Binomial Theorem Test

Result

Binomial Theorem Test - 67

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Weekly Quiz Competition

Question 1
1 / -0

The sum of the co-efficients of all odd degree terms in the expansion of $${ \left( x+\sqrt { { x }^{ 3 }-1 } \right) }^{ 5 }+{ \left( x-\sqrt { { x }^{ 3 }-1 } \right) }^{ 5 }$$, $$\left(x>1\right)$$

A
$$2$$

B
$$-1$$

C
$$0$$

D
$$1$$

Solution

$$\left(x+\sqrt{x^{3}-1}\right)^{5}+\left(x-\sqrt{x^{3}-1}\right)^{5}$$

$${ }^{5} c_{r} x^{5-r}\left(\sqrt{x^{3}-1}\right)^{r}=$$ General term

$${ }^{5} c_{r} x^{5-r}\left(-\sqrt{x^{3}-1}\right)^{r}=$$ General term

For degree terms

$$2\left[{ }^{5} C_{0} \cdot x^{5}+{ }^{5} c_{2} x^{3}\left(x^{3}-1\right)+{ }^{5} c_{4} x\left(x^{3}-1\right)^{2}\right.]$$

$$2\left(x^{5}+10 x^{6}-10 x^{3}+5 x^{7}-10 x^{4}+5 x\right]$$

Considering only odd degree term,

$$=2(1-10+5+5)$$

=2 Ans

option (a)
Question 2
1 / -0

If the number of terms in the expansion of $${ \left( 1-\dfrac { 2 }{ X } +\dfrac { 4 }{ { X }^{ 2 } } \right) }^{ n },x\neq 0,$$ is 28, then sum coefficients of all the terms in this expansion,is:

A
2187

B
243

C
729

D
64

Solution
Question 3
1 / -0

If $$\left | x \right |$$ < 1, then the coefficient of $$x^n$$ in the expansion of $$(1 + x + x^2 + x^3 + .....)^2$$ is

A
n

B
n 1

C
n + 2

D
n + 1

Solution
Question 4
1 / -0

Number of terms which are rational in the expansion of $$(\sqrt[4]{5}+\sqrt[3]{4})^{100}$$ is

A
$$10$$

B
$$8$$

C
$$9$$

D
$$11$$

Solution

General term

$$T_{r+1}={}^{100} C_{2}(5)^{\frac{100-r}{4}} \cdot(4)^{\frac{r}{3}}$$

In this case, for rational tems., power of $$5 and 4$$ should be integer
.
$$\therefore \frac{100-r}{4} and \frac{r}{3}$$ should be integers

For $$r=6,0,12 , 24,30,36,42,4.8,54,60$$

powers will be integer.

$$\therefore \quad Ans =10$$

option (a)
Question 5
1 / -0

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

A
$$12632$$

B
$$1260$$

C
$$126$$

D
none of these

Solution
Question 6
1 / -0

If $${ (1+x-{ 2x }^{ 2 }) }^{ 6 }=1+{ C }_{ 1 }x+{ C }_{ 2 }{ x }^{ 2 }+{ C }_{ 3 }{ x }^{ 3 }+...+{ C }_{ 12 }{ x }^{ 12 }$$, then the value of $${ C }_{ 2 }+{ C }_{ 4 }+{ C }_{ 6 }+...+{ C }_{ 12 }$$ is

A
30

B
32

C
31

D
None of these

Solution
Question 7
1 / -0

The sum of the coefficients of all the even power of $$x$$ in the expansion of $${(2{x^2} - 3x + 1)^{11}}$$

A
$${2.6^{10}}$$

B
$${3.6^{10}}$$

C
$${6^{11}}$$

D
None

Solution
Question 8
1 / -0

The number of terms which are free from radical signs in the expansion of $$\left( \frac { 1 }{ { y }^{ 4 } } +\frac { 1 }{ { y }^{ 8 } } \right) $$ is:

A
5

B
6

C
7

D
non of these

Solution
Question 9
1 / -0

The coefficient of $${ x }^{ 49 }$$ in the expansion of $$(x-1)(x-\frac { 1 }{ 2 } )(x-\frac { 1 }{ 2^{ 2 } } ).....(x-\frac { 1 }{ 2^{ 49 } } )$$ is equal to -

A
$$-2(1-\frac { 1 }{ { 2 }^{ 50 } } )$$

B
+ve coefficient of x

C
-ve coefficient of x

D
$$-2(1-\frac { 1 }{ { 2 }^{ 49 } } )$$

Solution
Question 10
1 / -0

If sum of the coefficient in the expression of $${ (-3{ x }^{ 2 }+\frac { 2 }{ x } ) }^{ 2n+1 }$$ is 'a' then the values of 'b' for which roots of the equation $${ x }^{ 2 }+bx+6a=0$$ are integral

A
{-7,-5,5,7}

B
{-7,-1,1,7}

C
{-5,-1,1,5}

D
none of these

Solution