AP EAPCET 2025 - 26th May Evening Shift

If the tangent of the curve $x y+a x+b y=0$ at $(1,1)$ makes an angle $\tan ^{-1} 2$ with $X$-axis, then $\frac{a b}{a+b}=$

If the function $f(x)=x^3+b x^2+c x-6$ satisfies all the conditions of Rolle's theorem in $[1,3]$ and $f^{\prime}\left(\frac{2 \sqrt{3}+1}{\sqrt{3}}\right)=0$, then $b c=$

If the displacement $S$ of a particle travelling along a straight line in $t$ seconds is given by $S=2 t^3+2 t^2-2 t-3$, then the time taken (in second) by the particle to change its direction is

The area of the region (in sq units) bounded by the curves $x^2+y^2=16$ and $y^2=6 x$ is

Let $S_1=\sum\limits_{j=1}^{10} j(j-1) \cdot{ }^{10} C_j, S_2=\sum\limits_{j=1}^{10} j \cdot{ }^{10} C_j$ and
\(S_3=\sum\limits_{j=1}^{10} j^2 \cdot{ }^{10} C_j\)
Assertion (A) $S_3=55 \times 2^9$
Reason (R) $S_1=90 \times 2^8$ and $S_2=1...

The coefficient of $x^{10}$ in the expansion of $\left(x+\frac{2}{x}-5\right)^{12}$ is

If $A(\cos \alpha, \sin \alpha), B(\sin \alpha,-\cos \alpha), C(1,2)$ are the vertices of a $\triangle A B C$, then the locus of its centroid is

The equation of the circle which cuts all the three circles $4(x-1)^2+4(y-1)^2=1,4(x+1)^2+4(y-1)^2$ and $4(x+1)^2+4(y+1)^2=1$ orthogonally is

If the chord joining the points $(1,2)$ and $(2,-1)$ on a circle subtends an angle of $\frac{\pi}{4}$ at any point on its circumference, then the equation of such a circle is

If $(\alpha, \beta)$ is the external centre of similitude of the circles $x^2+y^2=3$ and $x^2+y^2-2 x+4 y+4=0$, then $\frac{\beta}{\alpha}=$

The equation of the circle touching the lines $|x-2|+|y-3|=4$ is

A circle passing through origin cuts the coordinate axes is $A$ and $B$. If the straight line $A B$ passes through a fixed point $\left(x_1, y_1\right)$, then the locus of the centre of the circle is

If $x=3-2 \sqrt{3} \mathrm{i}$, then $x^4-12 x^3+54 x^2-108 x-54=$

The product of the four values of the complex number $(1+i)^{3 / 4}$ is

$z_1, z_2, z_3$ represent the vertices $A, B, C$ of a $\triangle A B C$ respectively in the argand plane. If $\left|z_1-z_2\right|=\sqrt{25-12 \sqrt{3}},\left|\frac{z_1-z_3}{z_2-z_3}\right|=\frac{3}{4}$ and $\angle A C B=30^{\circ}$, then t...

\(\int_0^\pi\left(\sin ^5 x \cos ^3 x+\sin ^4 x \cos ^4 x+\sin ^3 x \cos ^4 x\right) d x=\)

\(\int_0^1 \frac{x^4+1}{x^6+1} d x=\)

The general solution of the differential equation $\left(1+\sin ^2 x\right) \frac{d y}{d x}+y \sin 2 x=\cos x+\sin ^2 x \cos x$ is

If $a$ and $b$ are arbitrary constants, then the differential equation corresponding to the family of curves $y=\tan (a x+b)$ is

The general solution of the differential equation $x y(y+2) d y+\left(y^3-1\right) d x=0$ is

If $f(x)=x^{\sec ^{-1} x}$, then $f^{\prime}(2)=$

If a tangent having slope $\frac{1}{3}$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ is a normal to the circle $(x+1)^2+(y+1)^2=1$, then $a^2$ lies in the interval

If $P(\alpha, \beta)$ is a point on the curve $9 x^2+4 y^2=144$ in the first quadrant and the minimum area of the triangle formed by the tangent of the curve at $P$ with the coordinate axis is $S$, then

Let [ $x$ ] represent the greatest integer less than or equal to $x,\{x\}=x-[x] \sqrt{2}=1.414$ and $\sqrt{3}=1.732$. If $f(x)=\left\{x+\left[\frac{x}{1+x^2}\right]\right\}$ is a real valued function, then $f(\sqrt{2})+f(-\sqrt{3})=$

If the range of the function $f(x)=-3 x-3$ is $\{3,-6,-9,-18\}$, then which one of the following is not in the domain of $f$ ?

Let $P(a \sec \theta, b \tan \theta)$ and $Q(a \sec \phi, b \tan \phi)$, where $\theta+\phi=\frac{\pi}{2}$ be two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ If $(h, k)$ is the point of intersection of the normals drawn at $...

If the angle between the asymptotes of a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $2 \tan ^{-1}\left(\frac{2}{3}\right)$ and $a^2-b^2=45$, then $a b=$

\(\int(\log 2 x)^3 d x=\)

If $\int \frac{d x}{(x \tan x+1)^2}=f(x)+C$, then $\lim\limits_{x \rightarrow \frac{\pi}{2}} f(x)=$

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

\(\int \sin ^3 x \cos ^2 x d x=\)

If $\frac{2 x^4-3 x^2+4}{\left(x^2+1\right)\left(x^2+2\right)}=a+\frac{p x+q}{x^2+1}+\frac{m x+n}{x^2+2}$, then $\frac{n}{q}=$

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

\(\tanh ^{-1}\left(\frac{1}{3}\right)+\operatorname{coth}^{-1}(3)=\)

If $f(x)=\sec ^{-1}\left(\frac{1}{2 x^2-1}\right)$ and $g(x)=\tan ^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$, then the derivative of $f(x)$ with respect to $g(x)$ is

\(\tan \left(2 \tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{7}\right)\right)=\)

If $y=\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)$ and $\left(\frac{d^2 y}{d x^2}\right)_{x=2}=k$, then $25 k=$

\(\mathop {\lim }\limits_{x \to 0} \frac{x+2 \sin x+3 \tan x-\tan ^3 x}{\sqrt{x^2+2 \sin x+\tan x+3}-\sqrt{\sin ^2 x-2 \tan x-x+3}}\)

\(\mathop {\lim }\limits_{n \to \infty } \frac{\pi}{2 n}\left[\sin \frac{\pi}{2 n}+\sin \frac{2 \pi}{2 n}+\sin \frac{3 \pi}{2 n}+\ldots+\sin \frac{\pi}{2}\right]=\)

\(\mathop {\lim }\limits_{x \to \infty } \frac{(3-x)^{25}(6+x)^{35}}{(12+x)^{38}(9-x)^{22}}=\)

If a real valued function
$$ f(x)=\left\{\begin{array}{cc} \log (1+[x]), & x \geq 0 \\ \sin ^{-1}[x], & -1 \leq x<0 \\ k([x]+|x|), & x<-1 \end{array}\right. $$
is continuous at $x=-1$, then $k=$

If $A=\left[\begin{array}{lll}2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2\end{array}\right]$ and $\alpha, \beta, \gamma$ are the roots of the equation represented by $|A-x I|=0$, then $\alpha^2+\beta^2+\gamma^2=$

If $B$ is the inverse of a third order matrix $A$ and det $B=k$, then $(\operatorname{adj}(\operatorname{adj} \mathrm{A}))^{-1}=$

If the values of $x, y$ and $z$ which satisfy the equations $2 x-3 y+2 z+15=0,3 x+y-z+2=0$ and $x-3 y-3 z+8=0$ simultaneously are $\alpha, \beta$ and $\gamma$ respectively, then

If the normal chord drawn at the point $\left(\frac{15}{2}, \frac{15}{\sqrt{2}}\right)$ to the parabola $y^2=15 x$ subtends an angle $\theta$ at the vertex of the parabola, then $\sin \frac{\theta}{3}+\cos \frac{2 \theta}{3}-\sec \frac{4 \t...

The number of integers greater than 6000 that can be formed by using the digits $0,5,6,7,8$ and 9 without repetition is

The number of ways of dividing 15 persons into 3 groups containing 3,5 and 7 persons so that two particular persons are not included into the 5 persons groups is

A company representative is distributing 5 identical samples of a product among 12 houses in a row such that each house gets at most one sample. The probability that no two consecutive house get one sample is

Let $X \sim B(n, p)$ with mean $\mu$ and variance $\sigma^2$. If $\mu=2 \sigma^2$ and $\mu+\sigma^2=3$, then $P(X \leq 3)=$

$A$ and $B$ are two independent events of a random experiment and $P(A)>P(B)$.

If the probability that both $A$ and $B$ occurs is $\frac{1}{6}$ and neither of them occurs is $\frac{1}{3}$, then the probability of the occurance of $B$ is