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Worksheet: Finding a Specific Coefficient Using the Binomial Theorem

Q1:

Determine the coefficient of π‘Ž 2 in the expansion of ο€Ό π‘Ž 1 1 + 1 1 6 π‘Ž  1 2 .

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

Q2:

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

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

  • A 𝑛 = 1 6
  • B π‘˜ = 4 √ 1 0
  • C π‘˜ = √ 1 6 5
  • D π‘˜ = 4
  • E π‘˜ = 2

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

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

Q3:

Find the coefficient of in the expansion of , where .

  • A
  • B
  • C
  • D

Q4:

In the expansion of a binomial, determine which of the following is equivalent to the relation .

  • A
  • B
  • C
  • D

Q5:

Consider the expansion of ο€Ό π‘₯ 4 + 2 π‘₯  4 1 7 . Is the coefficient of π‘₯ 2 nonzero?

  • Ayes
  • Bno

Q6:

Find the coefficient of π‘₯ 8 in the expansion of  ( 3 + π‘₯ ) + 9 ( 3 + π‘₯ ) ( 6 + π‘₯ ) + 3 6 ( 3 + π‘₯ ) ( 6 + π‘₯ ) + β‹― + ( 6 + π‘₯ )  9 8 7 2 9 .

Q7:

In the expansion of in ascending powers of , denotes the th term.

If, in the expansion of , the coefficients of and are equal, which of the following describes the possible values of ?

  • A or
  • B or
  • C or
  • D or

Q8:

Find the coefficient of π‘₯ 7 in the expansion of ο€Ό 2 + 3 π‘₯ 5  1 1 .

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

Q9:

In an expansion, if coefficient of coefficient of coefficient of , find .

  • A56, 62
  • B47, 56
  • C82, 56
  • D47, 62

Q10:

Consider the expansion of . Find all possible values of such that the coefficients of and are equal.

  • A2, 4
  • B2, 1
  • C3, 1
  • D3, 2

Q11:

Find the coefficient of π‘₯ 5 in the expansion of ( 2 βˆ’ 5 π‘₯ ) 8 .

Q12:

If the order of the term free of π‘₯ in ο€Ό 2 π‘₯ βˆ’ 8 π‘₯  2 1 8 is equal to the term free of π‘₯ in ο€Ό π‘₯ βˆ’ 2 π‘₯  6 2 𝑛 , find 𝑛 .

Q13:

Find the coefficient of the term in the expansion of .

  • A
  • B
  • C
  • D
  • E

Q14:

Use Pascal’s triangle to determine the coefficients of the terms that result from the expansion of ( π‘₯ + 𝑦 ) 6 .

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

Q15:

Determine the coefficient of π‘₯ βˆ’ 6 in the expansion of ο€Ό π‘₯ + 1 π‘₯  2 6 .

Q16:

Determine the coefficient of π‘₯ βˆ’ 1 in the expansion of ο€Ό π‘₯ + 1 π‘₯  4 4 .

Q17:

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

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

Q18:

Find the coefficient of π‘₯ 5 in the expansion of ο€Ή 1 + π‘₯ βˆ’ π‘₯  ( 1 + π‘₯ ) 2 1 8 .

Q19:

Find the coefficient of ο€½ π‘₯ 𝑦  6 in the expansion of ο€½ 2 π‘₯ 𝑦 + 𝑦 2 π‘₯  1 0 .

Q20:

In the expansion of ο€Ό π‘₯ + 1 π‘Ž π‘₯  2 4 , if the coefficient of the middle term equals the coefficient of π‘₯ 5 , find the value of π‘Ž .

  • A3
  • B 2 3
  • C 1 3
  • D 3 2

Q21:

Find the coefficient of π‘ž 1 2 in 2 4 π‘ž ο€Ύ π‘ž 4 + 4 π‘ž  8 2 3 1 7 .

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

Q22:

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

Q23:

Find the coefficient of π‘₯ 2 in ( 1 βˆ’ π‘₯ ) ( 5 βˆ’ 2 π‘₯ ) 6 3 .

Q24:

Find the coefficient of π‘₯ 3 in the expansion of ( 2 + 3 π‘₯ ) 8 .

Q25:

The coefficient of π‘₯ 2 in the expansion of ( 1 + 2 π‘₯ ) 𝑛 is 144. Find the value of 𝑛 .

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