KC Sinha Mathematics Solution Class 10 Chapter 1 Real nuambers Exercise 1.4

Q1 A Q1 B Q1 C Q1 D Q1 E Q1 F Q1 G Q1 H Q1 I Q1 J Q1 K Q1 L Q2 A Q2 B Q2 C Q2 D Q2 E Q2 F Q2 G Q2 H Q2 I Q2 J Q2 K Q2 L Q3

Exercise 1.1 Exercise 1.2 Exercise 1.3 Exercise 1.4

Exercise 1.4

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Question 1 A 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\frac{17}{8}$
Sol :
Given rational number is $\frac{17}{8}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.
Firstly, we check co-prime
17 = 17 × 1
8 = 2 × 2 × 2
⇒17 and 8 have no common factors
Therefore, 17 and 8 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
8 = 2³
= 1 × 2³
= 5⁰ × 2³
So, denominator is of the form 2ⁿ5^m where n = 3 and m = 0
Thus, $\frac{17}{8}$ is a terminating decimal.

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Question 1 B 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.
$\frac{3}{8}$
Sol :
Given rational number is $\frac{3}{8}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.
Firstly, we check co-prime
3 = 3 × 1
8 = 2 × 2 × 2
⇒3 and 8 have no common factors
Therefore, 3 and 8 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
8 = 2³
= 1 × 2³
= 5⁰ × 2³
So, denominator is of the form 2ⁿ5^m where n = 3 and m = 0
Thus, $\frac{3}{8}$ is a terminating decimal.

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Question 1 C 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{29}{343}$
Sol :
Given rational number is $\frac{29}{343}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.
Firstly, we check co-prime
29 = 29 × 1
343 = 7 × 7 × 7
⇒29 and 343 have no common factors
Therefore, 29 and 343 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
343 = 7³
So, denominator is not of the form 2ⁿ5^m
Thus, $\frac{29}{343}$ is a non-terminating repeating decimal.

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Question 1 D 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{13}{125}$

Sol :
Given rational number is $\frac{13}{125}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.
Firstly, we check co-prime
13 = 13 × 1
125 = 5 × 5 × 5
⇒13 and 125 have no common factors
Therefore, 13 and 125 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
125 = 5³
= 1 × 2³
= 2⁰ × 5³
So, denominator is of the form 2ⁿ5^m where n = 0 and m = 3
Thus, $\frac{13}{125}$ is a terminating decimal.

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Question 1 E 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{27}{8}$

Sol :
Given rational number is $\frac{27}{8}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.
Firstly, we check co-prime
27 = 3 × 3 × 3
8 = 2 × 2 × 2
⇒27 and 8 have no common factors
Therefore, 27 and 8 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
8 = 2³
= 1 × 2³
= 5⁰ × 2³
So, denominator is of the form 2ⁿ5^m where n = 3 and m = 0
Thus, $\frac{27}{8}$ is a terminating decimal.

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Question 1 F 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{7}{80}$
Sol :
Given rational number is $\frac{7}{80}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.
Firstly, we check co-prime
7 = 7 × 1
80 = 2 × 2 × 2 × 2 × 5
⇒7 and 80 have no common factors
Therefore, 7 and 80 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
80 = 2⁴ × 5
So, denominator is of the form 2ⁿ5^m where n = 4 and m = 1
Thus, $\frac{7}{80}$ is a terminating decimal.

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Question 1 G 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{64}{455}$

Sol :
Given rational number is $\frac{64}{455}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.
Firstly, we check co-prime
64 = 2⁶
455 = 5 × 7 × 13
⇒64 and 455 have no common factors
Therefore, 64 and 455 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
455 = 5 × 7 × 13
So, denominator is not of the form 2ⁿ5^m
Thus, $\frac{64}{455}$ is a non-terminating repeating decimal.

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Question 1 H 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{6}{15}$
Sol :
Given rational number is $\frac{6}{15}$
$\frac{6}{15}=\frac{2}{5}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.
Firstly, we check co-prime
⇒2 and 5 have no common factor
Therefore, 2 and 5 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
5 = 5¹ × 1
= 5¹ × 2⁰
So, denominator is of the form 2ⁿ5^m where n = 0 and m = 1
Thus, $\frac{6}{15}$ is a terminating decimal.

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Question 1 I 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{35}{50}$
Sol :
Given rational number is $\frac{35}{50}$
$\frac{35}{50}=\frac{7}{10}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.
Firstly, we check co-prime
7 = 1 × 7
10 = 2 × 5
⇒ 7 and 10 have no common factor
Therefore, 7 and 10 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
10 = 5¹ × 2¹
So, denominator is of the form 2ⁿ5^m where n = 1 and m = 1
Thus, $\frac{35}{50}$ is a terminating decimal.

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Question 1 J 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{129}{2^{2} \times 5^{7} \times 7^{5}}$
Sol :
Given rational number is $\frac{129}{2^{2} \times 5^{7} \times 7^{5}}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.
Firstly, we check co-prime
129 = 3 × 43
Denominator = 2² ×5⁷ ×7⁵
⇒129 and 2² ×5⁷ ×7⁵ have no common factors
Therefore, 129 and 2² ×5⁷ ×7⁵ are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
Denominator = 2² ×5⁷ ×7⁵
So, denominator is not of the form 2ⁿ5^m
Thus, $\frac{129}{2^{2} \times 5^{7} \times 7^{5}}$ is a non-terminating repeating decimal.

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Question 1 K 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{2^{2} \times 7}{5^{4}}$
Sol :
Given rational number is $\frac{2^{2} \times 7}{5^{4}}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.
Firstly, we check co-prime
28 = 7 × 2²
625 = 5⁴
⇒ 28 and 625 have no common factors
Therefore, 28 and 625 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
625 = 5⁴ × 1
= 5⁴ × 2⁰
So, denominator is of the form 2ⁿ5^m where n = 0 and m = 4
Thus,$\frac{2^{2} \times 7}{5^{4}}$ is a terminating decimal.

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Question 1 L 

Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.

$\dfrac{29}{243}$
Sol :
Given rational number is $\frac{29}{243}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.
Firstly, we check co-prime
29 = 29 × 1
243 = 3⁵
⇒29 and 243 have no common factors
Therefore, 29 and 243 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
243 = 3⁵
So, the denominator is not of the form 2ⁿ5^m
Thus, $\frac{29}{243}$ is a non- terminating repeating decimal.

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Question 2 A 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{17}{8}$
Sol :
We know, $\frac{17}{8}=\frac{17}{2^{3} \times 5^{0}}$
Multiplying and dividing by 5³
$=\frac{17 \times 5^{3}}{2^{3} \times 5^{0} \times 5^{3}}$
$=\frac{17 \times 125}{2^{3} \times 1 \times 5^{3}}$
$=\frac{2125}{(2 \times 5)^{3}}$
$=\frac{2125}{(10)^{3}}$
$=\frac{2125}{1000}$
= 2.125

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Question 2 B 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{3}{8}$
Sol :
We know, $\frac{3}{8}=\frac{3}{2^{3} \times 5^{0}}$
Multiplying and dividing by 5³
$\frac{3 \times 5^{3}}{2^{3} \times 5^{0} \times 5^{3}}$
$=\frac{3 \times 125}{2^{3} \times 1 \times 5^{3}}$
$=\frac{375}{(2 \times 5)^{3}}$
$=\frac{375}{(10)^{3}}$
$=\frac{375}{1000}$
= 0.375

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Question 2 C 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{29}{343}$
Sol :
We know, $\frac{29}{343}=\frac{29}{7^{3}}$
Given rational number is $\frac{29}{343}$
$\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2ⁿ 5^m where n and m are non-negative integers.
Firstly we check co-prime
29 = 29 × 1
343 = 7 × 7 × 7
⇒29 and 343 have no common factors
Therefore, 29 and 343 are co-prime.
Now, we have to check that q is in the form of 2ⁿ5^m
343 = 7³
So, the denominator is not of the form 2ⁿ5^m
Thus, $\frac{29}{343}$ is a non-terminating repeating decimal.

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Question 2 D 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{13}{125}$
Sol :
We know, $\frac{13}{125}=\frac{13}{2^{0} \times 5^{3}}$
Multiplying and dividing by 2³
$=\frac{13 \times 2^{3}}{2^{0} \times 5^{3} \times 2^{3}}$
$=\frac{13 \times 8}{1 \times 2^{3} \times 5^{3}}$
$=\frac{104}{(2 \times 5)^{3}}$
$\frac{104}{(10)^{3}}$
$=\frac{104}{1000}$
= 0.104

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Question 2 E 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{27}{8}$

Sol :
We know, $\frac{27}{8}=\frac{3^{3}}{2^{3} \times 5^{0}}$
Multiplying and dividing by 5³
$\frac{27 \times 5^{3}}{2^{0} \times 2^{3} \times 5^{3}}$
$=\frac{27 \times 125}{1 \times 2^{3} \times 5^{3}}$
$=\frac{3375}{(2 \times 5)^{3}}$
$\frac{3375}{(10)^{3}}$
$=\frac{3375}{1000}$
= 3.375

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Question 2 F 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{7}{80}$
Sol :
We know, $\frac{7}{80}=\frac{7}{2^{4} \times 5^{1}}$
Multiplying and dividing by 5³
$=\frac{7 \times 5^{3}}{2^{4} \times 5^{1} \times 5^{3}}$
$=\frac{7 \times 125}{2^{4} \times 5^{4}}$
$=\frac{875}{(2 \times 5)^{4}}$
$=\frac{875}{(10)^{4}}$
$=\frac{875}{10000}$
= 0.0875

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Question 2 G 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{64}{455}$
Sol :
We know, $\frac{64}{455}=\frac{2^{6}}{5 \times 7 \times 13}$
Since the denominator is not of the form 2ⁿ5^m
$\frac{64}{455}$ has a non-terminating repeating decimal expansion.

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Question 2 H 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{6}{15}$
Sol :
We know, $\frac{6}{15}=\frac{2}{5}=\frac{2}{2^{0} \times 5}$
Multiplying and dividing by 2¹
$=\frac{2 \times 2}{2^{0} \times 5^{1} \times 2}$
$=\frac{4}{1 \times 10}$
$=\frac{4}{10}$
= 0.4

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Question 2 I 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{35}{50}$
Sol :
We know, $\frac{35}{50}$
$=\frac{7}{10}$
$=\frac{7}{2 \times 5}$
$=\frac{7}{10}$
= 0.7

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Question 2 J 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\frac{129}{2^{2} 5^{7} 7^{5}}$

Sol :
Given rational number is $\frac{129}{2^{2} \times 5^{7} \times 7^{5}}$
Since the denominator is not of the form 2ⁿ5^m
$\frac{129}{2^{2} \times 5^{7} \times 7^{5}}$has a non-terminating repeating decimal expansion.

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Question 2 K 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{2^{2} \times 7}{5^{4}}$
Sol :
We know, $\frac{2^{2} \times 7}{5^{4}}=\frac{2^{2} \times 7}{2^{0} \times 5^{4}}$
Multiplying and dividing by 2⁶
$\frac{2^{2} \times 7 \times 2^{6}}{2^{0} \times 5^{4} \times 2^{6}}$
$=\frac{2^{2} \times 7 \times 2^{6}}{1 \times 5^{4} \times 2^{6}}$
$=\frac{7 \times 2^{6}}{(2 \times 5)^{4}}$
$=\frac{448}{(10)^{4}}$
$=\frac{448}{10000}$
= 0.0448

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Question 2 L 

Write down the decimal expansions of the following numbers which have terminating decimal expansions.

$\dfrac{29}{243}$
Sol :
Given rational number is $\frac{29}{243}$
$\frac{29}{243}=\frac{29}{3^{5}}$
Since the denominator is not of the form 2ⁿ5^m.
$\frac{29}{243}$ has a non-terminating repeating decimal expansion.

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Question 3 

The following real numbers have decimal expansions as given below. In each case examine whether they are rational or not. If they are a rational number of the form p/q, what can be said about q?

(i) 7.2345
(ii) $5 . \overline{234}$
(iii) 23.245789
(iv) $7 . \overline{3427}$
(v) 0.120120012000120000…
(vi) 23.142857
(vii) 2.313313313331…
(viii) 0.02002000220002…
(ix) 3.300030000300003…
(x) 1.7320508…
(xi) 2.645713
(xii) 2.8284271…
Sol :

(i) 7.2345
Here, 7.2345 has terminating decimal expansion.
So, it represents a rational number.
i.e. 7.2345 $=\frac{7.2345}{10000}=\frac{p}{q}$
Thus, q = 10⁴, those factors are 2³ × 5³

(ii) $5 . \overline{234}$
$5 . \overline{234}$ is non-terminating but repeating.
So, it would be a rational number.
In a non-terminating repeating expansion of $\frac{p}{q}$ ,
q will have factors other than 2 or 5.

(iii) 23.245789
23.245789 is terminating decimal expansion
So, it would be a rational number.
i.e. 23.245789 $=\frac{23.245789}{1000000}=\frac{p}{q}$
Thus, q = 10⁶, those factors are 2⁵ × 5⁵
In a terminating expansion of $\frac{p}{q}$ , q is of the form 2ⁿ5^m
So, prime factors of q will be either 2 or 5 or both.

(iv) $7 . \overline{3427}$
$7 . \overline{3427}$ is non-terminating but repeating.
So, it would be a rational number.
In a non-terminating repeating expansion of $\frac{p}{q}$ ,
q will have factors other than 2 or 5.

(v) 0.120120012000120000…
0.120120012000120000… is non-terminating and non-repeating.
So, it is not a rational number as we see in the chart.

(vi) 23.142857
23.142857 is terminating expansion.
So, it would be a rational number.
i.e. 23.142857 $=\frac{23.142857}{1000000}=\frac{\underline{p}}{q}$
Thus, q = 10⁶, whose factors are 2⁵ × 5⁵
In a terminating expansion of $\frac{p}{q}$ , q is of the form 2ⁿ5^m
So, prime factors of q will be either 2 or 5 or both.

(vii) 2.313313313331…
2.313313313331… is non-terminating and non-repeating.
So, it is not a rational number as we see in the chart.

(viii) 0.02002000220002…
0.02002000220002… is non-terminating and non-repeating.
So, it is not a rational number as we see in the chart.

(ix) 3.300030000300003…
3.300030000300003… is non-terminating and non-repeating.
So, it is not a rational number as we see in the chart.

(x) 1.7320508…
1.7320508… is non-terminating and non-repeating.
So, it is not a rational number as we see in the chart.

(xi) 2.645713
2.645713 is terminating expansion
So, it would be a rational number.
i.e. 2.645713 $=\frac{2.645713}{1000000}=\frac{p}{q}$
Thus, q = 10⁶, those factors are 2⁵ × 5⁵
In a terminating expansion of $\frac{p}{q}$ , q is of the form 2ⁿ5^m
So, prime factors of q will be either 2 or 5 or both.

(xii) 2.8284271…
2.8284271… is non-terminating and non-repeating.
So, it is not a rational number as we see in the chart.

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S.no	Chapters	Links
1	Real numbers	Exercise 1.1 Exercise 1.2 Exercise 1.3 Exercise 1.4
2	Polynomials	Exercise 2.1 Exercise 2.2 Exercise 2.3
3	Pairs of Linear Equations in Two Variables	Exercise 3.1 Exercise 3.2 Exercise 3.3 Exercise 3.4 Exercise 3.5
4	Trigonometric Ratios and Identities	Exercise 4.1 Exercise 4.2 Exercise 4.3 Exercise 4.4
5	Triangles	Exercise 5.1 Exercise 5.2 Exercise 5.3 Exercise 5.4 Exercise 5.5
6	Statistics	Exercise 6.1 Exercise 6.2 Exercise 6.3 Exercise 6.4
7	Quadratic Equations	Exercise 7.1 Exercise 7.2 Exercise 7.3 Exercise 7.4 Exercise 7.5
8	Arithmetic Progressions (AP)	Exercise 8.1 Exercise 8.2 Exercise 8.3 Exercise 8.4
9	Some Applications of Trigonometry: Height and Distances	Exercise 9.1
10	Coordinates Geometry	Exercise 10.1 Exercise 10.2 Exercise 10.3 Exercise 10.4
11	Circles	Exercise 11.1 Exercise 11.2
12	Constructions	Exercise 12.1
13	Area related to Circles	Exercise 13.1
14	Surface Area and Volumes	Exercise 14.1 Exercise 14.2 Exercise 14.3 Exercise 14.4
15	Probability	Exercise 15.1