)=\left\{\begin{array}{cl}\frac{6^x-3^x-2^x+1}{1-\cos \left(\frac{x}{a}\right)} & \text { if } x \neq 0 \\ \log 3 \log 4 & \text { if } x=0\end{array}\right.$ is continuou...

If the augmented matrix corresponding to the system of equations $x+y-z=1,2 x+4 y-z=0$ and $3 x+4 y+5 z=18$ is transformed to $\left[\begin{array}{cccc}1 & a & 0 & -1 \\ 0 & 2 & 1 & b \\ 0 & 0 & c & 32\end{array}\right]$ then $\sqrt{a+b+c}=...

If $\left|\begin{array}{ccc}9 & 25 & 16 \\ 16 & 36 & 25 \\ 25 & 49 & 36\end{array}\right|=K$, then $K, K+1$ are the roots of the equation

$A=\left[\begin{array}{ccc}1 & -3 & -5 \\ -2 & 4 & -6 \\ 7 & -11 & 13\end{array}\right]$, then $\sqrt{|\operatorname{adj} A|}=$

If $\Delta_r=\left|\begin{array}{cc}\frac{1}{3 r-2} & \frac{2}{3 r-5} \\ 0 & \frac{3}{3 r+1}\end{array}\right|$ then $\sum\limits_{r=1}^{33} \Delta_r=$

If the normal drawn at $P(8,16)$ to the parabola $y^2=32 x$ meets the parabola again at $Q$, then the equation of the tangent drawn at $Q$ to the parabola is

The focal distance of a point $(5,5)$ on the parabola $x^2-2 x-4 y+5=0$ is

A student has to answer a multiple-choice question having 5 alternatives in which two or more than two alternatives are correct. Then, the number of ways in which the student can answer that question is

If all the letters of the word 'HANDLE' are permuted in all possible ways and the words (with or without meaning) thus formed are arranged in dictionary order, then the rank of the word 'HELAND' is

Number of triangles whose vertices are the points $(x, y)$ in the $X Y$-plane with integer coordinates satisfying $0 \leq x \leq 4$ and $0 \leq y \leq 4$ is

If three smallest squares are chosen at-random on a chess board, then the probability of getting them in such a way that they are all together in a row or in a column is

If three cards are drawn randomly from a pack of 52 playing cards then the probability of getting exactly, one spade card, exactly one king and exactly one card having a prime number is

Urn A contains 6 white and 2 black balls; run B contains 5 white and 3 black balls and urn C contains 4 white and 4 black balls. if an urn is chosen at random and a ball is drawn at random from it, then the probability that the ball drawn i...

If $p_1, p_2, p_3$ are the altitudes and $a=4, b=5, c=6$ are the sides of a $\triangle A B C$, then $\frac{1}{p_1^2}+\frac{1}{p_2^2}+\frac{1}{p_3^2}=$

Let the angles $A, B, C$ of a $\triangle A B C$ be in arithmetic progression. If the exradii $r_1, r_2, r_3$ of $\triangle A B C$ satisfy the condition $r_3^2=r_1 r_2+r_2 r_3+r_3 r_1$, then $b=$

The number of integral values of ' $a$ ' for which the quadratic equation $a x^2+a x+5=0$ cannot have real roots is

If the roots of the equation $32 x^3-48 x^2+22 x-3=0$ are in arithmetic progression, then the square of the common difference of the roots is

If the sum of two roots of the equation $x^4-2 x^3+x^2+4 x-6=0$ is zero, then the sum of the squares of the other two roots is

$1+(1+3)+(1+3+5)+(1+3+5+7)+\ldots$ to 10 terms $=$

The mean deviation from the median for the following data is
$$ \begin{array}{cllllll} x_i & 2 & 9 & 8 & 3 & 5 & 7 \\ \hline f_i & 5 & 3 & 1 & 6 & 6 & 1 \\ \hline \end{array} $$

If three dice are thrown, then the mean of the sum of the numbers appearing on them is