If N denote the set of all natural numbers and R be the relation on N $\times$ N defined by (a, b) R (c, d), if ad(b + c) = bc(a + d), then R is

• symmetric only

• reflexive only

• transitive only

• an equivalence relation

A cmplex number z is such that arg$\left(\frac{\mathrm{z} - 2}{\mathrm{z} + 2}\right) = \frac{\mathrm{z}}{3}$. The points representing this complex number will lie on

• an ellipse

• a parabola

• a circle

• a straight line

If a₁, a₂ and a₃ be any positive real numbers, then which of the following statement is not true?

• 3a₁a₂a₃ $\le {{\mathrm{a}}_1}^3 + {{\mathrm{a}}_2}^3 + {{\mathrm{a}}_3}^3$

• $\frac{{\mathrm{a}}_1}{{\mathrm{a}}_2} + \frac{{\mathrm{a}}_2}{{\mathrm{a}}_3} + \frac{{\mathrm{a}}_3}{{\mathrm{a}}_1} \ge 3$

• (a₁ + a₂ + a₃)$\left(\frac{1}{{\mathrm{a}}_1} + \frac{1}{{\mathrm{a}}_2} + \frac{1}{{\mathrm{a}}_3}\right) \ge 9$

• (a₁ . a₂ . a₃)$\left(\frac{1}{{\mathrm{a}}_1} + \frac{1}{{\mathrm{a}}_2} + \frac{1}{{\mathrm{a}}_3}\right) \ge 27$

If $\left|{\mathrm{x}}^2 - \mathrm{x} - 6\right| = \mathrm{x} + 2$, then the values of x are

• - 2, 2, - 4

• - 2, 2, 4

• 3, 2, - 2

• 4, 4, 3

The centres of a set of circles, each of radius 3, lie on the circles x² + y² = 25. the locus of any point in the set is

• $4 \le {\mathrm{x}}^2 +\mathrm{\_}\hspace{0.17em}{\mathrm{y}}^2 \le 64$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 \le 25$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 \ge 25$

• $3 \le {\mathrm{x}}^2 + {\mathrm{y}}^2 \le 9$

A tower AB leans towards West making an angle $\mathrm{\alpha }$ with the vertical. The angular elevation of B, the top most point of the tower is $\mathrm{\beta }$ as observed from a point C due East of A at a distance 'd' from A. If the angular elevation of B from a point D due East of C at a distance 2d from C is r, then 2$\tan \left(\mathrm{\alpha }\right)$ can be given as

• $3\cot \left(\mathrm{\beta }\right) - 2\cot \left(\mathrm{\gamma }\right)$

• $3\cot \left(\mathrm{\gamma }\right) - 2\cot \left(\mathrm{\beta }\right)$

• $3\cot \left(\mathrm{\beta }\right) - \cot \left(\mathrm{\gamma }\right)$

• $\cot \left(\mathrm{\beta }\right) - 3\cot \left(\mathrm{\gamma }\right)$

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are the roots of x² - ax + b = 0 and if ${\mathrm{\alpha }}^{\mathrm{n}} \mathrm{and} {\mathrm{\beta }}^{\mathrm{n}}$ = Vⁿ, then

• V_{n + 1} = aV_n + bV_{n - 1}

• V_{n + 1} = aV_n + aV_{n - 1}

• V_{n + 1} = aV_n - bV_{n - 1}

• V_{n + 1} = aV_{n - 1} + bV_n

The sum of the series

$\sum {\left(- 1\right)}^{\mathrm{r}}{}^{\mathrm{n}}\mathrm{C}_{\mathrm{r}}\left(\frac{1}{2^{\mathrm{r}}} + \frac{3^{\mathrm{r}}}{2^{2\mathrm{r}}} + \frac{7^{\mathrm{r}}}{2^{3\mathrm{r}}} + \frac{{15}^{\mathrm{r}}}{2^{4\mathrm{r}}} + ... \mathrm{m} \mathrm{terms}\right)$ is

• $\frac{2^{\mathrm{mn}} - 1}{2^{\mathrm{mn}}\left(2^{\mathrm{n}} - 1\right)}$

• $\frac{2^{\mathrm{mn}} - 1}{2^{\mathrm{n}} - 1}$

• $\frac{2^{\mathrm{mn}} + 1}{2^{\mathrm{n}} + 1}$

• None of these

The angle of intersection of the circles x² + y² - x + y - 8 = 0 and x² + y² + 2x + 2y - 11 = 0 is

• ${\tan }^{-1}\left(\frac{19}{9}\right)$

• ${\tan }^{-1}\left(19\right)$

• ${\tan }^{-1}\left(\frac{9}{19}\right)$

• ${\tan }^{-1}\left(9\right)$

10.

The number of integral values of K, for which the equation $7\cos \left(\mathrm{x}\right) + 5\sin \left(\mathrm{x}\right) = 2\mathrm{K} + 1$ has a solution

• 4

• 8

• 10

• 12