AP EAPCET 2024 - 21th May Evening Shift

The value of $c$ such that the straight line joining the points $(0,3)$ and $(5,-2)$ is tangent to the curve $y=\frac{c}{x+1}$ is

The equation of the tangent to the curve $y=x^3-2 x+7$ at the point $(1,6)$ is

If $x$ is real and $\alpha, \beta$ are maximum and minimum values of $\frac{x^2-x+1}{x^2+x+1}$ respectively, then $\alpha+\beta=$

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

If $a^2 x^4+b^2 y^4=c^6$, then maximum value of $x y$ is equal to

The distance ( s ) travelled by a particle in time $t$ is given by $S=4 t^2+2 t+3$. The velocity of the particle, when $t=3 \mathrm{sec}$ is

Area of the region enclosed by the curves $3 x^2-y^2-2 x y+4 x+1=0$ and $3 x^2-y^2-2 x y+6 x+2 y=0$ is

If the ratio of the terms equidistant from the middle term in the expansion of $(l+x)^{12}$ is $\frac{1}{256}(x \in N)$, then sum of all the terms of the expansion $(1+x)^{12}$ is

If the equation of the circle whose radius is 3 units and which touches internally the circle $x^2+y^2-4 x-6 y-12=0$ at the point $(-1,-1)$ is $x^2+y^2+p x+q y+r=0$, then $p+q-r=$

The pole of the straight line $9 x+y-28=0$ with respect to the circle $2 x^2+2 y^2-3 x+5 y-7=0$ is

The radical axis of the circle $x^2+y^2+2 g x+2 f y+c=0$ and $2 x^2+2 y^2+3 x+8 y+2 c=0$ touches the circle $x^2+y^2+2 x+2 y+1=0$. Then,

The equation of a circle which touches the straight lines $x+y=2, x-y=2$ and also touches the circle $x^2+y^2=1$ is

The perimeter of the locus of the point $P$ which divides the line segment QA internally in the ratio $1: 2$, where $A=(4,4)$ and $Q$ lies on the circle $x^2+y^2=9$, is

The equation of the circle touching the circle $x^2+y^2-6 x+6 y+17=0$ externally and to which the lines $x^2-3 x y-3 x+9 y=0$ are normal is

If $n$ is an integer and $Z=\cos \theta+i \sin \theta, \theta \neq(2 n+1) \frac{\pi}{2}$, then $\frac{1+Z^{2 n}}{1-Z^{2 n}}=$

Imaginary part of $\frac{(1-i)^3}{(2-i)(3-2 i)}$ is

The square root of $7+24 i$

\(\int\limits_0^{\pi /4} {{{{x^2}} \over {{{(x\,\sin \,x + \cos \,x)}^2}}}dx = }\)

$\int_1^5(|x-3|+|1-x|) d x=$

$\int_0^1 \frac{x}{(1-x)^{\frac{3}{4}}} d x=$

\(\int_{-1}^1\left(\sqrt{1+x+x^2}-\sqrt{1-x+x^2}\right) d x=\)

If $\cos x \frac{d y}{d x}-y \sin x=6 x,\left(0 < x < \frac{\pi}{2}\right)$ and $y\left(\frac{\pi}{3}\right)=0$, then $y\left(\frac{\pi}{6}\right)=$

$\frac{d y}{d x}=\frac{y+x \tan \frac{y}{x}}{x} \Rightarrow \sin \frac{y}{x}=$

The differential equation formed by eliminating arbitrary constants $A, B$ from the equation $y=A \cos 3 x+B \sin 3 x$ is

If $\frac{d}{d x}\left(\frac{1+x^2+x^4}{1+x+x^2}\right)=a x+b$, then $(a, b)=$

The product of perpendiculars from the two foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ on the tangent at any point on the ellipse is

If $3 f(x)-2 f(1 / x)=x$, then $f(2)=$

The real valued function $f: R \rightarrow\left[\frac{5}{2}, \infty\right)$ defined by $f(x)=|2 x+1|+|x-2|$ is

The descending order of magnitude of the eccentricities of the following hyperbolas is
A. A hyperbola whose distance between foci is three times the distance between its directrices.
B. Hyperbola in which the transverse axis is twice the co...

\(\int {\left( {\sum\limits_{r = 0}^\infty {{{{x^r}{3^r}} \over {r!}}} } \right)dx = }\)

$\int \frac{x^4+1}{x^6+1} d x=$

$\int e^x(x+1)^2 d x=$

$\int \frac{\sin ^6 x}{\cos ^8 x} d x=$

$\int \frac{x^5}{x^2+1} d x=$

If $\frac{x+2}{\left(x^2+3\right)\left(x^4+x^2\right)\left(x^2+2\right)}=\frac{A x+B}{x^2+3}+\frac{C x+D}{x^2+2}$ $+\frac{E x^3+F x^2+G x+H}{x^4+x^2}$, then $(E+F)(C+D)(A)=$

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

The real values of $x$ that satisfy the equation $\tan ^{-1} x+\tan ^{-1} 2 x=\frac{\pi}{4}$ is

If $y=\sin ^{-1} x$, then $\left(1-x^2\right) y_2-x y_1=$

$2 \operatorname{coth}^{-1}(4)+\sec h^{-1}\left(\frac{3}{5}\right)=$

\(\mathop {\lim }\limits_{x \to 1} \left( {{{x + {x^2} + {x^3} + ... + {x^n} - n} \over {x - 1}}} \right) =\)

If the function $f(x)=\frac{\sqrt{1+x}-1}{x}$ is continuous at $x=0$, then $f(0)=$

\(\mathop {\lim }\limits_{x \to 0} \left( {{{\sin (\pi {{\cos }^2}x} \over {{x^2}}}} \right) =\)

If $3 A=\left[\begin{array}{ccc}1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b\end{array}\right]$ and $A A^T=I$, then $\frac{a}{b}+\frac{b}{a}=$

$\left|\begin{array}{ccc}a+b+2 c & a & b \\ c & b+c+2 a & b \\ c & a & c+a+2 b\end{array}\right|=$

Assertion (A) : If $B$ is a $3 \times 3$ matrix and $|B|=6$, then $|\operatorname{adj}(B)|=36$
Reason (R) : If $B$ is a square matrix of order $n$, then $|\operatorname{adj}(B)|=|B|^n$

If the ordinates of points $P$ and $Q$ on the parabola $y^2=12 x$ are in the ratio $1: 2$. Then, the locus of the point of intersection of the normals to the parabola at $P$ and $Q$ is

There are 6 different novels and 3 different poetry books on a table. If 4 novels and 1 poetry book are to be selected and arranged in a row on a shelf such that the poetry book is always in the middle, then the number of such possible arr...

If a five-digit number divisible by 3 is to be formed using the numbers $0,1,2,3,4$ and 5 without repetition, then the total number of ways this can be done is

Four-digit numbers with all digits distinct are formed using the digits $1,2,3,4,5,6,7$ in all possible ways.If $p$ is the total number of numbers thus formed and $q$ is the number of numbers greater than 3400 among them, then $p: q=$

Two persons $A$ and $B$ throw a pair of dice alternately until one of them gets the sum of the numbers appeared on the dice as 4 and the person who gets this result first is declared as the winner. If $A$ starts the game, then the probabili...