AP EAPCET 2024 - 21th May Evening Shift

An urn contains 3 black and 5 red balls. If 3 balls are drawn at random from the urn, the mean of the probability distribution of the number of red balls drawn is

If all the letters of the word 'SENSELESSNESS' are arranged in all possible ways and an arrangement among them is chosen at random, then the probability that all the E's come together in that arrangement is

Bag $A$ contains 3 white and 4 red balls, bag $B$ contains 4 white and 5 red balls and bag $C$ is contains 5 white and 6 red balls. If one ball is drawn at random from each of these three bags, then the probability of getting one white and ...

If $X \sim B(5, p)$ is a binomial variate such that $P(X=3)=P(X=4)$, then $P(|X-3|<2)=$

If two numbers $x$ and $y$ are chosen one after the other at random with replacement from the set of number $\{1,2,3, \ldots \ldots 10\}$. Then, the probability that $\left|x^2-y^2\right|$ is divisible by 6 is

In $\triangle A B C$, if $a=13, b=14$ and $\cos \frac{C}{2}=\frac{3}{\sqrt{13}}$, then $2 r_1=$

In $\triangle A B C$, if $\left(r_2-r_1\right)\left(r_3-r_1\right)=2 r_2 r_3$, then $2(r+R)=$

If 7 and 8 are the length of two sides of a triangle and $a^{\prime}$ is the length of its smallest side. The angles of the triangle are in AP and ' $a$ ' has two values $a_1$ and $a_2$ satisfying this condition. If $a_1 < a_2$, then $2 a_...

If $\alpha$ is a common root of $x^2-5 x+\lambda=0$ and $x^2-8 x-2 \lambda=0(\lambda \neq 0)$ and $\beta, \gamma$ are the other roots of them, then $\alpha+\beta+\gamma+\lambda=$

The equation $x^4-x^3-6 x^2+4 x+8=0$ has two equal roots. If $\alpha, \beta$ are the other two roots of this equation, then $\alpha^2+\beta^2=$

In the expansion of $\frac{2 x+1}{(1+x)(1-2 x)}$ the sum of the coefficients of the first 5 odd powers of $x$ is

The condition that the roots of $x^3-b x^2+c x-d=0$ are in arithmetic progression is

If $1 \cdot 3 \cdot 5+3 \cdot 5 \cdot 7+5 \cdot 7 \cdot 9+\ldots n$ terms $=n(n+1) f(n)-3 n$, then $f(l)=$

The coefficient of variation for the frequency distribution is
$$ \begin{array}{|c|l|l|l|} \hline \boldsymbol{x}_{\boldsymbol{i}} & 4 & 3 & 1 \\ \hline \boldsymbol{f}_{\boldsymbol{i}} & 1 & 3 & 5 \\ \hline \end{array} $$

$P$ is a point on $x+y+5=0$, whose perpendicular distance from $2 x+3 y+3=0$ is $\sqrt{13}$, then the coordinates of $P$ are

Suppose the axes are to be rotated through an angle $\theta$ so as to remove the $x y$ form from the equation $3 x^2+2 \sqrt{3} x y+y^2=0$. Then, in the new coordinate system the equation $x^2+y^2+2 x y=2$ is transformed to

For $\lambda, \mu \in R,(x-2 y-1)+\lambda(3 x+2 y-11)=0$ and $(3 x+4 y-11)+\mu(-x+2 y-3)=0$ represent two families of lines. If the equation of the line common to both the families is $a x+b y-5=0$. Then, $2 a+b=$

If the pair of lines represented by $3 x^2-5 x y+P y^2=0$ and $6 x^2-x y-5 y^2=0$ have one line in common, then the sum of all possible value of $P$ is

If the plane $x-y+z+4=0$ divides the line joining the points $P(2,3,-1)$ and $Q(1,4,-2)$ in the ratio $l: m$, then $l+m$ is

If the line with direction ratios $(1, \alpha, \beta)$ is perpendicular to the line with direction ratios $(-1,2,1)$ and parallel to the line with direction ratios $(\alpha, 1, \beta)$ then $(\alpha, \beta)$ is

Angle between the planes, $\mathbf{r} \cdot(12 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})=5$ and, $\mathbf{r} \cdot(5 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}})=7$ is

The shortest distance between the skew lines $\mathbf{r}=(2 \hat{\mathbf{i}}-\hat{\mathbf{j}})+t(\hat{\mathbf{i}}+2 \hat{\mathbf{k}})$ and $\mathbf{r}=(-2 \hat{\mathbf{i}}+\hat{\mathbf{k}})+s(\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k...

Let $P\left(x_1, y_1, z_1\right)$ be the foot of perpendicular drawn from the point $Q(2,-2,1)$ to the plane $x-2 y+z=1$. If $d$ is the perpendicular from the point $Q$ to the plane and $l=x_1+y_1+z_1$, then $l+3 d^2$ is

Number of solutions of the trigonometric equation $2 \tan 2 \theta-\cot 2 \theta+1=0$ lying in the interval $[0, \pi]$ is

Assertion (A) : If $A=10^{\circ}, B=16^{\circ}$ and $C=19^{\circ}$, then $\tan 2 A \tan 2 B+\tan 2 B \tan 2 C+\tan 2 C \tan 2 A=1$
Reason (R) : If $A+B+C=180^{\circ}, \cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}$
$$ =\cot \frac{A}{2} ...

If $A, B, C$ are the angles of triangle, then $\sin 2 A-\sin 2 B+\sin 2 C=$

If $\alpha$ is in the 3rd quadrant, $\beta$ is in the 2nd quadrant such that $\tan \alpha=\frac{1}{7}, \sin \beta=\frac{1}{\sqrt{10}}$, then $\sin (2 \alpha+\beta)=$

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are 3 vectors such that $|\mathbf{a}|=5,|\mathbf{b}|=8,|\mathbf{c}|=11$ and $\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}$, then the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is

If $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-\hat{\mathbf{k}},-3 \hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ are the position vectors of three points, $A, B, C$ respectively, t...

If $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$ are position vectors of 4 points such that $2 a+3 b+5 c-10 d=0$, then the ratio in which the line joining $c$ and $d$ divides the line segment joining $a$ and $\mathbf{b}$ is