AP EAPCET 2025 - 24th May Morning Shift

A basket contains 5 apples and 7 oranges and another basket contains 4 apples and 8 oranges. If one fruit is picked out at random from each basket, then the probability of getting one apple and one orange is

Two cards are drawn from a pack of 52 playing cards one after the other without replacement. If the first card drawn is a queen, then the probability of getting a face card from a black suit in the second draw is

If the probability distribution of a random variable $X$ is as follows, then the mean of $X$ is
$$ \begin{array}{ccccc} \hline \boldsymbol{X}=\boldsymbol{x}_{\boldsymbol{i}} & -1 & 0 & 1 & 2 \\ \hline \boldsymbol{P}\left(\boldsymbol{X}=\bol...

In a school there are 3 sections $A, B$ and $C$. Section $A$ contains 20 girls and 30 boys, section $B$ contains 40 girls and 20 boys and section $C$ contains 10 girls and 30 boys. The probabilities of selecting the section $A, B$ and $C$ a...

If $X$ is a binomial variate with mean $\frac{16}{5}$ and variance $\frac{48}{25}$, then $P(X \leq 2)=$

In a $\triangle A B C, A, B$ and $C$ are in arithmetic progression, $r r_3=r_1 r_2$ and $c=10$, then $a^2+b^2+c^2=$

In a $\triangle A B C, \frac{2\left(r_1+r_3\right)}{a c(1+\cos B)}=$

In a $\triangle A B C$, if $a, b, c$ are in arithmetic progression and the angle $A$ is twice the angle $C$, then $\cos A: \cos B: \cos C=$

Two roots of the equation, $a x^4+b x^3+c x^2+d x+e=0$ are positive and equal. If the product of the other two real roots is 1 , then

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $2 x^3+3 x^2-5 x-7=0$, then $\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}=$

The number of solutions of the equation $\sqrt{3 x^2+x+5}=x-3$ is

The set of all real values of $x$ for which $\frac{x^2-1}{(x-4)(x-3)} \geq 1$ is

The mean deviation about the mean for the following data is
$$ \begin{array}{llllll} \hline \text { Class Interval } & 0-2 & 2-4 & 4-6 & 6-8 & 8-10 \\ \hline \text { Frequency } & 1 & 3 & 4 & 1 & 2 \\ \hline \end{array} $$

The point $P(4,1)$ undergoes the following transformations in succession :
(i) origin is shifted to the point $(1,6)$ by translation of axes.
(ii) translation through a distance of 2 units along the positive direction of $X$-axis.
(iii) rot...

If the pair of lines $a x^2-7 x y-3 y^2=0$ and $2 x^2+x y-6 y^2=0$ have exactly one line in common and ' $a$ ' is an integer, then the equation of the pair of bisectors of the angles between the lines $a x^2-7 x y-3 y^2=0$ is

If the angle between the pair of lines $2 x^2+2 h x y+2 y^2-x+y-1=0$ is $\tan ^{-1}\left(\frac{3}{4}\right)$ and $h$ is a positive rational number, then the point of intersection of these two lines is

If $(h, k)$ is the image of the point $(2,-3)$ with respect to the line $5 x-3 y=2$, then $h+k=$

$L_1 \equiv a x-3 y+5=0$ and $L_2 \equiv 4 x-6 y+8=0$ are two parallel lines. If $p, q$ are the intercepts made by $L_1=0$ and $m, n$ are the intercepts made by $L_2=0$ on the $X$, $Y$-coordinate axes respectively, then the equation of the ...

If $A(1,2,3), B(2,3,-1), C(3,-1,-2)$ are the vertices of a $\triangle A B C$, then the direction ratios of the bisector of $\angle A B C$ are

The locus of a point at which the line joining the points $(-3,1,2),(1,-2,4)$ subtends a right angle, is

Let $A=(2,0,-1), B=(1,-2,0), C=(1,2,-1)$ and $D=(0,-1,-2)$ be four points.
If $\theta$ is the acute angle between the plane determined by $A, B, C$ and the plane determined by $A, C, D$, then $\tan \theta=$

Assertion (A) For the lines $\mathbf{r}=\mathbf{a}+t \mathbf{b}$ and $\mathbf{r}=\mathbf{p}+s \mathbf{q}$, if $(\mathbf{a}-\mathbf{p}) \cdot(\mathbf{b} \times \mathbf{q}) \neq 0$, then the two lines are coplanar.
Reason $(\mathrm{R})|(\math...

\(\tan \frac{2 \pi}{7} \cdot \tan \frac{4 \pi}{7}+\tan \frac{4 \pi}{7} \cdot \tan \frac{\pi}{7}+\tan \frac{\pi}{7} \cdot \tan \frac{2 \pi}{7}=\)

The sum of the solutions of $\cos x \sqrt{16 \sin ^2 x}=1$ in $(0,2 \pi)$ is

\(\sin \frac{\pi}{5}+\sin \frac{2 \pi}{5}+\sin \frac{3 \pi}{5}+\sin \frac{4 \pi}{5}=\)

\(\cos 13^{\circ} \sin 17^{\circ} \sin 21^{\circ} \cos 47^{\circ}=\)

In a right angled triangle, if the position vector of the vertex having the right angle is $-3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and the position vector of the mid-point of its hypotenuse is $6 \hat{\mathbf{i}}+2 \hat{...

If the position vectors of the vertices $A, B, C$ of a triangle are $3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-\hat{\mathbf{k}}, \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, 5(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})$ respec...

Let $\mathbf{a}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}$ and $\mathbf{b}$ be two perpendicular vectors in the $X O Y$-plane. A vector $\mathbf{c}$ in the same plane and having projections 1 and 2 respectively on $\mathbf{a}$ and $\mathbf{b}$ ...

If the vectors $2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}},-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $p \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ are coplanar, then the unit vector in the directi...