AP EAPCET 2025 - 23rd May Morning Shift

The probability distribution of a discrete random variable $X$ is given below
$$ \begin{array}{lllll} \hline X=x & -1 & 0 & 1 & 2 \\ \hline P(X=x) & \frac{1}{3} & \frac{1}{6} & \frac{1}{6} & \frac{1}{3} \\ \hline \end{array} $$
Then, the va...

If the average number of accidents occurring at a particular junction on a highway in a week is 5 , then the probability that atmost one accident occurs in a particular week is

If 3 squares are chosen at random from the 64 squares of a chess board, then the probability that all of them lie along the same diagonal line is

In $\triangle A B C$, if $a: b: c=4: 5: 6$, then $\frac{\cos A+3 \cos C}{\cos B}=$

In $\triangle A B C$, if $a=6, b=8$ and $c=10$, then $\frac{2 r_2 r_3}{r r_1}=$

In $\triangle A B C$ if $\cos A \cos B+\sin A \sin B \sin C=1$, then $\sin A+\sin B+\sin C=$

If $\alpha, \beta$ are the roots of the equation $x^2+b x+c=0$ satisfying the conditions $\alpha+\beta=5$ and $\alpha^3+\beta^3=60$, then $3 c+2=$

If $\alpha, \beta, \gamma$ are the roots of the equation,
$$ \begin{aligned} & x^3+a x^2+b x+c=0, \text { then }(\alpha+\beta-2 \gamma) \\ & (\beta+\gamma-2 \alpha)(\gamma+\alpha-2 \beta)= \end{aligned} $$

If the sum of two roots of the equation $x^4+2 x^3-7 x^2-8 x+12=0$ is zero, then the sum of the squares of the other two roots is

The sum of all integers between 1 and 100 (both inclusive) which are divisible by 5 or 13 is

If $x>\sqrt{3}$ and $\frac{x^2+1}{\left(x^2+2\right)\left(x^2+3\right)}$ is expanded in terms of powers of $x$, then the coefficient of $x^{-8}$ is

If $t_n=\frac{1}{4}(n+2)(n+3), n \in N$, then which one of the following is true?
Assertion (A) $\frac{1}{t_1}+\frac{1}{t_2}+\ldots+\frac{1}{t_{2003}}=\frac{2003}{3009}$
Reason (R) $\frac{1}{t_1}+\frac{1}{t_2}+\ldots+\frac{1}{t_n}=\frac{4 n...

The mean and variance of the observations $x_1, x_2, x_3 \ldots x_{15}$ are respectively 2 and 4 . If the mean and variance of the observations $y_1, y_2 \ldots, y_{10}$ are respectively 2 and 5 , then the variance of the observations $x_1,...

If the perpendicular distances from the points $(2,3)$, $(4, a)$ and $(\alpha, \beta)$ on to the line $3 x+4 y-3=0$ are equal and $4 \alpha-3 \beta+1=0$, then sum of all possible values of $a, \alpha$ and $\beta$ is

Let $A(5,4)$ and $B(5,-4)$ be two points.
If $P$ is a point in the coordinate plane such that $\sqrt{A P B}=\frac{\pi}{4}$, then the point $P$ lies on the curve

The equation of the base of an equilateral triangle is $x+y=2$ and its opposite vertex is $(2,1)$. If $m_1, m_2$ are the slopes of the other two sides and the length of its side is $a$, then $\left|m_1-m_2\right|+a \sqrt{2}=$

The triangle formed by the lines $2 x^2+x y-6 y^2=0$ and $x+y-1=0$ is

If $\left(\frac{2}{3}, 0\right)$ is the centroid of the triangle formed by the lines $4 x^2-y^2=0$ and $l x+m y+n=0$, then, $l+m+n=$

A plane $\pi$ given by $a x+b y+11 z+d=0$ is perpendicular to the planes $2 x-3 y+z=4$, $3 x+y-z=5$ and the perpendicular distance from the origin to the plane $\pi$ is $\sqrt{6}$ units. If all the intercepts made by the plane $\pi$ on the ...

If $A(2,-1,1), B(2,5,1)$ and $C(0,-2,3)$ are the vertices of a triangle. If $D$ is the point of intersection of the side $B C$ and the internal angular bisector of angle $A$, then $A D=$

The number of solutions of the equation $\sec x \cdot \cos 5 x+1=0$ in the interval $[0,2 \pi]$ is

If $\sin x-\sin y=\frac{27}{65}$ and $\cos x-\cos y=\frac{-21}{65}$, then $\sin (x+y)=$

If $A$ and $B$ are positive acute angles satisfying $3 \cos ^2 A+2 \cos ^2 B=4$ and $\frac{3 \sin A}{\sin B}=\frac{2 \cos B}{\cos A}$, then $A+2 B=$

If $\alpha$ is the maximum value and $\beta$ is the minimum value of $\cos ^2 \frac{x}{4}+\sin \frac{x}{4}, x \in R$, then $\alpha-\beta=$

If the vectors $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+\hat{\mathbf{k}}, 4 \hat{\mathbf{i}}+p \hat{\mathbf{j}}+\hat{\mathbf{k}}$ are coplanar, then $p=$

A unit vector that is perpendicular to the vector $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and coplanar with the vectors $\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\m...

A line segment $P Q$ has the length 63 and direction ratios $(3,-2,6)$. If this line makes an obtuse angle with $X$-axis, then the components of the vector $\mathbf{P Q}$ are

If $\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}},-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{j}}+2 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ are the position vectors of four poin...

If the magnitudes of $\mathbf{a}, \mathbf{b}$ and $\mathbf{a}+\mathbf{b}$ are respectively 3,4 and 5 , then the magnitude of $\mathbf{a}-\mathbf{b}$ is

If the vectors $2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+l \hat{\mathbf{k}},-3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-4 l \hat{\mathbf{k}}$ and $\hat{\mathbf{i}}-\hat{\mathbf{j}}+3 / \hat{\mathbf{k}}$ form a right-angled triangle for a positive v...