AP EAPCET 2023 - 15th May Morning Shift

If four dice are thrown simultaneously, then the probability that none of the dice shows the number 1 on its face, is

The range of a random variable $X$ is $\{0,1,2\}$. If $P(X=0)=3 C^3, P(X=1)=4 C-10 C^2$ and $P(X=2)=5 C-1$, then the value of $C$ is

In a game, a pair of dice is rolled 24 times. If a person whis the game by not getting 6 on both the dice in any one of the 24 rolls, then the probability that a person wins the game is

In $\triangle A B C$, if $\frac{1}{r_1}, \frac{1}{r_2}$ and $\frac{1}{r_3}$ are in arithmetic progression, then $r_2: r=$

If $P_1, P_2$ and $P_3$ are the lengths of the altitudes drawn from the vertices $A, B$ and $C$ of $\triangle A B C$ respectively, then $\frac{\cos A}{P_1}+\frac{\cos B}{P_2}+\frac{\cos C}{P_3}=$

In $\triangle A B C$, if $\cot \frac{A}{2}: \cot \frac{B}{2}: \cot \frac{C}{2}=3: 7: 9$, then $a: b: c=$

If $(3+2 \sqrt{2})^{x^2-4}+(3-2 \sqrt{2})^{x^2-4}=6$, then $x^4+x^2+5=$

If the equation $x^4+a x^3+b x^2+c x+d=0$ has three equal roots, then that root is

If the values of $k$ for which the euqation $x^2+2(k+2) x+6 k+7=0$ has equal roots are $k_1$ and $k_2$, then $k_1^2+k_2^2=$

If -1 is a twice repeated root of the equation $a\left(x^3+x^2\right)+b x+c=0$, then $a: b: c=$

The variance of 20 observations is 5. If each one of the observations is multiplied by 2 , then the variance of the resulting observations is

If the area of the triangle formed by the lines $y=x+c$ and $2 x^2+5 x y+3 y^2=0$ is $\frac{1}{20}$ sq units, then $c=$

If the lines given by $\left(x^2+y^2\right) \sin ^2 \alpha=(x \cos \alpha-y \sin \alpha)^2$ are perpendicular to each other, then $\sin ^2 \alpha+\tan ^2 \alpha=$

In $\triangle A B C$ the coordinates of the vertex $A$ are $(-3,1)$. If the equation of the median through $B$ is $2 x+y-3=0$ and the equation of the bisector of $\angle C$ is $7 x-4 y-1=0$, then the equation of the side $B C$ is

The combined equation of the lines passing through the point $(3,4)$ and each making an angle $45^{\circ}$ with the line $x+y+1=0$ is-

The equal sides of an isosceles triangle are given by equations $7 x-y+3=0$ and $x+y-3=0$. If the slope $m$ of the third side is an integer, then $m=$

If the line passing through the points $(5,1, a)$ and $(3, b, 1)$ crosses the $Y Z$-plane at the point $\left(0, \frac{17}{2}, \frac{-13}{2}\right)$, then $a+b=$

$\triangle A B C$ is formed by $A(1,8,4), B(0,-11,4)$ and $C(2,-3,1)$. If $D$ is the foot of the perpendicular drawn from $A$ to $B C$, then the coordinates of $D$ are

Let $\pi_1$ be the plane determined by the vectors $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$ and $3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$. Let $\pi_2$ be the plane determined by the vectors $\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $3 \hat{\math...

The distance between two parallel planes $a x+b y+c z+d_1=0, a x+b y+c z+d_2=0$ is given by $\frac{\left|d_1-d_2\right|}{\sqrt{a^2+b^2+c^2}}$. If the plane $2 x-y+2 z+3=0$ has the distances $\frac{1}{3}$ and $\frac{2}{3}$ units from the pla...

If $A=\frac{\pi}{24}$, then $\frac{\cos A+\cos 3 A+\cos 5 A+\cos 7 A}{\sin A+\sin 3 A+\sin 5 A+\sin 7 A}=$

If $(1+\sqrt{1+a}) \sin \alpha=(1+\sqrt{1-a}) \cos \alpha$ and $0

$\sin 21^{\circ} \cos 9^{\circ}-\cos 84^{\circ} \cos 6^{\circ}=$

If $\cos \alpha+\cos \beta=a$ and $\sin \alpha+\sin \beta=b$, then match the items given in List-A with those of their values in List-B
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If $\alpha=\log _e(2+\sqrt{3})$, then $\frac{\cosh \alpha}{1-\tanh \alpha}+\frac{\sinh \alpha}{1-\operatorname{coth} \alpha}=$

If $\sec (\theta+\alpha), \sec \theta$ and $\sec (\theta-\alpha)$ are in arithmetic progression, then $\sin ^2 \theta=$

a and b are two vectors such that a is not parallel to b. If $\mathbf{p}=(x+2 y+3) \mathbf{a}+(5 x-y+2) \mathbf{b}$ and $\mathbf{q}=(2 x+3 y+5) \mathbf{a}+(x-5 y-2) \mathbf{b}$ are two vectors such that $\mathbf{p}=2 \mathbf{q}$, then $x-2 ...

Let $\mathbf{O A}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{O B}=\hat{\mathbf{i}}-4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ and $\mathbf{O C}=-3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ be the position vector...

$3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-\hat{\mathbf{k}},-2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $-\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ are the position vectors of the vertices $A, B$ and $C$ of a $\tr...

If $\mathbf{a}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}, \mathbf{b}=3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ and $\mathbf{c}=5 \hat{\mathbf{i}}+4 \hat{\mathbf{k}}$ are three vectors, then a vector which is perpendicular to $\mathbf{a}$ and $\mat...