TG EAPCET 2025 (Online) 4th May Evening Shift

Consider all functions given in List I in the interval [1,3]. The list II has the value of ' $c$ ' obtained by applying Lagrange's mean value theorem on the function of List I . Match the function and values of ' c '
$$ \begin{array}{llll} ...

The function $f(x)=2 x^3-9 a x^2+12 a^2 x+1$ where $a>0$ attains its local maximum and local minimum at $p$ and $q$ respectively. If $p^2=q$, then $a=$

If the percentage error in the radius of a circle is 3 , then the percentage error in its area is

If local maximum of $f(x)=\frac{a x+b}{(x-1)(x-4)}$ exists at $(2,-1)$, then $a+b=$

For the curve $\frac{x^n}{a^n}+\frac{y^n}{b^n}=2,(n \in N$ and $n>1)$ the line $\frac{x}{a}+\frac{y}{b}=2$ is

$f(x)=x^2-2(4 k-1) x+g(k)>0, \forall x \in R$ and for $k \in(a, b)$. If $g(k)=15 k^2-2 k-7$, then

The height of a cone with semi-vertical angle $\frac{\pi}{3}$ is increasing at the rate of 2 units $/ \mathrm{min}$. The rate at which the radius of the cone is to be decreased so as to have a fixed volume always is

The area of the region bounded by $y=x^3, X$-axis, $x=-2$ and $x=4$ is

Numerically greatest term in the expansion of $(3 x-4 y)^{23}$ when $x=\frac{1}{6}$ and $y=\frac{1}{8}$ is

Let $K$ be the number of rational terms in the expansion of $(\sqrt{2}+\sqrt[3]{3})^{6144}$. If the coefficient of $x^P(P \in N)$ in the expansion of $\frac{1}{(1+x)\left(1+x^2\right)\left(1+x^4\right)\left(1+x^8\right)\left(1+x^{16}\right)...

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...