31). Let N denotes the set of all natural numbers and R be the relation on \( \Large N \times N \) defined by \( \Large \left(a,\ b\right) R \left(c,\ d\right)\ if\ ad \left(b+c\right)=bc \left(a+d\right) \), then R is:
| A). symmetric only |
| B). reflexive only |
| C). transitive only |
| D). an equivalence relation |
|
32). Let \( \Large f : \left[ 0,\ 1 \right] \rightarrow \left[ 0,\ 1 \right] \) defined by \( \Large f \left(x\right)=\frac{1-x}{1+x},\ 0 \le x \le 1 \) and let \( \Large g : \left[ 0,\ 1 \right] \rightarrow \left[ 0,\ 1 \right] \) be defined by \( \Large g \left(x\right) = 4x \left(1-x\right),\ 0 \le x \le 1 \), then fog and gof is:
| A). \( \Large \frac{ \left(2x-1\right)^{2} }{1+4x+4x^{2}},\ \frac{8x \left(1-x\right) }{ \left(1+x\right)^{2} } \) |
| B). \( \Large \frac{ \left(2x-1\right) }{1+4x+4x^{2}},\ \frac{8 \left(1-x\right)x }{ \left(1+x\right)^{2} } \) |
| C). \( \Large \frac{ \left(2x+1\right)^{2} }{1+4x+4x^{2}},\ \frac{8}{ \left(1+x\right)^{2} } \) |
| D). \( \Large \frac{ \left(2x+1\right)^{2} }{ \left(1+x\right)^{2} },\ \frac{8 \left(1-x\right) }{ \left(1+x\right)^{2} } \) |
|
33). An integer m is said to be related to another integer n, if m is a multiple of n, Then the relation is:
| A). reflexive and symmetric |
| B). reflexive and transitive |
| C). symmetric and transitive |
| D). equivalence relation |
|
34). Which of the following is a null set?
| A). \( \Large \{ x : |x| < 1,\ x\ \epsilon\ N \} \) |
| B). \( \Large \{ x : |x| = 5,\ x\ \epsilon\ N \} \) |
| C). \( \Large \{ x : x^{2} = 1,\ \epsilon\ Z \} \) |
| D). \( \Large \{ x : x^{2}+2x+1=0,\ x\ \epsilon\ R \} \) |
|
35). A set contains n elements. Then the power set contains:
| A). \( \Large n^{2} elements \) |
| B). n elements |
| C). \( \Large \left(2^{n}-1\right) elements \) |
| D). \( \Large 2^{n} elements \) |
|
36). If \( \Large A=\{ 1, 3, 5, 7 \},\ B=\{ 1, 2, 4, 6, 8 \},\ C=\{ 1, 3, 6, 8 \} \) then finr \( \Large A\cap \left(B\cup C\right) \) and \( \Large \left(A\cap B\right)\cup \left(A\cap C\right) \)
| A). \( \Large \{ 1, 3 \} \) |
| B). \( \Large \{ 3, 5 \} \) |
| C). \( \Large \{ 6, 8 \} \) |
| D). None of these |
|
37). If \( \Large E=\{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \}; \) \( \Large A=\{ 1, 3, 5, 7 \},\ B=\{ 2, 4, 6, 8 \} \) then find the value of \( \Large \left(A\cup B\right)' \) and \( \Large A'\cap B' \)
| A). \( \Large \{ 6 \} \) |
| B). \( \Large \{ 9 \} \) |
| C). \( \Large \{ \} \) |
| D). None of these |
|
38). If \( \Large A=\{ 1, 2, 3, 4 \},\ B=\{ 3, 4, 5, 6 \},\ C=\{ 6, 7, 8, 9 \} \) then find the set \( \Large A\cap \left(B\cup C\right) \)
| A). \( \Large \{ 1, 2 \} \) |
| B). \( \Large \{ 2, 3 \} \) |
| C). \( \Large \{ 3, 4 \} \) |
| D). \( \Large \{ 6, 7 \} \) |
|
39). \( \Large A=\{ 1, 2, 3, 4 \},\ B=\{ 3, 4, 5, 6 \},\ C=\{ 6, 7, 8, 9 \} \) then find the set \( \Large A- \left(B\cup C\right)\ and\ \left(A-B\right)\cap \left(A-C\right) \)
| A). \( \Large \{ 1, 2 \} \) |
| B). \( \Large \{ 3, 4 \} \) |
| C). \( \Large \{ 6, 7 \} \) |
| D). \( \Large \{ 8, 9 \} \) |
|
40). \( \Large A=\{ 4, 9, 16, 25, 36 \},\ B=\{ 9, 25, 49, 81 \},\ C=\{ 16, 81, 256 \} \) then find \( \Large A- \left(B\cap C\right) \)
| A). \( \Large \{ 81 \} \) |
| B). \( \Large \{ 4, 9, 16, 25, 36 \} \) |
| C). \( \Large \{ 4, 36 \} \) |
| D). \( \Large \{ 4, 16, 36 \} \) |
|
