AP EAPCET 2025 - 21st May Evening Shift

A problem in Algebra is given to two students $A$ and $B$ whose chances of solving it are $\frac{2}{5}$ and $\frac{3}{4}$ respectively.
The probability that the problem is solved if both of them try independently is

Three dice are thrown simultaneously and the sum of the numbers appeared on them is noted. If $A$ is the event of getting a sum greater than 14 and $B$ is the event of getting a sum which is a multiple of 3 , then $P(A \cap \bar{B})+P(\bar{...

A manufacturing company of bulbs has 3 units $A, B$ and $C$ which produce $25 \%, 35 \%$ and $40 \%$ of the bulbs respectively. Out of the bulbs produced by $A, B, C$ units, $5 \%, 4 \%$ and $2 \%$ are defective, respectively. If a bulb is ...

The probability that a student gets distinction in a Mathematics test is $\frac{2}{3}$. If five such tests are conducted over a certain period of time, then the probability that he gets distinction in atleast 3 tests is

In a $\triangle A B C$ if $a+c=5 b$, then $\cot \frac{A}{2} \cot \frac{C}{2}=$

In a $\triangle A B C$, if $r_1=3, r_2=4, r_3=6$, then $b=$

In a $\triangle A B C$, if $\sin \frac{A}{2}=\frac{1}{4} \sqrt{\frac{3}{5}}, a=2, c=5$ and $b$ is an integer, then the area (in sq. units) of $\triangle A B C$ is

The roots $\alpha, \beta$ of the equation $x^2-6(k-1) x+4(k-2)=0$ are equal in magnitude but opposite in sign, if $\alpha>\beta$, then the product of the roots of the equation $2 x^2-\alpha x+6 \beta(\alpha+1)=0$

Given $f(x)=x^2-5 x+4$. Out of first 20 natural numbers, if a number $x$ is chosen at random, then the probability that the chosen $x$ satisfies the inequality $f(x)>10$ is

If $a x^2+b x+c<0 \forall x \in R$ and the expressions $c x^2+a x+b$ and $a x^2+b x+c$ have their extreme values at the same point $x$, then for the expression $c x^2+a x+b$

If $x^2-5 x+6$ is a factor of $f(x)=x^4-17 x^3+k x^2-247 x+210$, then the other quadratic factor of $f(x)$ is

If $2 \cdot 5+5 \cdot 9+8 \cdot 13+11 \cdot 17+\ldots$ to $n$ terms $=a n^3+b n^2+c n+d$, then $a-b+c-d=$

If the variance of the first $n$ natural numbers is 10 and the variance of the first $m$ even natural numbers is 16 , then $n: m=$

If the equation $3 x^2+4 y^2-x y+k=0$ is the transformed equation of $3 x^2+4 y^2-x y-5 x-7 y+2=0$ after shifting the origin to the point $(\alpha, \beta)$ by the translation of axes, then $\alpha+\beta-k=$

The equation $(2 p-3) x^2+2 p x y-y^2=0$ represents a pair of distinct lines

If $P$ is a variable point which is at a distance of 2 units. from the line $2 x-3 y+1=0$ and $\sqrt{13}$ units from the point $(5,6)$, then the equation of the locus of $P$ is

$A$ line $L$ passes through the point $P(1,2)$ and makes an angle of $60^{\circ}$ with $O X$ in the positive direction. $A$ and $B$ are two points lying on $L$ at a distance of 4 units from $P$. If $O$ is the origin, then the area of $\tria...

If the intercept of a straight line $L$ made between the straight lines $5 x-y-4=0$ and $3 x+4 y-4=0$ is bisected at the point $(1,5)$, then the equation of $L$ is

The point of intersection of the lines represented by $\mathbf{r}=(\hat{\mathbf{i}}-6 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})+\mathbf{t}(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}})$ and $\mathbf{r}=(4 \hat{\mathbf{j}}+\hat{\mathbf{k}...

If the four points $(6,2,4),(1,3,5),(1,-2,3)$ and $(6, k, 2)$ are coplanar, then $k=$

Number of solutions of the equation $\cos \theta+\cos 2 \theta-\sqrt{3}(\sin \theta+\sin 2 \theta)+1=0$ lying in the interval $(0,2 \pi)$ is

$$ \begin{aligned} \frac{1}{\sin 1^{\circ} \sin 2^{\circ}}+\frac{1}{\sin 2^{\circ} \sin 3^{\circ}}+\frac{1}{\sin 3^{\circ} \sin 4^{\circ}} & +\frac{1}{\sin 89^{\circ} \sin 90^{\circ}}= \end{aligned} $$

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

\(\cos ^3 \frac{\pi}{8} \cos \frac{3 \pi}{8}+\sin ^3 \frac{\pi}{8} \sin \frac{3 \pi}{8}=\)

$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are unit vectors. If $\mathbf{a}, \mathbf{b}$ are perpendicular vectors, $(\mathbf{a}-\mathbf{c}) \cdot(\mathbf{b}+\mathbf{c})=0$ and $\mathbf{c}=l \mathbf{a}+m \mathbf{b}+n(\mathbf{a} \times \mathbf{b})...

If the angles between the sides of the $\triangle A B C$ formed by $A(2,3,5), B(-1,3,2)$ and $C(3,5,-2)$ are $\alpha, \beta$ and $\gamma$, then $\sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma=$

$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are three vectors such that $|\mathbf{a}|=2,|\mathbf{b}|=3$, $|\mathbf{c}|=5,|\mathbf{a}+\mathbf{b}+\mathbf{c}|=\sqrt{69}$. If $(\mathbf{a} \cdot \mathbf{b})=(\mathbf{b} \cdot \mathbf{c})=\frac{\pi}{3}$,...

If the points $A, B, C, D$ with positions vectors $\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}, \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}, \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+\hat{\math...

If $O(0,0,0), A(1,2,1), B(2,1,3)$ and $C(-1,1,2)$ are the vertices of a tetrahedron, then the acute angle between its face $O A B$ and edge $B C$ is

Let the position vectors of the vertices of a $\triangle A B C$ be $\mathbf{a , b}, \mathbf{c}$. If on the plane of the triangle, $P$ is a point having position vector $\mathbf{x}$ such that $\mathbf{x} \cdot(\mathbf{c}-\mathbf{b})=\mathbf{...