AP EAPCET 2024 - 23th May Morning Shift

If the length of the sub-tangent at any $P$ on a curve is proportional to the abscissa of the point $P$, then the equation of that curve is ( $C$ is an arbitrary constant)

$A$ is a point on the circle with radius 8 and centre at $O$. A particle $P$ is moving on the circumference of the circle starting from $A . M$ is the foot of the perpendicular from $P$ on $O A$ and $\angle P O M=\theta$. When $O M$ $=4$ an...

If $(a, \beta)$ is the stationary point of the curve $y=2 x-x^2$, then the area bounded by the curves $y=2^x, y=2 x-x^2, x=0$ and $x=\alpha$ is

The independent term in the expansion of $\left(1+x+2 x^2\right)\left(\frac{3 x^2}{2}-\frac{1}{3 x}\right)^9$ is

For $|x|<\frac{1}{\sqrt{2}}$, the coefficient of $x$ in the expansion of $\frac{(1-4 x)^2\left(1-2 x^2\right)^{1 / 2}}{(4-x)^{3 / 2}}$ is

Consider the point $P(\alpha, \beta)$ on the line $2 x+y=1$. If the $P$ and $(3,2)$ are conjugate points with respect to the circle $x^2+y^2=4$, then $\alpha+\beta$ is equal to

If the circles $x^2+y^2+2 \alpha x+2 y-8=0$ and $x^2+y^2-2 x+a y-14=0$ intersect orthogonally, then the distance between their centres is

The circumference of a circle passing through the point $(4,6)$ with two normals represented by $2 x-3 y+4=0$ and $x+y-3=0$ is

If $(1,3)$ is the mid-point of a chord of the circle $x^2+y^2-4 x-8 y+16=0$, then the area of the triangle formed by that chord with the coordinate axes is

If the line through the point $P(5,3)$ meets the circle $x^2+y^2-2 x-4 y+\alpha=0$ at $A(4,2)$ and $B\left(x_1, y_1\right)$, then $P A \cdot P B$ is equal to

If the roots of the equation $z^3+i z^2+2 i=0$ are the vertices of a $\triangle A B C$, then that $\triangle A B C$ is

$\omega$ is a complex cube root of unity and if $z$ is a complex number satisfying $|z-1| \leq 2$ and $\left|\omega^2 z-1-\omega\right|=a$, then the set of possible values of $a$ is

$(r, \theta)$ denotes $r(\cos \theta+i \sin \theta)$. If $x=(1, \alpha), y=(1, \beta), z=(1, \gamma)$ and $x+y+z=0$, then $\Sigma \cos (2 \alpha-\beta-\gamma)$ is equal to

$\int_0^\pi \frac{x \sin x}{4 \cos ^2 x+3 \sin ^2 x} d x$ is equal to

If $\int_0^{2 \pi}\left(\sin ^4 x+\cos ^4 x\right) d x=K \int_0^\pi \sin ^2 x d x+L \int_0^{\frac{\pi}{2}} \cos ^2 x d x$ and $K, L \in N$, then the number of possible ordered pairs ( $K, L$ ) is

If $A=\int_0^{\infty} \frac{1+x^2}{1+x^4} d x, B=\int_0^1 \frac{1+x^2}{1+x^4} d x$, then

The solution of the differential equation $e^x y d x+e^x d y+x d x=0$ is

Among the options given below from which option a differential equation of order two can be formed ?

The differential equation for which $a x+b y=1$ is general solution is

If $y=(x-1)(x+2)\left(x^2+5\right)\left(x^4+8\right)$, then $\lim _{x \rightarrow-1}\left(\frac{d y}{d x}\right)$ is equal to

If $y=\left(\tan ^{-1} 2 x\right)^2+\left(\cot ^{-1} 2 x\right)^2$, then $\left(1+4 x^2\right)^2 y^{\prime \prime}-16$ is equal to

If $y=\sinh ^{-1}\left(\frac{1-x}{1+x}\right)$, then $\frac{d y}{d x}$ is equal to

Let $T_1$ be the tangent drawn at a point $P(\sqrt{2}, \sqrt{3})$ on the ellipse $\frac{x^2}{4}+\frac{y^2}{6}=1$. If ( $\alpha, \beta$ ) is the point where, $T_1$ intersects another tangent $T_2$ to the ellipse perpendicularly, then $\alpha...

Which of the following function are odd?
I. $f(x)=x\left(\frac{e^x-1}{e^x+1}\right)$
II. $f(x)=k^x+k^{-x}+\cos x$
III. $f(x)=\log \left(x+\sqrt{x^2+1}\right)$

If $A \subseteq Z$ and the function $f: A \rightarrow R$ is defined by $f(x)=\frac{1}{\sqrt{64-(0.5)^{24+x-x^2}}}$, then the sum of all absolute value of elements of $A$ is

If $y=x+\sqrt{2}$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{2}=1$, then equations of its directrices are

The area of the quadrilateral formed with the foci of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ and its conjugate hyperbola is (in sq units)

$\int(\log x)^m x^n d x$ is equal to

$\frac{4 x^2+5}{(x-2)^4}=\frac{A}{(x-2)}+\frac{B}{(x-2)^2}+\frac{C}{(x-2)^3}+\frac{D}{(x-2)^4}$, then $\sqrt{\frac{A}{C}+\frac{B}{C}+\frac{D}{C}}$ is equal to

$\int \sin ^{-1}\left(\sqrt{\frac{x-a}{x}}\right) d x$ is equal to

If $\int \frac{\sqrt[4]{x}}{\sqrt{x}+\sqrt[4]{x}} d x=$ $\frac{2}{3}\left[A \sqrt[4]{x^3}+B \sqrt[4]{x^2}+C \sqrt[4]{x}+D \log (1+\sqrt[4]{x})\right]+K$, then $\frac{2}{3}(A+B+C+D)$ is equal to

If $\int \frac{\sin x \cos x}{\sqrt{\cos ^4 x-\sin ^4 x}} d x=-\frac{f(x)}{2}+c$, then domain of $f(x)$ is

If $0 < x < \frac{1}{2}$ and $\alpha=\sin ^{-1} x+\cos ^{-1}\left(\frac{x}{2}+\frac{\sqrt{3-3 x^2}}{2}\right)$, then $\tan \alpha+\cot \alpha$ is equal to

In each of the following options, a function and an interval are given. Choose the option containing the function and the interval for which Lagrange's mean value theorem is not applicable

Let $f(x)=\left\{\begin{array}{cl}1+\frac{2 x}{a}, & 0 \leq x \leq 1 \\ a x, & 1 < x \leq 2\end{array}\right.$.If $\lim _{x \rightarrow 1} f(x)$ exists, then the sum of the cubes of the possible values of $a$ is

Let $[P]$ denote the greatest integer $\leq P$. If $0 \leq a \leq 2$, then the number of integral values of ' $a$ ' such that $\lim \limits_{x \rightarrow a}\left(\left[x^2\right]-[x]^2\right)$ does not exist is

If $f(x)=\left\{\begin{array}{cl}\frac{\sqrt{a^2-a x+x^2}-\sqrt{x^2+a x+a^2}}{\sqrt{a+x}-\sqrt{a-x}}, & x \neq 0 \text { is } \\ K & x=0\end{array}\right.$ continuous at $x=0$, then $K$ is equal to

If $f(x)=\left\{\begin{array}{cc}a x^2+b x-\frac{13}{8}, & x \leq 1 \\ 3 x-3, & 1 < x \leq 2 \text { is differentiable } \\ b x^3+1, & x > 2\end{array}\right.$ $\forall x \in R$, then $a-b$ is equal to

The function $f(x)=\left\{\begin{array}{cc}\frac{x-|x|}{x}, & x \neq 0 \\ 2, & x=0\end{array}\right.$

$\cosh (\log 4)$ is equal to

If $\left|\begin{array}{lll}a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c\end{array}\right|>0$, then $a b c>$

4. If $A=\left[\begin{array}{lll}83 & 74 & 41 \\ 93 & 96 & 31 \\ 24 & 15 & 79\end{array}\right]$, then $\operatorname{det}\left(A-A^T\right)$ is equal to

If the system of equations $a_1 x+b_1 y+c_1 z=0, a_2 x+b_2 y+c_2 z=0$ and $a_3 x+b_3 y+c_3 z=0$ has only trivial solution, then the rank of $\left[\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right]$ i...

If $P$ is a point which divides the line segment joining the focus of the parabola $y^2=12 x$ and a point on the parabola in the ratio $1: 2$. Then, the locus of $p$ is

The number of ways of arranging 2 red, 3 white and 5 yellow roses of different sizes into a garland such that no two yellow roses come together is

Among the 4 -digit numbers formed using the digits $0,1,2,3$ and 4 when repetition of digits allowed. Then, the number of numbers which are divisible by 4 is

Bag $A$ contains 2 white and 3 red balls and bag $B$ contains 4 white and 5 red balls. If one ball is drawn at random from one of the bags and is found to be red, then the probability that it was drawn from the bag $B$ is

In a binomial distribution $B(n, p)$ the sum and product of the mean and the variance are 5 and 6 respectively, then $6(n+p-q)$ is equal to

It is given that in a random experiment events $A$ and $B$ are such that $P(A)=\frac{1}{4}, P(A / B)=\frac{1}{2}$ and $P(B / A)=\frac{2}{3}$, then $P(B)$ is equal to

When two dice are thrown the probability of getting the sum of the values on them as 10 or 11 is