Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.

Lesson: Finding a Specific Coefficient Using the Binomial Theorem

Sample Question Videos

Worksheet • 25 Questions • 1 Video

Q1:

Determine the coefficient of ๐‘Ž 2 in the expansion of ๏€ผ ๐‘Ž 1 1 + 1 1 6 ๐‘Ž ๏ˆ 1 2 .

  • A 1 1 1 8 8
  • B 5 9 5 0 4
  • C0
  • D 7 7 1 2 8

Q2:

Answer the following questions for the expansion of ( 1 + ๐‘˜ ๐‘ฅ ) 5 .

Given that the coefficient of ๐‘ฅ 2 is 160, and ๐‘˜ is positive, find ๐‘˜ .

  • A ๐‘˜ = 4
  • B ๐‘˜ = 2
  • C ๐‘˜ = 4 โˆš 1 0
  • D ๐‘˜ = โˆš 1 6 5
  • E ๐‘› = 1 6

Hence, using your value of ๐‘˜ , work out the first three terms in ascending powers of ๐‘ฅ in the expansion.

  • A 1 + 2 0 ๐‘ฅ + 1 6 0 ๐‘ฅ 2
  • B 1 + 1 0 ๐‘ฅ + 4 0 ๐‘ฅ 2
  • C 1 โˆ’ 2 0 ๐‘ฅ + 1 6 0 ๐‘ฅ 2
  • D 1 + 8 0 ๐‘ฅ + 2 5 6 0 ๐‘ฅ 2
  • E 1 + 2 0 โˆš 1 0 ๐‘ฅ + 4 0 โˆš 1 0 ๐‘ฅ 2

Q3:

Find the coefficient of ๐‘Ž ๏Šญ ๏‰ in the expansion of ๏€ผ ๐‘Ž โˆ’ 1 2 ๐‘Ž ๏ˆ ๏Šฎ ๏Šจ ๏‰ , where ๐‘š โˆˆ ๐‘ ๏Šฐ .

  • A ๏Šจ ๏‰ ๏‰ ๏‰ ๐ถ ๏€ผ โˆ’ 1 2 ๏ˆ
  • B ๏Šจ ๏‰ ๏‰ ๏Šญ ๏‰ ๐ถ ๏€ผ โˆ’ 1 2 ๏ˆ
  • C ๏Šญ ๏‰ ๏‰ ๏Šฐ ๏Šจ ๏Šจ ๏‰ ๐ถ ๏€ผ โˆ’ 1 2 ๏ˆ
  • D ๏Šญ ๏‰ ๏‰ ๏‰ ๐ถ ๏€ผ โˆ’ 1 2 ๏ˆ

Q4:

In the expansion of a binomial, determine which of the following is equivalent to the relation 2 ( ๐‘‡ ) = ๐‘‡ + ๐‘‡ c o e ๏ฌƒ c i e n t o f c o e ๏ฌƒ c i e n t o f c o e ๏ฌƒ c i e n t o f 1 0 9 1 1 .

  • A 2 ( ๐‘‡ ) = ๐‘‡ + ๐‘‡ c o - c o - c o - 6 7 5
  • B 2 ( ๐‘‡ ) = ๐‘‡ + ๐‘‡ c o - c o - c o - 5 6 4
  • C 2 ( ๐‘‡ ) = ๐‘‡ + ๐‘‡ c o - c o - c o - 4 5 3
  • D 2 ( ๐‘‡ ) = ๐‘‡ + ๐‘‡ c o - c o - c o - 7 8 6

Q5:

Consider the expansion of ๏€ผ ๐‘ฅ 4 + 2 ๐‘ฅ ๏ˆ 4 1 7 . Is the coefficient of ๐‘ฅ 2 nonzero?

  • Ano
  • Byes

Q6:

Find the coefficient of ๐‘ฅ 8 in the expansion of ๏‘ ( 3 + ๐‘ฅ ) + 9 ( 3 + ๐‘ฅ ) ( 6 + ๐‘ฅ ) + 3 6 ( 3 + ๐‘ฅ ) ( 6 + ๐‘ฅ ) + โ‹ฏ + ( 6 + ๐‘ฅ ) ๏ 9 8 7 2 9 .

Q7:

In the expansion of ( 1 + ๐‘ฅ ) 3 ๐‘› in ascending powers of ๐‘ฅ , ๐‘‡ ๐‘˜ denotes the ๐‘˜ th term.

If, in the expansion of ( 1 + ๐‘ฅ ) 3 ๐‘› , the coefficients of ๐‘‡ ๐‘Ÿ + 2 and ๐‘‡ 2 ๐‘Ÿ โˆ’ 5 are equal, which of the following describes the possible values of ๐‘Ÿ ?

  • A ๐‘Ÿ = 7 or ๐‘Ÿ โˆ’ ๐‘› = 1
  • B ๐‘Ÿ = 7 or 3 ( ๐‘Ÿ + ๐‘› ) = 7
  • C ๐‘Ÿ = 7 or 3 ( ๐‘Ÿ โˆ’ ๐‘› ) = 7
  • D ๐‘Ÿ = 3 or 3 ( ๐‘Ÿ โˆ’ ๐‘› ) = 7

Q8:

Find the coefficient of ๐‘ฅ 7 in the expansion of ๏€ผ 2 + 3 ๐‘ฅ 5 ๏ˆ 1 1 .

  • A 2 3 0 9 4 7 2 1 5 6 2 5
  • B 2 1 8 7 7 8 1 2 5
  • C 9 2 3 7 8 8 8 1 5 6 2 5
  • D 8 4 4 8 1 5 6 2 5
  • E 4 6 1 8 9 4 4 0

Q9:

In an expansion, if 2 ( coefficient of ๐‘‡ ) = 2 8 coefficient of ๐‘‡ + 2 7 coefficient of ๐‘‡ 2 9 , find ๐‘› .

  • A47, 62
  • B47, 56
  • C82, 56
  • D56, 62

Q10:

Consider the expansion of ( 1 + ๐‘ฅ ) 2 3 . Find all possible values of ๐‘Ÿ such that the coefficients of ๐‘‡ 2 ๐‘Ÿ + 7 and ๐‘‡ ๐‘Ÿ + 9 are equal.

  • A3, 2
  • B2, 1
  • C3, 1
  • D2, 4

Q11:

Find the coefficient of ๐‘ฅ 5 in the expansion of ( 2 โˆ’ 5 ๐‘ฅ ) 8 .

Q12:

If the order of the term free of ๐‘ฅ in ๏€ผ 2 ๐‘ฅ โˆ’ 8 ๐‘ฅ ๏ˆ 2 1 8 is equal to the term free of ๐‘ฅ in ๏€ผ ๐‘ฅ โˆ’ 2 ๐‘ฅ ๏ˆ 6 2 ๐‘› , find ๐‘› .

Q13:

Find the coefficient of the ๐‘Ÿ t h term in the expansion of ๏€ฟ ๐‘ฅ โˆš 1 1 โˆ’ 1 ๐‘ฅ โˆš 1 1 ๏‹ 7 4 2 ๐‘› .

  • A ( โˆ’ 1 ) ร— ๐ถ ร— 1 1 ๐‘Ÿ โˆ’ 1 2 ๐‘› ๐‘Ÿ โˆ’ 1 ๐‘› โˆ’ ๐‘Ÿ + 1
  • B ( โˆ’ 1 ) ร— ๐ถ ร— ๏€ป โˆš 1 1 ๏‡ ๐‘Ÿ โˆ’ 1 2 ๐‘› ๐‘Ÿ โˆ’ 1 ๐‘› โˆ’ ๐‘Ÿ + 1
  • C 2 ๐‘› ๐‘Ÿ โˆ’ 1 ๐‘› โˆ’ ๐‘Ÿ + 1 ๐ถ ร— 1 1
  • D 2 ๐‘› ๐‘Ÿ ๐‘› โˆ’ ๐‘Ÿ ๐ถ ร— 1 1
  • E ( โˆ’ 1 ) ร— ๐ถ ร— 1 1 ๐‘Ÿ โˆ’ 1 2 ๐‘› ๐‘Ÿ โˆ’ 1 2 ๐‘› โˆ’ ๐‘Ÿ + 1

Q14:

Use Pascalโ€™s triangle to determine the coefficients of the terms that result from the expansion of ( ๐‘ฅ + ๐‘ฆ ) 6 .

  • A 1 , 6 , 1 5 , 2 0 , 1 5 , 6 , 1
  • B 1 , 3 , 6 , 1 0 , 1 5 , 2 1 , 2 8
  • C 1 , 5 , 1 0 , 1 0 , 5 , 1
  • D 1 , 6 , 6 , 6 , 6 , 6 , 1
  • E 1 , 6 , 7 , 1 3 , 7 , 6 , 1

Q15:

Determine the coefficient of ๐‘ฅ โˆ’ 6 in the expansion of ๏€ผ ๐‘ฅ + 1 ๐‘ฅ ๏ˆ 2 6 .

Q16:

Determine the coefficient of ๐‘ฅ โˆ’ 1 in the expansion of ๏€ผ ๐‘ฅ + 1 ๐‘ฅ ๏ˆ 4 4 .

Q17:

Consider the expansion of ( ๐‘Ž ๐‘ฅ + ๐‘ ) 4 in descending powers of ๐‘ฅ . Given that the coefficient of the third term is 2 7 9 8 , find all possible values of ๐‘Ž ๐‘ .

  • A 3 1 4 , โˆ’ 3 1 4
  • B 2 7 , โˆ’ 2 7
  • C 3 5 , โˆ’ 3 5
  • D 2 3 , โˆ’ 2 3
  • E 2 5 , โˆ’ 2 5

Q18:

Find the coefficient of ๐‘ฅ 5 in the expansion of ๏€น 1 + ๐‘ฅ โˆ’ ๐‘ฅ ๏… ( 1 + ๐‘ฅ ) 2 1 8 .

Q19:

Find the coefficient of ๏€ฝ ๐‘ฅ ๐‘ฆ ๏‰ 6 in the expansion of ๏€ฝ 2 ๐‘ฅ ๐‘ฆ + ๐‘ฆ 2 ๐‘ฅ ๏‰ 1 0 .

Q20:

In the expansion of ๏€ผ ๐‘ฅ + 1 ๐‘Ž ๐‘ฅ ๏ˆ 2 4 , if the coefficient of the middle term equals the coefficient of ๐‘ฅ 5 , find the value of ๐‘Ž .

  • A 3 2
  • B 2 3
  • C 1 3
  • D3

Q21:

Find the coefficient of ๐‘ž 1 2 in 2 4 ๐‘ž ๏€พ ๐‘ž 4 + 4 ๐‘ž ๏Š 8 2 3 1 7 .

  • A 4 6 4 1 1 6
  • B 1 5 4 7 2 5 6
  • C 4 6 4 1 3 2
  • D 1 5 4 7 1 2 8
  • E 7 2 9 3 1 6

Q22:

Find the coefficient of ๐‘ฅ 8 in the expansion of ๏€ผ ๐‘ฅ + 2 ๐‘ฅ ๏ˆ ๏€ผ ๐‘ฅ โˆ’ 2 ๐‘ฅ ๏ˆ 1 0 1 0 .

Q23:

Find the coefficient of ๐‘ฅ 2 in ( 1 โˆ’ ๐‘ฅ ) ( 5 โˆ’ 2 ๐‘ฅ ) 6 3 .

Q24:

Find the coefficient of ๐‘ฅ 3 in the expansion of ( 2 + 3 ๐‘ฅ ) 8 .

Q25:

The coefficient of ๐‘ฅ 2 in the expansion of ( 1 + 2 ๐‘ฅ ) ๐‘› is 144. Find the value of ๐‘› .

  • A ๐‘› = 9
  • B ๐‘› = 7
  • C ๐‘› = 6
  • D ๐‘› = 1 0
  • E ๐‘› = 8
Preview