Question types

Binomial Theorem question types

41 questions across 6 question groups — pick any mix to generate a MATHS paper with step-by-step answer keys.

41
Questions
6
Question groups
5
Question types
Sample Questions

Binomial Theorem questions

One sample from each question group in this chapter. Select any group above to see the full set with answer keys.

If the coefficients of 2nd, 3rd and the 4th terms in the expansion of (1 + x)n are in A.P., then value of n is:

  1. 2.
  2. 7.
  3. 11.
  4. 14.

[Hint: 2nC2 = nC1 + nC3 ⇒ n2 - 9n + 14 = 0 ⇒ n = 2 or 7.]

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The coefficient of xn in the expansion of (1 + x)2n and (1 + x)2n - 1 are in the ratio.

  1. 1 : 2.
  2. 1 : 3.
  3. 3 : 1.
  4. 2 : 1.

Hint: $^{2\text{n}}\text{C}_\text{n} : \ ^{2\text{n} - 1}\text{C}_\text{n}$

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Given the integers r > 1, n > 2, and coefficients of (3r)th and (r + 2)nd terms in the binomial expansion of (1 + x)2n are equal, then:

  1. n = 2r.
  2. n = 3r.
  3. n = 2r + 1.
  4. None of these.
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The two successive terms in the expansion of (1 + x)24 whose coefficients are in the ratio 1 : 4 are:

  1. 3rd and 4th.
  2. 4th and 5th.
  3. 5th and 6th.
  4. 6th and 7th.

$[\text{Hint}:\frac{^{24}\text{C}_\text{r}}{^{24}\text{C}_{\text{r}+1}}=\frac{1}{4}\ \frac{\text{r}+1}{24-\text{r}}\ \frac{1}{4}\Rightarrow4\text{r}+4=24-4\Rightarrow\text{r}=4]$

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Choose the correct answer.

The total number of terms in the expansion of (x + a)100 + (x - a)100 after simplification is:

  1. 50.
  2. 202.
  3. 51.
  4. None of these.
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The coefficient of a-6b4 in the expansion of $\Big(\frac{1}{\text{a}}-\frac{2\text{b}}{3}\Big)^{10}$ is ___________.
[Hint: $\text{T}_5=\ ^{10}\text{C}_4\Big(\frac{1}{\text{a}}\Big)^\text{b}\Big(\frac{-2\text{b}}{3}\Big)^4=\frac{1120}{27}\text{a}^{-6}\text{b}^4]$
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If the seventh terms from the beginning and the end in the expansion of $\Big(3\sqrt{2}+\frac{1}{3\sqrt{3}}\Big)^\text{n}$ are equal, then n equals _____________.
[Hint: $\text{T}_7=\text{T}_{\text{n}-7+2}\Rightarrow\ ^\text{n}\text{C}_6\Big(2^\frac{1}{3}\Big)^{\text{n}-6}\bigg(\frac{1}{3^\frac{1}{3}}\bigg)^6$ $=\ ^\text{n}\text{C}_{\text{n}-6}\Big(2^\frac{1}{3}\Big)^6\bigg(\frac{1}{3^\frac{1}{3}}\bigg)^{\text{n}-6}$
$\Rightarrow\Big(2^\frac{1}{3}\Big)^{\text{n}-12}=\bigg(\frac{1}{3^{\frac{1}{3}}}\bigg)^{\text{n}-12}\Rightarrow$ only problem when $\text{n}-12=0\Rightarrow\text{n}=12]$
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Q 173 Marks Question3 Marks
Show that the middle term in the expansion of $\Big(\text{x}-\frac{1}{\text{x}}\Big)^{2\text{n}}$ is $\frac{1\times3\times5\times....(2\text{n}-1)}{\text{n}!}\times(-2)^\text{n}.$
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Q 203 Marks Question3 Marks
Find n in the binomial $\Big(3\sqrt{2}+\frac{1}{3\sqrt{3}}\Big)^\text{n}$ if the ratio of 7th term from the beginning to the 7th term from the end is $\frac{1}{6}.$
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If xp occurs in the expansion of $\Big(\text{x}^2+\frac{1}{\text{x}}\Big)^{2\text{n}},$ prove that its coefficient is $\frac{2\text{n}!}{\Big(\frac{4\text{n}-\text{p}}{3}\Big)!\Big(\frac{2\text{n}+\text{p}}{3}\Big)!}.$
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Find the sixth term of the expansion $\Big(\text{y}^\frac{1}{2}+\text{x}^\frac{1}{3}\Big)^\text{n},$ if the binomial coefficient of the third term from the end is 45.
[Hint: Binomial coefficient of third term from the end = Binomial coefficient of third term from beginning = nC2.]
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