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.
Value of $\sum^{\infty}_{\text{k}=1}\sum^{\text{k}}_{\text{r}=0}\frac{1}{3^{\text{k}}}\big({^\text{k}}\text{C}_{\text{r}}\big)$ is:
- ✓
$2$
- B
$\frac{2}{3}$
- C
$\frac{1}{3}$
- D
$\text{None of these}$
Answer: A.
View full solution →If the $r^{th}$ term in the expansion of $\Big(\frac{\text{x}}{3}-\frac{2}{\text{x}^{2}}\Big)^{10}$ contains $x^4$, then $r$ is equal to:
Answer: A.
View full solution →Sum of the coefficients of $(1 - x)^{25}$ is:
Answer: C.
View full solution →The number of integral terms in the expansion of $\Big(3^{\frac{1}{8}}+5^{\frac{1}{4}}\Big)^{1024}$ is:
Answer: D.
View full solution →The $4^{th}$ term in the expansion of $\Big(\sqrt{\text{x}}+\frac{1}{\text{x}}\Big)^{12}$ is:
Answer: B.
View full solution →State which of the statement in True or False.
The last two digits of the numbers $3^{400}$ are $01.$
View full solution →State which of the statement in True or False.
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 →State which of the statement in True or False.
The expression $79 + 97$ is divisible by $64.$
Hint: $79 + 97 = (1 + 8)^7 - (1 – 8)^9$
View full solution →State which of the statement in True or False.
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 →State which of the statement in True or False.
The number of terms in the expansion of $[(2x + y^3 )^4]^7$ is $8.$
View full solution →Using binomial theorem, prove that $6^n–5n$ always leaves remainder $1$ when divided by $25.$
View full solution →Which is larger $(1.01)^{1000000}$ or $10,000?$
View full solution →Compute $(98)^5.$
View full solution →Expand $\left( x ^ { 2 } + \frac { 3 } { x } \right) ^ { 4 } , x \neq 0$
View full solution →Using binomial theorem, evaluate: $(99)^5$
View full solution →Using binomial theorem, evaluate: $(101)^4$
View full solution →Using binomial theorem, evaluate: $(102)^5$
View full solution →Using binomial theorem, evaluate: $(96)^3$
View full solution →Prove that ${\sum\limits_{r=0}^n3^r\;^nC_r=4^n}$
View full solution →Expand using binomial theorem ${\left[ {1 + \frac{x}{2} - \frac{2}{x}} \right]^4},x \ne 0$
View full solution →Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of ${\left( {\sqrt[4] 2 + \frac{1}{{\sqrt [4]{3} }}} \right)^n}$ is $\sqrt 6 :1$.
View full solution →Find the value of ${({a^2} + \sqrt {{a^2} - 1} )^4} + {({a^2} - \sqrt {{a^2} - 1} )^4}$
View full solution →Evaluate $(\sqrt{3}+\sqrt{2})^{6}-(\sqrt{3}-\sqrt{2})^{6}$.
View full solution →Expand the given expression ${\left( {x + \frac{1}{x}} \right)^6}$
View full solution →Fill in the blank.
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 →Fill in the blank.
The number of terms in the expansion of $(x + y + z)^n \_\_\_\_\_\_\_\_\_\_.$
$[$Hint: $(x + y + z)^n = [x + (y + z)]^n]$
View full solution →Fill in the blank.
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 →Fill in the blank.
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 →Fill in the blank.
Middle term in the expansion of $(a^3 + b^a)^{28}$ is $\_\_\_\_\_\_\_\_\_.$
View full solution →Find the coefficient of $x^5 $ in the product $(1 + 2x)^6 (1 -x)^7$ using binomial theorem.
View full solution →