# Mathematics Questions and Answers – Binomial Theorem for Positive Integral Index

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This set of Mathematics online quiz focuses on “Binomial Theorem for Positive Integral Index”.

1. What is the coefficient of x2y2 in (x + 1)2 . (x + 1)3?
a) 1
b) 5
c) 2
d) 10

Explanation: We know that (a + b)2 = a2 + 2ab + b2
(a + b)3 = a3 + 3ab2 + 3a2b + b2
Using these formulae, we get
P(x) = (x2 + 2xy + y2)(x3 + 3xy2 + 3x2y + y2)
P(x) = 3xy4 + 9x2y3 + 10x3y2 + 5x4y + x5 + y4 + 2xy3 + x2y2
The coefficient of x2y2 in (x + 1)2 . (x + 1)3 is 1.

2. What is the remainder when 848 is divided by 63?
a) 4
b) 2
c) 1
d) 7

Explanation: 858 can be written as (82)24.
848 = (64)24
848 = (63 + 1)24
We know that (60 + 1)24 = $$\Sigma_{r = 0}^{r = 24}$$(24Cr 6324 – r 1r)
= 24C0 6324 40 + 24C1 6323 41 +….+24C23 631 423 + 24C24 630 124
= 63 x k + 1
Therefore, the remainder will be 1.

3. What is the remainder when 4103 is divided by 17?
a) 10
b) 14
c) 13
d) 16

Explanation: 4103 = 4 x 4102
4103 = 4 x (42)51
4103 = 4 x (16)51
4103 = 4 x (17 – 1)51
4103 = 4 x $$\Sigma_{r = 0}^{r = 51}$$(51Cr 1724 – r (-1)r
4103 = 4 x [51C0 1751 (-1)0 + 51C1 1751 (-1)1 +….+ 51C50 171 (-1)50 + 51C51 170 (-1)51]
4103 = 4 x 17 x k – 1
The remainder = 17 – 1
Remainder = 16.

4. What is the integral part of (√3 + 1)8?
a) 1558
b) 1551
c) 1552
d) 1556

Explanation: By binomial expansion,
(√3 + 1)7 = $$\Sigma_{r = 0}^{r = 7}$$(7Cr √37 – r (1)r)
Whenever, r is an even number, 8 – r will also be even. Then √3 will also have an even power and thereby be integral.
Integral parts = 8C0 (√3)0 + 8C2 (√3)2 + 8C4 (√3)4 + 8C6 (√3)6 + 8C8 (√3)8
Integral parts = 1 + 28 x 3 + 70 x 9 + 28 x 27 + 1 x 81
Integral part = 1552.

5. What is the expansion of (x + y)1000?
a) $$\Sigma_{r = 0}^{r = 1000}$$(1000Cr xr – 1000 yr)
b) $$\Sigma_{r = 0}^{r = 1000}$$(100Cr x1000 – r yr)
c) $$\Sigma_{r = 0}^{r = 999}$$(1000Cr xr – 1000 yr)
d) $$\Sigma_{r = 0}^{r = 999}$$(1000Cr x1000 – r yr)

Explanation: The expansion can be done using binomial theorem.
(x + y)1000 = 1000C0 x1000 y0 + 1000C1 x999 y1 +….+ 1000C999 x1 y999 + 1000C1000 x0 y1000
This can also be written as,
(x + y)1000 = $$\Sigma_{r = 0}^{r = 1000}$$(100Cr x1000 – r yr).

6. What is the real part of (11 + i)3?
a) 1331
b) 1332
c) 1328
d) 1329

Explanation: (11 + i)3 = 113 + 3.112.i +3.i2.11 +i3
= 1331 + 363i – 3 – i
= 1328 + 365i.

7. What are the coefficients of the first and the last term of (a + b)n?
a) 2
b) 1
c) Coefficients depend on n
d) 3

Explanation: The coefficient of the first term and last term is same. The first term is nC1 an and the last term is nC0 bn unless, a and b are numbers that change the value of the coefficient.

8. What is the remainder when (4)2n + 1 is divided by 5?
a) 4
b) 1
c) 2
d) 3

Explanation: The powers of four follow the given order:
41 = 4
42 = 16
43 = 64
44 = 256
45 = 1024 and so on.
Odd powers of 4, have the number 4 in the units place. When 5 divides the nearest ten, 4 will be obtained as the remainder each time.

9. What is the expansion of the series (xy + 2)2?
a) x2 + y2 + 4
b) xy2 + 4 +2xy
c) x2y2 + 2xy + 4
d) x2y2 + 4xy + 4

Explanation: (a + b)2 can be expanded using binomial theorem to get:
(a + b)2 = a2 + 2ab + b2
Here, a = xy and b = 2
Therefore, (xy + 2)2 = (xy)2 + 2(xy)(2) + (2)2
(xy + 2)2 = x2y2 + 4xy + 4.

10. What is the answer of $$\frac{x^2+y^2+2xy}{x^2-y^2}$$?
a) (x – y) (x + y)-2
b) (x + y) (x – y)-2
c) (x + y) (x – y)-1
d) (x – y) (x + y)-1

Explanation: x2 + y2 + 2xy is the expansion of (x + y)2
x2 – y2 can be written as (x – y)(x + y)
Substituting in the fraction we get, $$\frac{(x + y)^2}{(x – y)(x + y)}$$.
After cancelling the terms we get, $$\frac{x^2+y^2+2xy}{x^2-y^2}$$ = (x + y) (x – y)-1.

11. What is the value of $$\frac{7^3+2^3+84}{7^2-2^2}$$ ?
a) 9 $$\frac{1}{2}$$
b) 9 $$\frac{2}{3}$$
c) 9 $$\frac{1}{3}$$
d) 9 $$\frac{1}{4}$$

Explanation: Using binomial theorem we know that (a + b)3 = a3 + 3ab2 + 3a2b + b3
Therefore, (7 + 2)3 = 73 + 23 + (3 x 7 x 22) + (3 x 2 x 72)
(9)3 = 73 + 23 + 84 + (3 x 2 x 72)
729 = 73 + 23 + 84 + 294
73 + 23 + 84 = 435
Also 72 – 22 = (7 – 2)(7 + 2)
72 – 22 = (5)(9)
72 – 22 = 45
So $$\frac{7^3 + 2^3 + 84}{7^2-2^2}$$ = 435 / 45
= 9 $$\frac{30}{45}$$
= 9 $$\frac{2}{3}$$.

12. What is the value of $$\frac{101^3-99^3+2969703–3029697}{ 101^2 – 99^2}$$?
a) 1
b) 1/200
c) 1/100
d) 1/50

Explanation: The numerator when simplified is of the form (101 – 99)3
The denominator can be simplified as (101 – 99)(101 + 99)
When we substitute in the numerator and denominator we get (2 x 2 x 2) / (2 x 200)
= 1/50.

13. What is the quotient when x4 + 4x3y + 6x2y + 4xy3 + y4 is divided by (x + y)?
a) (x + y)3
b) x2 + y2
c) (x + y)2
d) (x + y)

Explanation: Using binomial expansions properties, x4 + 4x3y + 6x2y + 4xy3 + y4 can be written as
= 4C0x4y0 + 4C1x3y1 + 4C2x2y2 + 4C3x1y2 + 4C4x0y4
= (x + y)4
When divided by (x + y), we get (x + y)3.

14. What is the real part of (9 + 3i)2?
a) 81
b) 90
c) 54
d) 72

Explanation: Using binomial theorem (9 + 3i)2 = 81 + 54i + 9i2
We know that i2 = –1
Therefore, (9 + 3i)2 = 81 + 54i – 9
(9 + 3i)2 = 72 + 54i
Real part = 72.

Sanfoundry Global Education & Learning Series – Mathematics – Class 11.

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