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Worksheet: Middle Term in the Binomial Theorem

Q1:

Find the middle term in the expansion of ( 5 π‘Ž + 7 ) 2 .

  • A βˆ’ 7 0 π‘Ž
  • B 1 2 π‘Ž
  • C βˆ’ 1 2 π‘Ž
  • D 7 0 π‘Ž

Q2:

Find the middle term in the expansion of ( 1 5 π‘Ž + 3 𝑏 ) 2 .

  • A 4 5 π‘Ž 𝑏
  • B βˆ’ 9 0 π‘Ž 𝑏
  • C βˆ’ 4 5 π‘Ž 𝑏
  • D 9 0 π‘Ž 𝑏

Q3:

For which values of π‘₯ are the two middle terms of ο€Ύ 1 6 π‘₯ + π‘₯ 6 2 5  3 1 9 equal?

  • A 2 5 , βˆ’ 2 5
  • B 1 0 0 , βˆ’ 1 0 0
  • C 5 0 , βˆ’ 5 0
  • D 1 0 , βˆ’ 1 0

Q4:

Given that the middle term in the expansion of ο€» π‘₯ + π‘š π‘₯  8 is 3 5 8 , find ( 2 π‘š βˆ’ 2 ) 6 .

Q5:

Consider the expansion of ο€Ό 2 π‘₯ + 1 4 π‘₯  2 2 3 . Are the middle terms of the expansion equal when π‘₯ = 1 6 ?

  • Ano
  • Byes

Q6:

Given that the middle term in the expansion of ( 6 + π‘₯ ) 2 4 is equal to 2 Γ— π‘Ž 1 4 , find the value of π‘₯ .

  • A13
  • B 1 3 2
  • C 3 3 1 4
  • D 1 3 4
  • E 3 3 7

Q7:

Find the result of dividing the lower-order middle term by the higher one in the expansion of ο€Ό 6 π‘₯ + 1 3 π‘₯  3 5 .

  • A 1 9 π‘₯ 2 0
  • B 1 2 π‘₯ 4
  • C 1 2 π‘₯ 2 0
  • D 1 8 π‘₯ 4

Q8:

In the expansion of ( π‘Ž π‘₯ + 𝑏 ) 2 𝑛 + 9 , the two middle terms are equal when π‘₯ = 9 . Which of the following represents the correct relation between π‘Ž and 𝑏 ?

  • A π‘Ž 𝑏 = 9
  • B π‘Ž = 9 𝑏
  • C π‘Ž 𝑏 = 1 9
  • D 𝑏 = 9 π‘Ž

Q9:

Determine the value of π‘₯ if the value of the middle term in the expansion of ο€Ό 6 π‘₯ + 1 1 3 π‘₯  2 6 is 4 320.

Q10:

Find the middle term in the expansion of ο€Ή 8 π‘₯ + 1 3 π‘₯  4 βˆ’ 4 4 .

Q11:

Consider the expansion of ( π‘₯ βˆ’ 𝑦 ) ( 𝑛 βˆ’ 5 ) . Find the value of 𝑛 given that the middle terms of the expansion are π‘Ž 1 0 and π‘Ž 1 1 .

Q12:

Consider the expansion of ( 5 π‘₯ βˆ’ 9 ) 2 𝑛 + 1 . Find the value of π‘₯ given that the middle terms of the expansion are equal.

  • A1
  • B βˆ’ 5 9
  • C 9 5
  • D βˆ’ 9 5

Q13:

Consider the expansion of ο€Ύ π‘₯ 9 βˆ’ 8 6 4 π‘₯  4 3 5 in descending powers of π‘₯ . For which values of π‘₯ is the sum of the two middle terms equal to zero?

  • A19
  • B36
  • C38
  • D6

Q14:

What is the middle term in the following expansion:

  • A
  • B
  • C
  • D

Q15:

Find the two middle terms in the expansion of ο€Ό 1 4 π‘₯ + 1 3 π‘₯  5 3 .

  • A 1 9 6 π‘₯ 9 , 1 4 9 π‘₯ 3
  • B 1 9 6 3 π‘₯ 9 , 1 4 9 π‘₯ 3
  • C 1 9 6 3 π‘₯ 9 , 1 4 3 π‘₯ 3
  • D 1 9 6 π‘₯ 9 , 1 4 3 π‘₯ 3

Q16:

Find the two middle terms in the expansion of ο€Ό 2 π‘₯ + 1 1 2 π‘₯  4 2 5 .

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

Q17:

Given that the two middle terms are equal in the expansion find all possible real values of π‘₯ .

  • A βˆ’ 3 2 ,
  • B 6 βˆ’ 5 ,
  • C βˆ’ 6 5 ,
  • D 3 βˆ’ 2 ,

Q18:

Determine if is the middle term in the binomial expansion of .

Q19:

What is the value of π‘˜ ∈ β„€ + that makes the middle term of the binomial expansion of the following independent of π‘₯ ?

Q20:

Consider the expansion of ( 1 + π‘₯ ) 6 in ascending powers of π‘₯ where π‘₯ β‰  0 . If the middle term in the expansion is five times the fifth term, what is the value of π‘₯ ?

  • A 5 4
  • B 1 5 4
  • C 4 5
  • D 4 1 5