AP EAPCET 2024 - 22th May Evening Shift

$A$ and $B$ throw a pair of dice alternately and they note the sum of the numbers appearing on the dice. $A$ wins if he throws 6 before $B$ throws 7 and $B$ wins if he throws 7 before $A$ throws 6 . If $A$ begins then, the probability of hi...

If a random variable $X$ has the following probability distribution, then its variance is nearly
$$ \begin{array}{clllllll} \hline X=x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline P(X=x) & 0.05 & 0.1 & 2 K & 0 & 0.3 & K & 0.1 \\ \hline \end{arr...

$E_1$ and $E_2$ are two independent events of a random experiment such that $P\left(E_1\right)=\frac{1}{2}$ and $P\left(E_1 \cup E_2\right)=\frac{2}{3}$. Then, match the items of List I with the items of List II.
$$ \begin{array}{lll} \hlin...

A bag contains 4 red and 5 black balls. Another bag contains 3 red and 6 black balls. If one ball is drawn from first bag and two balls from the second bag at random. The probability that out of the three, two are black and one is red, is

In $a \triangle A B C$ if $r: R: r_2=1: 3: 7$, then $\sin (A+C)+\sin B$ is equal to

In a $\triangle A B C$, if $a=13, b=14$ and $c=15$, then $r_1=$

In $\triangle A B C,\left(r_1+r_2\right) \operatorname{cosec}^2 \frac{C}{2}$ is equal to

If ' $a$ ' is a rational number, then the roots of the equation $x^2-3 a x+a^2-2 a-4=0$ are

The set of all real values ' $a$ ' for which $-1<\frac{2 x^2+a x+2}{x^2+x+1}<3$ holds for all real values of $x$ is

The quotient, when $3 x^5-4 x^4+5 x^3-3 x^2+6 x-8$ is divided by $x^2+x-3$ is

Based on the following statements, choose the correct option.
Statement I The variance of the first $n$ even natural numbers is $\frac{n^2-1}{4}$.
Statement II The difference between the variance of the first 20 even natural numbers and the...

If the slope of one of the lines in the pair of lines $8 x^2+a x y+y^2=0$ is thrice the slope of the second line, then $a$ is equal to

$A$ line $L$ passing through the point $P(-5,-4)$ cuts the lines $x-y-5=0$ and $x+3 y+2=0$ respectively at $Q$ and $R$ such that $\frac{18}{P Q}+\frac{15}{P R}=2$, then slope of line $L$ is

The locus of a variable point which forms a triangle of fixed area with two fixed points is

The equation of a line which makes an angle of $45^{\circ}$ with each of the pair of lines $x y-x-y+1=0$ is

If the reflection of a point $A(2,3)$ in $X$-axis is $B$, reflection of $B$ in the line $x+y=0$ is $C$ and the reflection of $C$ in $x-y=0$ is $D$, then the point of intersection of the lines $C D, A B$ is

If $\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}$ are the vertices of a tetrahedron, t...

If a line $L$ makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with $Y$-axis and $Z$-axis respectively, then the angle between $L$ and another line having direction ratio $1,1,1$ is

If $l, m$ and $n$ are the direction cosines of a line that is perpendicular to the lines having the direction ratios $(1,2,-1)$ and $(1,-2,1)$, then $(l+m+n)^2$ is equal to

The foot of the perpendicular drawn from a point $A(1,1,1)$ on to a plane $\pi$ is $P(-3,3,5)$.If the equation of the plane parallel to the plane of $\pi$ and passing through the mid-point of $A P$ is $a x-y+c z+d=0$, then $a+c-d$ is equal ...

If $A=(1,2,3), B=(3,4,7)$ and $C=(-3,-2,-5)$ are three points, then the ratio in which the point $C$ divides $A B$ externally is

The values of $x$ in $(-\pi, \pi)$, which satisfy the equation $8^{1+\cos ^2 x+\cos ^4 x+\ldots \ldots}=4^3$ are

$\cos 6^{\circ} \sin 24^{\circ} \cos 72^{\circ}$ is equal to

$\tan \alpha+2 \tan 2 \alpha+4 \tan 4 \alpha+8 \cot 8 \alpha$ is equal to

If $\sinh x=\frac{\sqrt{21}}{2}$, then $\cosh 2 x+\sinh 2 x$ is equal to

$\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}$ is equal to

If $\cos \alpha+4 \cos \beta+9 \cos \gamma=0$ and $\sin \alpha+4 \sin \beta+9 \sin \gamma=0$, then 81 $\cos (2 \gamma-2 \alpha)-16 \cos (2 \beta-2 \alpha)$ is equal to

If the vectors $a \hat{\mathbf{i}}+\mathbf{j}+3 \hat{\mathbf{k}}, 4 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $4 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}$ are coplanar, then $a$ is equal to

Let $|\hat{\mathbf{a}}|=2=|\hat{\mathbf{b}}|=3$ and the angle between $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$ be $\frac{\pi}{3}$.
If a parallelogram is constructed with adjacent sides $2 \hat{\mathbf{a}}+3 \hat{\mathbf{b}}$ and $\hat{\mat...

The values of $x$ for which the angle between the vectors $x^2 \hat{\mathbf{i}}+2 x \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+x \hat{\mathbf{k}}$ is obtuse lie in the interval