Binomial Theorem Test

Result

Binomial Theorem Test - 48

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Weekly Quiz Competition

Question 1
1 / -0

The sum of coefficients of the two middle terms in the expansion of $$( 1 + x ) ^ { 2 n - 1 }$$ is equal to

A
$$( 2 n - 1 ) C _ { n }$$

B
$$^ { ( 2 n - 1 ) } C _ { n + 1 }$$

C
$$^ { 2 n } C _ { n - 1 }$$

D
$$^ { 2 n } C _ { n }$$

Solution

$$(1+x)^{2n-1}$$ has $$2n$$ terms.
Hence there are $$2$$ middle terms, the $$n^{th}$$ term and the $$(n+1)^{th}$$ term.
$$t_n=(2n-1)C_{n-1}\cdot x^{n-1}$$
$$t_{n+1}=(2n-1)C_{n+1-1}x^{n+1-1}=2nC_nx^n$$
Sum of coefficients$$=(2n-1)C_{n-1}+(2n-1)C_n=2nC_n$$.
Question 2
1 / -0

If the coefficients of $$x^{-7}$$ and $$x^{-8}$$ to the expansion of $$\left(2+\dfrac {1}{3x}\right)^{n}$$ are equal then $$n=$$

A
$$56$$

B
$$15$$

C
$$45$$

D
$$55$$

Solution
Question 3
1 / -0

The coefficient of $$x^{4}$$ in the expansion of $$\left(\dfrac{x}{2}-\dfrac{3}{x^{2}}\right)^{10}$$, is

A
$$\dfrac{405}{256}$$

B
$$\dfrac{504}{259}$$

C
$$\dfrac{450}{263}$$

D
$$none\ of\ these$$

Solution
Question 4
1 / -0

The coefficient of $$x^{5}$$ in the expansion of $$(1+x)^{21}+(1+x)^{22}+....+(1+x)^{30}$$ is

A
$$^{51}C_{5}$$

B
$$^{9}C_{5}$$

C
$$^{31}C_{6}-^{21}C_{6}$$

D
$$^{30}C_{5}-^{20}C_{5}$$
Question 5
1 / -0

Find the middle term in the expansion of $${ \left( \dfrac { { 2x }^{ 2 } }{ 3 } +\dfrac { 3 }{ { 2x }^{ 2 } } \right) }^{ 10 }$$

A
$$252$$

B
$$248$$

C
$$230$$

D
$$200$$

Solution
Question 6
1 / -0

The coefficient of $${x}^{3}$$ in the expression of $${(1+{x}^{2}+{x}^{3})}^{10}$$ is

A
$$10$$

B
$$220$$

C
$$211$$

D
$$None \ of \ these$$

Solution
Question 7
1 / -0

If $$(1+ax)^n=1+8x+24x^2+.....$$ then

A
$$a=3$$

B
$$a=4$$

C
$$a=2$$

D
$$a=5$$

Solution
Question 8
1 / -0

The coefficient of $$x^4$$ in the expansion of $$(1-2x+3x^2+4x^3)(1-x)^{-8}$$ is:

A
$$232$$

B
$$231$$

C
$$230$$

D
None of these
Question 9
1 / -0

The coefficient of $$x^6 y^5 z^3$$ in the expansion of $$(3xy - 2xz + zy)^7$$ is

A
$$\dfrac {7 !}{4! 2! 1!}\cdot 3^4 \cdot 2^2$$

B
$$-\dfrac {7 !}{4! 2! 1!}\cdot 3^4 \cdot 2^2$$

C
$$-\dfrac {7 !}{4! 2! 1!}\cdot 3^2 \cdot 2^4$$

D
$$\dfrac {7 !}{4! 2! 1!}\cdot 3^2 \cdot 2^4$$

Solution
Question 10
1 / -0

The coefficient of $$x^{4}$$ in the expansion $$\left(1+5x+9x^{2}+.+\left(4k+1\right)x^{k}+..\right)\left(1+x^{2}\right)^{11}$$ is

A
$$^{11}C_{2}+4 ^{11}C_{1}+3$$

B
$$^{11}C_{2}+3 ^{11}C_{1}+4$$

C
$$3 ^{11}C_{2}+4 ^{11}C_{1}+3$$

D
$$171$$

Solution