AP EAPCET 2024 - 22th May Morning Shift

The value of Lagrange's mean value theorem for $f(x)=e^x+24$ in $[0,1]$ is

Equation of the normal to the curve $y=x^2+x$ at the point $(1,2)$ is

Displacement $s$ of a particle at time $t$ is expressed as $s=2 t^3-9 t$. Find the acceleration at the time when $b^{t 5}$ velocity vanishes.

If a running track of 500 ft is to be laid out enclosing a playground the shape of which is a rectangle with a semi-circle at each end, then the length of the rectangular portion such that the area of the rectangular portion is to be maximu...

The area (in sq units) bounded by the curves $x=y^2$ and $x=3-2 y^2$ is

If the $2 \mathrm{nd}, 3 \mathrm{rd}$ and 4 th terms in the expansion of $(x+a)^n$ are $96,216,216$ respectively and $n$ is a positive integer, then $a+x=$

If $|x|<1$, then the number of terms in the expansion of $\left[\frac{1}{2}\left(1 \cdot 2+2 \cdot 3 x+3 \cdot 4 x^2+\ldots . \infty\right)\right]^{-25}$

From a point $(1,0)$ on the circle $x^2+y^2-2 x+2 y+1=0$ if chords are drawn to this circle, then locus of the poles of these chords with respect the circle $x^2+y^2=4$ is

If $A$ and $B$ are the centres of similitude with respect to the circles $x^2+y^2-14 x+6 y+33=0$ and $x^2+y^2+30 x-2 y+1=0$, then mid-point of $A B$ is

$C_1$ is the circle with centre at $O(0,0)$ and radius $4, C_2$ is a variable circle with centre at $(\alpha, \beta)$ and radius 5 . If the common chord of $C_1$ and $C_2$ has slope $\frac{3}{4}$ and of maximum length, then one of the possi...

If the pair of tangents drawn to the circle $x^2+y^2=a^2$ from the point $(10,4)$ are perpendicular. then $a=$

If $x-4=0$ is the radical axis of two orthogonal cirlces out of which one is $x^2+y^2=36$, then the centre of the other circle is

If $Z$ is a complex number such that $|Z| \leq 3$ and $\frac{-\pi}{2} \leq \operatorname{amp} Z \leq \frac{\pi}{2}$, then the area of the region formed by locus of $Z$ is

The locus of the complex number $Z$ such that $\arg \left(\frac{Z-1}{Z+1}\right)=\frac{\pi}{4}$ is

All the values of $(8 i)^{\frac{1}{3}}$ are

If the number of real roots of $x^9-x^5+x^4-1=0$ is $n$, the number of complex roots having argument on imaginary axis is $m$ and the number of complex roots having argument in 2nd quadrant is $K, m \cdot n \cdot k=$

\(\int_0^{\pi / 4} \log (1+\tan x) d x=\)

\(\int\limits_\pi ^\pi {}\frac{x \sin x}{1+\cos ^2 x} d x=\)

The general solution of the differential equation $(1+\tan y)(d x-d y)+2 x d y=0$ is

The general solution of the differential equation $x d y-y d x=\sqrt{x^2+y^2} d x$ is

The sum of the order and degree of differential equation $x\left(\frac{d^2 y}{d x^2}\right)^{1 / 2}=\left(1+\frac{d y}{d x}\right)^{4 / 3}$

If $y=\sqrt{\sin x+\sqrt{\sin x+\sqrt{\sin x+\ldots \infty}}}$, then the value of $\frac{d^2 y}{d x^2}$ at the point $(\pi, 1)$ is

64. If $f(0)=0, f^{\prime}(0)=3$, then the derivative of $y=f(f(f(f(f(x)))))$ at $x=0$ is

The domain of the real valued function $f(x)=\sqrt{9-\sqrt{x^2-144}}$ is

The transformed equation of $x^2-y^2+2 x+4 y=0$ when the origin is shifted to the point $(-1,2)$ is

If the ellipse $4 x^2+9 y^2=36$ is confocal with a hyperbola whose length of the transverse axis is 2 , then the points of intersection of the ellipse and hyperbola lie on the circle

If the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $\sec \alpha$, then area of the triangle formed by the asymptotes of the hyperbola with any of its tangent is

If $e_1$ and $e_2$ are respectively the eccentricities of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and its conjugate hyperbola, then the line $\frac{x}{2 e_1}+\frac{y}{2 e_2}=1$ touches the circle having centre at the origin, then ...

\(\int \frac{x^2-1}{x^3 \sqrt{2 x^4-2 x^2+1}} d x\)

\(\int \frac{x^3 \tan ^{-1} x^4}{1+x^8} d x=\)

\(\int \frac{2}{1+x+x^2} d x=\)

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

\(\int \frac{\sin 7 x}{\sin 2 x \sin 5 x} d x=\)

The value of $x$ such that $\sin \left(2 \tan ^{-1} \frac{3}{4}\right)=\cos \left(2 \tan ^{-1} x)\right.$

\(\mathop {\lim }\limits_{x \to o} \left[\frac{1}{x}-\frac{1}{e^x-1}\right]=\)

Let $f(x)=\left\{\begin{array}{cl}0, & x=0 \\ 2-x, & \text { for } 0 < x < 1 \\ 2, & \text { for } x=1 \\ \frac{1}{2}-x, & \text { for } 1 < x < 2 \\ \frac{-3}{2}, & \text { for } x \geq 2\end{array}\right.$
then which of the following is t...

If $f(x)=\left(\frac{1+x}{1-x}\right)^{\frac{1}{x}}$ is continuous at $x=0$, then $f(0)=$

The function $f(x)=|x-24|$ is

\(\mathop {\lim }\limits_{n \to \infty }\left(\frac{1}{\sqrt{n^2}}+\frac{1}{\sqrt{n^2-1}}+\ldots+\frac{1}{\sqrt{n^2-(n-1)^2}}\right)=\)

If the set of equations $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+a z=b$ has unique solution, then

If $P$ and $Q$ are two $3 \times 3$ matrices such that $|P Q|=1$ and $|P|=9$, then the determinant of adjoint of the matrix $P$. $\operatorname{adj} 3 Q$ is

If $A=\left[\begin{array}{lll}a & 1 & 2 \\ 1 & 2 & b \\ c & 1 & 3\end{array}\right]$ and $\operatorname{adj} A=\left[\begin{array}{ccc}7 & -1 & -5 \\ -3 & 9 & 5 \\ 1 & -3 & 5\end{array}\right]$, then $a^2+b^2+c^2=$

If the normal chord drawn at $(2 a, 2 a \sqrt{2})$ on the parabola $y^2=4 a x$ subtends an angle $\theta$ at its vertex, then $\theta=$

All the letters of the word 'TABLE' are permuted and the strings of letters (may or may not have meaning) thus formed are arranged in dictionary order. Then, the rank of the word 'TABLE' counted from the rank of the word 'BLATE' is

5 boys and 6 girls are arranged in all possible ways. Let $X$ denote the number of linear arrangements in which no two boys sit together and $Y$ denote the number of linear arrangements in which no two girls sit together. If $Z$ denote the ...

The number of ways of distributing 15 apples to three persons $A, B, C$ such that $A$ and $C$ each get at least 2 apples and $B$ gets at most 5 apples, is

Three numbers are chosen at random from 1 to 20 , then the probability that the sum of three numbers is divisible by 3 is

Two persons $A$ and $B$ throw three unbiased dice one after the another. If $A$ gets the sum 13, then the probability that $B$ gets higher sum is

8 teachers and 4 students are sitting around a circular table at random, then the probability that no two students sit together is

A bag contains 6 balls. If three balls are drawn at a time and all of them are found to be green, then the probability that exactly 5 of the balls in the bag are green is