How to Use This Page
1. Solve first, then verify. Attempt every exercise problem on your own before checking the solutions here. Struggling through the problem is how you learn. 2. Compare your method with the step-by-step working shown. If your answer differs, rework from the step where you diverged — do not simply copy the correct answer. 3. Memorise the formulas in Key Concepts first — they are the foundation for every problem type in this chapter. 4. Pay close attention to the Common Mistakes section — these are the exact errors examiners mark down for in TN Board exams. 5. Use the Exam Checklist before your test to confirm you have covered every topic in the chapter.
Key Concepts (Quick Recap)
Sets
| Concept | Definition / Formula | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Set | A well-defined collection of distinct objects | ||||||||
| Subset | A ⊆ B if every element of A is also in B | ||||||||
| Proper subset | A ⊂ B if A ⊆ B and A ≠ B | ||||||||
| Universal set | U — the set containing all elements under consideration | ||||||||
| Union | A ∪ B = {x : x ∈ A or x ∈ B} | ||||||||
| Intersection | A ∩ B = {x : x ∈ A and x ∈ B} | ||||||||
| Difference | A − B = {x : x ∈ A and x ∉ B} | ||||||||
| Complement | A′ = U − A | ||||||||
| Power set | P(A) = set of all subsets of A; \ | P(A)\ | = 2^n where n = \ | A\ | |||||
| Cardinality formula | \ | A ∪ B\ | = \ | A\ | + \ | B\ | − \ | A ∩ B\ |
De Morgan's Laws:
Relations
| Type | Condition |
|---|---|
| Reflexive | (a, a) ∈ R for every a ∈ A |
| Symmetric | (a, b) ∈ R ⇒ (b, a) ∈ R |
| Transitive | (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R |
| Equivalence | Reflexive + Symmetric + Transitive |
Functions
| Type | Condition |
|---|---|
| One-one (Injective) | f(a) = f(b) ⇒ a = b (distinct inputs give distinct outputs) |
| Onto (Surjective) | For every y ∈ B, ∃ x ∈ A such that f(x) = y |
| Bijective | Both one-one and onto |
| Constant function | f(x) = c for all x in the domain |
| Identity function | f(x) = x |
Sequences and Series — Arithmetic Progression (AP)
Sequences and Series — Geometric Progression (GP)
Permutations and Combinations
Binomial Theorem
Logarithms
I. Choose the Best Answer (MCQs)
Explanation:
Explanation:
II. Fill in the Blanks
1. If A = {a, b, c}, then the number of proper subsets of A is 7 (2^3 − 1 = 7; exclude the set itself).
2. If f(x) = 2x + 3, then f(4) = 11 (2 × 4 + 3 = 11).
3. The common difference of the AP 5, 8, 11, 14, … is 3 (8 − 5 = 3).
4. The sum of the first 20 natural numbers is 210 (Sn = n(n + 1)/2 = 20 × 21/2 = 210).
5. The common ratio of the GP 2, 6, 18, 54, … is 3 (6/2 = 3).
6. The value of 5P0 is 1 (nPr = n!/(n−r)!; 5P0 = 5!/5! = 1).
7. The value of nC0 is 1 for any positive integer n (choosing 0 items from n items can be done in exactly 1 way).
8. The 5th term in the expansion of (a + b)^7 is 35a^3 b^4 (T5 = 7C4 · a^3 · b^4 = 35 a^3 b^4).
9. log_10 1000 = 3 (10^3 = 1000).
10. If A ∩ B = ∅, then A and B are disjoint sets (they have no common elements).
11. The sum to infinity of the GP 1, 1/2, 1/4, 1/8, … is 2 (S∞ = a/(1 − r) = 1/(1 − 1/2) = 2).
12. The number of combinations of n objects taken r at a time where r > n is 0 (you cannot choose more objects than are available).
III. True or False
1. Statement: Every subset of a set is a proper subset.
False. A set is a subset of itself (A ⊆ A), but it is not a proper subset. A proper subset must be strictly contained: A ⊂ B means A ⊆ B and A ≠ B.
2. Statement: Every one-one function is onto.
False. A one-one (injective) function maps distinct inputs to distinct outputs, but the range may be smaller than the codomain. For example, f: R → R defined by f(x) = e^x is one-one but not onto, because negative values are never attained.
3. Statement: The sum of the first n terms of an AP is always given by Sn = n/2 [2a + (n − 1)d].
True. This is the standard formula for the sum of an AP. Here a is the first term and d is the common difference. It can also be written as Sn = n/2 (a + l) where l is the last term.
4. Statement: nCr + nC(r−1) = (n+1)Cr.
True. This is Pascal's identity, a fundamental property of combinations. It is used in constructing Pascal's triangle and in many combinatorial proofs.
5. Statement: The binomial expansion of (a + b)^n has n + 2 terms.
False. The expansion of (a + b)^n has exactly n + 1 terms, corresponding to r = 0, 1, 2, …, n.
IV. Match the Following
| Column A | Column B | ||
|---|---|---|---|
| 1. A ∪ A′ | (d) Universal set U | ||
| 2. nPr / r! | (f) nCr | ||
| 3. Tn of AP | (a) a + (n − 1)d | ||
| 4. S∞ of GP (when \ | r\ | < 1) | (e) a/(1 − r) |
| 5. General term in binomial expansion | (c) nCr a^(n−r) b^r | ||
| 6. log_a a | (b) 1 |
Matches: 1 → d, 2 → f, 3 → a, 4 → e, 5 → c, 6 → b
V. Answer Briefly (Two-Mark Questions)
VI. Answer in Detail (Five-Mark Questions)
1. Worked AP/GP Sum Problem
Question: Find the sum of the first 15 terms of the AP 5, 9, 13, 17, … Also, find the sum of the first 6 terms of the GP 3, 6, 12, 24, … Hence find the combined total.
2. Worked Permutation/Combination Counting Problem
Question: A committee of 5 is to be formed from 7 men and 6 women. In how many ways can this be done if (i) there is no restriction, (ii) the committee must include exactly 3 women, (iii) the committee must include at least 2 men?
(i) No restriction:
(ii) Exactly 3 women (and therefore 2 men):
(iii) At least 2 men (i.e., 2, 3, 4, or 5 men):
3. Worked Binomial Expansion Problem
Question: Expand (2x + 3y)^5 using the binomial theorem and find the coefficient of x^3 y^2.
Full expansion:
Coefficient of x^3 y^2:
Common Mistakes to Avoid
1. Confusing nPr with nCr. Permutations (nPr) count arrangements where order matters; combinations (nCr) count selections where order does not. Always ask: "Does the order matter?" before choosing the formula. Arranging people in a row = permutation; choosing a committee = combination.
2. Forgetting that 0! = 1. Many students write 0! = 0. This is wrong. By convention and definition, 0! = 1. This directly affects calculations in nPr and nCr.
3. Misapplying the AP sum formula. Remember Sn = n/2 [2a + (n − 1)d]. A common error is using n instead of (n − 1) in the formula, or forgetting to divide by 2. Double-check your substitution.
4. Using the wrong GP sum formula based on r. When r > 1, use Sn = a(r^n − 1)/(r − 1). When 0 < r < 1, use Sn = a(1 − r^n)/(1 − r). Mixing these up gives a negative denominator, leading to wrong answers.
| 5. **Ignoring the condition | r | < 1 for infinite GP sum.** The formula S∞ = a/(1 − r) is valid only when | r | < 1. If | r | ≥ 1, the infinite sum diverges — do not apply this formula. |
|---|
6. Wrong middle term in binomial expansion. When n is even, there is one middle term: the (n/2 + 1)th term. When n is odd, there are two middle terms. Students often forget to add 1 to n/2 for the even case.
7. Confusing the general term index. In the binomial expansion, T(r+1) = nCr · a^(n−r) · b^r. The index starts from r = 0 (the first term), not r = 1. So the 3rd term uses r = 2, not r = 3.
8. Forgetting De Morgan's laws in set problems. When asked to find the complement of a union or intersection, apply (A ∪ B)′ = A′ ∩ B′ or (A ∩ B)′ = A′ ∪ B′. Do not distribute the complement to only one set — it applies to both.
Exam Checklist
- [ ] I can define sets, subsets, power sets, and universal sets and state their properties
- [ ] I can perform union, intersection, difference, and complement operations on sets
- [ ] I can state and apply De Morgan's laws to simplify set expressions
- [ ] I can identify whether a relation is reflexive, symmetric, transitive, or an equivalence relation
- [ ] I can classify functions as one-one, onto, or bijective and give examples of each type
- [ ] I can find the nth term and sum of n terms of an AP using the correct formula
- [ ] I can find the nth term and sum of n terms of a GP, and the sum to infinity when |r| < 1
- [ ] I can distinguish between permutations and combinations and compute nPr and nCr correctly
- [ ] I can expand (a + b)^n using the binomial theorem and identify any term or coefficient
- [ ] I can apply logarithm properties to simplify expressions and solve logarithmic equations