AP EAPCET 2025 - 22nd May Morning Shift

If the probability distribution of a random variable $X$ is as follows, then $P(X \leq 2)=$
$$ \begin{array}{cccccc}\hline x_i & 0 & 1 & 2 & 3 & 4 \\ \hline P\left(X=x_i\right) & 3 k & 5 k & 3 k^2 & 4 k^2+k & 3 k^2 \\ \hline \end{array} $$

The probability that a person $A$ completes a work in a given time is $\frac{2}{3}$ and the probability that another person
$B$ completes the same work in the same time is $\frac{3}{4}$.
If both $A$ and $B$ start doing this work at the same...

If $l, m$ represent any two elements (identical or different) of the set $\{1,2,3,4,5,6,7\}$, then the probability that $l x^2+m x+1>0 \forall x \in R$ is

$A$ and $B$ are playing chess game with each other. The probability that $A$ wins the game is 0.6 . the probability that he loses is 0.3 and the probability its draw is 0.1 . If they played three games, then the probability that $A$ wins at...

$U_1, U_2, U_3$ are three urns. $U_1$ contains 5 red, 3 white, 2 back balls: $U_2$ contains 4 red 4 white, 2 black balls and $U_3$ contains 3 red. 4 white, 3 black balls. If a ball is chosen at random from an urn chosen at random, then the ...

If $X$ follows poisson distribution with variance 2 , then $P(X \geq 3)=$

In $\triangle A B C, \sqrt{\frac{r \cdot r_2}{r_3 r_1}}=$

In a $\triangle A B C, A-B=120^{\circ}, R=8 r$, then $\frac{1+\cos C}{1-\cos C}=$

If $A(0,0,0) B(3,4,0)$ and $C(0,12,5)$ are the vertices of a $\triangle A B C$, then the $x$-coordinate of its incentre is

The equation $6 x^4-5 x^3+13 x^2-5 x+6=0$ will have

If $\alpha$ is a repeated root of multiplicity 2 of the equation $18 x^3-33 x^2+20 x-4=0$, then

If $(2 k-1) x^2-2(3 k-2) x+4 k>0$ for every $x \in R$, then the sum of all possible integral values of $k$ is

For all $n \in N, \frac{3^n-1}{2} \geq$

The following data represents the frequency distribution of 20 observations
$$ \begin{array}{ccccccc} \hline x_i & 3 & 4 & 5 & 8 & 10 & 11 \\ \hline f_i & \alpha+2 & (\alpha-1)^2 & 4 & \alpha-1 & 2 & \alpha \\ \hline \end{array} $$
Then, it...

One line of the pair of lines $x^2+x y-2 y^2=0$ is perpendicular to one line of the pair of lines $3 y^2-5 x y-2 x^2=0$ If the combined equation of the two lines other than those two perpendicular lines is $a x^2+2 h x y+b y^2=0$, then $a+2...

If the angle between the lines joining the origin to the points of intersection of $x+2 y+\lambda=0$ and $2 x^2-2 x y+3 y^2+2 x-y-1=0$ is $\frac{\pi}{2}$, then a value of $\lambda$. is

If the lines $x+2 a y+a=0, x+3 b y+b=0$, $x+4 c y+c=0$ are concurrent, then $a, b, c$ are in

A straight line passing through a fixed point $(2,3)$ intersects the coordinate axes at points $P$ and $Q$. If $O$ is the origin and $R$ is a variable point such that $O P R Q$ is a rectangle, then the locus of $R$ is

If $M$ is the foot of the perpendicular drawn from the origin to the line $x-2 y+3=0$ which meets the $X$ and $Y$-axes at $A$ and $B$, respectively, then $A M=$

$\hat{\mathbf{i}}-2 \hat{\mathbf{j}}$ is a point on the line parallel to the vector $2 \hat{\mathbf{i}}+\hat{\mathbf{k}}$. If $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$ is a point on the plane parallel to the vectors $2 \hat{\mathbf{j}}-\hat{\ma...

A plane $\pi$ is passing through the points $A(1,-2,3)$ and $B(6,4,5)$. If the plane $\pi$ is perpendicular the plane $3 x-y+z=2$, then the perpendicular distance from $(0,0,0)$ to the plane $\pi$ is

Angle between a diagonal of a cube and a diagonal of its face which are coterminus is

The number of solutions of $\sin 2 x+\cos 4 x=2$ in the interval $[-\pi, \pi]$ is

If $A+B=\frac{\pi}{4}$, then $\frac{\cos B-\sin B}{\cos B+\sin B}=$

If $7 \cos \theta-\sin \theta=5$ and $\tan \theta>0$, then $\tan \theta=$

\(\sin ^3 10^{\circ}+\sin ^3 50^{\circ}-\sin ^3 70^{\circ}=\)

Points $P$ and $Q$ are given by $\mathbf{O P}=\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{O Q}=-\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$. A line along the vector $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}$...

For $a \in R$, if the vectors $\mathbf{p}=(a+1) \hat{\mathbf{i}}+a \hat{\mathbf{j}}+a \hat{\mathbf{k}}$, $\mathbf{q}=a \hat{\mathbf{i}}+(a+1) \hat{\mathbf{j}}+a \hat{\mathbf{k}}$ and $\mathbf{r}=a \hat{\mathbf{i}}+a \hat{\mathbf{j}}+(a+1) \...

If $\mathbf{a}=\hat{\mathbf{i}}+4 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}, \mathbf{b}=-2 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{c}=3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}$ are three vectors such th...

If $A=(0,4,-3), B=(5,0,12)$ and $C=(7,24,0)$, then $\sqrt{B A C}=$