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
View Answer

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

Answer: c
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
View Answer

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

Answer: c
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)
View Answer

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

Answer: c
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
View Answer

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

Answer: a
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
View Answer

Answer: d
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.
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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
View Answer

Answer: c
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 \(\frac{1}{4}\)
View Answer

Answer: b
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
View Answer

Answer: d
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)
View Answer

Answer: a
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
View Answer

Answer: d
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.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter