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NCERT Exemplar · Class 11 Mathematics Mathematical Reasoning

37 questions · 37 still being checked

This chapter is from the older syllabus and is not in the current NCERT textbook, so you may skip it.

EXERCISE 14.3 21–30 (part 3 of 4)

  1. Choose the correct answer out of the four options given against each of the Exercises $\displaystyle 17$ to $\displaystyle 36$ (M.C.Q.).

    Exercise 21

    The negation of the statement "A circle is an ellipse" is
    (A)
    An ellipse is a circle.
    (B)
    An ellipse is not a circle.
    (C)
    A circle is not an ellipse.
    (D)
    A circle is an ellipse.

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    NCERT’s answer
    C
    (C) A circle is not an ellipse.\[p:\ \text{A circle is an ellipse} \] \[\sim p:\ \text{A circle is not an ellipse} \]The negation denies \(\displaystyle p\) itself; (B) speaks of an ellipse being a circle, which is a different statement.

  2. Exercise 22

    The negation of the statement "7\displaystyle 7 is greater than 8\displaystyle 8" is
    (A)
    7\displaystyle 7 is equal to 8.
    (B)
    7\displaystyle 7 is not greater than 8.
    (C)
    8\displaystyle 8 is less than 7.
    (D)
    none of these

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    NCERT’s answer
    B
    (B) $\displaystyle 7$ is not greater than 8.\[p:\ 7>8 \] \[\sim p:\ 7 \not> 8 \]The negation must be true exactly when \(\displaystyle p\) is false. (A) is too narrow, since \(\displaystyle 7<8\) also makes \(\displaystyle p\) false; (C) says the same as \(\displaystyle p\).

  3. Exercise 23

    The negation of the statement "72\displaystyle 72 is divisible by 2\displaystyle 2 and 3\displaystyle 3" is
    (A)
    72\displaystyle 72 is not divisible by 2\displaystyle 2 or 72\displaystyle 72 is not divisible by 3.
    (B)
    72\displaystyle 72 is not divisible by 2\displaystyle 2 and 72\displaystyle 72 is not divisible by 3.
    (C)
    72\displaystyle 72 is divisible by 2\displaystyle 2 and 72\displaystyle 72 is not divisible by 3.
    (D)
    72\displaystyle 72 is not divisible by 2\displaystyle 2 and 72\displaystyle 72 is divisible by 3.

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    NCERT’s answer
    A
    (A) $\displaystyle 72$ is not divisible by $\displaystyle 2$ or $\displaystyle 72$ is not divisible by 3.\[p:\ 72 \text{ is divisible by } 2, \qquad q:\ 72 \text{ is divisible by } 3 \] \[\sim(p \wedge q) \equiv \sim p \vee \sim q \]Negating "and" turns it into "or".

  4. Exercise 24

    The negation of the statement "Plants take in CO2\displaystyle \mathrm{CO}_2 and give out O2\displaystyle \mathrm{O}_2" is
    (A)
    Plants do not take in CO2\displaystyle \mathrm{CO}_2 and do not give out O2\displaystyle \mathrm{O}_2.
    (B)
    Plants do not take in CO2\displaystyle \mathrm{CO}_2 or do not give out O2\displaystyle \mathrm{O}_2.
    (C)
    Plants take in CO2\displaystyle \mathrm{CO}_2 and do not give out O2\displaystyle \mathrm{O}_2.
    (D)
    Plants take in CO2\displaystyle \mathrm{CO}_2 or do not give out O2\displaystyle \mathrm{O}_2.

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    NCERT’s answer
    B
    (B) Plants do not take in \(\displaystyle \mathrm{CO}_2\) or do not give out \(\displaystyle \mathrm{O}_2\).\[p:\ \text{Plants take in } \mathrm{CO}_2, \qquad q:\ \text{Plants give out } \mathrm{O}_2 \] \[\sim(p \wedge q) \equiv \sim p \vee \sim q \]Negating "and" turns it into "or".

  5. Exercise 25

    The negation of the statement "Rajesh or Rajni lived in Bangalore" is
    (A)
    Rajesh did not live in Bangalore or Rajni lives in Bangalore.
    (B)
    Rajesh lives in Bangalore and Rajni did not live in Bangalore.
    (C)
    Rajesh did not live in Bangalore and Rajni did not live in Bangalore.
    (D)
    Rajesh did not live in Bangalore or Rajni did not live in Bangalore.

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    NCERT’s answer
    C
    (C) Rajesh did not live in Bangalore and Rajni did not live in Bangalore.Let \(\displaystyle p\): Rajesh lived in Bangalore; \(\displaystyle q\): Rajni lived in Bangalore. The statement is \(\displaystyle p \vee q\).\[\sim(p \vee q) \equiv (\sim p) \wedge (\sim q) \quad \text{(De Morgan)} \]Answer: (C)

  6. Exercise 26

    The negation of the statement "101\displaystyle 101 is not a multiple of 3\displaystyle 3" is
    (A)
    101\displaystyle 101 is a multiple of 3.
    (B)
    101\displaystyle 101 is a multiple of 2.
    (C)
    101\displaystyle 101 is an odd number.
    (D)
    101\displaystyle 101 is an even number.

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    NCERT’s answer
    A
    (A) $\displaystyle 101$ is a multiple of 3.Let \(\displaystyle p\): $\displaystyle 101$ is a multiple of 3. The statement is \(\displaystyle \sim p\).\[\sim(\sim p) \equiv p \]Answer: (A)

  7. Exercise 27

    The contrapositive of the statement "If 7\displaystyle 7 is greater than 5\displaystyle 5, then 8\displaystyle 8 is greater than 6\displaystyle 6" is
    (A)
    If 8\displaystyle 8 is greater than 6\displaystyle 6, then 7\displaystyle 7 is greater than 5.
    (B)
    If 8\displaystyle 8 is not greater than 6\displaystyle 6, then 7\displaystyle 7 is greater than 5.
    (C)
    If 8\displaystyle 8 is not greater than 6\displaystyle 6, then 7\displaystyle 7 is not greater than 5.
    (D)
    If 8\displaystyle 8 is greater than 6\displaystyle 6, then 7\displaystyle 7 is not greater than 5.

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    NCERT’s answer
    C
    (C) If $\displaystyle 8$ is not greater than $\displaystyle 6$, then $\displaystyle 7$ is not greater than 5.Let \(\displaystyle p\): $\displaystyle 7$ is greater than $\displaystyle 5$; \(\displaystyle q\): $\displaystyle 8$ is greater than 6.\[p \Rightarrow q \quad \text{(given)} \]\[\sim q \Rightarrow \sim p \quad \text{(contrapositive)} \]Answer: (C)

  8. Exercise 28

    The converse of the statement "If x>y\displaystyle x>y, then x+a>y+a\displaystyle x+a>y+a" is
    (A)
    If x<y\displaystyle x<y, then x+a<y+a\displaystyle x+a<y+a.
    (B)
    If x+a>y+a\displaystyle x+a>y+a, then x>y\displaystyle x>y.
    (C)
    If x<y\displaystyle x<y, then x+a>y+a\displaystyle x+a>y+a.
    (D)
    If x>y\displaystyle x>y, then x+a<y+a\displaystyle x+a<y+a.

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    NCERT’s answer
    B
    (B) If \(\displaystyle x+a>y+a\), then \(\displaystyle x>y\).The converse of \(\displaystyle p \Rightarrow q\) is \(\displaystyle q \Rightarrow p\), with\[p:\ x>y, \qquad q:\ x+a>y+a \]\[q \Rightarrow p:\ x+a>y+a \Rightarrow x>y \]Answer: (B)

  9. Exercise 29

    The converse of the statement "If sun is not shining, then sky is filled with clouds" is
    (A)
    If sky is filled with clouds, then the sun is not shining.
    (B)
    If sun is shining, then sky is filled with clouds.
    (C)
    If sky is clear, then sun is shining.
    (D)
    If sun is not shining, then sky is not filled with clouds.

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    NCERT’s answer
    A
    (A) If sky is filled with clouds, then the sun is not shining.Let \(\displaystyle p\): sun is not shining; \(\displaystyle q\): sky is filled with clouds.\[p \Rightarrow q \quad \text{(given)} \]\[q \Rightarrow p \quad \text{(converse)} \]Answer: (A)

  10. Exercise 30

    The contrapositive of the statement "If p\displaystyle p, then q\displaystyle q", is
    (A)
    If q\displaystyle q, then p\displaystyle p.
    (B)
    If p\displaystyle p, then ∼q\displaystyle \sim q.
    (C)
    If ∼q\displaystyle \sim q, then ∼p\displaystyle \sim p.
    (D)
    If ∼p\displaystyle \sim p, then ∼q\displaystyle \sim q.

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    NCERT’s answer
    C
    (C) If \(\displaystyle \sim q\), then \(\displaystyle \sim p\).\[p \Rightarrow q \equiv \sim q \Rightarrow \sim p \]Option (A) is the converse and (D) is the inverse.Answer: (C)