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NCERT Exemplar · Class 11 Mathematics Sets

58 questions · 58 still being checked

EXERCISE 1.3 41–50 (part 5 of 6)

  1. Choose the correct answers from the given four options in each Exercises $\displaystyle 29$ to $\displaystyle 43$ (M.C.Q.).

    Exercise 41

    If A={1,3,5,7,9,11,13,15,17}B={2,4,…,18}\displaystyle \mathrm{A}=\{1,3,5,7,9,11,13,15,17\} \mathrm{B}=\{2,4, \ldots, 18\} and N\displaystyle \mathbf{N} the set of natural numbers is the universal set, then A′∪(A∪B)∩B′)\displaystyle \left.\mathrm{A}^{\prime} \cup(\mathrm{A} \cup \mathrm{B}) \cap \mathrm{B}^{\prime}\right) is
    (A)
    ϕ\displaystyle \phi
    (B)
    N\displaystyle \mathbf{N}
    (C)
    A (D) B

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    NCERT’s answer
    B
    (B) \(\displaystyle \mathbf{N}\). Odd and even numbers do not overlap, so \(\displaystyle A\cap B=\phi\) and \(\displaystyle A\cap B'=A\).\[(A\cup B)\cap B' = (A\cap B')\cup(B\cap B') \] \[= A\cup\phi = A \] \[A'\cup\big((A\cup B)\cap B'\big) = A'\cup A = \mathbf{N} \]

  2. Exercise 42

    Let S={x∣x\displaystyle \mathrm{S}=\{x \mid x is a positive multiple of 3\displaystyle 3 less than 100\displaystyle 100}\displaystyle \} P={x∣x\displaystyle \mathrm{P}=\{x \mid x is a prime number less than 20\displaystyle 20}\displaystyle \}. Then n( S)+n(P)\displaystyle n(\mathrm{~S})+n(\mathrm{P}) is
    (A)
    34\displaystyle 34
    (B)
    31\displaystyle 31
    (C)
    33\displaystyle 33
    (D)
    30\displaystyle 30

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    NCERT’s printed answer for this exercise does not match its own question. This working follows the question as printed.

    None of the options: \(\displaystyle n(\mathrm{S})+n(\mathrm{P})=41\), and the printed options ($\displaystyle 34$, $\displaystyle 31$, $\displaystyle 33$, $\displaystyle 30$) do not include 41.\[\mathrm{S}=\{3,6,9,\ldots,99\}=\{3k : k=1,2,\ldots,33\} \quad (3\times 33=99<100<102=3\times 34) \] \[n(\mathrm{S})=33 \] \[\mathrm{P}=\{2,3,5,7,11,13,17,19\} \] \[n(\mathrm{P})=8 \] \[n(\mathrm{S})+n(\mathrm{P})=33+8=41 \]NCERT prints: (B) $\displaystyle 31$ -- the counts $\displaystyle 33$ and $\displaystyle 8$ sum to $\displaystyle 41$, not 31.

  3. Exercise 43

    If X and Y are two sets and X′\displaystyle \mathrm{X}^{\prime} denotes the complement of X, then X∩(X∪Y)′\displaystyle \mathrm{X} \cap(\mathrm{X} \cup \mathrm{Y})^{\prime} is equal to
    (A)
    X (B) Y (C) ϕ\displaystyle \phi
    (D)
    X∩Y\displaystyle \mathrm{X} \cap \mathrm{Y}

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    NCERT’s answer
    C
    (C) \(\displaystyle \phi\)\[X\cap(X\cup Y)' = X\cap(X'\cap Y') \quad \text{(De Morgan)} \] \[= (X\cap X')\cap Y' \] \[= \phi\cap Y' = \phi \]
  4. Fill in the blanks in each of the Exercises from $\displaystyle 44$ to $\displaystyle 51$ :

    Exercise 44

    The set {x∈R:1≤x<2}\displaystyle \{x \in \mathbf{R}: 1 \leq x<2\} can be written as ____\displaystyle \_\_\_\_.

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    \(\displaystyle \mathbf{[1,2)}\)Closed at $\displaystyle 1$ (\(\displaystyle \le\)), open at $\displaystyle 2$ (\(\displaystyle <\)).\[\{x\in\mathbf{R}: 1\le x<2\} = [1,2) \]

  5. Exercise 45

    When A=ϕ\displaystyle \mathrm{A}=\phi, then number of elements in P(A)\displaystyle \mathrm{P}(\mathrm{A}) is ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    $\displaystyle 1$
    $\displaystyle 1$\[n(A)=0 \Rightarrow n(P(A)) = 2^{n(A)} = 2^{0} = 1 \] \[P(\phi)=\{\phi\} \]

  6. Exercise 46

    If A and B are finite sets such that A⊂B\displaystyle \mathrm{A} \subset \mathrm{B}, then n( A∪ B)=\displaystyle n(\mathrm{~A} \cup \mathrm{~B})= ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    \(\displaystyle n\) (B)
    \(\displaystyle \mathbf{n(B)}\)\[A\subset B \Rightarrow A\cup B = B \] \[n(A\cup B) = n(B) \]

  7. Exercise 47

    If A and B are any two sets, then A−B\displaystyle \mathrm{A}-\mathrm{B} is equal to ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    \(\displaystyle \mathrm{A} \cap \mathrm{B}^{\prime}\)
    \(\displaystyle \mathbf{A\cap B'}\)\[A-B=\{x: x\in A \text{ and } x\notin B\} \] \[= \{x: x\in A \text{ and } x\in B'\} = A\cap B' \]

  8. Exercise 48

    Power set of the set A={1,2}\displaystyle \mathrm{A}=\{1,2\} is ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    \(\displaystyle \{\phi,\{1\},\{2\},\{1,2\}\)
    \(\displaystyle \mathbf{\{\phi,\{1\},\{2\},\{1,2\}\}}\)\[n(A)=2 \Rightarrow n(P(A)) = 2^{2} = 4 \] \[P(A) = \{\phi,\ \{1\},\ \{2\},\ \{1,2\}\} \]

  9. Exercise 49

    Given the sets A={1,3,5}.B={2,4,6}\displaystyle \mathrm{A}=\{1,3,5\} . \mathrm{B}=\{2,4,6\} and C={0,2,4,6,8}\displaystyle \mathrm{C}=\{0,2,4,6,8\}. Then the universal set of all the three sets A,B\displaystyle \mathrm{A}, \mathrm{B} and C can be ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    \(\displaystyle \{0,1,2,3,4,5,6,8\}\)
    \(\displaystyle \{0,1,2,3,4,5,6,8\}\)\[A\cup B\cup C=\{1,3,5\}\cup\{2,4,6\}\cup\{0,2,4,6,8\}=\{0,1,2,3,4,5,6,8\} \]\[A,\,B,\,C\subseteq U \iff A\cup B\cup C\subseteq U \]\[U=A\cup B\cup C=\{0,1,2,3,4,5,6,8\} \]

  10. Exercise 50

    If U={1,2,3,4,5,6,7,8,9,10},A={1,2,3,5},B={2,4,6,7}\displaystyle \mathrm{U}=\{1,2,3,4,5,6,7,8,9,10\}, \mathrm{A}=\{1,2,3,5\}, \mathrm{B}=\{2,4,6,7\} and C={2,3,4,8}\displaystyle \mathrm{C}=\{2,3,4,8\}. Then
    (i)
    (B∪C)′\displaystyle (\mathrm{B} \cup \mathrm{C})^{\prime} is ____\displaystyle \_\_\_\_.
    (ii)
    (C−A)′\displaystyle (\mathrm{C}-\mathrm{A})^{\prime} is ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    (i)
    \(\displaystyle \{1,5,9,10\}\)
    (ii)
    \(\displaystyle \{1,2,3,5,6,7,9,10\}\)
    (i) \(\displaystyle \{1,5,9,10\}\); (ii) \(\displaystyle \{1,2,3,5,6,7,9,10\}\).\[B\cup C=\{2,3,4,6,7,8\} \]\[(B\cup C)'=U-(B\cup C)=\{1,5,9,10\} \]\[C-A=\{2,3,4,8\}-\{1,2,3,5\}=\{4,8\} \]\[(C-A)'=U-\{4,8\}=\{1,2,3,5,6,7,9,10\} \]