TS EAMCET 2022 (Online) 20th July Evening Shift

In $\triangle A B C$, if $a=7, b=8, \tan C=\frac{3 \sqrt{5}}{2}$ and $C$ is an acute angle, then $c=$

Let $a$ be a common root of the equations $x^3-2 x-25 \lambda=0,3 x^3-8 x-\frac{175}{3} \lambda=0$ and $\lambda>0$. Then, $\lambda=$

If the sum of two roots of the equation $x^3-7 p x^2+5 q x-6 r=0$ is zero, then

Statement I The set of solutions of $|x|^2-4|x|+3<0$ is the interval $(-3,3)$
Statement II If $x<3$ or $x>5$, then $x^2-8 x+15>0$
Which of the above statements is (are) true?

The roots of a cubic equation $f(x)=0$ are diminished by $\frac{-3}{2}$ so, as to remove the term containing $x^2$ and the transformed equation is $8 x^3-54 x-78=0$. Then, the equation $f(x)=0$ is

If $\alpha$ and $\beta$ are the irrational roots of the equation $3 p^2 x^3+p x^2+q x+3=0$ when $p=1$ and $q=-7$, then $|\alpha-\beta|=$

If $6 x-x^2+12$ attains its extreme value $\beta$ at $x=\alpha$, then $\beta=$

Statement I The range of the ungrouped data does not change even if certain intermediate observations are removed
Statement II The value of the mean deviation of an ungrouped data about the median is always less than or equal to the value o...

By rotating the axes through an angle of $30^{\circ}$ in the anti-clockwise direction about the origin, the equation $4 x^2+12 x y+9 y^2+6 x+9 y+2=0$ becomes $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ becomes, then

If $\theta$ is the acute angle between the pair of lines $12 x^2+2 h x y+7 y^2=0$ and $\tan \theta=\frac{8}{19}$, then $h=$

The line $x+2 y-c=0$ meets the curve $x^2+y^2-3 x-6 y+3=0$ at two points $P$ and $Q$ and $\angle P O Q=\frac{\pi}{2}$, where $O$ is the origin. Then, $2 c^2-15 c=$

If the lengths of the perpendiculars drawn from a point $(a, b)$ to the lines $2 x+3 y+4=0$ and $3 x-2 y+4=0$ are same, then the point $(a, b)$ lies on the line

If $3 x+6 y+2=0, x+y+1=0,2 x-y+3=0$ are three given lines, then the point $\left(\frac{-4}{3}, \frac{1}{3}\right)$ is

If the lines $L_1 \equiv x-2 y+3=0, L_2 \equiv 2 x+y+1=0$ and $L_3 \equiv 3 x+y+c=0$ are concurrent and $\theta$ is the acute angle between the lines $L_1=0$ and $L_3=0$, then $\tan \theta=$

The number of real values of $\alpha$ for which the pair of lines represented by $\left(\alpha^2+12|\alpha|\right) x^2+6 x y+(18-21|\alpha|) y^2=0$ are at right angles to each other, is

In an isosceles triangle the ends of its base are $(2 a, 0),(0, a)$ and one of its two other sides is a horizontal line other than $X$-axis. If the third vertex is $\left(x_1, y_1\right)$, then $x_1+y_1=$

$A(-4,0)$ and $B(4,0)$ are two fixed points. $C$ and $D$ are two points on $Y$ - axis such that $C D=4$ and $C$ is a point below $D$. Then, the locus of the point of intersection of the lines $A C$ and $B D$ is

Let $\pi_1$ be a plane passing through the point $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and perpendicular to the vector $-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$. Let the line $L$ passing through the points $3 \hat{\mathbf{i}}-2 ...

If the d.r.'s of two lines are connected by the relations $a-b+c=0, a^2-b^2+2 c^2=0$ and $\theta$ is the angle between these lines, then $\cos \theta=$

If $\mathbf{r}=(2-\lambda+\mu) \hat{\mathbf{i}}+(1-\mu) \hat{\mathbf{j}}+(2-3 \lambda+2 \mu) \hat{\mathbf{k}}$ is the vector equation of a plane, then the equivalent cartesian equation of the plane is

If $l, m$ and $n$ are the d.c.'s of a normal to the plane passing through the points $(0,1,2)$, $(3,0,2)$ and $(4,5,0)$, then $|I|+|m|+|n|=$

Let $A=(1,2,0), B=(2,0,-1), C=(0,-2,3)$ and $D=(-1,2,-3)$ be four points in the space. Let $G_1$ be the centroid of $\triangle A B C$ and $G_2$ be the centroid of tetrahedron $A B C D$. If $P$ divides, $G_1 G_2$ in the ratio $4: 3$ internal...

If $\frac{5 \sinh 2 x}{7+6 \cosh 2 x}=\frac{3}{2}$, then $3 \tanh ^2 x+20 \tanh x=$

If $a \tan \alpha+b \tan \beta=(a+b) \tan \left(\frac{\alpha+\beta}{2}\right)$ and $\alpha-\beta \neq 2 n \pi$ then $\frac{\cos \beta}{\cos \alpha}=$

If $\sin \theta-\cos \theta=\frac{1}{\sqrt{3}}$, then $\sin (2 \theta)+\cos (4 \theta)+\sin (6 \theta)=$

If $\sin A=\frac{-7}{25}, \cos B=\frac{8}{17}, A$ does not lie in the 3rd quadrant and $B$ does not lie in the 1st quadrant, then $8 \tan A-5 \cot B=$

If $\mathbf{a}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \mathbf{x}=\left(\frac{\mathbf{a b}}{|\mathbf{b}|^2}\right) \mathbf{b}, \mathbf{y}=\left(\frac{\mathbf{a b...

If $\alpha, \beta$ and $\gamma$ are real numebrs such that
$$ \begin{aligned} & \left(\frac{7}{3}+\beta\right) \hat{\mathbf{i}}-\hat{\mathbf{j}}+(\alpha+\gamma) \hat{\mathbf{k}} \\ & =\frac{5}{3}(\alpha \hat{\mathbf{i}}+\hat{\mathbf{j}}-\ha...

Three non-coplanar vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are the coterminous edges of a parallelopiped. If $\mathbf{a}$ and $\mathbf{b}$ determine the base of the parallelopiped, then its height is

Let $A B C$ be a triangle and $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ be the position vectors of $A, B$ and $C$, respectively. Let $D$ divides $B C$ in the ratio $3: 1$ internally and $E$ divides $A D$ in the ratio $4: 1$ internally. Let ...