Consider the following statements for the two non-empty sets A and B :

$$\begin{gathered} 1. \left(\mathrm{A} \cap \mathrm{B}\right) \cup \left(\mathrm{A} \cap \overline{\mathrm{B}}\right) \cup \left(\overline{\mathrm{A}} \cap \mathrm{B}\right) = \mathrm{A} \cap \mathrm{B} \\ 2. \left(\mathrm{A} \cup \left(\overline{\mathrm{A}} \cap \overline{\mathrm{B}}\right)\right) = \mathrm{A} \cup \mathrm{B} \end{gathered}$$

Which of the above statements is/are correct ?

• 1 only

• 2 Only

• Both 1 and 2

• Neither 1 nor 2

$\mathrm{If} \mathrm{A} = \left\{\mathrm{\lambda }, \left\{\mathrm{\lambda }, \mathrm{\mu }\right\}\right\}, \mathrm{then} \mathrm{the} \mathrm{power} \mathrm{set} \mathrm{of} \mathrm{A} \mathrm{is} :$

• $\left\{\mathrm{\varphi }, \left\{\mathrm{\varphi }\right\}, \left\{\mathrm{\lambda }\right\}, \left\{\mathrm{\lambda }, \mathrm{\mu }\right\}\right\}$

• $\left\{\mathrm{\varphi }, \left\{\mathrm{\lambda }\right\}, \left\{\left\{\mathrm{\lambda }, \mathrm{\mu }\right\}\right\}, \left\{\mathrm{\lambda }, \left\{\mathrm{\lambda }, \mathrm{\mu }\right\}\right\}\right\}$

• $\left\{\mathrm{\varphi }, \left\{\mathrm{\lambda }\right\}, \left\{\mathrm{\lambda }, \mathrm{\mu }\right\}, \left\{\lambda , \left\{\mathrm{\lambda }, \mathrm{\mu }\right\}\right\}\right\}$

• $\left\{\left\{\mathrm{\lambda }\right\}, \left\{\mathrm{\lambda }, \mathrm{\mu }\right\}, \left\{\mathrm{\lambda }, \left\{\mathrm{\lambda }, \mathrm{\mu }\right\}\right\}\right\}$

Let X be a non-empty set and let A, B, C be sub sets of X. Consider the following statements :

$1.\mathrm{A} \subset \mathrm{C} \Rightarrow \left(\mathrm{A} \cap \mathrm{B}\right) \subset \left(\mathrm{C} \cap \mathrm{B}\right), \left(\mathrm{A} \cup \mathrm{B}\right) \subset \left(\mathrm{C} \cup \mathrm{B}\right)$

2. A ∩ B ⊂ C ∩ B for all sets B ⇒ A ⊂ C

3. A ∪ B ⊂ C ∪ B for all sets B ⇒ A ⊂ C

Which of the statements are correct ?

• $1 \mathrm{and} 2 \mathrm{only}$

• 2 and 3 only

• 1 and 3 only

• 1, 2 and 3

4.

Consider the following statements:

1. The algebraic sum of deviations of a set of values from their arithmetic mean is always zero.
2. Arithmetic mean > Median > Mode for a symmetric distribution.                                                          Which of the above statements is/are correct ?

• 1 Only

• 2 Only

• Both 1 and 2

• Neither 1 nor 2

$$\begin{gathered} \mathrm{Let} \mathrm{A} \mathrm{and} \mathrm{B} \mathrm{be} \mathrm{subsets} \mathrm{of} \mathrm{X} \mathrm{and} \mathrm{C} = (\mathrm{A} \cap \mathrm{B}\text{'}) \cup (\mathrm{A}\text{'} \cap \mathrm{B}), \mathrm{where} \mathrm{A}\text{'} \mathrm{and} \mathrm{B}\text{'} \mathrm{are} \\ \mathrm{complements} \mathrm{of} \mathrm{A} \mathrm{and} \mathrm{B} \mathrm{respectively} \mathrm{in} \mathrm{X}. \mathrm{What} \mathrm{is} \mathrm{C} \mathrm{equal} \mathrm{to}? \end{gathered}$$

• $(A \cup B\text{'}) - (A \cap B\text{'})$

• $\left(\mathrm{A}\text{'} \cup \mathrm{B}\right) - \left(\mathrm{A}\text{'} \cap \mathrm{B}\right)$

• $\left(\mathrm{A} \cup \mathrm{B}\right) - \left(\mathrm{A} \cap \mathrm{B}\right)$

• $\left(\mathrm{A}\text{'} \cup \mathrm{B}\text{'}\right) - \left(\mathrm{A}\text{'} \cap \mathrm{B}\text{'} \right)$

If A, B and C are subsets of a Universal set, then which one of the following is not correct ?

• $\mathrm{A} \cup \left(\mathrm{B} \cap \mathrm{C}\right) = \left(\mathrm{A} \cup \mathrm{B}\right) \cap \left(\mathrm{A} \cup \mathrm{C}\right)$

• $\mathrm{A}\text{'} \cup \left(\mathrm{A} \cup \mathrm{B}\right) = \left(\mathrm{B}\text{'} \cap \mathrm{A}\right)\text{'} \cup \mathrm{A}$

• $\mathrm{A}\text{'} \cup \left(B \cup C\right) = \left(C\text{'} \cap B\right)\text{'} \cup \mathrm{A}\text{'}$

• $\left(\mathrm{A} \cap \mathrm{B}\right) \cup \mathrm{C} = \left(\mathrm{A} \cup \mathrm{C}\right) \cap \left(\mathrm{B} \cup \mathrm{C}\right)$

$$\begin{gathered} \mathrm{Let} \mathrm{A} = \left[\mathrm{x}\in \mathrm{R} : - 1 \le \mathrm{x} \le 1\right], \\ \mathrm{B} = \left[\mathrm{y} \in \mathrm{R} : - 1 \le \mathrm{y} \le 1\right] \mathrm{and} \mathrm{S} \mathrm{be} \mathrm{the} \mathrm{subset} \mathrm{of} \mathrm{A} \times \mathrm{B}, \mathrm{defined} \mathrm{by} \\ \mathrm{S} = \left[\left(\mathrm{x}, \mathrm{y}\right) \in \mathrm{A} \times \mathrm{B} : {\mathrm{x}}^2 + {\mathrm{y}}^2 = 1\right] \\ \mathrm{Which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ? \end{gathered}$$

• S is a one-one function from A into B

• S is a many-one function from A into B

• S is a bijective mapping from A into B

• S is not a function

Two integers x and y are chosen with replacement from the set [0, 1, 2, ..., 10]. The probability that $\left|\mathrm{x} - \mathrm{y}\right|$ > 5 is

• $\frac{6}{11}$

• $\frac{35}{121}$

• $\frac{30}{121}$

• $\frac{25}{121}$

Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Then, the number of subsets of A containing two or three elements is

• 45

• 120

• 165

• 330

$$\begin{gathered} \mathrm{Consider} \mathrm{the} \mathrm{following} \mathrm{in} \mathrm{respect} \mathrm{of} \mathrm{sets} \mathrm{A} \mathrm{and} \mathrm{B} : \\ 1. \left(\mathrm{A} \cup \mathrm{B}\right) \cup \mathrm{B} = \mathrm{A} \\ 2. \left(\mathrm{A} - \mathrm{B}\right) \cup \mathrm{A} = \mathrm{A} \\ 3. \left(\mathrm{A} - \mathrm{B}\right) \cap \mathrm{B} = \mathrm{\varphi } \\ 4. \mathrm{A}\subseteq \mathrm{B} \Rightarrow \mathrm{A} \cup \mathrm{B} = \mathrm{B} \\ \mathrm{Which} \mathrm{of} \mathrm{the} \mathrm{above} \mathrm{are} \mathrm{correct} ? \end{gathered}$$

• 1, 2 and 3

• 2, 3 and 4

• 1, 3 and 4

• 1, 2 and 4