Vidaara.orgClass 11 · Mathematics
CodeVID-M11-16-CT
Mathematical Reasoning — Full Chapter Test
Name: ____________________
Roll No.: __________
Date: ____________
General Instructions
- This is a full-length test covering the whole chapter — every topic is included.
- 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.
Which is a statement?
- A.Open the gate.
- B.$7$ is an even number.
- C.What time is it?
- D.Hurrah!
2.
The symbol $\forall$ means:
- A.there exists
- B.for all
- C.implies
- D.not
3.
To disprove a "for all" statement you need:
- A.a direct proof
- B.one counter-example
- C.a contradiction
- D.the converse
4.
$p \wedge q$ is false when:
- A.both true
- B.at least one false
- C.both false only
- D.never
5.
The contrapositive of $p \Rightarrow q$ is:
- A.$q \Rightarrow p$
- B.$\sim p \Rightarrow \sim q$
- C.$\sim q \Rightarrow \sim p$
- D.$p \Rightarrow \sim q$
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Write the negation of "$\sqrt{3}$ is irrational".
7.
Negate: "Every real number is positive."
8.
Disprove: "Every odd number is prime."
9.
State the truth value of "$2 > 3$ or $4 > 1$".
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Break "$36$ is divisible by $4$ and by $9$" into components and give its truth value.
11.
For "If a number is a multiple of $9$, then it is a multiple of $3$", write the converse and say whether it is true.
12.
Prove by contrapositive: "If $3n + 2$ is odd, then $n$ is odd".
13.
Negate "For all integers $n$, $n^2 \ge n$".
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
Construct the truth table for $\sim p$, $p \wedge q$ and $p \vee q$ for all four combinations of $p,q$.
15.
Prove by contradiction that $\sqrt{2}$ is irrational, stating each step.
Answer Key
Section A — Multiple Choice Questions
- (B) $7$ is an even number.
- (B) for all
- (B) one counter-example
- (B) at least one false
- (C) $\sim q \Rightarrow \sim p$
Section B — Short Answer (2 marks)
- "$\sqrt{3}$ is not irrational" (i.e. it is rational).
- "There exists a real number that is not positive."
- $9$ is odd but not prime — a counter-example.
- True — the second component is true (inclusive or).
Section C — Short Answer (3 marks)
- $p$: "$36$ is divisible by $4$" (true), $q$: "$36$ is divisible by $9$" (true); $p \wedge q$ is true.
- Converse: "If a number is a multiple of $3$, then it is a multiple of $9$" — false ($6$ is a counter-example).
- If $n$ is even, $n = 2k$, then $3n + 2 = 2(3k+1)$ is even — establishing the contrapositive.
- "There exists an integer $n$ such that $n^2 < n$."
Section D — Long Answer (5 marks)
- (T,T): F, T, T. (T,F): F, F, T. (F,T): T, F, T. (F,F): T, F, F.
- 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.
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