TS EAMCET 2022 (Online) 20th July Morning Shift

$y=x^2$ is the given curve. Imagine that this curve is dragged along the positive $X$-axis to a distance of ' $a$ ' units. If the acute angle between the curves at two positions is $\theta$, then

The approximate value of $\sqrt[3]{28}$ rounded up to 3 decimal places is

If $x$ and $y$ are two positive integers such that $x+2 y=10$ and $x^2 y^3$ is maximum, then $x^2+2 y^3=$

$\frac{1}{8}-\frac{7}{8 \cdot 12}+\frac{7 \cdot 10}{8 \cdot 12 \cdot 16}-\ldots=$

The straight line $x+2 y=1$ cuts the $X$-axis at $A$ and $Y$-axis at $B, A$ circle is drawn through $A, B$ and the origin. The sum of the perpendicular distances from $A$ and $B$ on to the tangent drawn at origin to the circle $S$ is

The equation of the direct common tangent of the circles $x^2+y^2-6 x-4 y-23=0$ and $x^2+y^2+2 x+2 y+1=0$ is

The length of the common chord of the two circles $x^2+y^2-4 x-8 y+4=0$ and $x^2+y^2-8 x-12 y+16=0$ is

Let the centre of the circle $S=0$ lie on the line $x+y-5=0$ and also lie in the first quadrant. If this circle touches both the lines $x-2=0$ and $y-5=0$, then the area of the circle is

$A\left(x_1, y_1\right)$ is the internal centre of similitude and $B\left(x_2, y_2\right)$ is the external centre of similitude of two circles $C_1$ and $C_2$ whose centes are $P(\alpha, \beta)$ and $Q(\gamma, \delta)$, respectively. If $P ...

Let $P$ and $Q$ be two external points of the circle $S=x^2+y^2-a^2=0$. Let the chord of contact of the point $P$ with respect to the circle $S=0$ passes through $Q$. If $l_1$ and $l_2$ are the lengths of the tangents drawn from $P$ and $Q$...

If $|x+i y|=\sqrt{x^2+y^2}$, then $\left|(1-\sqrt{3} i)^9+(\sqrt{3}+i)^9\right|=$

$\{x \in[0,2 \pi] / \sin x+i \cos 2 x$ and $\cos x-i \sin 2 x$ are conjugate to each other} $=$

If $1, \omega, \omega^2$ are the cube roots of unity and $1, \alpha, \alpha^2, \alpha^3$ are the fourth roots of unity in usual notation, then $\alpha+\alpha \omega-\alpha^3 \omega^2=$

If $f(x)=\left|\begin{array}{ccc}2 \cos ^2 x & \sin 2 x & \sin x \\ \sin 2 x & 2 \sin ^2 x & -\cos x \\ \sin x & -\cos x & 0\end{array}\right|$, then
\(\left.\int_0^{\pi / 4}|2| f(x) \mid+5 f^{\prime}(x)\right) d x=\)

Given that $\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=1}^{n p} f\left(\frac{r}{n}\right)=\int_0^p f(x) d x$. If $f: R \rightarrow R$ is defined by $f(x)=x^2+2$, then

$$ \lim _{n \rightarrow \infty} \frac{3}{n}\left[f\left(\frac{7}{...

Assertion (A) The degree of the differential equation $y^{\prime \prime}+2 x y^{\prime}+\log _e\left(\frac{d y}{d x}\right)=0$ is 2 .
Reason (R) The degree of a differential equation is the highest degree of the highest order derivative occ...

Let $S$ be the family of curves given by the general solution of the differential equation $\frac{y^2 e^{-1 / y}}{\sqrt{x}} d x-2 \sec \sqrt{x} d y=0$. Then, the equation of the curve belonging to $S$ and passing through $\left(\pi^2, 1\rig...

The number of arbitrary constants that appear in the general solution of the differential equation $\left(\frac{d^4 y}{d x^4}+\frac{d^2 y}{d x^2}\right)^{3 / 2}=5 \frac{d^3 y}{d x^3}$ is

If $f(x)=\frac{e^{-x} \sin x}{\log _e x}$ and $f^{\prime}(x)=f(x) \cdot g(x)$, then $g^{\prime}(e)=$

If $y=\frac{e^{\sin x}+\sinh ^3 x}{\cosh x-\tan x}$, then $y^{\prime}(0)=$

The length of the latusrectum of an ellipse is 6 units and the distance between a focus and its nearest vertex on the major-axis is $5 / 3$ units. If $e$ is the eccentricity of this ellipse, then $e$ satisfies the equation

If the line $2 x-3 y+4=0$ cuts the ellipse $x=3 \cos \theta, y=5 \sin \theta$ in $A$ and $B$ and $(\alpha, \beta)$ is the mid-point of $A B$, then $3 \beta-2 \alpha=$

The range of the real valued function $f(x)=|x-2|+|x-3|$ is

The domain of the real valued function $f(x)=\sqrt{\frac{2 x^2-7 x+5}{3 x^2-5 x-2}}$ is

Let $e_1$ be the eccentricity of a hyperbola for which distance between its focii is 2 times the distance between its directrices and $e_2$ be the eccentricity of another hyperbola for which the length of its transverse axis is twice the le...

Assertion (A) The distance between the points $p\left(\frac{\pi}{4}\right)$ and $p\left(\frac{\pi}{3}\right)$ on the hyperbola $9 x^2+16 y^2=9$ is

\(\frac{1}{2 \sqrt{2}} \sqrt{66-33 \sqrt{2}-9 \sqrt{3}}\)
Reason (R) $x=a \cosh t, y=b \s...

If $\frac{3 \pi}{4}

If $\tan \alpha=\frac{4}{3}$, then $\int \frac{1}{3 \cos x-4 \sin x} d x=$

If $\frac{3 x^2+a x+3}{(2 x+3)\left(x^2+2\right)}=\frac{3}{2 x+3}+\frac{B x+C}{x^2+2}$, then $a(B+C)=$

If $x \neq(2 n+1) \frac{\pi}{2}$, then $\int \frac{\cos ^3 x}{(1+\sin x)^4} d x=$

If $\frac{2 x^2-3 x+5}{(x-7)^3}=\frac{A}{x-7}+\frac{B}{(x-7)^2}+\frac{C}{(x-7)^3}$, then $2 A-3 B+C=$

Let $f(x)$ be a differentiable function such that $f(0)=0$ and $f^{\prime}(0)=20$. For $x \in\left(0, \frac{\pi}{2}\right]$, if
$A(x)=2 f(x) \operatorname{cosec} 4 x+4 f(x)\left(\cos ^2 x+1\right)-4 \cos ^2 x$, then $\mathop {\lim }\limits_...

Let [ $x$ ] denote the greatest integer less than or equal to $x$ and $f(x)=2 x-[2 x]$. If $\mathop {\lim }\limits_{x \to {2^ - }} f(x)=l_1$ and $\mathop {\lim }\limits_{x \to {2^ + }} f(x)=l_2$, then $l_1+l_2=$

\(\mathop {\lim }\limits_{x \to 0} \frac{\left(2^x-1\right)(1+\sin x)^{\frac{2}{\sin x}}}{\log (1+2 x)}=\)

$A=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 3 & 2\end{array}\right]$, then $\left(A+A^T\right)\left(A-A^T\right)=$

If $f(x)=\left|\begin{array}{ccc}x & x+1 & x+3 \\ x+2 & x+4 & x+7 \\ x+6 & x+9 & x+13\end{array}\right|$, then $f(5)=$

Let $A=\left[\begin{array}{lll}2 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 2\end{array}\right]$. If $A^{-1}=\alpha A^2+\beta A+\gamma I$, where $\alpha, \beta$ and $\gamma$ are real numbers and $I$ is a $3 \times 3$ identity matrix, then $17 \alpha+5...

For a system of simultaneous linear equations, if $A X=\left[\begin{array}{l}1 \\ 1 \\ 2\end{array}\right], \operatorname{Adj} A=\left[\begin{array}{ccc}1 & -1 & -1 \\ 1 & 1 & -1 \\ 1 & 1 & 1\end{array}\right]$ and $\operatorname{det} A>0$,...

If the focal chord drawn through the point $(1,2)$ to the parabola $y^2=8 x$ meets this parabola in $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$, then $x_1+x_2=$

If $\left(2 t^2, 4 t\right)$ is a point on the parabola $y^2=8 x$ such that its focal distance is 3 , then $t=$

Let $a, b, c \in N$ and $a+b+c=5$. Let $L, M$ be the least and greatest values of $2^a 3^b 5^c$, respectively. Then $M-L=$

The number of positive divisors of 360 which are multiples of 3 is

The probabilities of two persons to hit a target are $1 / 4$ and $1 / 5$ respectively. The probability that the target is being hit when both of them attempt independently is

The probability of getting a success in a trail is five times that of a failure. The probability of getting atmost one success in 5 trails, is

If the probability distribution of a random variable $X$ is given by
$$ \begin{array}{|c|c|c|c|c|c|c|} \hline X=x & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline P(X=x) & 0 & k & 2 k & 5 k^2 & 2 k^2 & 3 k \\ \hline \end{array} $$
then the mean of $X$ is

When 3 dice are thrown at a time, the sum of the numbers appeared on 3 dice were found to be 15 . Then, the probability that the number 5 does not appear on any one of the dice is

A bag contains 9 identical black balls numbered 1 to 9 . and 4 identical white balls numbered 1 to 4 . If 3 balls are drawn at a time randomly from that bag, then the probability of getting atleast one white ball is

In a $\triangle A B C$, if $a=3, b=7$ and $c=8$, then $\sin \frac{B}{2} \tan \frac{C-A}{2}=$

In $\triangle A B C$, if $A=\frac{\pi}{3}$ and $B=\frac{\pi}{4}$, then $\frac{a^2-b^2}{c^2}=$

If $a, b$ and $c$ are the sides of $a \triangle A B C$ and $\left|\begin{array}{lll}b & 1 & a \\ a & 1 & c \\ c & 1 & b\end{array}\right|=0$, then $2(\cos A+\cos B+\cos C)=$