The set of all values of a such that both the points (1, 2) and (3, 4) lie on the same side of the line 3x - 5y + a = 0

• $\left\{\mathrm{x} \in \mathrm{IR} \mathrm{x} > 11\right\}$

• $\left\{\mathrm{x} \in \mathrm{IR} \mathrm{x} > 11\right\} \cup \left\{\mathrm{x} \in \mathrm{IR} \mathrm{x} < 7\right\}$

• $\left\{\mathrm{x} \in \mathrm{IR} \mathrm{x} < 7\right\}$

• $\mathrm{\varphi }$

Let S be the set of all $\mathrm{\lambda } \in \mathrm{R}$ for which the system of linear equations

2x – y + 2z = 2

x – 2y + $\mathrm{\lambda }$z = – 4

x + $\lambda$y + z = 4

has no solution. Then, the set S

• is a singleton

• contains exactly two elements

• contains more than two elements

• is an empty set

The set of all real values of $\mathrm{\lambda }$ for which the quadratic equation ($\mathrm{\lambda }$² + 1) x² – 4$\mathrm{\lambda }$x + 2 = 0 always have exactly one root in the interval (0, 1) is :

• ( - 3, - 1)

• (0, 2)

• (1, 3)

• (2, 4)

214.

Let p, q, r be three statements such that the truth value of $\left(\mathrm{p} \wedge \mathrm{q}\right) \to \left(~ \mathrm{q} \vee \mathrm{r}\right)$ is F. Then the truth values of p, q, r are respectively :

• T, T, F

• T, T, T

• T, F, T

• F, T, F

$$\begin{gathered} \mathrm{Let} \underset{\mathrm{i} = 1}{\overset{50}{\cup }}\mathrm{X} = \underset{\mathrm{i} = 1}{\overset{\mathrm{n}}{\cup }}{\mathrm{y}}_{\mathrm{i}} = \mathrm{T}, \mathrm{where} \mathrm{each} {\mathrm{X}}_{\mathrm{i}} \mathrm{contains} 10 \mathrm{elements} \mathrm{and} \mathrm{each} {\mathrm{Y}}_{\mathrm{i}} \mathrm{contains} 5 \mathrm{elements}. \mathrm{If} \mathrm{each} \\ \mathrm{element} \mathrm{of} \mathrm{the} \mathrm{set} \mathrm{T} \mathrm{is} \mathrm{an} \mathrm{element} \mathrm{of} \mathrm{exactly} 20 \mathrm{of} \mathrm{sets} {\mathrm{X}}_{\mathrm{i}}\text{'}\mathrm{s} \mathrm{and} \mathrm{exactly} 6 \mathrm{of} \mathrm{sets} {\mathrm{Y}}_{\mathrm{i}}\text{'}\mathrm{s} \mathrm{then} \mathrm{n} \mathrm{is} \mathrm{equal} \mathrm{to} : \end{gathered}$$

• 45

• 15

• 30

• 50

Set A has m elements and Set B has n elements. If the total number of subsets of A is 112 more than the total number of subsets of B, then the value of m - n is _____

$\mathrm{If} \left(\mathrm{x} + \mathrm{y}\right) = \mathrm{f}\left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{y}\right) \mathrm{and} \sum _{\mathrm{x} = 1}^{\infty } \mathrm{f}\left(\mathrm{x}\right) = 2, \mathrm{x}, \mathrm{y} \in \mathrm{N} \mathrm{is} \mathrm{the} \mathrm{set} \mathrm{of} \mathrm{all} \mathrm{natural} \mathrm{numbers}, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \frac{\mathrm{f}\left(4\right)}{\mathrm{f}\left(2\right)} \mathrm{is} :$

• $\frac{2}{3}$

• $\frac{1}{3}$

• $\frac{1}{9}$

• $\frac{4}{9}$