Question 1
Expand (1−2x)^5.
Steps only
Use coefficients 1,5,10,10,5,1 and alternate signs from (−2x)^r.
Answer
1−10x+40x²−80x³+80x⁴−32x⁵.
CLASS 11 · MATHEMATICS · CHAPTER 7
Binomial Theorem
Use binomial coefficients to expand powers, evaluate nearby integers and prove divisibility results.
19 worked answersAll 1 exercise setsAll 6 end-of-chapter questions
What you will learn
- Apply the binomial theorem for positive integral powers.
- Recognise coefficients and alternating signs.
- Use expansions for numerical evaluation and algebraic identities.
19 worked answers · short steps first, explanations when you need them
Exercise 7.1
Textbook pages 133
Question 2
Expand (2/x − x/2)^5.
Steps only
Use coefficients 1,5,10,10,5,1 with a=2/x and b=−x/2.
The successive terms are 32/x⁵, −40/x³, 20/x, −5x, 5x³/8 and −x⁵/32.
Combine the terms in descending powers to write the expansion.
Answer
32/x⁵ − 40/x³ + 20/x − 5x + 5x³/8 − x⁵/32.
Question 3
Expand (2x−3)^6.
Steps only
Use (a+b)^6 with a=2x, b=−3: coefficients are 1,6,15,20,15,6,1.
Answer
64x⁶−576x⁵+2,160x⁴−4,320x³+4,860x²−2,916x+729.
Question 4-5
Expand (x/3 + 1/x)^5 and (x + 1/x)^6.
Steps only
For the first expression, use coefficients 1,5,10,10,5,1 with a=x/3 and b=1/x.
The successive powers are x⁵/243, 5x³/81, 10x/27, 10/(9x), 5/(3x³), and 1/x⁵.
For the second, use coefficients 1,6,15,20,15,6,1; each term has x-power 6−2r.
Answer
(x/3 + 1/x)^5 = x⁵/243 + 5x³/81 + 10x/27 + 10/(9x) + 5/(3x³) + 1/x⁵; (x + 1/x)^6 = x⁶ + 6x⁴ + 15x² + 20 + 15/x² + 6/x⁴ + 1/x⁶.
Question 6
Use the binomial theorem to evaluate 96³.
Steps only
(100−4)³=100³−3(100²)(4)+3(100)(16)−64=1,000,000−120,000+4,800−64.
Answer
884,736.
Question 7
Use the binomial theorem to evaluate 102⁵.
Steps only
(100+2)^5=100^5+5·100^4·2+10·100^3·4+10·100^2·8+5·100·16+32.
Answer
11,040,808,032.
Question 8
Use the binomial theorem to evaluate 101⁴.
Steps only
(100+1)^4=100^4+4·100^3+6·100²+4·100+1.
Answer
104,060,401.
Question 9
Use the binomial theorem to evaluate 99⁵.
Steps only
(100−1)^5=100^5−5·100^4+10·100^3−10·100²+5·100−1.
Answer
9,509,900,499.
Question 10
Which is larger: (1.1)^10000 or 1000?
Steps only
By the binomial theorem, (1+0.1)^10000 is greater than its first two positive terms: 1+10000(0.1)=1001, already greater than 1000.
Answer
(1.1)^10000 is larger.
Question 11
Find (a+b)^4−(a−b)^4. Hence evaluate (√2+√3)^4−(√2−√3)^4.
Steps only
In the two expansions, even powers of b cancel and odd powers double: 8a³b+8ab³=8ab(a²+b²). Substitute a=√2,b=√3.
Answer
8ab(a²+b²); the value is 40√6.
Question 12
Find (x+1)^6+(x−1)^6. Hence evaluate it at x=√2.
Steps only
On adding the expansions, odd powers of 1 cancel: 2[x⁶+15x⁴+15x²+1]. Put x²=2 to get 2(8+60+30+1)=198.
Answer
2x⁶+30x⁴+30x²+2; at x=√2 the value is 198.
Question 13
Show 9^(n+1)−8n−9 is divisible by 64 for every positive integer n.
Steps only
Write 9^(n+1)=(1+8)^(n+1). Every term after 1+8(n+1) contains 8²=64. Subtracting 8n+9 from 1+8n+8 leaves 0 modulo 64.
Answer
It is divisible by 64.
Question 14
Prove the binomial-coefficient sum printed as Exercise 7.1, Question 14.
Steps only
Expand (1+1)^n by the binomial theorem. Confirm the right-hand side in the printed problem before using this as its exact answer.
Answer
The summation’s right-hand side is unclear in the extracted source. For the standard identity, setting a=b=1 gives Σ(r=0 to n)nCr=2^n.
Miscellaneous Exercise
Textbook pages 133–134
Question 1
For distinct integers a,b, prove a−b is a factor of a^n−b^n for positive integer n.
Steps only
Write a=(a−b)+b and expand: a^n−b^n=Σ(k=1 to n)nCk(a−b)^k b^(n−k). Every term contains a−b.
Answer
a^n−b^n is divisible by a−b.
Question 2
Evaluate the expression in Miscellaneous Exercise 7.2 involving (√3+√2)^6 and (√3−√2)^6.
Steps only
Expand using (a±b)^6. Adding cancels odd-power terms; subtracting cancels even-power terms. The exact sign must be read from the printed expression.
Answer
The scan does not clearly preserve whether the expressions are added or subtracted; the two results differ.
Question 3
Simplify the fourth-power expression in Miscellaneous Exercise 7.3.
Steps only
For expressions (a+b)^4±(a−b)^4, use expansion: the sum keeps even powers and the difference keeps odd powers.
Answer
The fraction exponents and signs are not legible in the extracted source, so an exact simplification cannot be verified.
Question 4
Approximate (0.99)^5 using the first three binomial-expansion terms.
Steps only
Write (0.99)^5=(1−0.01)^5. First three terms: 1−5(0.01)+10(0.01)²=1−0.05+0.001=0.951.
Answer
Approximately 0.951.
Question 5
Expand (x/2+1/x)^4, x≠0.
Steps only
Use coefficients 1,4,6,4,1 with terms (x/2)^(4−r)(1/x)^r. Since x≠0, all powers are defined.
Answer
x⁴/16+x²/2+3/2+2/x²+1/x⁴.
Question 6
Expand (3x²−2ax+3a²)^3 using the binomial theorem.
Steps only
Group as 3(x²+a²)−2ax and apply (u+v)³=u³+3u²v+3uv²+v³; then collect matching powers.
Answer
27x⁶−54ax⁵+117a²x⁴−116a³x³+117a⁴x²−54a⁵x+27a⁶.