NCERT Solutions for Class 12 Maths Chapter 1 – Relations and Functions Exercise 1.1

Flashcard for Question 1

Quick Tip:
Check properties by definition with “test pairs”:
• Reflexive ⇒ verify (a, a) ∈ R for every a in the set.
• Symmetric ⇒ whenever (a, b) ∈ R, confirm (b, a) ∈ R.
• Transitive ⇒ whenever (a, b) and (b, c) ∈ R, confirm (a, c) ∈ R. Use direct algebra (substitute and simplify) or simple counterexamples from the smallest elements.

Common Mistake:
Forgetting to test all elements for reflexivity, assuming symmetry from one example, or declaring transitivity without checking the needed chained pairs; also mixing up the universe/set (e.g., N vs Z vs a finite A), which can change whether a pair belongs to R.

Flashcard for Question 2

Quick Tip: Disprove each property with a single clear counterexample; pick small numbers like 0, ±1, 1/2.

Common Mistake:
Choosing examples that accidentally satisfy the property (e.g., symmetric with 2 and −2); verify both directions carefully.

Exam Insight:
For relations defined by inequalities, reflexivity usually fails if the inequality isn’t always true at x = x; symmetry often fails by swapping a “small” and a “large” value; transitivity can fail by chaining through an intermediate value.

Flashcard for Question 3

Quick Tip:
If a relation shifts values (like b = a + 1), reflexivity almost always fails, symmetry is impossible unless the shift is zero, and transitivity can fail unless the shift “adds up” correctly.

Common Mistake:
Students sometimes check symmetry by picking pairs that accidentally satisfy b = a + 1 in both orders double-check the definition for all elements, not just special cases.

Flashcard for Question 4

Quick Tip:
Use the inequality properties directly: check a ≤ a for reflexive, use a simple counterexample (e.g. 0 and 1) for symmetry, and apply transitivity of ≤ for the transitive check.

Common Mistake:
Reversing the inequality when testing symmetry (thinking a ≤ b implies b ≤ a) or forgetting to provide a concrete counterexample.

Flashcard for Question 7

Quick Tip:
For “same number of pages,” check equality-based reasoning – equality is always reflexive, symmetric, and transitive.

Common Mistake:
Overcomplicating the proof by comparing page counts indirectly instead of directly using equality properties.

Flashcard for Question 9

Quick Tip:
When checking if a relation is an equivalence relation, focus on proving reflexive, symmetric, and transitive systematically. For the “related to 1” part, just substitute a=1 into the condition.

Common Mistake:
For modulus-based conditions (like “multiple of 4”), students sometimes miss negative differences or forget that 0 is also a multiple of 4.

Flashcard for Question 11

Quick Tip:
Show reflexive/symmetric/transitive directly using distances: let r = distance(OP). Then (P,P) holds since OP = OP; if OP = OQ then OQ = OP (symmetry); if OP = OQ and OQ = OR then OP = OR (transitivity). The equivalence class of P (P ≠ (0,0)) is {Q : OQ = OP}, i.e. the circle centered at the origin with radius OP

Common Mistake:
Forgetting the special case P=(0,0) (its class is the single point {(0,0)}), or confusing “same distance from origin” with “same coordinates” — the related points form a circle (not a line or set of identical points).

Flashcard for Question 11

Quick Tip:
Check reflexive (polygon has same number of sides as itself), symmetric (if P1 and P2 have same number, P2 and P1 do too), and transitive (if P1 and P2 match, and P2 and P3 match, then P1 and P3 match). The class of the 3–4–5 right triangle is all triangles (3-sided polygons).

Common Mistake:
Mixing up “same shape” or “same side lengths” with “same number of sides” – the relation is about side count only, so size and angles don’t matter.