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Lesson: The Binomial Theorem

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14:12

Sample Question Videos

Worksheet • 25 Questions • 1 Video

Q1:

Expand ( 7 + 2 π‘₯ ) 3 .

  • A 8 π‘₯ + 8 4 π‘₯ + 2 9 4 π‘₯ + 3 4 3 3 2
  • B π‘₯ + 2 1 π‘₯ + 1 4 7 π‘₯ + 3 4 3 3 2
  • C βˆ’ π‘₯ + 2 1 π‘₯ βˆ’ 1 4 7 π‘₯ + 3 4 3 3 2
  • D βˆ’ 8 π‘₯ + 8 4 π‘₯ βˆ’ 2 9 4 π‘₯ + 3 4 3 3 2

Q2:

Find the coefficient of π‘₯ 1 0 in the expansion of ο€Ή 1 + π‘₯ βˆ’ π‘₯  2 8 .

Q3:

Consider the expansion of ο€Ή π‘₯ + π‘₯  6 βˆ’ 6 5 in descending powers of π‘₯ . What are the possible values of it, if the third term in this expansion is equal to 640?

  • A 2 , βˆ’ 2
  • B12
  • C 4 , βˆ’ 4
  • D10

Q4:

Answer the following questions for the expansion of ( 1 βˆ’ 3 π‘₯ ) 𝑛 .

Given that the coefficient of π‘₯ 2 is 189, find 𝑛 .

  • A 𝑛 = 7
  • B 𝑛 = 8
  • C 𝑛 = 1 0
  • D 𝑛 = 9
  • E 𝑛 = 6

Hence, work out the value of the coefficient of π‘₯ 5 .

Q5:

Use the binomial theorem to find the expansion of ( 1 + π‘₯ ) 4 .

  • A 1 + 4 π‘₯ + 6 π‘₯ + 4 π‘₯ + π‘₯ 2 3 4
  • B 1 + 4 π‘₯ + 6 π‘₯ + 4 π‘₯ + 4 π‘₯ 2 3 4
  • C 1 + π‘₯ 4
  • D 1 + 3 π‘₯ + 6 π‘₯ + 1 0 π‘₯ + 1 5 π‘₯ 2 3 4
  • E π‘₯ + 4 π‘₯ + 4 π‘₯ + π‘₯ 2 3 4

Q6:

Given that and 2 π‘Ž = 3 π‘Ž 1 2 , find the values of 𝑛 and 𝑐 where 𝑐 β‰  0 .

  • A 𝑛 = 3 , 𝑐 = 2
  • B 𝑛 = 4 , 𝑐 = 2
  • C 𝑛 = 4 , 𝑐 = 3
  • D 𝑛 = 3 , 𝑐 = 3

Q7:

Answer the following questions for the expansion of ( 2 + π‘˜ π‘₯ ) 6 .

Given that the coefficient of π‘₯ 2 is 60, and π‘˜ is positive, find π‘˜ .

  • A π‘˜ = 1 2
  • B π‘˜ = √ 1 5 4
  • C π‘˜ = 2
  • D π‘˜ = 1 4
  • E π‘˜ = 1

Hence, using your value of π‘˜ , work out the coefficient of π‘₯ 5 in the expansion.

  • A 3 8
  • B 1 5
  • C384
  • D 3 2 5 6
  • E12

Q8:

Find the third term in the expansion of ο€Ό 1 0 π‘₯ + 2 3 π‘₯  2 4 .

  • A 8 0 0 3 π‘₯ 2
  • B 8 0 0 3 π‘₯ 4
  • C 4 0 0 9 π‘₯ 4
  • D 4 0 0 9 π‘₯ 2

Q9:

Find the third term in the expansion of ο€Ό 3 π‘₯ + 7 6 π‘₯  2 4 .

  • A 1 4 7 2 π‘₯ 2
  • B 1 4 7 2 π‘₯ 4
  • C 4 9 4 π‘₯ 4
  • D 4 9 4 π‘₯ 2

Q10:

Expand ο€Ό 6 π‘₯ βˆ’ 1 3 π‘₯  2 2 .

  • A 3 6 π‘₯ βˆ’ 4 π‘₯ + 1 9 π‘₯ 4 2
  • B 3 6 π‘₯ βˆ’ 1 2 π‘₯ + 1 π‘₯ 4 2
  • C 3 6 π‘₯ + 4 π‘₯ + 1 9 π‘₯ 4 2
  • D π‘₯ βˆ’ 2 π‘₯ 3 + 1 9 π‘₯ 4 2

Q11:

Expand ( 5 π‘₯ + 4 𝑦 ) 4 .

  • A 6 2 5 π‘₯ + 2 0 0 0 π‘₯ 𝑦 + 2 4 0 0 π‘₯ 𝑦 + 1 2 8 0 π‘₯ 𝑦 + 2 5 6 𝑦 4 3 2 2 3 4
  • B π‘₯ + 4 π‘₯ 𝑦 + 6 π‘₯ 𝑦 + 4 π‘₯ 𝑦 + 𝑦 4 3 2 2 3 4
  • C π‘₯ + 1 6 π‘₯ 𝑦 + 9 6 π‘₯ 𝑦 + 2 5 6 π‘₯ 𝑦 + 2 5 6 𝑦 4 3 2 2 3 4
  • D 6 2 5 π‘₯ + 5 0 0 π‘₯ 𝑦 + 1 5 0 π‘₯ 𝑦 + 2 0 π‘₯ 𝑦 + 𝑦 4 3 2 2 3 4

Q12:

Expand ( π‘₯ + 2 𝑦 ) 2 2 .

  • A π‘₯ + 4 π‘₯ 𝑦 + 4 𝑦 4 2 2
  • B π‘₯ + 4 π‘₯ 𝑦 + 4 𝑦 2 2
  • C π‘₯ + 2 π‘₯ 𝑦 + 𝑦 2 2
  • D π‘₯ + 2 π‘₯ 𝑦 + 𝑦 4 2 2

Q13:

Expand ο€Ό π‘₯ 4 βˆ’ 1 π‘₯  5 .

  • A π‘₯ 1 0 2 4 βˆ’ 5 π‘₯ 2 5 6 + 5 π‘₯ 3 2 βˆ’ 5 8 π‘₯ + 5 4 π‘₯ βˆ’ 1 π‘₯ 5 3 3 5
  • B π‘₯ βˆ’ 5 π‘₯ + 1 0 π‘₯ βˆ’ 1 0 π‘₯ + 5 π‘₯ βˆ’ 1 π‘₯ 5 3 3 5
  • C π‘₯ βˆ’ 2 0 π‘₯ + 1 6 0 π‘₯ βˆ’ 6 4 0 π‘₯ + 1 2 8 0 π‘₯ βˆ’ 1 0 2 4 π‘₯ 1 0 7 4 2 5
  • D π‘₯ βˆ’ 2 0 π‘₯ + 1 6 0 π‘₯ βˆ’ 6 4 0 π‘₯ + 1 2 8 0 π‘₯ βˆ’ 1 0 2 4 1 0 8 6 4 2

Q14:

Answer the following questions for the expansion of ( 2 + 4 π‘₯ ) 𝑛 .

Given that the coefficient of π‘₯ 2 is 3 8 4 0 , find 𝑛 .

  • A 𝑛 = 6
  • B 𝑛 = 9
  • C 𝑛 = 7
  • D 𝑛 = 5
  • E 𝑛 = 8

Hence, work out the value of the coefficient of π‘₯ 5 .

Q15:

Consider the expansion of Given that the constant of this expansion is 720, find all the possible values of π‘Ž .

  • A 4 , βˆ’ 4
  • B 1 6 , βˆ’ 1 6
  • C 8 , βˆ’ 8
  • D 2 , βˆ’ 2

Q16:

Use the binomial theorem to find the expansion of ( π‘Ž + 2 𝑏 ) 4 .

  • A π‘Ž + 8 π‘Ž 𝑏 + 2 4 π‘Ž 𝑏 + 3 2 π‘Ž 𝑏 + 1 6 𝑏 4 3 2 2 3 4
  • B π‘Ž + 4 π‘Ž 𝑏 + 6 π‘Ž 𝑏 + 4 π‘Ž 𝑏 + 𝑏 4 3 2 2 3 4
  • C π‘Ž + 4 π‘Ž 𝑏 + 2 4 π‘Ž 𝑏 + 3 2 π‘Ž 𝑏 + 1 6 𝑏 4 3 2 2 3 4
  • D π‘Ž + 8 π‘Ž 𝑏 + 2 4 π‘Ž 𝑏 + 3 2 π‘Ž 𝑏 + 6 4 𝑏 4 3 2 2 3 4
  • E π‘Ž + 1 6 𝑏 4 4

Q17:

Find π‘₯ given that the ratio of the middle terms in the expansion of ( 1 + π‘₯ ) 3 is 1 ∢ 2 .

Q18:

Find the two middle terms in the expansion of ( 1 4 π‘₯ + 𝑦 ) 3 .

  • A 5 8 8 π‘₯ 𝑦 2 , 4 2 π‘₯ 𝑦 2
  • B 1 9 6 π‘₯ 𝑦 2 , 4 2 π‘₯ 𝑦 2
  • C 1 9 6 π‘₯ 𝑦 2 , 1 4 π‘₯ 𝑦 2
  • D 2 7 4 4 π‘₯ 𝑦 2 , 1 4 π‘₯ 𝑦 2

Q19:

Find the coefficient of the fourth term in the expansion of ο€Ό π‘₯ + 1 π‘₯  4 .

  • A4
  • B8
  • C6
  • D14

Q20:

Which of the following is equal to

  • A 1 4 Γ— 2 1 3
  • B 1 3 Γ— 2 1 4
  • C 1 4 Γ— 2 1 4
  • D 2 1 4
  • E 2 1 3

Q21:

Which of the following is equal to

  • A 1 0 Γ— 2 9
  • B 9 Γ— 2 1 0
  • C 1 0 Γ— 2 1 0
  • D 2 1 0
  • E 2 9

Q22:

Evaluate ο€» √ 3 + 1  + ο€» √ 3 βˆ’ 1  3 3 using the binomial expansion theorem.

  • A 1 2 √ 3
  • B27
  • C12
  • D 2 7 √ 3
  • E36

Q23:

Use the binomial theorem to expand ( 2 π‘₯ βˆ’ 3 𝑦 ) 3 .

  • A 8 π‘₯ βˆ’ 3 6 π‘₯ 𝑦 + 5 4 π‘₯ 𝑦 βˆ’ 2 7 𝑦 3 2 2 3
  • B 8 π‘₯ βˆ’ 1 2 π‘₯ 𝑦 + 1 8 π‘₯ 𝑦 βˆ’ 2 7 𝑦 3 2 2 3
  • C 8 π‘₯ + 3 6 π‘₯ 𝑦 βˆ’ 5 4 π‘₯ 𝑦 + 2 7 𝑦 3 2 2 3
  • D 8 π‘₯ βˆ’ 3 6 π‘₯ 𝑦 βˆ’ 5 4 π‘₯ 𝑦 βˆ’ 2 7 𝑦 3 2 2 3
  • E 8 π‘₯ + 3 6 π‘₯ 𝑦 + 5 4 π‘₯ 𝑦 + 2 7 𝑦 3 2 2 3

Q24:

Use the binomial theorem to find the expansion of ( π‘Ž βˆ’ 𝑏 ) 5 .

  • A π‘Ž βˆ’ 5 π‘Ž 𝑏 + 1 0 π‘Ž 𝑏 βˆ’ 1 0 π‘Ž 𝑏 + 5 π‘Ž 𝑏 βˆ’ 𝑏 5 4 3 2 2 3 4 5
  • B π‘Ž + 5 π‘Ž 𝑏 βˆ’ 1 0 π‘Ž 𝑏 + 1 0 π‘Ž 𝑏 βˆ’ 5 π‘Ž 𝑏 + 𝑏 5 4 3 2 2 3 4 5
  • C π‘Ž + 5 π‘Ž 𝑏 + 1 0 π‘Ž 𝑏 + 1 0 π‘Ž 𝑏 + 5 π‘Ž 𝑏 + 𝑏 5 4 3 2 2 3 4 5
  • D π‘Ž βˆ’ 5 π‘Ž 𝑏 βˆ’ 1 0 π‘Ž 𝑏 βˆ’ 1 0 π‘Ž 𝑏 βˆ’ 5 π‘Ž 𝑏 βˆ’ 𝑏 5 4 3 2 2 3 4 5
  • E 5 π‘Ž βˆ’ 5 π‘Ž 𝑏 + 1 0 π‘Ž 𝑏 βˆ’ 1 0 π‘Ž 𝑏 + 5 π‘Ž 𝑏 βˆ’ 𝑏 5 4 3 2 2 3 4 5

Q25:

Write the coefficients of the terms that result from the expansion of ( π‘₯ + 𝑦 ) 4 .

  • A 1 , 4 , 6 , 4 , 1
  • B 1 , 2 , 1
  • C 1 , 5 , 1 0 , 5 , 1
  • D 1 , 3 , 3 , 1
  • E 1 , 4 , 4 , 1
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