TG EAPCET 2025 (Online) 4th May Evening Shift

If a tangent of the circle $x^2+y^2+2 x+2 y+1=0$ is radical axis of the circles $x^2+y^2+2 g x+2 f y+c=0$ and $2 x^2+2 y^2+3 x+8 y+2 c=0$, then

If the circle $S=0$ intersect the three circle
$$ \begin{aligned} & S_1 \equiv x^2+y^2+4 x-7=0 \\ & S_2 \equiv x^2+y^2+y=0 \text { and } S_3 \equiv x^2+y^2+\frac{3}{2} x+\frac{5}{2} y-\frac{9}{2}=0 \end{aligned} $$
orthogonally, then radica...

If $P\left(\frac{7}{5}, \frac{6}{5}\right)$ is the inverse point of $A(1,2)$ with respect to a circle with centre $C(2,0)$, then the radius of that circle is

Among the chords of the circle $x^2+y^2=75$, the number of chords having their mid-points on the line $x=8$ and having their slopes as integers is

The equation of the circle which touches the circle $S \equiv x^2+y^2-10 x-4 y+19=0$ at the point $(2,3)$ internally and having radius equal to half of the radius of the circle $S=0$ is

The radius of the circle having three chords along Y-axis, the line $y=x$ and the line $2 x+3 y=10$

If $Z=r(\cos \theta+i \sin \theta),\left(\theta \neq-\frac{\pi}{2}\right)$ is solution of $x^3=i$, then $r^9(\cos \theta+i \sin \theta)^9=x^{3-}=i$

If the eight vertices of a regular octagon are given by the complex number $\frac{1}{x_j-2 i}(j=1,2,3,4,5,6,7,8)$, then the radius of the circumcircle of the octagon is

If $\left|Z_1-3-4 i\right|=5$ and $\left|Z_2\right|=15$, then the sum of the maximum and minimum values of $\left|Z_1-Z_2\right|$ is

If $\omega \neq 1$ is a cube root of unity, then one root among the 7th roots of $(1+\omega)$ is

If $1+2 i$ is a root of the equation $x^4-3 x^3+8 x^2-7 x+5=0$, then sum of the squares of the other roots is

If $\int_0^{\frac{\pi}{2}} \tan ^{14}\left(\frac{x}{2}\right) d x=2\left[\sum_{n=1}^7 f(n)-\frac{\pi}{4}\right]$, then $f(n)=$

\(\lim _{n \rightarrow \infty} \frac{\left(2 n(2 n-1) \ldots .(n+2)(n+1)^{1 / n}\right.}{n}=\)

If the general solution of $\left(1+y^2\right) d x=\left(\tan ^{-1} y-x\right) d y$ is $x=f(y)+c e^{-\tan ^{-1} y}$, then $f(y)=$

The differential equation of the family of all circles of radius ' $a$ ' is

If $x=t-\sin t, y=1-\cos t$ and $\frac{d^2 y}{d x^2}=-1$ at $t=k, k>0$ then $\lim _{i \rightarrow K} \frac{y}{x}=$

When the coordinate axes are rotated about the origin through an angle $\frac{\pi}{4}$ in the positive direction, the equation $a x^2+2 h x y+b y^2=c$ is transformed to $25 x^2+9 y^2=225$, then $(a+2 h+b-\sqrt{c})^2=$

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(b>a)$ is an ellipse with eccentricity $\frac{1}{\sqrt{2}}$. If the angle of intersection between the ellipse and parabola $y^2=4 a x$ is $\theta$, then the coordinates of the point $\frac{2 \theta}{3}$ on...

The circumcenter of the equilateral triangle having the three points $\theta_1, \theta_2, \theta_3$ lying on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ as its vertices is $(r, s)$. Then, the average of $\cos \left(\theta_1-\theta_2\rig...

If $2^{4 n+3}+3^{3 n+1}$ is divisible by $P$ for all natural numbers $n$, then $P$ is

Consider the following statements
Statement $\mathrm{I} \cosh ^{-1} x=\tanh ^{-1} x$ has no solution
Statement II $\cosh ^{-1} x=\operatorname{coth}^{-1} x$ has only one solution
The correct answer is

A function $f: R \rightarrow R$ defined by
$$ f(x)=\left\{\begin{array}{c} 2 x+3, x \leq \frac{4}{3} \\ -3 x^2+8 x, x>\frac{4}{3} \end{array}\right. \text { is } $$

The domain and range of $f(x)=\frac{1}{\sqrt{|x|-x^2}}$ are $A$ and $B$ respectively. Then $A \cup B=$

The number of common tangents that can drawn to the curves $\frac{x^2}{16}-\frac{y^2}{9}=1$ and $x^2+y^2=16$ is

$$ \begin{aligned} I_1 & =\int \frac{e^x}{e^{4 x}+e^{2 x}+1} d x, I_2 \\ & =\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x, \text { then } I_2-I_1= \end{aligned} $$

$$ \begin{aligned} & \int \frac{3 x+2}{4 x^2+4 x+5} d x=A \log \\ & \left(4 x^2+4 x+5\right)+B \tan ^{-1}\left(\frac{2 x+1}{2}\right)+C, \text { then } A+B= \end{aligned} $$

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

If $\int \frac{1}{x} \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}} d x=2 f(x)-2 \sin ^{-1} \sqrt{x}+c$, then $f(x)=$

$$ \begin{aligned} &\text { If } y=f(x)^{g(x)} \text { and } \frac{d y}{d x}=y\left[H(x) f^{\prime}(x)+G(x) g^{\prime}(x)\right] \text {, then }\\ &\int \frac{G(x) H(x) f^{\prime}(x)}{g(x)} d x= \end{aligned} $$

Consider the following
Assertion
$$ \begin{aligned} & \text { (A) } \begin{array}{r} \sqrt{x-3}\left(\sin ^{-1}(\log x)+\cos ^{-1}\right. \\ (\log x) d x=\frac{\pi}{3}(x-3)^{3 / 2}+c \end{array} \end{aligned} $$
Reason $(\mathrm{R}) \sin ^{...

The number of real solution of $\tan ^{-1} x+\tan ^{-1} 2 x=\frac{\pi}{4}$ is

Let $f:[-1,2] \rightarrow R$ be defined by $f(x)=\left[x^2-3\right]$ where $[$. denotes greatest integer function, then the number of points of discontinuity for the function $f$ in $(-1,2)$ is

If $f(x)=\left\{\begin{array}{cc}x^2\left|\cos \frac{\pi}{2}\right|, & x \neq 0 \\ 0, & x=0\end{array}\right.$, then at $x=2, f(x)$ is

The set of all values of $x$ for which $f(x)=\| x|-1|$ is differentiable is

\(\mathop {\lim }\limits_{x \to 0} \frac{\sqrt{\cos x}-\sqrt[3]{\cos x}}{\sin ^2 x}=\)

If $A=\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & x & 1\end{array}\right], A^{-1}=\frac{1}{2}\left[\begin{array}{ccc}1 & -1 & 1 \\ -8 & 6 & 2 y \\ 5 & -3 & 1\end{array}\right]$, then the point $(x, y)$ lies on the curve represente...

A is a $3 \times 3$ matrix satisfying $A^3-5 A^2+7 A+I=0$ If $A^5-6 A^4+12 A^3-6 A^2+2 A+2 I=l A+m I$, then $l+m=$

The number of real values of ' $a$ ' for which the system of equations $2 x+3 y+a z=0, x+a y-2 z=0$ and $3 x+y+3 z=0$ has non-trivial solution is

Consider a homogeneous system of three linear equations in three unknowns represented by $A X=0$.
If $X=\left[\begin{array}{c}l \\ m \\ 0\end{array}\right], l \neq 0, m \neq 0, l, m \in R$ represents an infinite number of solutions of this ...

The normal at a point on the parabola $y^2=4 x$ passes through a point $P$. Two more normals to this parabola also pass through $P$. If the centroid of the triangle formed by the feet of these three normals is $G(2,0)$, then the abscissa of...

If the angle between the tangents drawn to the parabola $y^2=4 x$ from the points on the line $4 x-y=0$ is $\frac{\pi}{3}$, then the sum of the abscissae of all such points is

The number of positive integral solutions of $x y z=60$ is

The number of positive integral solution of $\frac{1}{x}+\frac{1}{y}=\frac{1}{2025}$ is

All the letters of the word MOTHER are arranged in all possible ways and the resulting words (may or may not have meaning) are arranged as in the dictionary. The number of words that appear after the word MOTHER is

Functions are formed from the set $A=\left\{a_1, a_2, a_3\right\}$ to another set $B=\left\{b_1, b_2, b_3, b_4, b_5\right\}$. If a function is selected at random, then probability, that it is a non-one function is

In a shelf there are three mathematics and two physics books. A student takes a book randomly. If he randomly takes, successively for three time by replacing the book already taken every time, then the mean of the number of mathematics book...

$A$ and $B$ are two events of a random experiment such that $P(B)=0.4, P(A \cap \bar{B})=0.5, P(A \cup B)+P\left(\frac{B}{A \cup \bar{B}}\right)=1.15$ then $P(A)=$

In possion distribution, if $\frac{P(x=5)}{P(X=2)}=\frac{1}{7500}$ and $\frac{P(X=5)}{P(X=3)}=\frac{1}{500}$, then the mean of the distribution is

There are two boxes each containing 10 balls. In each box, few of them are black balls and rest are white. A ball is drawn at random from one of the boxes and found that it is black. If the probability that the black ball drawn is from the ...

In a $\triangle A B C, a=5, b=4$ and $\tan \frac{C}{2}=\sqrt{\frac{7}{9}}$, then its inradius $r=$