TS EAMCET 2020 (Online) 14th September Morning Shift

If the curves $a x^2+b y^2=1$ and $c x^2+d y^2=1$ intersect orthogonally, then $\frac{b-a}{d-c}=$

A vessel in the shape of an inverted cone of height 10 ft and semi vertical angle $30^{\circ}$ is full of water. Due to a hole at the vertex, the slant height of the water in the vessel is decreasing at a constant rate of $\frac{1}{\sqrt{3}...

The area (in sq. units) of the triangle formed by the tangent and normal drawn to the curve $\left(\frac{x}{3}\right)^n+\left(\frac{y}{4}\right)^n=2$ at $(3,4)$ and $x$-axis is

If the area lying in the first quadrant and bounded by the circle $x^2+y^2-4 x=0$, the parabola $y^2=x$ and the $X$-axis is $A$, then $6 A-9 \sqrt{3}=$

$p, q$ are two prime numbers. For $n=p q$, if the expansion $\left(\sqrt[4]{x^{-5}}+2 \sqrt[5]{x^5}\right)^n$ contains non-zero coefficient of $x^{-n}$ and $x^0$, then the least value of such $n$ is

The binomial expansion $(7+3 x)^{-2 / 5}$ is valid for all $x$ in the interval $\left(\frac{-7}{3}, \frac{7}{3}\right)$ and if the 4 th term of its expansion is $k x^3$, then $\left(7^{12 / 5} k\right)=$

If the straight lines $3 x-4 y+4=0$ and $6 x-8 y-7=0$ are the tangents to the same circle, then the area of that circle (in square units) is

In the List-I each item contains equations of two circles, List-II contains the number of common tangents for each pair of circles given in List-I. Match the items of List-I with those of the items of List-II


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$\left(0, \frac{3}{4}\right)$ is the radical centre of the circles $S_1: x^2+y^2-2 x+6 y=0, S_2: x^2+y^2+2 g x-2 y+6=0$ and $S_3: x^2+y^2-12 x+2 f y+3=0$. If $S_2$ and $S_3$ intersect orthogonally, then $(g, f)=$

For the circles $(x-a)^2+y^2=a^2$ and $x^2+(y-a)^2=a^2$, where $a>0$, which one of the following is not true?

If the four points $A, B, C, D$ in the Argand plane represented respectively by the complex numbers $2+i, 4+3 i, 2+5 i, 3 i$ lie on a circle, then the centre of the circle is

The centre and radius of the circumcircle of the triangle formed by the lines $2 x+3 y=10, y=x$ and the $X$-axis are respectively.

$z_1, z_2$ are two fixed points on the Argand plane. If $z$ is a complex number such that $\left|z-z_1\right|+\left|z-z_2\right|=\lambda$, then the locus of $z$ is

The roots of the equation $(x-1)^5=32(x+1)^5$ are

If $\omega$ is a non-real cube root of unity and $x=\omega^2-\omega-3$, then the value of $x^4+6 x^3+10 x^2-12 x-19$ is

Sum of the modulii of the complex roots of the equation $\left(x^2+\frac{1}{x^2}\right)-5\left(x+\frac{1}{x}\right)+6=0$ is

Assertion (A) If $a_1, a_2, \ldots, a_n$ are the $n$ distinct roots of the equation $x^n-2=0$, then $1+\left(1-a_1\right)\left(1-a_2\right) \ldots \left(1-a_{n-1}\right)\left(1-a_n\right)=0$
Reason (R) If $\alpha_1, \alpha_2, \ldots, \alpha...

$$ \begin{array}{r}\mathop {\lim }\limits_{n \to \infty }\left[\frac{n^{3 / 2}}{n^{5 / 2}}-\frac{n^{1 / 2}}{n^{3 / 2}}+\frac{n^{3 / 2}}{(n+2)^{5 / 2}}-\frac{n^{1 / 2}}{(n+3)^{3 / 2}}\right. \\ +\frac{n^{3 / 2}}{(n+4)^{5 / 2}}-\frac{n^{1 / 2...

\(\lim _{n \rightarrow \infty}\left[\frac{n+3}{n^2+1^2}+\frac{n+6}{n^2+2^2}+\frac{n+9}{n^2+3^2}+\ldots+\frac{2}{n}\right]=\)

If the order and degree of the differential equation corresponding to the family of curves $(x-2)^2+(y-a)^2=b^2$, (where $a$ and $b$ are parameters) are $m$ and $n$ respectively, then $m^2+n=$

Consider the differential equation $\frac{d y}{d x}=\frac{1}{a x+4 y+7}$ and the following statements
A. The given differential equation is linear in $x$.
B. The given differential equation is not linear in $y$.
C. The given differential eq...

The solution of the differential equation $x \cos x \frac{d y}{d x}+(x \sin x+\cos x) y=1$ is

If $\operatorname{Lt}_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=e^x(x+1)$ and $f(0)=0$, then $\frac{d}{d x}\left(f(x) e^{-x}\right)+\frac{d}{d x}\left(\frac{f(x)}{x}\right)=$

If $y(x)=\tan ^{-1}\left(\frac{\sqrt{1+a^2 x^2}-1}{a x}\right)$ and $\left(1+a^2 x^2\right) y^{\prime \prime}+g(x) y^{\prime}=0$ then, the sum of the roots of the equation $1+a^2 x^2+g(x)=0$ is

Equation of a common tangent to the circle $x^2+y^2=4$ and to the ellipse $2 x^2+25 y^2=50$ is

The $\theta$ is the angle made by the common tangent to the circle $x^2+y^2=16$ and the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ with positive $X$-axis, then $\cos 2 \theta=$

Given that for any $n \in \mathbf{N}$ there exist an odd integer $q$ and a non-negative integer $r$ such that, $n$ can be written uniquely as $n=q \times 2^r$.
Let $f: \mathbf{N} \rightarrow \mathbf{N} \times \mathbf{N}$ be function defined...

Let $H(x)=3 x^4+6 x^3-2 x^2+1$ and $g(x)$ be a polynomial of degree one. If
$\frac{H(x)}{(x-1)(x+1)(x-2)}=f(x)+\frac{g(x)}{(x-1)(x+1)(x-2)}$ then
$H(-1)+2 H(2)-3 H(1)=$

If $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined by $f(x)=x+2|x+1|+2|x-1|$, then the element in the co-domain, which has unique pre image in the domain is

Suppose the axes $X$ and $Y$ are obtained by rotating the axes $x$ and $y$ an angle $\theta$. If the equation $x^2+2 \sqrt{3} x y-y^2=4 a^2$ is transformed to $X^2-Y^2=2 a^2$ with respect to the $X Y$-axes, then $\theta$ is equal to

For the hyperbola $x^2-y^2-4 x+2 y+c=0$, if the focus is $S(2+2 \sqrt{2}, k)$ and the directrix that is adjacent to $S$ is $x=2+\sqrt{2}$, then $c=$

If $5(f(x))^2=x f(x)+30$ and
$$ \begin{aligned} & \int \frac{\left(3 x^3+\left(1-30 x^2\right) f(x)\right)}{(10 f(x)-x)\left(x^3-f(x)\right)^2} d x \\ & =\frac{A}{B x^3+D f(x)}+C \text { then } A+B+D= \end{aligned} $$

If $\int x[\log (1+x)]^3 d x=\frac{(1+x)^2}{16}(f(x))+(1+x)(g(x))$, then
\(f(x)+g(x)=\)

\(\int \frac{\left(x+\sqrt{1+x^2}\right)^2}{\sqrt{1+x^2}} d x=\)

\(\int\left[\frac{x^4-x}{x^{20}}\right]^{1 / 4} d x=\)

If $\sinh \left(2 \tanh ^{-1} x\right)=\frac{11}{60}$, then $x=$

If $f: \mathbf{R} \rightarrow \mathbf{R}$ and $g: \mathbf{R} \rightarrow \mathbf{R}$ be defined by $f(x)=\left\{\begin{array}{cc}x+2, & x>0 \\ 2-x, & x \leq 0\end{array}\right.$ and $g(x)=\left\{\begin{array}{cc}x^2-2 x-2, & 1 \leq x<2 \\ x...

Define $f: R \rightarrow R$ by $f(x)= \begin{cases}(x-a) \frac{e^{\frac{1}{(x-a)}}-1}{\frac{1}{(x-a)}}+1 & \text { for } x \neq a \\ 0, \quad \text { at } x=a\end{cases}$
Then which one of the following is true?

Let $f, g: \mathbf{R} \rightarrow \mathbf{R}$ be functions defined by
$$ f(x)=\left\{\begin{array}{cc} x \sin \left(\frac{1}{x}\right), & \text { for } x \neq 0 \\ 0, & \text { for } x=0 \end{array}\right. $$
and $g(x)=x f(x)$
Consider the ...

For a square matrix $B$ of order 3 , if $B^T=B^{-1}$ and $|B|=1$, then $|B-I|=$

For $\alpha, \beta \in[0,2 \pi]$ and $\gamma \in[0, \pi)$ consider the system of equations
$$ \begin{aligned} & 2 \sin \alpha-\cos \beta+3 \tan \gamma=3 \\ & 4 \sin \alpha+2 \cos \beta-2 \tan \gamma=2 \\ & 6 \sin \alpha-3 \cos \beta+\tan \g...

$$ \text { The rank of } A=\left[\begin{array}{ccc} 1 & x & x+1 \\ 2 x & x^2-x & x^2+x \\ 3 x(x-1) & x\left(x^2-3 x+2\right) & x\left(x^2-1\right) \end{array}\right] \text { is } $$

If $S(a, b)$ is a fixed point and $P(\alpha, \beta)$ is such a variable point that $4\left[(x-a)^2+(y-b)^2\right]=(\alpha x+\beta y+7)^2$ represents a parabola, then the locus of $P(\alpha, \beta)$ is

If $P(-3,2)$ is an end point of the focal chord $P Q$ of the parabola $y^2+4 x+4 y=0$, then the slope of the normal drawn at $Q$ is

Let $S_r=\{x, y, z) / x+y+z=11, x \geq r, y \geq r$, $z \geq r, x, y, z, r$ are integers $\}$ and $n\left(S_r\right)$ represents the number of elements in $S_r$. Then $n\left(S_{2)}+n\left(S_3\right)+n\left(S_4\right)=\right.$

A certain question paper contains three parts $A, B, C$ with four questions in part $A$, five questions in part $B$ and six questions in part $C$. A student is required to answer seven questions choosing at least two questions from each par...

A person tossing a biased coin indefinitely wins the game by getting head for the first time. The probability that he wins the game in odd number of tosses is $3 / 4$. If 5 such coins are tossed at a time then the probability that head appe...

Let $B(\alpha, \beta, \gamma)$ represents that a bag $B$ contains $\alpha$ red balls, $\beta$ green balls and $\gamma$ blue balls. Given $B_1(2,3,2), B_2(3,2,2), B_3(2,2,3)$. A die is rolled. If the die shows up 2 or 3 or 5 , then a ball wi...

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 random variable $X$ has the following probability distribution
$$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline X=x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline P(X=x) & 0.15 & 0.23 & K & 0.10 & 0.20 & 0.08 & 0.07 & 0.05 \\ \hline \end{array} $$...