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⚡ Surprise Test — Determinants

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⚡ Surprise Test · Determinants
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takeatest.net · ISC Class XII
Chapter 5 — Determinants
4 Days · 4 Periods · 16–23 Apr 2026 | ISC Mathematics
Teacher: Ravishankar
Campus: CAMP / KLN / UNDRI
Board Weightage: ~10 marks
Year: 2026–27
Class
Division
Subject
Class XII · Mathematics
Chapter 01
Days 4
Periods 6 × 40 min
QR Slots / Day 4
Solution Mode Manual Click · Auto Play
Clip Upload MP4 / MOV per question
Overview
Day 1 · Types
Day 2 · RST
Day 3 · Equivalence
Day 4 · ℕ×ℕ & Review
Lesson Overview
Subject
Mathematics
Chapter
1 — Relations
Reference
NCERT
Dates
16–19 Mar 2026
How to Use Each Question
📝 Tab 1 — Workspace
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⚠ Common Misconceptions
✗Confusing symmetric with antisymmetric
✗Empty relation is vacuously S & T — but not Reflexive unless A = ∅
✗Transitivity — only pairs already in R need checking
✗Equivalence class [a] = set of elements, not ordered pairs
DAY 1 · MON 23 FEB 2026
Day 1 — Types of Relations
Period 1–2 · 80 min
Binary · Empty · Universal · Identity
Types of Relations
BinaryAny subset R ⊆ A × A
IdentityIₐ = { (a,a) : a ∈ A }
EmptyR = ∅ — vacuously Symmetric & Transitive
UniversalR = A×A — RST all hold
CountTotal relations from A to B: 2^(n(A)·n(B))
✎Concept Board — Day 1Click to open · Draw diagrams & explain concepts▶
Q1 Write the Cartesian product A×B and find n(A×B), given A={1,2,3} and B={4,5}.
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1
A×B = {(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)}
2
n(A)=3, n(B)=2 → n(A×B) = 3×2 = 6
3
∴ A×B has 6 pairs. Total possible relations from A to B = 2⁶ = 64.
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Q2 Write the Identity relation on A={p,q,r}. Is this relation reflexive? Justify.
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1
Identity Relation: Iₐ = {(p,p),(q,q),(r,r)}
2
Check Reflexive: (p,p),(q,q),(r,r) all ∈ Iₐ — every element related to itself ✓
3
∴ Yes, Identity relation is Reflexive.
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Q3 Let A={2,3,5,7}. Is R={(a,b): a²=b, a,b ∈ A} an empty relation? Justify.
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1
Need (a,b) with a²=b, both in A={2,3,5,7}
2
2²=4∉A, 3²=9∉A, 5²=25∉A, 7²=49∉A
3
No pair (a,b) satisfies a²=b with both in A
4
∴ R=∅. Yes, R is an Empty Relation on A.
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Q4 Ex 1(A) Q1(a): For the relation 'is greater than', state which properties apply from: S (Symmetric), T (Transitive), R (Reflexive), E (Equivalence), N (None).
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Q5 For the relation 'is similar to' on all triangles, state all applicable properties from {S,T,R,E,N}.
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1
Reflexive: Every △ is similar to itself ✓
2
Symmetric: If △A~△B then △B~△A ✓
3
Transitive: If △A~△B and △B~△C then △A~△C ✓
4
∴ 'is similar to' is R, S, T → it is an Equivalence Relation. Answer: R, S, T, E
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Day 1 — Homework Questions
HW 1 Write the Universal relation on A = {1, 3, 5}. Verify that it satisfies a − b < 5 for all pairs. How many ordered pairs does it contain?
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HW 2 How many relations are possible from A = {1, 2} to B = {a, b, c}? Use the formula 2^(mn), where |A| = m, |B| = n.
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Day 1 — Homework
HW 1
Write Universal relation on A={1,3,5}; verify using a−b<5.
HW 2
How many relations possible from A={1,2} to B={a,b,c}?
📎 Assignments — Day 1 (Optional · Viewable by students)
Day 1 — Video Resources
Exit Ticket
1Write one example each: empty relation and universal relation on A={1,2,3}
2Is Identity relation on A={a,b} an Equivalence Relation?
ISC Board Tags — Day 1
DAY 2 · TUE 24 FEB 2026
Day 2 — Reflexive, Symmetric & Transitive
Period 3–4 · 80 min
RST · Matrix · Arrow Diagrams
RST Definitions
Reflexive(a,a) ∈ R ∀ a ∈ A | Count: 2^(n²−n)
Symmetric(a,b)∈R ⟹ (b,a)∈R ⟺ R=R⁻¹ | Count: 2^((n²+n)/2)
Transitive(a,b)∈R and (b,c)∈R ⟹ (a,c)∈R
✎Concept Board — Day 2Click to open · Draw diagrams & explain concepts▶
Ex 3 R = {(a, b) : b = a + 1} on {1, 2, 3, 4, 5, 6}. Check R for Reflexive, Symmetric, and Transitive properties.
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1
Write R: y=3x → R={(1,3),(2,6),(3,9),(4,12)}
2
Reflexive? Need 3a−a=0 → 2a=0 → a=0 ∉ A ✗
3
Symmetric? (1,3)∈R but 3(3)−1=8≠0, so (3,1)∉R ✗
4
Transitive? (1,3)∈R,(3,9)∈R but 3(1)−9=−6≠0 so (1,9)∉R ✗
5
∴ R is neither Reflexive, nor Symmetric, nor Transitive.
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Ex 4 R = {(x, y) : x and y work at the same place} on a set of persons. Check R for Reflexive, Symmetric, and Transitive.
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1
Reflexive: a|a for every a → (a,a)∈R ✓
2
Symmetric? (1,2)∈R but is 1 divisible by 2? No ✗
3
Transitive: x|y and y|z ⟹ x|z (standard divisibility) ✓
4
∴ R is Reflexive and Transitive but NOT Symmetric.
✓ Complete
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Ex1A · Q6 R={(a,b): a ≤ b} on ℝ. Show R is Reflexive and Transitive but NOT Symmetric.
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1
Reflexive: a≤a is always true ∴ (a,a)∈R ∀a∈ℝ ✓
2
Symmetric? Let a=1,b=2: (1,2)∈R but 2≤1 is false → (2,1)∉R ✗
3
Transitive: (a,b)∈R and (b,c)∈R → a≤b and b≤c → a≤c → (a,c)∈R ✓
4
∴ R is Reflexive and Transitive but NOT Symmetric.
✓ Complete
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Day 2 — RST Examples (Set 1)
Ex 1 R = {(x, y) : 3x − y = 0} on A = {1, 2, …, 14}. Check whether R is Reflexive, Symmetric, and Transitive.
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Ex 2 R = {(x, y) : y is divisible by x} on A = {1, 2, 3, 4, 5, 6}. Determine which of the properties R / S / T hold.
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Day 2 — RST Examples (Set 3)
Ex 5 R = {(x, y) : x is exactly 7 cm taller than y} on a set of persons. Determine whether R is Reflexive, Symmetric, or Transitive.
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Ex 6 For each, check R / S / T: (a) R = {(x, y) : x is wife of y} (b) R = {(x, y) : x is father of y}.
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Day 2 — Construct Relations with Specific Properties
Ex 7 Construct a relation on a suitable set that is Symmetric but NEITHER Reflexive NOR Transitive. Verify each property explicitly.
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Ex 8 Construct a relation on a suitable set that is Transitive but NEITHER Reflexive NOR Symmetric. Verify each property explicitly.
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Ex 9 Construct a relation on {1, 2, 3} that is Reflexive and Symmetric but NOT Transitive. Display as a set of ordered pairs.
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Ex 10 Construct a relation on a suitable set that is Reflexive and Transitive but NOT Symmetric. Verify each property.
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Ex 11 Construct a relation on {1, 2} that is Symmetric and Transitive but NOT Reflexive. Verify each property.
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Day 2 — Exercise 1(A)
Q 4 Ex 1(A) Q4: Show that R = {(1, 2), (2, 1)} on {1, 2, 3} is Symmetric but neither Reflexive nor Transitive.
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Q 5 R on {1,2,3,4} = {(1,2),(2,2),(1,1),(4,4),(1,3),(3,3),(3,2)}. Choose the correct property from options (a)–(d).
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Q 9 Is the relation 'is the square of' defined on ℕ an Equivalence Relation? Justify with proof or counterexample.
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Day 2 — Homework Questions
HW Q1 Ex 1(A): (a) For 'is the cube of' on ℝ — state all applicable properties. (b) For 'is the sister of' on a set of children — state all applicable properties.
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HW Q2 Write a relation which is: (a) only Transitive (b) only Symmetric (c) only Reflexive and Transitive (d) only Symmetric and Reflexive.
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HW Q7 R = {(a, b) : a ≤ b²} on ℝ. Show that R is neither Reflexive, nor Symmetric, nor Transitive.
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Day 2 Homework
HW 1
R={(a,b): a≤b²} on ℝ. Show R is neither R, S, nor T.
HW 2
Write a relation which is: (a) only T (b) only S (c) only R and T (d) only S and R.
📎 Assignments — Day 2 (Optional · Viewable by students)
Day 2 — Video Resources
Exit Ticket
1Symmetric but neither R nor T — give one example.
2R={(1,2),(2,1)} on {1,2,3}: Is R symmetric? Reflexive? Transitive?
Mark Scheme Note
State property → assume (a,b)∈R → show consequence → conclude. Every step earns marks.
DAY 3 · WED 25 FEB 2026
Day 3 — Equivalence Relations & Classes
Period 5 · 40 min
ER · Equivalence Classes · Partition
Equivalence Relation
DefinitionR is ER ⟺ Reflexive AND Symmetric AND Transitive
Class [a]{b ∈ A : (b,a) ∈ R} — set of all elements related to a
PartitionClasses are mutually disjoint; their union = A
ISC NoteAlways write conclusion sentence — 1 mark awarded by ISC for it
✎Concept Board — Day 3Click to open · Draw diagrams & explain concepts▶
Ex 7 R={(a,b): (a−b) divisible by 5} on ℤ. Prove R is ER. Write equivalence classes [0],[1],[2].
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1
Reflexive: a−a=0=5×0, so 5|(a−a) ∴ (a,a)∈R ✓
2
Symmetric: 5|(a−b) ⟹ 5|−(a−b)=(b−a) ∴ (b,a)∈R ✓
3
Transitive: 5|(a−b) and 5|(b−c) ⟹ 5|(a−b+b−c)=5|(a−c) ✓
4
Conclusion: Since R is Reflexive, Symmetric and Transitive → R is an Equivalence Relation.
5
[0]={…,−10,−5,0,5,10,…} [1]={…,−4,1,6,11,…} [2]={…,−3,2,7,12,…}
6
There are 5 distinct classes [0],[1],[2],[3],[4] — they partition ℤ.
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CBSE 2024 R on A={−4,…,4}, R={(x,y): x+y divisible by 2}. Show R is ER; write equivalence class [2].
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1
Reflexive: x+x=2x, divisible by 2 ✓
2
Symmetric: 2|(x+y) ⟹ 2|(y+x) ✓
3
Transitive: 2|(x+y) and 2|(y+z) ⟹ 2|(x+z+2y) ⟹ 2|(x+z) ✓
4
∴ R is an Equivalence Relation.
5
[2] = all x where x+2 divisible by 2, i.e. x even: [2]={−4,−2,0,2,4}
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Day 3 — Equivalence Relations (Examples Set 1)
Ex 4 R = {(x, y) : x and y have the same number of pages} on all books in a library. Show that R is an Equivalence Relation.
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Ex 5 The congruence relation '≅' on the set of all triangles in Euclidean geometry. Show that it is an Equivalence Relation.
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Day 3 — Equivalence Relations (Examples Set 2)
Ex 6 The subset relation '⊂' with respect to sets. Show that '⊂' is NOT an Equivalence Relation. Identify which property fails.
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Ex 8 S = {(a, b) : |a − b| is divisible by 4} on A = {x ∈ ℤ : 0 ≤ x ≤ 12}. Prove S is an ER. Find all elements related to 1.
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Ex 9 R = {(P, Q) : OP = OQ} in a plane (O = origin). Show R is an ER. Describe the set of all points related to P ≠ (0, 0).
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Day 3 — Equivalence Relations (Examples Set 3 & 4)
Ex 10 Show that the relation '≥' on ℝ is NOT an Equivalence Relation. Identify which property fails and give a counterexample.
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Ex 12 (i) R = {(T₁, T₂) : T₁ is similar to T₂} — show R is an ER. (ii) Among T₁(3,4,5), T₂(5,12,13), T₃(6,8,10) — which triangles are related?
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Day 3 — Exercise 1(A)
Q 3 Prove that the relation 'is brother of' on the set of all family members is a Transitive relation.
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Q 11 R = {(P₁, P₂) : P₁ and P₂ have the same number of sides} on all polygons. Show R is an ER. Find the equivalence class of right-triangle T(3, 4, 5).
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Q 12 R = {(L₁, L₂) : L₁ ∥ L₂} on all lines in the XY-plane. Show R is an ER. Find the set of all lines related to y = 2x + 4.
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Q 15 R = {(a, b) : |a − b| is even} on A = {1, 2, 3, 4, 5}. Prove that R is an Equivalence Relation.
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Q 16 aRb iff (a + b) is even, a, b ∈ ℤ. Prove that R is an Equivalence Relation.
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Day 3 — Homework Questions
HW Ex11 For each relation, write YES or NO for being an Equivalence Relation: (a) parallel to (b) perpendicular to (c) greater than (d) factor of (e) multiple of.
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Day 3 Homework
HW 1
Prove R={(L₁,L₂): L₁∥L₂} on all lines in XY-plane is an ER. Find all lines related to y=2x+4.
HW 2
aRb iff (a+b) is even, a,b∈ℤ. Prove R is an ER.
📎 Assignments — Day 3 (Optional · Viewable by students)
Day 3 — Video Resources
Exit Ticket
1What is [3] under R={(a,b): 3|(a−b)} on ℤ?
2Can two equivalence classes overlap? Prove or disprove.
6M Mark Scheme
1M Reflexive + 1M Symmetric + 1M Transitive + 1M Conclusion + 2M for Equivalence Class
DAY 4 · THU 26 FEB 2026
Day 4 — ℕ×ℕ Equivalence & Review
Period 6 · 40 min
ℕ×ℕ · VSA · True/False
Equivalence on ℕ×ℕ
Standard(a,b)R(c,d) ⟺ ad=bc (cross-multiplication form of a/b=c/d)
ISC 2024(a,b)R(c,d) iff a−c=b−d → same as a−b=c−d
Class[(2,6)]={(1,3),(2,6),(3,9),…} all pairs in ratio 1:3
✎Concept Board — Day 4Click to open · Draw diagrams & explain concepts▶
Ex 13 R on ℕ×ℕ: (a,b)R(c,d) ⟺ ad=bc. Prove R is ER. Find [(2,6)].
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Reflexive: (a,b)R(a,b)? Need ab=ba ✓ (commutative)
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Symmetric: ad=bc ⟹ cb=da ∴ (c,d)R(a,b) ✓
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Transitive: (a,b)R(c,d): ad=bc. (c,d)R(e,f): cf=de.
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Multiply: (ad)(cf)=(bc)(de) ⟹ acdf=bcde ⟹ af=be (divide by cd) ∴ (a,b)R(e,f) ✓
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∴ R is an Equivalence Relation on ℕ×ℕ.
6
[(2,6)]: need 2b=6a → a/b=1/3 ∴ [(2,6)]={(1,3),(2,6),(3,9),(4,12),…}
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Ex 19 R on A×A (A={1..9}): (a,b)R(c,d) iff a+d=b+c. Show R is ER. Find [(2,5)].
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Note: a+d=b+c ⟺ a−b=c−d (same difference)
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Reflexive: a+b=b+a ✓
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Symmetric: a+d=b+c ⟹ c+b=d+a ✓
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Transitive: a+d=b+c and c+f=d+e → adding → a+f=b+e ✓
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∴ R is an Equivalence Relation. Class depends on a−b=2−5=−3.
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[(2,5)]={(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)}
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Day 4 — Equivalence on ℕ×ℕ (Examples Set 1 & 2)
Ex 14 R = {(a, b) : a − b < 5} on ℝ. Is R an Equivalence Relation? Justify by checking all three properties.
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Ex 15 R = {(a, b) : a − b is even, a, b ∈ ℤ}. Show that R is an Equivalence Relation.
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Ex 17 (a, b) R (c, d) iff ad(b + c) = bc(a + d) on ℕ×ℕ. Show that R is an Equivalence Relation.
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Ex 18 R = {(a, b) : a = b, 0 ≤ a ≤ 12} on A = {0, 1, …, 12}. Show R is an ER. Find all elements related to 1.
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Day 4 — CBSE 2024 & Exercise 1(A)
CBSE Q2 CBSE 2024: (a, b) R (c, d) iff a − c = b − d on ℕ×ℕ. Show that R is an Equivalence Relation.
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Q 10 Ex 1(A) Q10(a)–(e): State S / T / R / E / N for each: (a) smaller than (b) father of (c) parallel to (d) multiple of (e) congruent to.
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Q 14 Ex 1(A) Q14: R = {(x, y) : x + y = 10, x, y ∈ ℕ}. Is R Reflexive, Symmetric, Transitive? Examine each property separately.
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Day 4 — Chapter Review: Very Short Answer (VSA)
VSA Q1 Fill in the blank: 'A relation R in set A is called _______ if (a₁, a₂) ∈ R ⟹ (a₂, a₁) ∈ R.'
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VSA Q6 Write the smallest Reflexive relation on A = {a, b, c}. How many ordered pairs does it contain?
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VSA Q8 How many Reflexive relations are possible in a set A with n(A) = 3? Use the formula 2^(n²−n).
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VSA Q9 R = {(1,1),(1,2),(2,2),(3,3)} on S = {1,2,3}. Which element(s) must be removed from R to make R an Equivalence Relation?
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VSA Q11 R = {(x, y) : x + 2y = 8} on ℕ. Write all ordered pairs in R and find the Range of R.
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VSA Q12 R = {(a, a³) : a is a prime number less than 5}. List all ordered pairs in R and find the Range of R.
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Day 4 — True or False
T/F Q16 True or False: R = {(3,1),(1,3),(3,3)} on A = {1,2,3} is Symmetric and Transitive but NOT Reflexive. Justify your answer.
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T/F Q17 True or False: R = {(1,1),(1,2),(2,1),(3,3)} on {1,2,3} is an Equivalence Relation. Identify any missing ordered pairs.
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T/F Q19 True or False: Every relation that is Symmetric and Transitive is also Reflexive. Prove or give a counterexample.
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Day 4 — Homework & Self-Study
HW 1 NCERT Exemplar: For all a, b ∈ ℤ, define aRb iff (a − b) is divisible by n. Show that R is an Equivalence Relation. Also complete all pending Ex 1(A) questions.
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HW 2 Attempt Assertion–Reason questions Q21(i) to Q21(vi) from the textbook.
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HW 3 Read Case Study Q24 (General Elections context) from the textbook and answer ALL parts of the case study carefully.
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Day 4 Homework
HW 1
Complete all pending Ex1(A). NCERT Exemplar: aRb iff (a−b) divisible by n on ℤ — show R is ER.
HW 2
Attempt Assertion–Reason Q21(i)–(vi) from textbook.
📎 Assignments — Day 4 (Optional · Viewable by students)
HW 3
Read Case Study Q24 (General Elections context) and answer all parts.
Day 4 — Video Resources
Final Exit Ticket
1In one sentence: what makes a relation an equivalence relation?
2Write [(1,2)] under (a,b)R(c,d) iff a+d=b+c on ℕ×ℕ.
3How many equivalence relations on {1,2,3}? (Bell number)
Chapter Summary
R(a,a)∈R ∀a
S(a,b)∈R ⟹ (b,a)∈R
T(a,b),(b,c)∈R ⟹ (a,c)∈R
ER+S+T simultaneously
[a]Disjoint partition of A
ISC Class XII Mathematics · Chapter 1: Relations · 16–19 Mar 2026