We define a binary relation $~$ on the set of all 3 x 3 real matrices as A $~$ B,if and only if there exist invertible matrices P and Q such that B = PAQ^-1 .The binary relation $~$ is

• neither reflexive nor symmetric

• reflexive and symmetric but not transitive

• symmetric and transitive but not reflexive

• an equivalence relation

The angle of intersection between the curves $\mathrm{y} = \left[\left|\sin \left(\mathrm{x}\right)\right| + \left|\cos \left(\mathrm{x}\right)\right|\right]$ and x² + y² = 10, where [x] denotes the greatest integer $\le \mathrm{x}$, is

• ${\tan }^{-1}\left(3\right)$

• ${\tan }^{-1}\left(- 3\right)$

• ${\tan }^{-1}\left(\sqrt{3}\right)$

• ${\tan }^{-1}\left(1/\sqrt{3}\right)$

$$\begin{gathered} \mathrm{Let} \mathrm{f}\left(\mathrm{x}\right) = \left\{{\int }_0^{\mathrm{x}}\left|1 - \mathrm{t}\right|d\mathrm{t}, \mathrm{x} > 0 \\ \mathrm{x} - \frac{1}{2}, \mathrm{x} \le 1\right\} \end{gathered}$$. Then

• f(x) is continuous at x = 1

• f(x) is not continuous at x = 1

• f(x) is differentiable at x = 1

• f(x) is not differentiable at x = 1

If f(x) = $$\begin{gathered} \left\{2{\mathrm{x}}^2 + 1, \mathrm{x} \le 1 \\ 4{\mathrm{x}}^3 - 1, \mathrm{x} > 1\right\} \end{gathered}$$, then ${\int }_0^2\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$ is

• 47/3

• 50/3

• 1/3

• 47/2

The integrating factor of the differential equation

$\left(1 + {\mathrm{x}}^2\right)\frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{y} = {\mathrm{e}}^{{\tan }^{-1}\left(\mathrm{x}\right)}$ is

• ${\tan }^{-1}\left(\mathrm{x}\right)$

• 1 + x²

• ${\mathrm{e}}^{{\tan }^{-1}\left(\mathrm{x}\right)}$

• ${\log }_{\mathrm{e}}\left(1 + {\mathrm{x}}^2\right)$

66.

If $\sqrt{\mathrm{y}} = {\cos }^{-1}\left(\mathrm{x}\right)$, then it satisfies the differential equation

$\left(1 - {\mathrm{x}}^2\right)\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} - \mathrm{x}\frac{d\mathrm{y}}{d\mathrm{x}} = \mathrm{c}$, where c equal to 

• 0

• 3

• 1

• 2

The area of the region bounded by the curves y = x² and x = y² is

• 1/3

• 1/2

• 1/4

• 3

If $\mathrm{I} = {\int }_0^2{\mathrm{e}}^{{\mathrm{x}}^4}\left(\mathrm{x} - \mathrm{\alpha }\right)d\mathrm{x} = 0$, then $\mathrm{\alpha }$ lies in the interval

• (0, 2)

• (- 1, 0)

• (2, 3)

• (- 2, - 1)

The solution of the differential equation $\mathrm{y}\frac{d\mathrm{y}}{d\mathrm{x}} = \mathrm{x}\left[\frac{{\mathrm{y}}^2}{{\mathrm{x}}^2} + \frac{\mathrm{\varphi }\left(\frac{{\mathrm{y}}^2}{{\mathrm{x}}^2}\right)}{\mathrm{\varphi }\text{'}\left(\frac{{\mathrm{y}}^2}{{\mathrm{x}}^2}\right)}\right]$ is (where, c is a constant)

• $\mathrm{\varphi }\left(\frac{{\mathrm{y}}^2}{{\mathrm{x}}^2}\right) = \mathrm{cx}$

• $\mathrm{x\varphi }\left(\frac{{\mathrm{y}}^2}{{\mathrm{x}}^2}\right) = \mathrm{c}$

• $\mathrm{\varphi }\left(\frac{{\mathrm{y}}^2}{{\mathrm{x}}^2}\right) = {\mathrm{cx}}^2$

• ${\mathrm{x}}^2\mathrm{\varphi }\left(\frac{{\mathrm{y}}^2}{{\mathrm{x}}^2}\right) = \mathrm{c}$

There is a group of 265 persons who like either singing or dancing or painting. In this group 200 like singing, 110 like dancing and 55 like painting. If 60 persons like both singing and dancing, 30 like both singing and painting and 10 like all three activities, then the number of persons who like only dancing and painting is

• 10

• 20

• 30

• 40