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Lesson: General Term in the Binomial Theorem

Sample Question Videos

Worksheet • 25 Questions • 1 Video

Q1:

Find the third term in the expansion of ( 4 π‘₯ + 3 ) 3 .

  • A 1 0 8 π‘₯
  • B 2 7 π‘₯
  • C 2 7 π‘₯ 2
  • D 1 0 8 π‘₯ 2

Q2:

Find the value of π‘₯ that satisfies

Q3:

Consider the expansion of ( 1 + π‘₯ ) 𝑛 in ascending powers of π‘₯ . Given that the coefficient of π‘₯ 1 4 is equal to the coefficient of 𝑇 1 8 , determine the value of 𝑛 .

Q4:

Find 𝑇 4 in the expansion of ο€Ώ 5 √ π‘₯ + √ π‘₯ 5  9 .

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

Q5:

Find 𝑇 3 in the expansion of ο€Ώ 2 √ π‘₯ + √ π‘₯ 2  1 3 .

  • A 3 9 9 3 6 π‘₯ βˆ’ 9 2
  • B 7 8 π‘₯ 9 2
  • C 3 9 9 3 6 π‘₯ 9 2
  • D 5 1 2 π‘₯ 9 2
  • E 5 1 2 π‘₯ βˆ’ 9 2

Q6:

Find the third term in the expansion of ο€Ώ 2 π‘₯ + 5 √ π‘₯  5 .

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

Q7:

Find 𝑇 𝑛 + 3 in the expansion of ο€Ό 9 π‘₯ βˆ’ 1 π‘₯  8 3 𝑛 + 9 .

  • A ( βˆ’ 1 ) Γ— 𝐢 Γ— 9 Γ— π‘₯ 𝑛 + 2 3 𝑛 + 9 𝑛 + 2 2 𝑛 + 7 βˆ’ 6 𝑛 βˆ’ 9
  • B 3 𝑛 + 9 𝑛 + 2 2 𝑛 + 7 βˆ’ 6 𝑛 βˆ’ 9 𝐢 Γ— 9 Γ— π‘₯
  • C ( βˆ’ 1 ) Γ— 𝐢 Γ— 9 Γ— π‘₯ 𝑛 + 3 3 𝑛 + 9 𝑛 + 3 2 𝑛 + 7 βˆ’ 5 𝑛 βˆ’ 1 7
  • D ( βˆ’ 1 ) Γ— 𝐢 Γ— 9 Γ— π‘₯ 𝑛 + 3 3 𝑛 + 9 𝑛 + 3 2 𝑛 + 7 βˆ’ 6 𝑛 βˆ’ 9
  • E ( βˆ’ 1 ) Γ— 𝐢 Γ— 9 Γ— π‘₯ 𝑛 + 2 3 𝑛 + 9 𝑛 + 2 2 𝑛 + 7 βˆ’ 5 𝑛 βˆ’ 2 5

Q8:

Find 𝑇 𝑛 βˆ’ 3 in the expansion of ο€Ό 3 π‘₯ βˆ’ 1 π‘₯  2 4 𝑛 + 9 .

  • A ( βˆ’ 1 ) Γ— 𝐢 Γ— 3 Γ— π‘₯ 𝑛 βˆ’ 4 4 𝑛 + 9 𝑛 βˆ’ 4 3 𝑛 + 1 3 𝑛 + 2 1
  • B 4 𝑛 + 9 𝑛 βˆ’ 4 3 𝑛 + 1 3 𝑛 + 2 1 𝐢 Γ— 3 Γ— π‘₯
  • C ( βˆ’ 1 ) Γ— 𝐢 Γ— 3 Γ— π‘₯ 𝑛 βˆ’ 3 4 𝑛 + 9 𝑛 βˆ’ 3 3 𝑛 + 1 3 2 𝑛 + 1 9
  • D ( βˆ’ 1 ) Γ— 𝐢 Γ— 3 Γ— π‘₯ 𝑛 βˆ’ 3 4 𝑛 + 9 𝑛 βˆ’ 3 3 𝑛 + 1 3 𝑛 + 2 1
  • E ( βˆ’ 1 ) Γ— 𝐢 Γ— 3 Γ— π‘₯ 𝑛 βˆ’ 4 4 𝑛 + 9 𝑛 βˆ’ 4 3 𝑛 + 1 3 2 𝑛 + 1 7

Q9:

If the coefficient of the third term in the expansion of ο€Ό π‘₯ βˆ’ 1 4  𝑛 is 3 3 8 , determine the middle term in the expansion.

  • A 2 3 1 1 0 2 4 π‘₯ 6
  • B 9 9 5 1 2 π‘₯ 6
  • C 9 9 5 1 2 π‘₯ 7
  • D 2 3 1 1 0 2 4 π‘₯ 7

Q10:

If the ratio between the fourth term in the expansion of ο€Ό π‘₯ + 1 π‘₯  9 and the third term in the expansion of ο€Ό π‘₯ βˆ’ 1 π‘₯  2 8 equals 7 ∢ 1 2 , find the value of π‘₯ .

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

Q11:

Consider the binomial expansion of ( 1 + π‘₯ ) 𝑛 in ascending powers of π‘₯ . Given that 𝑇 = 𝑇 8 6 when π‘₯ = 1 √ 5 , find the value of 𝑛 .

Q12:

In the binomial expansion of ( 1 + π‘₯ ) 𝑛 , 𝑛 is a positive, whole number and 𝑇 π‘Ÿ is the π‘Ÿ th term, or the term which contains π‘₯ π‘Ÿ βˆ’ 1 .

If 8 ( 𝑇 ) = 2 7 𝑇 Γ— 𝑇 6 2 4 8 , what is the value of 𝑛 ?

Q13:

Find the coefficient of 𝑇 5 in the expansion of ( 9 π‘₯ + 2 ) 6 .

Q14:

Find the coefficient of 𝑇 3 in the expansion of ( 1 4 π‘₯ + 3 ) 3 .

Q15:

Find the third term in the expansion of ο€½ π‘Ž + 𝑏 π‘Ž  1 9 1 1 1 4 1 9 βˆ’ 2 8 .

  • A 3 7 8 π‘Ž 𝑏 8 3 1 1 7
  • B π‘Ž 𝑏 8 3 1 1 7
  • C π‘Ž 𝑏 2 6 9 1 1 7
  • D 3 7 8 π‘Ž 𝑏 2 6 9 1 1 7

Q16:

Consider the binomial expansion of ( 2 π‘₯ βˆ’ 𝑦 ) 9 in ascending powers of π‘₯ . What is the seventh term?

  • A βˆ’ 5 3 7 6 π‘₯ 𝑦 6 3
  • B 6 7 2 π‘₯ 𝑦 3 6
  • C βˆ’ 6 7 2 π‘₯ 𝑦 3 6
  • D 5 3 7 6 π‘₯ 𝑦 6 3

Q17:

In a binomial expansion, where the general term is 1 5 π‘Ÿ 1 8 βˆ’ 9 π‘Ÿ 𝐢 π‘₯ , determine the position of the term containing π‘₯ 9 .

  • A 𝑇 2
  • B 𝑇 1
  • C 𝑇 4
  • D 𝑇 3

Q18:

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

  • A 3 6 4 π‘₯ 8
  • B 7 2 8 π‘₯ 8
  • C π‘₯ 4
  • D π‘₯ 8

Q19:

Let 𝑇 π‘˜ be the π‘˜ t h term in the expansion of ( 1 + π‘₯ ) 3 4 in increasing powers of π‘₯ . Find all nonzero values of π‘₯ for which 2 𝑇 = 𝑇 + 𝑇 2 1 2 0 2 2 .

  • A2, 1
  • B2, 14
  • C2, βˆ’ 1
  • D4, 12

Q20:

Given that the sum of the first, middle, and last terms in the expansion of ( π‘₯ βˆ’ 1 ) 6 is 42 337, find all possible real values of π‘₯ .

  • A π‘₯ = 6 , π‘₯ = βˆ’ √ 1 9 6 3
  • B π‘₯ = 2 1 6 , π‘₯ = βˆ’ 1 9 6
  • C π‘₯ = 6 , π‘₯ = √ 1 9 6 3
  • D π‘₯ = βˆ’ 6 , π‘₯ = √ 1 9 6 3
  • E π‘₯ = βˆ’ 6 , π‘₯ = βˆ’ √ 1 9 6 3

Q21:

Find π‘Ž 6 in the expansion of ο€Ό 2 4 π‘₯ + 𝑦 4  7 .

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

Q22:

Find π‘Ž 6 in the expansion of ο€Ό 6 π‘₯ + 𝑦 3  8 .

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

Q23:

Find the second-to-last term in ( 2 + π‘₯ ) 3 4 .

  • A 6 8 π‘₯ 3 3
  • B 6 8 π‘₯
  • C 3 4 π‘₯
  • D 3 4 π‘₯ 3 3

Q24:

Let 𝑇 π‘˜ be the π‘˜ th term in the expansion of ( π‘₯ βˆ’ 2 ) 1 9 in descending powers of π‘₯ . Find all the nonzero values of π‘₯ for which 6 𝑇 βˆ’ 5 𝑇 + 𝑇 = 0 8 9 1 0 .

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

Q25:

Find the general term in ο€Ό 6 π‘₯ βˆ’ 1 6 π‘₯  𝑛 + 7 .

  • A ( βˆ’ 1 ) Γ— 𝐢 Γ— 6 Γ— π‘₯ π‘Ÿ 𝑛 + 7 π‘Ÿ 𝑛 βˆ’ 2 π‘Ÿ + 7 𝑛 βˆ’ 2 π‘Ÿ + 7
  • B ( βˆ’ 1 ) Γ— 𝐢 Γ— 6 Γ— π‘₯ π‘Ÿ + 1 𝑛 + 7 π‘Ÿ 𝑛 βˆ’ 2 π‘Ÿ + 7 𝑛 βˆ’ 2 π‘Ÿ + 7
  • C 𝑛 + 7 π‘Ÿ 𝑛 βˆ’ 2 π‘Ÿ + 7 𝑛 βˆ’ 2 π‘Ÿ + 7 𝐢 Γ— 6 Γ— π‘₯
  • D ( βˆ’ 1 ) Γ— 𝐢 Γ— 6 Γ— π‘₯ π‘Ÿ 𝑛 + 7 π‘Ÿ + 1 𝑛 βˆ’ 2 π‘Ÿ + 7 𝑛 βˆ’ 2 π‘Ÿ + 7
  • E ( βˆ’ 1 ) Γ— 𝐢 Γ— 6 Γ— π‘₯ π‘Ÿ 𝑛 + 7 π‘Ÿ 𝑛 βˆ’ π‘Ÿ + 7 𝑛 βˆ’ π‘Ÿ + 7
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