TG EAPCET 2024 (Online) 9th May Morning Shift

The radius of a sphere is 7 cm . If an error of 0.08 sq cm is made in measuring it, then the approximate error (in cubic cm ) found in its volume is

If the slope of the tangent drawn at any point $(x, y)$ on the curve $y=f(x)$ is $\left(6 x^2+10 x-9\right)$ and $f(2)=0$, then $f(-2)=$

If the interval in which the real valued function $f(x)=\log \left(\frac{1+x}{1-x}\right)-2 x-\frac{x^3}{1-x^2}$ is decreasing in $(a, b)$, where $|b-a|$ is maximum, then $\frac{a}{b}=$

The curve $y=x^3-2 x^2+3 x-4$ intersects the horizontal line $y=-2$ at the point $P(h, k)$. If the tangent drawn to this curve at $P$ meets the $X$-axis at $\left(x_1, y_1\right)$, then $x_1=$

If $f(x)=(2 x-1)(3 x+2)(4 x-3)$ is a real valued function defined on $\left[\frac{1}{2}, \frac{3}{4}\right]$, then the value(s) of $c$ as defined in the statement of Rolle's theorem

Area of the region enclosed between the curves $y^2=4(x+7)$ and $y^2=5(2-x)$ is

For $n \in N$ the largest positive integer that divides $81^n+20 n-1$ is $k$. If $S$ is the sum of all positive divisors of $k$, then $S-k=$

If $(1,1),(-2,2)$ and $(2,-2)$ are 3 points on a circle $S$, then the perpendicular distance from the centre of the circle $S$ to the line $3 x-4 y+1=0$ is

The centroid of a variable $\triangle A B C$ is at the distance of 5 units from the origin. If $A=(2,3)$ and $B=(3,2)$, then the locus of $C$ is

If the line $4 x-3 y+p=0(p+3>0)$ touches the circle $x^2+y^2-4 x+6 y+4=0$ at the point $(h, k)$, then $h-2 k=$

If the inverse point of the point $P(3,3)$ with respect to the circle $x^2+y^2-4 x+4 y+4=0$ is $Q(a, b)$, then $a+5 b=$

A circle $S \equiv x^2+y^2+2 g x+2 f y+6=0$ cuts another circle $x^2+y^2-6 x-6 y-6=0$ orthogonally. If the angle between the circles $S=0$ and $x^2+y^2+6 x+6 y+2=0$ is $60^{\circ}$, then the radius of the circle $S=0$ is

If the equation of the transverse common tangent of the circles $x^2+y^2-4 x+6 y+4=0$ and $x^2+y^2+2 x-2 y-2=0$ is $a x+b y+c=0$, then $\frac{a}{c}=$

If $m_1$ and $m_2$ are the slopes of the direct common tangents drawn to the circles $x^2+y^2-2 x-8 y+8=0$ and $x^2+y^2-8 x+15=0$, then $m_1+m_2=$

$x$ and $y$ are two complex numbers such that $|x|=|y|=1$.
If $\arg (x)=2 \alpha, \arg (y)=3 \beta$ and $\alpha+\beta=\frac{\pi}{36}$, then $x^6 y^4+\frac{1}{x^6 y^4}=$

If $\sqrt{5}-i \sqrt{15} \doteqdot r(\cos \theta+i \sin \theta),-\pi<\theta<\pi$, then $r^2\left(\sec \theta+3 \operatorname{cosec}^2 \theta\right)=$

One of the roots of the equation $x^{14}+x^9-x^5-1=0$ is

The point $P$ denotes the complex number $z=x+i y$ in the argand plane. If $\frac{2 z-i}{z-2}$ is a purely real number, then the equation of the locus of $P$ is

If $f(x)=\int \frac{\sin 2 x+2 \cos x}{4 \sin ^2 x+5 \sin x+1} d x$ and $f(0)=0$, then $f(\pi / 6)=$

$\int_{-2}^2 x^4\left(4-x^2\right)^{\frac{7}{2}} d x=$

The general solution of the differential equation $\left(3 x^2-2 x y\right) d y+\left(y^2-2 x y\right) d x=0$ is

A particle moving from a fixed point on a straight line travels a distance $S$ metres in $t \mathrm{sec}$. If $S=t^3-t^2-t+3$, then the distance (in mts) travelled by the particle when it comes to rest, is

If $x=\frac{9 t^2}{1+t^4}$ and $y=\frac{16 t^2}{1-t^4}$ then $\frac{d y}{d x}=$

If $y=\sin a x+\cos b x$, then $y^{\prime \prime}+b^2 y=$

If $2 x^2-3 x y+4 y^2+2 x-3 y+4=0$, then $\left(\frac{d y}{d x}\right)_{(3,2)}=$

If $6 x-5 y-20=0$ is a normal to the ellipse $x^2+3 y^2=K$, then $K=$

If $f(x)=\frac{2 x-3}{3 x-2}$ and $f_n(x)=($ fofofo .......n times) $(x)$, then $f_{32}(x)=$

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

The point of intersection of two tangents drawn to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{4}=1$ lie on the circle $x^2+y^2=5$. If these tangents are perpendicular to each other, then $a=$

If $\frac{x^4}{\left(x^2+1\right)(x-2)}=f(x)+\frac{A x+B}{x^2+1}+\frac{C}{x-2}$, then $f(14)+2 A-B=$

$\int \frac{\left(1-4 \sin ^2 x\right) \cos x}{\cos (3 x+2)} d x=$

$\int(\sqrt{1-\sin x}+\sqrt{1+\sin x}) d x=f(x)+c$, where $c$ is the constant of integration. If $\frac{5 \pi}{2}$<$x<\frac{7 \pi}{2}$ and \(f\left(\frac{8 \pi}{3}\right)=-2, \text { then } f^{\prime}\left(\frac{8 \pi}{3}\right)=\)

$\int \frac{\left(1-4 \sin ^2 x\right) \cos x}{\cos (3 x+2)} d x=$

$\int \frac{1}{x^{m \sqrt[m]{m}} x^{m}+1} d x=$

If $\cosh ^{-1}\left(\frac{5}{3}\right)+\sinh ^{-1}\left(\frac{3}{4}\right)=k$, then $e^k=$

$\cos ^{-1} \frac{3}{5}+\sin ^{-1} \frac{5}{13}+\tan ^{-1} \frac{16}{63}=$

If $ f(x)=\left\{\begin{array}{ll}\frac{8}{x^{3}}-6 x & \text {, if } 0 < x \leq 1 \\\\ \frac{x-1}{\sqrt{x}-1} & \text {,if } x > 1\end{array}\right. $ is a real valued function, then at $ x=1, f $ is

$\lim _{x \rightarrow 0} \frac{3^{\sin x}-2^{\tan x}}{\sin x}=$

$\lim \limits_{n \rightarrow \infty}\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{4}{n^2}\right)\left(1+\frac{9}{n^2}\right) \ldots .(2)\right]^{1 / n}=$

If the function
$$ f(x)=\left\{\begin{array}{cc} \frac{\cos a x-\cos 9 x}{x^2} & \text {, if } x \neq 0 \\ 16 & \text {, if } x=0 \end{array}\right. $$
is continuous at $x=0$, then $a=$

The system of simultaneous linear equations
$$ \begin{aligned} & x-2 y+3 z=4,3 x+y-2 z=7 \\ & 2 x+3 y+z=6 \text { has } \end{aligned} $$

If $A$ is square matrix and $A^2+I=2 A$, then $A^9=$

$\operatorname{det}\left[\begin{array}{ccc}\frac{a^2+b^2}{c} & c & c \\\\ a & \frac{b^2+c^2}{a} & a \\\ b & b & \frac{c^2+a^2}{b}\end{array}\right]=$

$A, B, C$ and $D$ are square matrices such that $A+B$ is symmetric, $A-B$ is skew-symmetric and $D$ is the transpose of $C$.
If $A=\left[\begin{array}{ccc}-1 & 2 & 3 \\\\ 4 & 3 & -2 \\\\ 3 & -4 & 5\end{array}\right]$ and
$C=\left[\begin{arr...

The equation of the common tangent to the parabola $y^2=8 x$ and the circle $x^2+y^2=2$ is $a x+b y+2=0$. If $-\frac{a}{b}>0$, then $3 a^2+2 b+1=$

Consider the parabola $25\left[(x-2)^2+(y+5)^2\right]=(3 x+4 y-1)^2$, match the characteristic of this parabola given in List I with its corresponding item in List II.
$$ \begin{array}{lll} \hline & \text { List I } & \text { List II } \\\...

If $(2,3)$ is the focus and $x-y+3=0$ is the directrix of a parabola, then the equation of the tangent drawn at the vertex of the parabola is

The sum of all the 4 -digit numbers formed by taking all the digits from $2,3,5,7$ without repetition, is

If ${ }^n c_r=c_r$ and $2 \frac{c_1}{c_0}+4 \frac{c_2}{c_1}+6 \frac{c_3}{c_2}+\ldots .+2 n \frac{c_n}{c_{n-1}}=650$, then
${ }^n C_2=$ $\qquad$

The number of ways in which 15 identical gold coins can be distributed among 3 persons such that each one gets atleast 3 gold coins, is