IMO Practice Test — Binomial Theorem
6 Questions • 20 min • Olympiad level
20:00
Question 1 of 6
hard
The coefficient of x³ in (1 + x)⁶ + (1 + x)⁵ is:
30
35
25
40
Explanation: ⁶C₃+⁵C₃=20+10=30.
Question 2 of 6
hard
The constant term in (x + 2/x)⁴ is:
24
16
32
6
Explanation: Exponent = 4−2r=0 → r=2. Constant term = ⁴C₂×2²=6×4=24.
Question 3 of 6
hard
What is the constant term in the expansion of (2x − 1/x)⁸?
1120
−1120
17920
−17920
Explanation: T_(r+1) = 8Cr (2x)^(8−r) (−1/x)ʳ. For constant term, 8−2r = 0 → r = 4. 8C4 (2)⁴ (−1)⁴ = 70 × 16 × 1 = 1120.
Question 4 of 6
hard
A constant voltage in a circuit is modeled by the term independent of t in (√t − 1/t²)¹⁰. What is this voltage magnitude?
45
120
210
252
Explanation: T_(r+1) = 10Cr (t^(1/2))^(10−r) (−t⁻²)ʳ = 10Cr t^(5 − r/2 − 2r). For t⁰, 5 − 2.5r = 0 → r = 2. 10C2 (−1)² = 45.
Question 5 of 6
hard
The ratio of the greatest binomial coefficient in (1+x)²ⁿ to the greatest binomial coefficient in (1+x)^(2n−1) is:
1:1
2:1
n:1
2n:1
Explanation: Max in 2n is (2n)Cn. Max in 2n−1 is (2n−1)Cn. Ratio = (2n)!/(n!n!) × (n!(n−1)!)/(2n−1)! = 2n/n = 2. So 2:1.
Question 6 of 6
hard
The coefficient of x³ in (1 + x)⁷ − (1 + x)⁶ is:
15
20
35
5
Explanation: 35−20=15.