TS EAMCET 2023 (Online) 14th May Morning Shift

The number of diagonals of a polygon is 35 . If $A$ and $B$ are two distinct vertices of this polygon, then the number of all those triangles formed by joining three vertices of the polygon having $A B$ as one of its sides is

If the coefficients $a$ and $b$ of a quadratic expression $x^2+a x+b$ are chosen from the sets $A=\{3,4,5\}$ and $B=\{1,2,3,4\}$ respectively, then the probability that the equation $x^2+a x+b=0$ has real roots is

A bag contains four balls. Two balls are drawn randomly and found them to be white. The probability that all the balls in the bag are white is

5 persons entered a lift cabin in the cellar of a 7 floor building apart from cellar. If each of them independently and with equal probability can leave the cabin at any floor out of the 7 floors beginning with the first, then the probabili...

A random variable $X$ has the following probability distribution
$$ \begin{array}{|c|l|l|l|l|l|l|l|l|} \hline \boldsymbol{X}=\boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) & 0.15 & 0.2...

The perimeter of a $\triangle A B C$ is 6 times the arithmetic mean of the values of the sine of its angles. If its side $B C$ is of unit length, then $\angle A=$

In $\triangle A B C$, if $a: b: c=4: 5: 6$, then the ratio of the circumradius to its inradius is

The roots of the equation $x^4+x^3-4 x^2+x+1=0$ are diminished by $h$ so that, the transformed equation does not contain $x^2$ term. If the values of such $h$ are $\alpha$ and $\beta$, then $12(\alpha-\beta)^2=$

If $x^2+2 p x-2 p+8>0$ for all real values of $x$, then the set of all possible values of $p$ is

If $R-(\alpha, \beta)$ is the range of $\frac{x+3}{(x-1)(x+2)}$, then the sum of the intercepts of the line $\alpha x+\beta y+1=0$ on the coordinate axes is

If $\sin 2 \theta$ and $\cos 2 \theta$ are solutions of $x^2+a x-c=0$, then

If $f(x)$ is a function such that $f(x+y)=f(x)+f(y)$ and $f(1)=7$, then $\sum_{r=1}^n f(r)=$

If $i=\sqrt{-1}$, then $\sum_{n=0}^{\infty}\left(\frac{i}{3}\right)^n=$

The variance of 50 observations is 7 . Suppose that each observation in this data is multiplied by 6 and then 5 is subtracted from it. Then, the variance of that new data is

Let $d$ be the distance between the parallel lines $3 x-2 y+5=0$ and $3 x-2 y+5+2 \sqrt{13}=0$.
Let $L_1 \equiv 3 x-2 y+k_1=0\left(k_1>0\right)$ and $L_2 \equiv 3 x-2 y+k_2=0\left(k_2>0\right)$ be two lines that are at the distance of $\fra...

For $l \in R$, the equation $(2 l-3) x^2+2 l x y-y^2=0$ represents a pair of distinct lines

The orthocenter of the triangle whose sides are given by $x+y+10=0, x-y-2=0$ and $2 x+y-7=0$ is

A straight line parallel to the line $y=\sqrt{3} x$ passes through $Q(2,3)$ and cuts the line $2 x+4 y-27=0$ at $P$. Then, the length of the line segment $P Q$ is

If $(h, k)$ is the image of the point $(3,-4)$ with respect to the line $2 x-3 y-5=0$ and $(l, m)$ is the foot of the perpendicular from $(h, k)$ on to the line $3 x+2 y+12=0$, then $l h+m k+1=$

If a line $a x+2 y=k$ forms a triangle of area 3 sq. units with the coordinate axis and is perpendicular to the line $2 x-3 y+7=0$, then the product of all the possible values of $k$ is

If $l, m, n$ and $a, b, c$ are direction cosines of two lines, then

If $(2,-1,3)$ is the foot of the perpendicular drawn from the origin to a plane, then the equation of that plane is

In a $\triangle A B C$, if the mid-points of sides $A B, B C$ and $C A$ are $(3,0,0),(0,4,0)$ and $(0,0,5)$ respectively, then $A B^2+B C^2+C A^2=$

If $A(1,2,3), B(3,7,-2), C(6,7,7)$ and $D(-1,0,-1)$ are points in a plane, then the vector equation of the line passing through the centroids of $\triangle A B D$ and $\triangle A C D$ is

If $\cos \theta=\frac{-3}{5}$ and $\pi<\theta<\frac{3 \pi}{2}$, then $\tan \frac{\theta}{2}+\sin \frac{\theta}{2}+2 \cos \frac{\theta}{2}=$

The quadratic equation whose roots are $\sin ^2 18^{\circ}$ and $\cos ^2 36^{\circ}$ is

Let $\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{b}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ be two vectors. If the orthogonal projection vector of $\mathbf{a}$ on $\mathbf{b}$ is $\mathbf{...

If $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}},-12 \hat{\mathbf{i}}-\hat{\mathbf{j}}-3 \hat{\mathbf{k}},-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}$ and $\lambda \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ ...

If $\mathbf{a}+\mathbf{b}+\mathbf{c}=0,|\mathbf{a}|=3,|\mathbf{b}|=5,|\mathbf{c}|=7$, then the angle between $\mathbf{a}$ and $\mathbf{b}$ is

If $|\mathbf{a}|=4,|\mathbf{b}|=5$ and $|\mathbf{a}-\mathbf{b}|=3$ and $\theta$ is the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$, then $\cot ^2 \theta=$