Binomial Theorem for Positive Integral Index - Mathematics Multiple Choice Questions

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.

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 63^{24 – r} 1^r)
= 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 17^{24 – 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 √3^{7 – 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 x^{r – 1000} y^r)
b) \(\Sigma_{r = 0}^{r = 1000}\)(100Cr x^{1000 – r} y^r)
c) \(\Sigma_{r = 0}^{r = 999}\)(1000Cr x^{r – 1000} y^r)
d) \(\Sigma_{r = 0}^{r = 999}\)(1000Cr x^{1000 – r} y^r)
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 x^{1000 – r} y^r).

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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.

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.

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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