Binomial Theorem Test

Result

Binomial Theorem Test - 62

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Weekly Quiz Competition

Question 1
1 / -0

Find the middle terms(s) in the expansion of $$\left ( 3x-\dfrac{2}{x^{2}} \right )^{15}$$.

A
$$\dfrac {-6435\times 3^7\times 2^7}{x^6}, \dfrac {6437\times 3^7 \times 2^8}{x^9}$$

B
$$\dfrac {-6435\times 3^8\times 2^7}{x^6}, \dfrac {6437\times 3^7 \times 2^8}{x^9}$$

C
$$\dfrac {-6435\times 3^8\times 2^7}{x^6}, \dfrac {6437\times 3^7 \times 2^7}{x^9}$$

D
$$\dfrac {-6435\times 3^8\times 2^7}{x^6}, \dfrac {6437\times 3^8 \times 2^8}{x^9}$$

Solution
Question 2
1 / -0

The value of $$\dfrac{1}{12!}+\dfrac{1}{10!2!}+\dfrac{1}{8!4!}+...+\dfrac{1}{12!}$$

A
$$\dfrac{{2}^{12}}{12!}$$

B
$$\dfrac{{2}^{11}}{12!}$$

C
$$\dfrac{{2}^{11}}{11!}$$

D
None of these

Solution
Question 3
1 / -0

Coefficient of $$x^{25}$$ in $$(1+x+x^{2}+x^{3}+....+x^{10})^{7}$$ is

A
$$^{31}C_{15}-7.^{20}C_{14}$$

B
$$^{31}C_{14}-7.^{20}C_{14}$$

C
$$31$$

D
$$None\ of\ these$$
Question 4
1 / -0

The greatest terms of the expansion $$(2x+5y)^{13}$$ when $$x=10$$, $$y=2$$ is?

A
$$^{13}C_5\cdot 20^8\cdot 10^5$$

B
$$^{13}C_6\cdot 20^7\cdot 10^4$$

C
$$^{13}C_4\cdot 20^9\cdot 10^4$$

D
None of these
Question 5
1 / -0

The coefficient of $$x^{8}$$ in $$(1+2x^{2}-x^{3})^{9}$$ is

A
$$1680$$

B
$$2140$$

C
$$2520$$

D
$$2730$$

Solution
Question 6
1 / -0

The coefficient of x$$^9$$ in (x - 1) (x - 4) (x - 9)........(x - 100) is

A
-235

B
235

C
385

D
None of these

Solution
Question 7
1 / -0

Find the value of $$\dfrac{1}{\left(n-1\right)!}+\dfrac{1}{\left(n-3\right)!3!}+\dfrac{1}{\left(n-5\right)!5!}+...$$

A
$$\dfrac{{2}^{n-1}}{(n-1)!}$$

B
$$\dfrac{{2}^{n}}{n!}$$

C
$$\dfrac{{2}^{n-1}}{n!}$$

D
None of these

Solution
Question 8
1 / -0

The constant term in the expansion of $${ (1+x) }^{ n }{ (1+\frac { 1 }{ x } ) }^{ n }$$ is

A
$${ C }_{ 0 }^{ 2 }+2{ C }_{ 1 }^{ 2 }+3{ C }_{ 2 }^{ 2 }+......+(n+1){ C }_{ n }^{ 2 }$$

B
$${ ({ C }_{ 0 }+{ C }_{ 1 }+.....+{ C }_{ n }) }^{ 2 }$$

C
$${ C }_{ 0 }^{ 2 }+{ C }_{ 1 }^{ 2 }+.....+{ C }_{ n }^{ 2 }$$

D
None of these

Solution
Question 9
1 / -0

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

A
Less than $$200$$

B
Less than $$400$$ but greater than $$200$$

C
$$1400$$

D
$$1500$$

Solution
Question 10
1 / -0

The largest coefficient in the expansion of $$\left(4+3x\right)^{25}$$ is

A
$$^{25}C_{11}3^{25}\left(\dfrac{4}{3}\right)^{14}$$

B
$$^{25}C_{11}4^{25}\left(\dfrac{3}{4}\right)^{11}$$

C
$$^{25}C_{14}4^{14}3^{11}$$

D
$$^{25}C_{14}4^{11}.3^{14}$$

Solution

$$=(4+3 x)^{25}\\$$

$$\left({}^{25} c_{0}(4)+{ }^{25} c_{1}(4)^{2}(3 x)+\cdots^{25} c_{25}\right.\\$$

$$(4+3 x)^{25}\\$$

$$\text { Let Tr the term is largest }\\$$

$$\therefore \quad T_{r-1}<T_{r}>T_{r+1}\\$$

$$=\frac{T_{2}}{T_{r_{-1}}}>1 \quad \frac{T_{r+1}}{T_{r}}<1$$

$$T_{r+1^{}}= {}^{25} C_{r}(4)^{25-r}(3 x)^{k}$$

$$T_{r-1}=25_{C_{r-2}}(4)^{25-r+2}(3 x)^{r-2}$$

$$T_{r}={ }^{25} C_{r-1}\left(4\right)^{26-r}(3 x)^{r-1}$$

$$\therefore \frac{25 C_{r-1}(4)^{26-r}(3 x)^{r-1}}{25 C_{r-2}(4)^{27-r}(3 x)^{r-2}}>1$$

$$\frac{(25-r)}{(x-1)} \times \frac{3}{4}>1$$

$${}75-3 r > 4 r -4\\ $$

$$79>7 r\\$$

$$11.1>r$$

$$\Rightarrow r=12$$

largest term = $$\left({ }^{25}c_{11}(3)^{25}\left(\frac{4}{3}\right)^{14}\right.)$$