AP EAPCET 2023 - 15th May Evening Shift

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96 is

If $A$ and $B$ are events of a random experiment with $P(A)=0.5, P(B)=0.4$ and $P(A \cap B)=0.3$, then the probability that neither $A$ nor $B$ occurs is

In $\triangle A B C$, if $a \cos ^2 \frac{C}{2}+c \cos ^2 \frac{A}{2}=\frac{3 b}{2}$, then $a+c: b=$

In $\triangle A B C$, if $\sin ^2 B=\sin C$ and $3 \cos ^2 B=2 \cos ^2 C$, then $\triangle A B C$ is

If $\triangle A B C$ is a right angled isosceles triangle and $\angle C=90^{\circ}$, then $r: r_3=$

If one root of the equation $a x^3+b x+c=0$ is twice another root, then

$\alpha$ and $\beta$ are the roots of the equation $x^2-a x+b=0$. If $\alpha^2+\beta^2$ and $\alpha^3+\beta^3$ are the roots of the equation $A x^2+B x+C=0$, then $C=$

If -1 is a twice repeated root of the equation $a x^3+b x^2+c x+1=0$, then

The minimum value of $f(x)=\frac{x^2-2 x+3}{x^2-4 x+7}$ is

The mean of 5 observations is 4.4 and their variance is 8.24. If three of those observations are 1,2 and 6 , then the other two observations are

Assertion (A) The difference of the slopes of the lines represented by $y^2-2 x y \sec ^2 \alpha+\left(3+\tan ^2 \alpha\right)$ $\left(-1+\tan ^2 \alpha\right) x^2=0$ is 4
Reason (R) The difference of the slopes represented by $a x^2+2 h x ...

The lines $p\left(p^2+1\right) x-y+q=0$ and
$\left(p^2+1\right)^2 x+\left(p^2+1\right) y+2 q=0$ are perpendicular to a line $L$ for

If $2 x^2-3 x y+y^2=0$ represents two sides of a triangle and $x+y-1=0$ is its third side, then the distance between the orthocentre and the circumcentre of that triangle is

If a line is moving between the coordinate axes such that the sum of the intercepts made by it on the coordinate axes is always 12, then the equation of that line which forms a triangle of maximum area with the coordinate axes is

Let $P=(-1,0), Q=(0,0)$ and $R=(3,3 \sqrt{3})$ be three Points. Then, the equation of the bisector of the $\angle P Q R$ is

The diagonals $A C$ and $B D$ of a rhombus $A B C D$ intersect at the point $(3,4)$. If $B D=2 \sqrt{2}, A=(1,2), B=(\alpha, \beta)$, $C(5,6), D=(\gamma, \delta)$ and $\alpha<\delta<\gamma<\beta$, then $\beta+\gamma-\delta=$

Let $A B C D$ be a tetrahedron in which the coordinates of each of its vertices are in arithmetic progression with same common difference. If the centroid $G$ of the tetrahedron is $(2,3, k)$, then the distance of $G$ from the origin is

If $S$ is the set of all real values of ' $a$ ' such that a plane passing through the points $\left(-a^2, 1,1\right),\left(1,-a^2, 1\right)$ and $\left(1,1,-a^2\right)$ also passes through the point $(-1,-1,1)$, then $S=$

The distance of a point $\mathbf{a}$ from the plane $\mathbf{r} \cdot \mathbf{m}=q$ is given by $\frac{|\mathbf{a} \cdot \mathbf{m}-9|}{|\mathbf{m}|}$. If the distance of the point $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ fr...

If $A(3,-1,11), B(0,2,3)$ and $C(4,8,11)$ are three points, then the coordinates of the foot of the perpendicular drawn from the point $A$ to the line joining the points $B$ and $C$ is

If $\cot x \cot y=a$ and $x+y=\frac{\pi}{6}$, then the quadratic equation satisfying $\cot x$ and $\cot y$ is

If $\cos h x=\frac{5}{4}$, then $\tan h 3 x=$

The range of $\frac{1}{\sin ^2 x+3 \sin x \cos x+5 \cos ^2 x}$ is

If $\tan A+\tan B=x$ and $\cot A+\cot B=y$, then $\tan (A+B)=$

\(\frac{1}{\cos 290^{\circ}}+\frac{1}{\sqrt{3} \sin 250^{\circ}}=\)

$$ \begin{aligned} & \text { If }\left[1-\cos \left(\frac{\pi}{2}+\alpha\right)+\sin \left(\frac{3 \pi}{2}+\alpha\right)\right]^2 +\left[1-\sin \left(\frac{3 \pi}{2}-\alpha\right)-\cos \left(\frac{3 \pi}{2}+\alpha\right)\right]^2 \\ & =a+b ...

Let $(\mathbf{a}, \mathbf{b})$ denote the angle between vectors $\mathbf{a}$ and $\mathbf{b}$. If $\mathbf{a}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}, \mathbf{a} \cdot \mathbf{b}=4$ and $(\mathbf{a}, \mathbf{b})=\cos ^{-1}\...

If $7 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ is the position vector of the vertex $A$ of a tetrahedron $A B C D$ and $-\hat{\mathbf{i}}+4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ is the position vector of the centroid of the $\...

Let $\mathbf{O A}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{O B}=-2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}$ be the position vectors of two points $A$ and $B$. If $C$ is a point on the bisector $\an...

If $\mathbf{a}$ and $\mathbf{b}$ are two vectors such that $|\mathbf{a}|=|\mathbf{b}|=\sqrt{14}$ and $\mathbf{a} \cdot \mathbf{b}=-7$, then $\frac{|\mathbf{a} \times \mathbf{b}|}{|\mathbf{a} \cdot \mathbf{b}|}=$