Vidaara.orgClass 11 · Mathematics
CodeVID-M11-16-MR-03
Validating Statements — Assignment
Name: ____________________
Roll No.: __________
Date: ____________
General Instructions
- All questions are compulsory.
- Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
- Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions
5 × 1 = 5 marks
1.
To disprove a "for all" statement you need:
- A.a direct proof
- B.one counter-example
- C.a contradiction
- D.the converse
2.
Reductio ad absurdum means proof by:
- A.direct method
- B.contrapositive
- C.contradiction
- D.example
3.
$\sqrt{2}$ is irrational is usually proved by:
- A.counter-example
- B.contradiction
- C.direct method
- D.converse
4.
To prove "$p$ iff $q$" you must prove:
- A.only $p \Rightarrow q$
- B.only $q \Rightarrow p$
- C.both directions
- D.a counter-example
5.
The contrapositive method proves $p \Rightarrow q$ by deducing:
- A.$q$ from $p$
- B.$\sim p$ from $\sim q$
- C.$\sim q$ from $\sim p$
- D.$p$ from $q$
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Disprove: "Every odd number is prime."
7.
State which method you would use to prove "$\sqrt{3}$ is irrational".
8.
Write the contrapositive used to prove "If $n^2$ is odd then $n$ is odd".
9.
Disprove: "For all real $x$, $x^2 \ge x$."
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Prove by the direct method: "The product of two odd integers is odd."
11.
Prove by contrapositive: "If $3n + 2$ is odd, then $n$ is odd."
12.
Disprove: "For all integers $n$, $2^n + 1$ is prime."
13.
By contradiction, show there is no smallest positive real number.
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
Prove by contradiction that $\sqrt{2}$ is irrational, stating each step.
15.
Prove the biconditional "$n$ is even $\Leftrightarrow n^2$ is even" by proving both directions.
Answer Key
Section A — Multiple Choice Questions
- (B) one counter-example
- (C) contradiction
- (B) contradiction
- (C) both directions
- (B) $\sim p$ from $\sim q$
Section B — Short Answer (2 marks)
- $9$ is odd but not prime — a counter-example.
- Proof by contradiction.
- "If $n$ is even, then $n^2$ is even."
- $x = \tfrac{1}{2}$ gives $x^2 = \tfrac{1}{4} < \tfrac{1}{2}$.
Section C — Short Answer (3 marks)
- $(2m+1)(2n+1) = 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1$, which is odd.
- Contrapositive: if $n$ is even, $n = 2k$, then $3n + 2 = 6k + 2 = 2(3k+1)$ is even.
- $n = 3$ gives $2^3 + 1 = 9 = 3 \times 3$, not prime.
- If $r > 0$ were smallest, $r/2$ is a smaller positive real — contradiction.
Section D — Long Answer (5 marks)
- Assume $\sqrt{2} = a/b$ in lowest terms; then $a^2 = 2b^2$, so $a$ even, $a = 2c$, giving $b^2 = 2c^2$, so $b$ even — contradicting "lowest terms". Hence $\sqrt{2}$ is irrational.
- Forward: $n = 2k \Rightarrow n^2 = 2(2k^2)$ even. Backward (contrapositive): $n$ odd $\Rightarrow n^2$ odd. Both hold, so the biconditional is valid.
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