if x denotes the fractional part of x then 3 2n 8 n in n is

If $$\left\{ x \right\}$$ denotes the fractional part of $$x$$ then $$\left\{ {\frac{{{3^{2n}}}}{8}} \right\},n \in N,$$ is

A. $$\frac{3}{8}$$

B. $$\frac{7}{8}$$

C. $$\frac{1}{8}$$

D. None of these

Answer : $$\frac{1}{8}$$

Solution :
$$\eqalign{ & {3^{2n}} = {\left( {1 + 8} \right)^n} = {\,^n}{C_0} + {\,^n}{C_1} \cdot 8 + {\,^n}{C_2} \cdot {8^2} + ..... + {\,^n}{C_n}{8^n} \cr & \therefore \,\,\frac{{{3^{2n}}}}{8} = \frac{1}{8} + \left( {^n{C_1} + {\,^n}{C_2} \cdot 8 + ..... + {\,^n}{C_n} \cdot {8^7}} \right) = \frac{1}{8} + {\text{integer}}{\text{.}} \cr} $$

Releted MCQ Question on
Algebra >> Binomial Theorem

Releted Question 1

Given positive integers $$r > 1, n > 2$$ and that the co - efficient of $${\left( {3r} \right)^{th}}\,{\text{and }}{\left( {r + 2} \right)^{th}}$$ terms in the binomial expansion of $${\left( {1 + x} \right)^{2n}}$$ are equal. Then

A. $$n = 2r$$

B. $$n = 2r + 1$$

C. $$n = 3r$$

D. none of these

View Answer

Releted Question 2

The co-efficient of $${x^4}$$ in $${\left( {\frac{x}{2} - \frac{3}{{{x^2}}}} \right)^{10}}$$ is

A. $$\frac{{405}}{{256}}$$

B. $$\frac{{504}}{{259}}$$

C. $$\frac{{450}}{{263}}$$

D. none of these

View Answer

Releted Question 3

The expression $${\left( {x + {{\left( {{x^3} - 1} \right)}^{\frac{1}{2}}}} \right)^5} + {\left( {x - {{\left( {{x^3} - 1} \right)}^{\frac{1}{2}}}} \right)^5}$$ is a polynomial of degree

A. 5

B. 6

C. 7

D. 8

View Answer

Releted Question 4

If in the expansion of $${\left( {1 + x} \right)^m}{\left( {1 - x} \right)^n},$$ the co-efficients of $$x$$ and $${x^2}$$ are $$3$$ and $$- 6\,$$ respectively, then $$m$$ is

A. 6

B. 9

C. 12

D. 24

View Answer

If $$\left\{ x \right\}$$ denotes the fractional part of $$x$$ then $$\left\{ {\frac{{{3^{2n}}}}{8}} \right\},n \in N,$$ is

Releted MCQ Question on
Algebra >> Binomial Theorem

Given positive integers $$r > 1, n > 2$$ and that the co - efficient of $${\left( {3r} \right)^{th}}\,{\text{and }}{\left( {r + 2} \right)^{th}}$$ terms in the binomial expansion of $${\left( {1 + x} \right)^{2n}}$$ are equal. Then

The co-efficient of $${x^4}$$ in $${\left( {\frac{x}{2} - \frac{3}{{{x^2}}}} \right)^{10}}$$ is

The expression $${\left( {x + {{\left( {{x^3} - 1} \right)}^{\frac{1}{2}}}} \right)^5} + {\left( {x - {{\left( {{x^3} - 1} \right)}^{\frac{1}{2}}}} \right)^5}$$ is a polynomial of degree

If in the expansion of $${\left( {1 + x} \right)^m}{\left( {1 - x} \right)^n},$$ the co-efficients of $$x$$ and $${x^2}$$ are $$3$$ and $$- 6\,$$ respectively, then $$m$$ is

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If $$\left\{ x \right\}$$ denotes the fractional part of $$x$$ then $$\left\{ {\frac{{{3^{2n}}}}{8}} \right\},n \in N,$$ is

Releted MCQ Question on Algebra >> Binomial Theorem

Given positive integers $$r > 1, n > 2$$ and that the co - efficient of $${\left( {3r} \right)^{th}}\,{\text{and }}{\left( {r + 2} \right)^{th}}$$ terms in the binomial expansion of $${\left( {1 + x} \right)^{2n}}$$ are equal. Then

The co-efficient of $${x^4}$$ in $${\left( {\frac{x}{2} - \frac{3}{{{x^2}}}} \right)^{10}}$$ is

The expression $${\left( {x + {{\left( {{x^3} - 1} \right)}^{\frac{1}{2}}}} \right)^5} + {\left( {x - {{\left( {{x^3} - 1} \right)}^{\frac{1}{2}}}} \right)^5}$$ is a polynomial of degree

If in the expansion of $${\left( {1 + x} \right)^m}{\left( {1 - x} \right)^n},$$ the co-efficients of $$x$$ and $${x^2}$$ are $$3$$ and $$- 6\,$$ respectively, then $$m$$ is

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