IB · Mathematics: Analysis & Approaches SL & HL · Number & algebra

Binomial theorem: IB Mathematics: Analysis & Approaches questions with answers

Free Binomial theorem questions for IB Mathematics: Analysis & Approaches (SL & HL), part of Number & algebra. Each question has the answer and the marking points an examiner would look for.

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Binomial theorem practice questions

Question 1Structured · 6 marksA* challenge

In the expansion of (1 + ax)n, where n ∈ ℤ+, the coefficient of x is −20 and the coefficient of x2 is 180.

Find the value of n and the value of a.

[6]
Show answers and mark scheme
PartAnswer and marking pointsMarks
n = 10, a = −2
na = −20 A1
n(n − 1)2a2 = 180 A1
attempt to eliminate a, e.g. n(n − 1)2·400n2 = 180 M1
n − 1n = 910 (A1)
n = 10 A1
a = −2 A1
6
Question 2Structured · 5 marks

Find the constant term in the expansion of (x + 1x2)9.

[5]
Show answers and mark scheme
PartAnswer and marking pointsMarks
84
general term 9Cr(x)9−r(1x2)r (M1)
power of x: (9 − r) − 2r = 0 (M1)
r = 3 A1
9C3(1)6(1)3 (A1)
84 A1
5
Question 3Structured · 5 marks
(a)

Find the first three terms in the expansion of (9 − 2x)1/2 in ascending powers of x.

[4]
(b)

State the set of values of x for which the expansion is valid.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)3 − 13x − 154x2
91/2(1 − 29x)1/2 (M1)
use of (1 + u)n = 1 + nu + n(n − 1)2!u2 + … (M1)
3 − 13x A1
−154x2 A1
4
(b)|x| < 92
|x| < 92 A1
1
Question 4Structured · 6 marksA* challenge

In the expansion of (1 + ax)n, where n ∈ ℤ+, the coefficient of x is 30 and the coefficient of x2 is 405.

Find the value of n and the value of a.

[6]
Show answers and mark scheme
PartAnswer and marking pointsMarks
n = 10, a = 3
na = 30 A1
n(n − 1)2a2 = 405 A1
attempt to eliminate a, e.g. n(n − 1)2·900n2 = 405 M1
n − 1n = 910 (A1)
n = 10 A1
a = 3 A1
6
Question 5Structured · 5 marks

Find the constant term in the expansion of (3x2 − 1x)9.

[5]
Show answers and mark scheme
PartAnswer and marking pointsMarks
2268
general term 9Cr(3x2)9−r(−1x)r (M1)
power of x: 2(9 − r) − r = 0 (M1)
r = 6 A1
9C6(3)3(−1)6 (A1)
2268 A1
5
Question 6Structured · 5 marks
(a)

Find the first three terms in the expansion of (1 − x)−2 in ascending powers of x.

[4]
(b)

State the set of values of x for which the expansion is valid.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)1 + 2x + 3x2
(1 − x)−2 (M1)
use of (1 + u)n = 1 + nu + n(n − 1)2!u2 + … (M1)
1 + 2x A1
3x2 A1
4
(b)|x| < 1
|x| < 1 A1
1

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Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in