Mathematical Reasoning — Class 11 MCQs with Answers

Mathematical Reasoning — important questions & MCQs with answers (Class 11 Mathematics)

6 solved questions from this chapter's difficulty levels, each with its answer and explanation. Tap an option to check yourself.

0 of 6 answered

1. Q1 of 6Easy

   A mathematical statement is:

   AA sentence that is either true or false (not both)BA questionCA commandDA wish
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   Answer: A sentence that is either true or false (not both)

   Explanation: A statement is a declarative sentence with a definite truth value — true or false, never both — unlike questions, commands or wishes, which have no truth value to test at all.

2. Q2 of 6Easy

   Negation of "x is even":

   Ax is odd plus 1Bx is not evenCx is positiveDx = 0
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   Answer: x is not even

   Explanation: Negation ~p simply denies p — "x is even" becomes "x is not even." Don't over-specify with "x is odd," which silently assumes x is an integer that the statement never stated.

3. Q3 of 6Medium

   Negation of 'All triangles have 3 sides':

   AAll triangles have 4 sidesBNo triangle has 3 sidesCSome triangle does not have 3 sides (or there exists a triangle...)DSome triangles have 3 sides
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   Answer: Some triangle does not have 3 sides (or there exists a triangle...)

   Explanation: Negating "All triangles have 3 sides" (∀x, P(x)) gives ∃x, ¬P(x): "Some triangle does not have 3 sides." Keeping the universal quantifier, as in "No triangle has 3 sides," is the wrong flip.

4. Q4 of 6Medium

   Compound statement: 'It is sunny, but cold'. Connective is:

   AIf-thenBOrCAnd (conjunction)DNot
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   Answer: And (conjunction)

   Explanation: "But" logically behaves exactly like "and" — it still requires both parts to hold, just with an implied contrast in tone. Symbolically it's still conjunction, p ∧ q.

5. Q5 of 6Hard

   Contrapositive of 'If a triangle is equilateral, then it is isosceles':

   AIf a triangle is not equilateral, then it is not isoscelesBIf a triangle is isosceles, then it is equilateralCIf a triangle is equilateral, then it is not isoscelesDIf a triangle is not isosceles, then it is not equilateral
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   Answer: If a triangle is not isosceles, then it is not equilateral

   Explanation: The contrapositive negates and swaps both parts: "equilateral → isosceles" becomes "not isosceles → not equilateral," guaranteed to share the original's truth value — the converse and inverse are not.

6. Q6 of 6Hard

   'A function is bijective if and only if it is one-one and onto'. This is:

   AExistentialBConditionalCUniversalDBiconditional statement
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   Answer: Biconditional statement

   Explanation: "If and only if" signals a biconditional, p ↔ q — bijective holds exactly when both one-one and onto hold, and each direction of the claim must be separately true.

Mathematical Reasoning notes: key concepts (Class 11 Mathematics)

Mathematics rests on precise logic. This chapter defines a statement as a sentence that is definitely true or false, builds compound statements with the connectives 'and', 'or' and 'not', introduces the quantifiers 'for all' and 'there exists', and studies implication and the methods for validating statements.

Statement

A declarative sentence that is either true or false but not both; questions, commands and opinions are not statements in logic.

Negation

The negation ~p reverses the truth value of p; read 'it is not the case that p', it is true exactly when p is false.

Compound statement

Two or more simple statements joined by connectives; its truth depends on the truth of its components and the connective used.

Conjunction (and)

p ∧ q is true only when both p and q are true; a single false component makes the whole 'and' statement false.

Disjunction (or)

p ∨ q, the inclusive 'or', is false only when both parts are false; the exclusive 'or' additionally rules out the both-true case.

Quantifiers

'For all' (∀) claims a property holds for every element; 'there exists' (∃) claims at least one does, turning open sentences into statements.

Implication

'If p then q' (p → q) is false only when p is true and q is false; p is the hypothesis and q the conclusion.

Converse

The converse of p → q is q → p; it swaps hypothesis and conclusion and is not logically equivalent to the original.

Contrapositive

The contrapositive of p → q is ~q → ~p; it is logically equivalent to the original, so proving one proves the other.

Inverse

The inverse of p → q is ~p → ~q; it negates both parts and, like the converse, is not equivalent to the original statement.

Biconditional

'p if and only if q' (p ↔ q) is true when p and q share the same truth value; it bundles p → q with its converse q → p.

Validating statements

To prove p → q use a direct proof, its contrapositive, or contradiction — assume p and ~q together and derive an absurdity.

Counter-example

A single instance where a 'for all' claim fails disproves it outright; you can never prove such a statement by examples alone.