AP EAPCET 2024 - 23th May Morning Shift

The probability that $A$ speaks truth is $75 \%$ and the probability that $B$ speaks truth is $80 \%$. The probability that they contradict each other when asked to speak on a fact is

If the probability distribution of a random variable $X$ is as follows, then $k$ is equal to
$$ \begin{array}{c|l|l|l|l} \hline X=x & 1 & 2 & 3 & 4 \\ \hline P(X=x) & 2 k & 4 k & 3 k & k \\ \hline \end{array} $$

\(\text { In } \triangle A B C, \frac{r_2\left(r_1+r_3\right)}{\sqrt{r_1 r_2+r_2 r_3+r_3 r_1}} \text { is equal to }\)

In $\triangle A B C,\left(r_2+r_3\right) \operatorname{cosec}^2 \frac{A}{2}$ is equal to

In $\triangle A B C, a^2 \sin 2 B+b^2 \sin 2 A$ is equal to

The set of all real values of $x$ satisfying the inequality $\frac{7 x^2-5 x-18}{2 x^2+x-6}<2$ is

$\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3+3 x^2-10 x-24=0$ If $\alpha(\beta+\gamma), \beta(\gamma+\alpha)$ and $\gamma(\alpha+\beta)$ are the roots of the equation $x^3+p x^2+q x+r=0$, then $q$ is equal to

The set of all values of $k$ for which the inequality $x^2-(3 k+1) x+4 k^2+3 k-3>0$ is true for all real values of $x$, is

The cubic equation whose roots are the square of the roots of the equation is
\(12 x^3-20 x^2+x+3=0\)

The number of ways of selecting- 3 numbers that are in GP from the set $\{1,2,3$, $100\}$ is

The $n$th term of the series $1+(3+5+7)+(9+11+13+15+17)+\ldots$ is

If the mean of the data $7,8,9,7,8,7, \lambda$ and 8 is 8 , then variance of the data is equal to

The locus of the mid-point of the portion of the line $x \cos \alpha+y \sin \alpha=p$ intercepted by the coordinate axes, where $p$ is a constant, is

The origin is shifted to the point $(2,3)$ by translation of axes and then the coordinate axes are rotated about the origin through an angle $\theta$ in the counter - clockwise sense. Due to this if the equation $3 x^2+2 x y+3 y^2-18 x-22 y...

If the straight line passing through $P(3,4)$ makes an angle $\frac{\pi}{6}$ with the positive $X$-axis in anti-clockwise direction and meets the line $12 x+5 y+10=0$ at $Q$, then the length of the segment $P Q$ is

The equation of the perpendicular bisectors of the sides $A B$ and $A C$ of $\triangle A B C$ are $x-y+5=0$ and $x+2 y=0$ respectively, If the coordinates of $A$ are $(1,-2)$, then the equal of the line $B C$ is

The combined equation of the bisectors of the angles between the lines joining the origin to the points of intersection of the curve $x^2+y^2+x y+x+3 y+1=0$ and the line $x+y+2=0$ is

A pair of lines drawn through the origin forms a right angled isosceles triangle with right angle at the origin with the line $2 x+3 y=6$. The area (in sq units) of the triangle thus formed is

If the direction cosines of lines are given by $l+m+n=0$ and $m n-2 l m-2 n l=0$, then the acute angle between those lines is

If the angle $\theta$ between the line $\frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2}$ and the plane $2 x-y+\sqrt{\lambda} z+4=0$ is such that $\sin \theta=\frac{1}{3^{\prime}}$ then the value of $\lambda=$

The length of the internal bisector of angle $A$ in $\triangle A B C$ with vertices $A(4,7,8), B(2,3,4)$ and $C(2,5,7)$ is

The general solution of $\cot \frac{x}{2}-\cot x=\operatorname{cosec} \frac{x}{2}$ is

\(\tan ^2 \frac{\pi}{16}+\tan ^2 \frac{2 \pi}{16}+\tan ^2 \frac{3 \pi}{16}+\tan ^2 \frac{4 \pi}{16}\)
$+\tan ^2 \frac{5 \pi}{16}+\tan ^2 \frac{6 \pi}{16}+\tan ^2 \frac{7 \pi}{16}$ is equal to

If $A B$ and $C$ are the angles of a triangle, then $\frac{\sin A+\sin B+\sin C}{\sin ^2 \frac{A}{2}-\sin ^2 \frac{B}{2}+\sin ^2 \frac{C}{2}-1}$ is equal to

$$ \begin{aligned} & \sin ^2 18^{\circ}+\sin ^2 24^{\circ}+\sin ^2 36^{\circ}+\sin ^2 42^{\circ}+\sin ^2 78^{\circ} \\ & +\sin ^2 90^{\circ}+\sin ^2 96^{\circ}+\sin ^2 102^{\circ}+\sin ^2 138^{\circ}+\sin ^2 162^{\circ} \text { is } \\ & \t...

If $\mathbf{A B}=2 \mathbf{i}+3 \mathbf{j}-6 \mathbf{k}, \mathbf{B C}=6 \mathbf{i}-2 \mathbf{j}+3 \mathbf{k}$ are the vectors along two sides of a $\triangle A B C$. Then, perimeter of $\triangle A B C$ is

The orthogonal projection vector of $a=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ on $\mathbf{b}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ is

A unit vector perpendicular to the vectors $a=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ and $\mathbf{b}=3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ is

If the vectors $a \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+b \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+\hat{\mathbf{j}}+c \hat{\mathbf{k}}$ $(a \neq b \neq c \neq 1)$ are coplanar, then $\frac{1}{1...

If $\mathbf{a}=-4 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ and $\mathbf{b}=\sqrt{2} \hat{\mathbf{i}}-\sqrt{2} \hat{\mathbf{j}}$ are two vectors, then angle between the vectors $2 \mathbf{a}$ and $\frac{\mathbf{b}}{2}$ is