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10th Maths Answers Chapter 2 Numbers and Sequences Ex 2.3
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1.Find the least positive value of x such that
(i) 71 ≡ x (mod 8)
(ii) 78 + x ≡ 3 (mod 5)
(iii) 89 ≡ (x + 3) (mod 4)
(iv) 96 = `\frac{x}{7}` (mod 5)
(v) 5x ≡ 4 (mod 6)
Answer:
To find the least value of x such that
(i) 71 ≡ x (mod 8)
71 ≡ 7 (mod 8)
∴ x = 7.[ ∵ 71 – 7 = 64 which is divisible by 8]
(ii) 78 + x ≡ 3 (mod 5)
⇒ 78 + x – 3 = 5n for some integer n.
75 + x = 5 n
75 + x is a multiple of 5.
75 + 5 = 80. 80 is a multiple of 5.
Therefore, the least value of x must be 5.
(iii) 89 ≡ (x + 3) (mod 4)
89 – (x + 3) = 4n for some integer n.
86 – x = 4 n
86 – x is a multiple of 4.
∴ The least value of x must be 2 then
86 – 2 = 84.
84 is a multiple of 4.
∴ x value must be 2.
(iv) 96 ≡ `\frac{x}{7}` (mod 5)
96 – `\frac{x}{7}` = 5n for some integer n.
`\frac{672 - x}{7}` = 5n
672 – x = 35n.
672 – x is a multiple of 35.
∴ The least value of x must be 7 i.e. 665 is a multiple of 35.
(v) 5x ≡ 4 (mod 6)
5x – 4 = 6M for some integer n.
5x = 6n + 4
x = `\frac { 6n+4 }{ 5 }`
When we put 1, 6, 11, … as n values in x = `\frac { 6n+4 }{ 5 }` which is divisible by 5.
When n = 1, x = `\frac { 10 }{ 5 }` = 2
When n = 6, x = `\frac { 36+4 }{ 5 }`= `\frac { 40 }{ 5 }` = 8 and so on.
∴ The Answers are 2, 8, 14…..
∴ Least value is 2.
2.If x is congruent to 13 modulo 17 then 7x – 3 is congruent to which number modulo 17?
Answer:
Given x ≡ 13 (mod 17) ……(1)
7x – 3 ≡ a (mod 17) ……..(2)
From (1) we get
x- 13 = 17 n (n may be any integer)
x – 13 is a multiple of 17
∴ The least value of x = 30
From (2) we get
7(30) – 3 ≡ a(mod 17)
210 – 3 ≡ a(mod 17)
207 ≡ a (mod 17)
207 ≡ 3(mod 17)
∴ The value of a = 3
3.Solve 5x ≡ 4 (mod 6)
Answer:
5x ≡ 4 (mod 6)
5x – 4 = 6M for some integer n.
5x = 6n + 4
x = `\frac { 6n+4 }{ 5 }` where n = 1, 6, 11,…..
∴ x = 2, 8, 14,…
4.Solve 3x – 2 ≡ 0 (mod 11)
Answer:
3x – 2 ≡ 0 (mod 11)
3x – 2 = 11 n for some integer n.
3x = 11n + 2
5.What is the time 100 hours after 7 a.m.?
Answer:
100 ≡ x (mod 12) (∵7 comes in every 12 hrs)
100 ≡ 4 (mod 12) (∵ Least value of x is 4)
∴ The time 100 hrs after 7 O’ clock is 7 + 4 = 11 O’ clock i.e. 11 a.m
6.What is time 15 hours before 11 p.m.?
Answer:
15 ≡ x (mod 12)
15 ≡ 3 (mod 12)
The value of x must be 3.
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The time 15 hours before 11 o’clock is (11 – 3) 8 pm
7.Today is Tuesday. My uncle will come after 45 days. In which day my uncle will be coming?
Answer:
No. of days in a week = 7 days.
45 ≡ x (mod 7)
45 – x = 7n
45 – x is a multiple of 7.
∴ Value of x must be 3.
∴ Three days after Tuesday is Friday. Uncle will come on Friday.
8.Prove that 2ⁿ + 6 × 9n is always divisible by 7 for any positive integer n.
Answer:
9 = 2 (mod 7)
9ⁿ = 2ⁿ (mod 7) and 2ⁿ = 2ⁿ (mod 7)
2ⁿ + 6 × 9ⁿ = 2ⁿ (mod 7) + 6 [2ⁿ (mod 7)]
= 2ⁿ (mod 7) + 6 × 2ⁿ (mod 7)
7 × 2ⁿ (mod 7)
It is always divisible for any positive integer n
9.Find the remainder when 281 is divided by 17.
Answer:
2⁸¹ ≡ x (mod 17)
2⁴⁰ × 2⁴⁰ × 2⁴¹ ≡ x (mod 17)
(2⁴)¹⁰ × (2⁴)¹⁰ × 2¹ ≡ x (mod 17)
(16)¹⁰ × (16)¹⁰ × 2 ≡ x(mod 17)
(16⁵)² × (16⁵)²× 2
(16⁵) ≡ 16 (mod 17)
(165)2 ≡ 162 (mod 17)
(16⁵)² ≡ 256 (mod 17)
≡ 1 (mod 17) [∵ 255 is divisible by 17]
(16⁵)² × (16⁵)² × 2 ≡ 1 × 1 × 2 (mod 17)
∴ 2⁸¹ ≡ 2(mod 17)
∴ x = 2
10.The duration of flight travel from Chennai to London through British Airlines is approximately 11 hours. The aeroplane begins its journey on Sunday at 23:30 hours. If the time at Chennai is four and a half hours ahead to that of London’s time, then find the time in London, when will the flight lands at London Airport.
Answer:
The duration of the flight from Chennai to London is 11 hours.
Starting time at Chennai is 23.30 hrs. = 11.30 p.m.
Travelling time = 11.00 hrs. = 22.30 hrs = 10.30 a.m.
Chennai is 4 `\frac{1}{2}` hrs ahead to London.
= 10.30 – 4.30 = 6.00
∴ At 6 a.m. on Monday the flight will reach at London Airport.
