11.

Ler R₁ and R₂ be two relations defined as follows :

R₁ = {(a, b) $\in$ R₂ : a₂ + b₂ $\in$ Q} and R₂ = {(a, b) $\in$R2: a₂ + b₂ $\in$Q}, where Q is the set of all rational numbers, then

• R₁ is transitive but R₂ is not transitive.

• R₂ is transitive but R₁ is not transitive.

• Neither R₁ nor R₂ is transitive

• R₁ and R₂ are both transitive.

Suppose f(x) is a polynomial of degree four, having critical points at – 1, 0, 1. If T = {x $\in$ R |f(x) = f(0)}, then the sum of squares of all the elements of T is :

• 4

• 6

• 2

• 8

$\mathrm{If} {\mathrm{x}}^3\mathrm{dy} + \mathrm{xydx} = {\mathrm{x}}^2\mathrm{dy} + 2\mathrm{ydx}; \mathrm{y}\left(2\right) = \mathrm{e} \mathrm{and} \mathrm{x} > 1, \mathrm{then} \mathrm{y}\left(4\right) = ?$

• $\frac{\sqrt{\mathrm{e}}}{2}$

• $\frac{1}{2} + \sqrt{\mathrm{e}}$

• $\frac{3}{2}\sqrt{\mathrm{e}}$

• $\frac{3}{2} + \sqrt{\mathrm{e}}$

$$\begin{gathered} \mathrm{If} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{the} \mathrm{series} 20 + 19\frac{3}{5} + 19\frac{1}{5} + 18\frac{4}{5} + ... \mathrm{upto} {\mathrm{n}}^{\mathrm{th}} \mathrm{term} \mathrm{is} 488 \mathrm{and} \\ \mathrm{the} {\mathrm{n}}^{\mathrm{th}} \mathrm{term} \mathrm{is} \mathrm{negative} \end{gathered}$$

• ${\mathrm{n}}^{\mathrm{th}} \mathrm{term} \mathrm{is} - 4\frac{2}{5}$

• n = 41

• n^th term is - 4

• n = 60

The probability that a randomly chosen 5-digit number is made from exactly two digits is :

• $\frac{135}{{10}^4}$

• $\frac{150}{{10}^4}$

• $\frac{134}{{10}^4}$

• $\frac{121}{{10}^4}$

$$\begin{gathered} \mathrm{Let} \mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathrm{R} \mathrm{be} \mathrm{such} \mathrm{that} {\mathrm{a}}^2 + {\mathrm{b}}^2 + {\mathrm{c}}^2 = 1. \mathrm{If} \\ \mathrm{acos}\left(\mathrm{\theta }\right) = \mathrm{bcos}\left(\mathrm{\theta } + \frac{2\mathrm{\pi }}{3}\right) = \mathrm{ccos}\left(\mathrm{\theta } + \frac{4\mathrm{\pi }}{3}\right), \mathrm{where} \mathrm{\theta } = \frac{\mathrm{\pi }}{9}, \\ \mathrm{then} \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{the} \mathrm{vectors} \mathrm{a}\hat{\mathrm{i}} + \mathrm{b}\hat{\mathrm{j}} + \mathrm{c}\hat{\mathrm{k}} \mathrm{and} \mathrm{b}\hat{\mathrm{i}} + \mathrm{c}\hat{\mathrm{j}} +\hspace{0.17em}\hat{\mathrm{k}} \mathrm{is}: \end{gathered}$$

• 0

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{9}$

If the surface area of a cube is increasing at a rate of 3.6 cm²/sec, retaining its shape; then the rate of change of its volume (in cm³/sec.), when the length of a side of the cube is 10 cm, is

• 20

• 10

• 18

• 9

$$\begin{gathered} \mathrm{If} \int {\sin }^{-1}\left(\sqrt{\frac{\mathrm{x}}{1 + \mathrm{x}}}\right)\mathrm{dx} = \mathrm{A}\left(\mathrm{x}\right){\tan }^{-1}\left(\sqrt{\mathrm{x}}\right) + \mathrm{Bx} + \mathrm{C}, \mathrm{where} \mathrm{C} \mathrm{is} \mathrm{a} \mathrm{constant} \\ \mathrm{of} \mathrm{integration}, \mathrm{then} \mathrm{ordered} \mathrm{pair} \left(\mathrm{A}\left(\mathrm{x}\right), \mathrm{B}\left(\mathrm{x}\right)\right) \mathrm{can} \mathrm{be} : \end{gathered}$$

• $\left(\mathrm{x} - 1, \sqrt{\mathrm{x}}\right)$

• $\left(\mathrm{x} - 1, - \sqrt{\mathrm{x}}\right)$

• $\left(\mathrm{x} + 1, \sqrt{\mathrm{x}}\right)$

• $\left(\mathrm{x} + 1, - \sqrt{\mathrm{x}}\right)$

$\mathrm{If} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{the} \mathrm{integral} {\int }_0^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{{\mathrm{x}}^2}{{\left(1 - {\mathrm{x}}^2\right)}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\mathrm{dx} \mathrm{x} = \sin \left(\mathrm{\theta }\right); \mathrm{dx} = \cos \left(\mathrm{\theta }\right)\mathrm{d\theta }$

• $2\sqrt{3} + \mathrm{\pi }$

• $2\sqrt{3} - \mathrm{\pi }$

• $3\sqrt{2} + \mathrm{\pi }$

• $3\sqrt{2} - \mathrm{\pi }$

Let the latus rectum of the parabola y² = 4x be the common chord to the circles C₁ and C₂ each of them having radius 25. Then, the distance between the centres of the circles C₁ and C₂ is :

• 12

• 8

• $8\sqrt{5}$

• $4\sqrt{5}$