Sample QuestionsBinomial Theorem questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The two successive terms in the expansion of $(1 + x)^{24}$ whose coefficients are in the ratio $1 : 4$ are: $[$Hint:$\frac{^{24}\text{C}_\text{r}}{^{24}\text{C}_{\text{r}+1}}=\frac{1}{4}\ \frac{\text{r}+1}{24-\text{r}}\ \frac{1}{4}\Rightarrow4\text{r}+4=24-4\Rightarrow\text{r}=4]$
- A
$3^{rd} and 4^{th}.$
- B
$4^{th} and 5^{th}.$
- ✓
$5^{th} and 6^{th}.$
- D
$6^{th} and 7^{th}.$
Answer: C.
View full solution →If the coefficients of $2^{nd}, 3^{rd}$ and the $4^{th}$ terms in the expansion of $(1 + x)^n$ are in $A.P.,$ then value of n is: $[$Hint: $2^nC_2 = ^nC_1 + ^nC_3 \Rightarrow n^2 - 9n + 14 = 0 \Rightarrow n = 2\ $or $7.]$
Answer: B.
View full solution →Choose the correct answer.
The total number of terms in the expansion of $(x + a)^{100} + (x - a)^{100}$ after simplification is:
Answer: C.
View full solution →The coefficient of $x^n$ in the expansion of $(1 + x)^{2n} and (1 + x)^{2n - 1}$ are in the ratio. $[$Hint: $^{2\text{n}}\text{C}_\text{n} : \ ^{2\text{n} - 1}\text{C}_\text{n}]$
- A
$1 : 2.$
- B
$1 : 3.$
- C
$3 : 1.$
- ✓
$2 : 1.$
Answer: D.
View full solution →If A and B are coefficient of $x^n$ in the expansions of $(1 + x)^{2n}$ and $(1 + x)^{2n – 1}$ respectively, then $\frac{\text{A}}{\text{B}}$ equals:
$[$Hint: $\frac{\text{A}}{\text{B}}=\frac{^{2\text{n}}\text{C}_\text{n}}{^{2\text{n}-1}\text{C}_\text{n}}=2]$
- A
$1.$
- ✓
$2.$
- C
$\frac{1}{2}.$
- D
$\frac{1}{\text{n}}.$
Answer: B.
View full solution →The last two digits of the numbers $3^{400}$ are $01.$
View full solution →The sum of coefficients of the two middle terms in the expansion of $(1 + x)^{2n - 1}$ is equal to $^{2n - 1}C_n.$
View full solution →The expression $79 + 97$ is divisible by $64.$
$[$Hint: $79 + 97 = (1 + 8)^7 - (1 – 8)^9]$
View full solution →Number of terms in the expansion of $(a + b)\ n$ where $\text{n}\in\text{N}$ is one less than the power $n$.
View full solution →The number of terms in the expansion of $[(2x + y^3 )^4]^7$ is $8.$
View full solution →Find the coefficient of $x$ in the expansion of $(1 - 3x + 7x^2) (1 - x)^{16}.$
View full solution →Find the middle term $($terms$)$ in the expansion of:
$\Big(\frac{\text{x}}{\text{a}}-\frac{\text{a}}{\text{x}}\Big)^{10}$
View full solution →Show that the middle term in the expansion of $\Big(\text{x}-\frac{1}{\text{x}}\Big)^{2\text{n}}$ is $\frac{1\times3\times5\times....(2\text{n}-1)}{\text{n}!}\times(-2)^\text{n}.$
View full solution →Find the coefficient of $\frac{1}{\text{x}^{17}}$ in the expansion of $\Big(\text{x}^4-\frac{1}{\text{x}^3}\Big)^{15}.$
View full solution →Find n in the binomial $\Big(3\sqrt{2}+\frac{1}{3\sqrt{3}}\Big)^\text{n}$ if the ratio of $7^{th}$ term from the beginning to the $7^{th}$ term from the end is $\frac{1}{6}.$
View full solution →Find the coefficient of $x^{15}$ in the expansion of $(x - x^2)^{10}.$
View full solution →The coefficient of $a^{-6}b^4$ in the expansion of $\Big(\frac{1}{\text{a}}-\frac{2\text{b}}{3}\Big)^{10}$ is ___________.
$[$Hint: $\text{T}_5=\ ^{10}\text{C}_4\Big(\frac{1}{\text{a}}\Big)^\text{b}\Big(\frac{-2\text{b}}{3}\Big)^4=\frac{1120}{27}\text{a}^{-6}\text{b}^4]$
View full solution →The number of terms in the expansion of $(x + y + z)^n$ __________.
$[$Hint: $(x + y + z)^n = [x + (y + z)]^n]$
View full solution →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 $\text{n}-12=0\Rightarrow\text{n}=12]$
View full solution →The position of the term independent of x in the expansion of $\Big(\sqrt{\frac{\text{x}}{3}}+\frac{3}{2\text{x}^2}\Big)^{10}$ is __________.
View full solution →Middle term in the expansion of $(a^3 + b^a)^{28}$ is _________.
View full solution →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 $= ^nC_2.]$
View full solution →If the term free from x in the expansion of $\sqrt{\text{x}}-\frac{\text{k}}{\text{x}^2}^{10}$ is 405, find the value of k.
View full solution →Find the term independent of x in the expansion of $(1+\text{x}+2\text{x}^3)\Big(\frac{3}{2}\text{x}^2-\frac{1}{3\text{x}}\Big)^9.$
View full solution →In the expansion of $(x + a)^n$ if the sum of odd terms is denoted by $O$ and the sum of even term by $E$.
Then prove that,
- $\text{O}^2 – \text{E}^2 = (\text{x}^2 – \text{a}^2 )^\text{n}$
- $4\text{OE} = (\text{x} + \text{a})^{2\text{n}} – (\text{x} – \text{a})^{2\text{n}}$
View full solution →Find the term independent of x in the expansion of, $3\text{x}-\frac{2}{\text{x}^2}^{15}.$
View full solution →