If $\frac{1}{{}^5\mathrm{C}_{\mathrm{r}}} + \frac{{\displaystyle 1}}{{\displaystyle {}^6\mathrm{C}_{\mathrm{r}}}} = \frac{{\displaystyle 1}}{{\displaystyle {}^4\mathrm{C}_{\mathrm{r}}}}$, then the value of r is

• 4

• 2

• 5

• 3

12.

For positive integer n, n³ + 2n is always divisible by

• 3

• 7

• 5

• 6

In the expansion of (x - 1) (x - 2) ... (x - 18), the coefficient of x¹⁷ is

• 684

• - 171

• 171

• - 342

$1 + {}^{\mathrm{n}}\mathrm{C}_1\cos \left(\mathrm{\theta }\right) + {}^{\mathrm{n}}\mathrm{C}_2\cos \left(2\mathrm{\theta }\right) + ... + {}^{\mathrm{n}}\mathrm{C}_{\mathrm{n}}\cos \left(\mathrm{n\theta }\right)$ equals

• ${\left(2\cos \left(\frac{\mathrm{\theta }}{2}\right)\right)}^{\mathrm{n}}\cos \left(\frac{\mathrm{n\theta }}{2}\right)$

• $2{\cos }^2\left(\frac{\mathrm{n\theta }}{2}\right)$

• $2{\cos }^{2\mathrm{n}}\left(\frac{\mathrm{\theta }}{2}\right)$

• ${\left(2{\cos }^2\left(\frac{\mathrm{\theta }}{2}\right)\right)}^{\mathrm{n}}$

Let R be a relation defined on the set Z of all integers and xRy, when x + 2y is divisible by 3, then

• A is not transitive

• R is symmetric only

• R is an equivalence relation

• R is not an equivalence relation

If A = $\left\{5^{\mathrm{n}} - 4\mathrm{n} - 1: \mathrm{n} \in \mathrm{N}\right\}$ and $\mathrm{B} = \left\{16\left(\mathrm{n} - 1: \mathrm{n} \in \mathrm{N}\right)\right\}$, then

• A = B

• $\mathrm{A} \cap \mathrm{B} = \mathrm{\phi }$

• $\mathrm{A} \subseteq \mathrm{B}$

• $\mathrm{B} \subseteq \hspace{0.17em}\mathrm{A}$

Standard deviation of n observations a₁, a₂, a₃, ... , a_n is 0. Then, the standard deviation of the observations ${\mathrm{\lambda a}}_1, {\mathrm{\lambda a}}_2, ... , {\mathrm{\lambda a}}_{\mathrm{n}}$ is

• $\mathrm{\lambda \sigma }$

• $- \mathrm{\lambda \sigma }$

• $\left|\lambda \right|\sigma$

• ${\mathrm{\lambda }}^{\mathrm{n}}\mathrm{\sigma }$

Let A and B be two events such that $\mathrm{P}\left(\mathrm{A} \cap \mathrm{B}\right) = \frac{1}{6}, \mathrm{P}(\mathrm{A} \cup \mathrm{B}) = \frac{31}{45} \mathrm{and} \mathrm{P}\left(\overline{\mathrm{B}}\right) = \frac{7}{10}$ then

• A and B are independent

• A and B are mutually exclusive

• $\mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right) < \frac{1}{6}$

• $\mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}}\right) < \frac{1}{6}$

The value of $\cos \left(15°\right)\cos \left(7\frac{1°}{2}\right)\sin \left(7\frac{1°}{2}\right)$ is

• $\frac{1}{2}$

• $\frac{1}{8}$

• $\frac{1}{4}$

• $\frac{1}{16}$

If in a $∆$ABC, AD, BE and CF are the altitudes and R is the circumradius, then the radius of the circumcircle of $∆$DEF is

• $\frac{\mathrm{R}}{2}$

• $\frac{2\mathrm{R}}{3}$

• $\frac{\mathrm{R}}{3}$

• None of these