Let a relation R be defined on set of all real numbers by a R b if and only if 1 + ab > 0. Then, R is

• reflexive, transitive but not symmetric

• reflexive, symmetric but not transitive

• symmetric, transitive but not reflexive

• an equivalence relation

If $\frac{\mathrm{xy}}{\mathrm{x} + \mathrm{y}} = \frac{2}{3}, \frac{{\displaystyle \mathrm{yz}}}{{\displaystyle \mathrm{y} + \mathrm{z}}} = \frac{{\displaystyle 6}}{{\displaystyle 5}}, \frac{{\displaystyle \mathrm{xz}}}{{\displaystyle \mathrm{x} + \mathrm{z}}} = \frac{{\displaystyle 3}}{{\displaystyle 4}}$, then (x, y, z) is equal to

• (1, 2, 3)

• (2, 1, 3)

• (3, 1, 2)

• (3, 2, 1)

The value of ${\tan }^{-1}\left(\frac{1}{2}\right) + {\tan }^{-1}\left(\frac{1}{3}\right) + {\tan }^{-1}\left(\frac{7}{8}\right)$ is

• ${\tan }^{-1}\left(\frac{7}{8}\right)$

• ${\cot }^{-1}\left(15\right)$

• ${\tan }^{-1}\left(15\right)$

• ${\tan }^{-1}\left(\frac{25}{24}\right)$

The smallest circle with centre on y-axis and passing through the point (7, 3) has radius

• $\sqrt{58}$

• 7

• 3

• 4

If sum of two numbers is 6, the minimum value of the sum of their reciprocals is

• $\frac{6}{5}$

• $\frac{3}{4}$

• $\frac{2}{3}$

• $\frac{1}{2}$

The normal to the curve x = $\mathrm{a}\left(\cos \left(\mathrm{\theta }\right) + \sin \left(\mathrm{\theta }\right)\right)$, y = $\mathrm{a}\left(\sin \left(\mathrm{\theta }\right) - \mathrm{\theta }\cos \left(\mathrm{\theta }\right)\right)$ at any point $\mathrm{\theta }$ is such that

• it makes a constant angle with x-axis

• it passes through origin

• it is at a constant distance from origin

• None of the above

The function f(x) = $\log \left(1 + \mathrm{x}\right) - \frac{2\mathrm{x}}{2 + \mathrm{x}}$ is increasing on

• $\left(- 1, \infty \right)$

• $\left(- \infty , 0\right)$

• $\left(- \infty , \infty \right)$

• None of these

An orthogonal matrix is

• $\left[\begin{array}{cc}\cos \left(\mathrm{\alpha }\right)& 2\sin \left(\mathrm{\alpha }\right)\\ - 2\sin \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)\end{array}\right]$

• $\left[\begin{array}{cc}\cos \left(\mathrm{\alpha }\right)& \sin \left(\mathrm{\alpha }\right)\\ - \sin \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)\end{array}\right]$

• $\left[\begin{array}{cc}\cos \left(\mathrm{\alpha }\right)& \sin \left(\mathrm{\alpha }\right)\\ \sin \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)\end{array}\right]$

• $\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$

19.

Differential equation of those circles which passes through origin and their centres lie on y-axis will be

• $\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} +\hspace{0.17em}2\mathrm{xy} = 0$

• $\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} =\hspace{0.17em}2\mathrm{xy}$

• $\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} =\hspace{0.17em}\mathrm{xy}$

• $\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} +\hspace{0.17em}\mathrm{xy} = 0$

${\int }_0^{\mathrm{\pi }}{\mathrm{xsin}}^4\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\frac{3\mathrm{\pi }}{16}$

• $\frac{3{\mathrm{\pi }}^2}{16}$

• $\frac{16\mathrm{\pi }}{3}$

• $\frac{16{\mathrm{\pi }}^2}{3}$