Binomial Theorem Test

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Binomial Theorem Test - 54

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Weekly Quiz Competition

Question 1
1 / -0

The coefficient if $$ x^6 $$ in the expansion of $$ ( 3x^2 - \frac {1}{3x})^9 $$ is

A
378

B
756

C
189

D
567

Solution
Question 2
1 / -0

The largest term in the expansion of $$\left(\frac{b}{2} + \frac{b}{2}\right)^{100}$$ is

A
$$b^{100}$$

B
$$\left(\frac{b}{2}\right)^{100}$$

C
$$^{100}C_{50} \left(\frac{b}{2}\right)^{100} $$

D
None of these

Solution
Question 3
1 / -0

Which of the following expansion will have term containing $$x^2$$

A
$$\displaystyle(x^{-1/5} + 2x^{3/5})^{25}$$

B
$$\displaystyle(x^{3/5} + 2x^{-1/5})^{24}$$

C
$$\displaystyle (x^{3/5} - 2x^{-1/5})^{23}$$

D
$$\displaystyle (x^{3/5} + 2x^{-1/5})^{22}$$

Solution
Question 4
1 / -0

If the second term in the expansion $$\displaystyle \lgroup ^{13}\sqrt{a} + \frac{a}{\sqrt{a^{-1}}} \rgroup^n$$ is $$\displaystyle 14a^{5/2}$$, then the value of $$\displaystyle ^nC_3/^nC_2$$ is

A
4

B
3

C
12

D
6

Solution
Question 5
1 / -0

If $$\displaystyle ( 1+ 2x + 3x^2)^{10} = a_0 + a_1x + a_2x^2 + \dots + a_{20}x^{20}$$ then $$a_1$$equals

A
10

B
20

C
210

D
420

Solution
Question 6
1 / -0

If the fourth term of $$\displaystyle \left(\sqrt{_x\left(\frac{1}{1 + \log_{10} x}\right)} + ^{12}\sqrt{x} \right)^6$$ is equal to 200 and x > 1, then x is equal to

A
$$\displaystyle 10\sqrt{2}$$

B
$$\displaystyle 10 $$

C
$$\displaystyle 10^4$$

D
None of these

Solution

Question 7

1 / -0

The sum of rational term in $$(\sqrt 2+\sqrt [3]{3}+\sqrt[6]{5})^{10}$$ is equal to

$$12632$$

$$1260$$

$$1236$$

none of these

Solution

General term in the expansion of $$( \sqrt 2+ \sqrt[3]{3}+\sqrt[6]{5})^{10}$$ is $$\dfrac{10!}{a!b!c!}(\sqrt 2)^a (\sqrt[3]{3})^b ( \sqrt[6]{5})^c$$ where $$a+b+c=10$$.
For rational term, we have the following:

Value of $$a, b, c$$	Value of term
$$a=4, b=0, c=6$$	$$\dfrac{10!}{4!0!6!}( \sqrt 2)^4 (\sqrt[3]{3})^0( \sqrt [6]{5})^6=4200$$
$$a=10, b=0, c=0$$	$$\dfrac{10!}{10!0!0!}(\sqrt 2)^{10}( \sqrt[3]{3})^0 (\sqrt [6]{5})^0=32$$
$$a=4, b=6, c=0$$	$$\dfrac{10!}{4!6!0!}(\sqrt 2)^{4}( \sqrt[3]{3})^6 (\sqrt [6]{5})^0=7560$$

Question 8
1 / -0

The number of distinct terms in the expansion of $$\left( x+\dfrac{1}{x}+x^2+\dfrac{1}{x^2}\right)^{15}$$ is/are ( with respect to different power of $$x$$ )

A
$$255$$

B
$$61$$

C
$$127$$

D
none of these
Question 9
1 / -0

The number of terms in the expansion of $$\bigg(\dfrac{a}{x}+bx\bigg)^{12}$$ are

A
$$11$$

B
$$13$$

C
$$10$$

D
$$14$$

Solution

Terms of R.H.S. of the expansion of $$(x + a)^n$$ is finite for the positive values of x and the number of terms is $$(n + 1)$$.
So, value of n is $$12$$ in the given expression.
Hence, number of total terms $$= 12 + 1 = 13.$$
Question 10
1 / -0

The general term in the expansion of $$(x + a)^n$$

A
$$\,^nC_r\,x^{n-r} .a^r$$

B
$$\,^nC_r\,x^{r} .a^r$$

C
$$\,^nC_{n-r}\,x^{n-r} .a^r$$

D
$$\,^nC_{n-r}\,x^{r} .a^{n-r}$$

Solution

The general term in the expansion of $$(x + a)^n$$ is$$\,^nC_r\,x^{n-r} .a^r$$

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Re-Attempt Weekly Quiz Competition

Binomial Theorem Test - 54

Result

Binomial Theorem Test - 54

Score

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Rank

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SHARING IS CARING

Question 1

378

756

189

567

Solution

Question 2

$$b^{100}$$

$$\left(\frac{b}{2}\right)^{100}$$

$$^{100}C_{50} \left(\frac{b}{2}\right)^{100} $$

None of these

Solution

Question 3

$$\displaystyle(x^{-1/5} + 2x^{3/5})^{25}$$

$$\displaystyle(x^{3/5} + 2x^{-1/5})^{24}$$

$$\displaystyle (x^{3/5} - 2x^{-1/5})^{23}$$

$$\displaystyle (x^{3/5} + 2x^{-1/5})^{22}$$

Solution

Question 4

4

3

12

6

Solution

Question 5

10

20

210

420

Solution

Question 6

$$\displaystyle 10\sqrt{2}$$

$$\displaystyle 10 $$

$$\displaystyle 10^4$$

None of these

Solution

Question 7

$$12632$$

$$1260$$

$$1236$$

none of these

Solution

Question 8

$$255$$

$$61$$

$$127$$

none of these

Question 9

$$11$$

$$13$$

$$10$$

$$14$$

Solution

Question 10

$$\,^nC_r\,x^{n-r} .a^r$$

$$\,^nC_r\,x^{r} .a^r$$

$$\,^nC_{n-r}\,x^{n-r} .a^r$$

$$\,^nC_{n-r}\,x^{r} .a^{n-r}$$

Solution

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