Chapter 1 – Number Systems

Class 9 Mathematics · 46 questions · 32 with answers

Solved examples

Ex. 1Short answer

Which of the following is not equal to ? 6

Ex. 1Short answerExercise 1.1

Are there two irrational numbers whose sum and product both are rationals? Justify.

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Solution : Yes.

Ex. 2True / FalseExercise 1.1

State whether the following statement is true: There is a number x such that x2 is irrational but x4 is rational. Justify your answer by an example.

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Solution : True. Let us take x = 4 2 Now, x2 = ( 4 2 ) = 2 , an irrational number. x4 = ( 4 2 ) = 2 , a rational number. So, we have a number x such that x2 is irrational but x4 is rational.

Ex. 1Short answerExercise 1.2

Locate 13 on the number line.

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Solution : We write 13 as the sum of the squares of two natural numbers :

Ex. 2Short answerExercise 1.2

Express 0.123 in the form , where p and q are integers and q ≠ 0.

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Solution : Let x = 0.123 so, 10x = 1.23 or 10x – x = 1.23 – 0.123 = 1.2333 ... – 0.12333 ... or 9x = 1.11 1.11 111 or x= = 9 900 111 37 Therefore, 0.123 = 900 = 300

Ex. 3Short answerExercise 1.2

Simplify : ( 3 5 − 5 2 ) ( 4 5 + 3 2 ) . ( )(

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Solution : 3 5 – 5 2 4 5 + 3 2 ) = 12 × 5 − 20 2 × 5 + 9 5 × 2 – 15 × 2 = 60 − 20 10 + 9 10 – 30 = 30 − 11 10

Ex. 4Short answerExercise 1.2

Find the value of a in the following : = 3 2−a 3 3 2 −2 3 6 6 3 2+2 3

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Solution : = × 3 2 −2 3 3 2 −2 3 3 2+2 3 6 (3 2 + 2 3 ) 6 (3 2 + 2 3 ) 6 (3 2 + 2 3 ) = 2 2 = = (3 2 ) − ( 2 3 ) 18 − 12 6 = 3 2+2 3 Therefore, 3 2+2 3 = 3 2−a 3 or a=–2

Ex. 5Short answerExercise 1.2

Simplify : 13 1 34 ( 5 8 + 27 3 )

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Solution : 1 1 1 3 = 3 4 ( 1 5 8 3 + 27 3 ) ( 1 5 (23 ) 3 + (33 ) 3 ) NUMBER SYSTEMS 9 = 5 ( 2 + 3)3 4 = 5 ( 5 )3 4 = [5 4 ] 4 = 5

Ex. 1Short answerExercise 1.3

If a = 5 + 2 6 and b = , then what will be the value of a2 + b2 ?

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Solution : a = 5 + 2 6

Questions

Q1Multiple choice

1 1 6 1 1 – – 5 5 6 5 5 6 30 5 30

(A)
(B)
(C)
(D)6 6 5 6 Solution : Answer (A)

Q1Multiple choiceExercise 1.1

Every rational number is

(A)a natural number
(B)an integer
(C)a real number
(D)a whole number NUMBER SYSTEMS 3

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Q2Multiple choiceExercise 1.1

Between two rational numbers

(A)there is no rational number
(B)there is exactly one rational number
(C)there are infinitely many rational numbers
(D)there are only rational numbers and no irrational numbers

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Q3Multiple choiceExercise 1.1

Decimal representation of a rational number cannot be

(A)terminating
(B)non-terminating
(C)non-terminating repeating
(D)non-terminating non-repeating

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(D) non-terminating non-repeating

Q4Multiple choiceExercise 1.1

The product of any two irrational numbers is

(A)always an irrational number
(B)always a rational number
(C)always an integer
(D)sometimes rational, sometimes irrational

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(D) sometimes rational, sometimes irrational

Q5Multiple choiceExercise 1.1

The decimal expansion of the number 2 is

(A)a finite decimal
(B)1.41421
(C)non-terminating recurring
(D)non-terminating non-recurring

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(D) non-terminating non-recurring

Q6Multiple choiceExercise 1.1

Which of the following is irrational? 4 12

(A)
(B)
(C)7
(D)81

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Q9Multiple choiceExercise 1.1

3 7. Which of the following is irrational?

(A)0.14
(B)0.1416
(C)0.1416
(D)0.4014001400014... 8. A rational number between 2 and 3 is 2+ 3 2⋅ 3 (A) (B) (C) 1.5 (D) 1.8 2 2 9. The value of 1.999... in the form q , where p and q are integers and q ≠ 0 , is 19 1999 1 (A) (B) (C) 2 (D)

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Q10Multiple choiceExercise 1.1

1000 9 10. 2 3 + 3 is equal to

(A)2 6
(B)6
(C)3 3
(D)4 6

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Q11Multiple choiceExercise 1.1

10 × 15 is equal to

(A)6 5
(B)5 6
(C)25
(D)10 5

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(B) 5 6

Q12Multiple choiceExercise 1.1

The number obtained on rationalising the denominator of is 7 –2 7+2 7 –2 7+2 7+2

(A)
(B)
(C)
(D)3 3 5 45

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(A)

Q13Short answerExercise 1.1

is equal to 9– 8

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(D)

Q1Multiple choiceExercise 1.1

(A)(3–2 2 )
(B)3+ 2 2
(C)3– 2 2
(D)3+ 2 2 14. After rationalising the denominator of , we get the denominator as 3 3–2 2 (A) 13 (B) 19 (C) 5 (D) 35 32 + 48 15. The value of is equal to 8 + 12 (A) 2 (B) 2 (C) 4 (D) 8 2 –1 16. If 2 = 1.4142, then is equal to 2 +1 NUMBER SYSTEMS 5 (A) 2.4142 (B) 5.8282 (C) 0.4142 (D) 0.1718 17. 4 3 22 equals

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Q1Multiple choiceExercise 1.1

(A)−
(B)2– 6
(C)
(D)26

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(C)

Q2Multiple choiceExercise 1.1

6 26 18. The product 3 2 ⋅ 4 2 ⋅ 12 32 equals

(A)
(B)2
(C)
(D)12 2 12 2 32 −2 19. Value of 4 (81) is

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(C)

Q1Multiple choiceExercise 1.1

1 1

(A)
(B)
(C)9
(D)9 3 81 20. Value of (256)0.16 × (256)0.09 is (A) 4 (B) 16 (C) 64 (D) 256.25 21. Which of the following is equal to x? 1 2 12 5 12 7 (A) 7 x –x 7 (B) 12 (x ) 4 3 (C) ( ) x3 3 (D) x 7 × x12

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Q3Short answer (reasoning)Exercise 1.1

+ 2 and 3 − 2 are two irrational numbers. ( 3 + 2 ) + (3 − 2 ) = 6 , a rational number. ( 3 + 2 ) × ( 3 − 2 ) = 7 , a rational number. So, we have two irrational numbers whose sum and product both are rationals.

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(D)

Q1Short answerExercise 1.2

Let x and y be rational and irrational numbers, respectively. Is x + y necessarily an irrational number? Give an example in support of your answer.

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Yes. Let x = 21, y = 2 be a rational number. Now x + y = 21 + 2 = 21 + 1.4142 ... = 22.4142 ... Which is non-terminating and non-recurring. Hence x + y is irrational.

Q2Short answerExercise 1.2

Let x be rational and y be irrational. Is xy necessarily irrational? Justify your answer by an example.

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No. 0 × 2 = 0 which is not irrational . 2 p

Q3Multiple choiceExercise 1.2

State whether the following statements are true or false? Justify your answer.

(i)is a rational number.
(ii)There are infinitely many integers between any two integers.
(iii)Number of rational numbers between 15 and 18 is finite.
(iv)There are numbers which cannot be written in the form q , q ≠ 0 , p, q both are integers. (v) The square of an irrational number is always rational. (vi) is not a rational number as 12 and 3 are not integers. 15 p (vii) is written in the form , q ≠ 0 and so it is a rational number. 3 q

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(i) is a rational number.

(ii) There are infinitely many integers between any two integers.

(iii) Number of rational numbers between 15 and 18 is finite.

(iv) There are numbers which cannot be written in the form q , q ≠ 0 , p, q both are integers. (v) The square of an irrational number is always rational. (vi) is not a rational number as 12 and 3 are not integers. 15 p (vii) is written in the form , q ≠ 0 and so it is a rational number. 3 q

Q4Multiple choiceExercise 1.2

Classify the following numbers as rational or irrational with justification : 9 28

(i)196
(ii)3 18
(iii)
(iv)27 343 NUMBER SYSTEMS 7 (v) – 0.4 (vi) (vii) 0.5918 (viii) (1 + 5 ) – ( 4 + 5 ) (ix) 10.124124... (x) 1.010010001...

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(i) 196

(ii) 3 18

(iii)

(iv) 27 343 NUMBER SYSTEMS 7 (v) – 0.4 (vi) (vii) 0.5918 (viii) (1 + 5 ) – ( 4 + 5 ) (ix) 10.124124... (x) 1.010010001...

Q13Short answerExercise 1.2

= 9 + 4 = 32 + 22 On the number line, take OA = 3 units. Draw BA = 2 units, perpendicular to OA. Join OB (see Fig.1.1). By Pythagoras theorem, OB = 13 Using a compass with centre O and radius OB, draw an arc which intersects the number line at the Fig. 1.1 point C. Then, C corresponds to 13 . Remark : We can also take OA = 2 units and AB = 3 units.

This question refers to a figure in the original PDF.

Q1Multiple choiceExercise 1.3

Find which of the variables x, y, z and u represent rational numbers and which irrational numbers :

(i)x2 = 5
(ii)y2 = 9
(iii)z2 = .04
(iv)u2 =

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(ii) y2 = 9

(iii) z2 = .04

(i) x2 = 5

(iv) u2 =

Q2Multiple choiceExercise 1.3

Find three rational numbers between

(i)–1 and –2
(ii)0.1 and 0.11

Q5Short answerExercise 1.3

6 1 1 (iii) and (iv) and

Q7Multiple choiceExercise 1.3

7 4 5 3. Insert a rational number and an irrational number between the following : 1 1

(i)2 and 3
(ii)0 and 0.1
(iii)and 3 2 –2 1
(iv)and (v) 0.15 and 0.16 (vi) 2 and 3 5 2 (vii) 2.357 and 3.121 (viii) .0001 and .001 (ix) 3.623623 and 0.484848 (x) 6.375289 and 6.375738 4. Represent the following numbers on the number line : –3 –12 7, 7.2, , 2 5 5. Locate 5, 10 and 17 on the number line. 6. Represent geometrically the following numbers on the number line : (i) 4.5 (ii) 5.6 (iii) 8.1 (iv) 2.3 7. Express the following in the form q , where p and q are integers and q ≠ 0 : (i) 0.2 (ii) 0.888... (iii) 5.2 (iv) 0.001 (v) 0.2555... (vi) 0.134 (vii) .00323232... (viii) .404040...

Q8Short answerExercise 1.3

Show that 0.142857142857... =

Q9Multiple choiceExercise 1.3

Simplify the following: 24 54

(i)45 – 3 20 + 4 5
(ii)+ 8 9
(iii)12 × 7 6
(iv)4 28 ÷ 3 7 ÷ 3 7 7 2 (v) 3 3 + 2 27 + (vi) ( 3 – 2) 3 1 (vii) 4 81 – 8 3 216 + 15 5 32 + 225 (viii) + 8 2 2 3 3 (ix) – 3 6

Q10Multiple choiceExercise 1.3

Rationalise the denominator of the following: 2 40 3+ 2

(i)
(ii)
(iii)3 3 3 4 2 16 2+ 3 6
(iv)(v) (vi) 41 – 5 2– 3 2+ 3 3+ 2 3 5+ 3 4 3+5 2 (vii) (viii) (ix) 3– 2 5– 3 48 + 18

Q11Multiple choiceExercise 1.3

Find the values of a and b in each of the following: 5+ 2 3

(i)=a−6 3 7+4 3 NUMBER SYSTEMS 11 3– 5 19
(ii)=a 5– 3+ 2 5 11 2+ 3
(iii)= 2–b 6 3 2 –2 3 7+ 5 7– 5 7
(iv)– =a+ 5b 7– 5 7+ 5 11

Q12Short answerExercise 1.3

If a = 2 + 3 , then find the value of a – .

Q13Multiple choiceExercise 1.3

Rationalise the denominator in each of the following and hence evaluate by taking 2 = 1.414 , 3 = 1.732 and 5 = 2.236 , upto three places of decimal. 4 6 10 – 5

(i)
(ii)
(iii)3 6 2 2 1
(iv)(v) 2+ 2 3+ 2

Q14Short answerExercise 1.3

Simplify : 4 −12 6

Q1Short answerExercise 1.3

3 8 32 (i) (1 + 2

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Rational numbers: (ii), (iii) Irrational numbers: (i), (iv)

Q3Short answerExercise 1.3

3 + 33 2 ) (ii) 5 5 5 1 2 1 − − 4 −2 2 (iii)

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(i) 2.1, 2.040040004 ... (ii) 0.03, 0.007000700007, ... (iii) , 0.414114111 ... (iv) 0, 0.151151115 ... (v) 0.151, 0.151551555 ... (vi) 1.5, 1.585585558 ... (vii) 3, 3.101101110 ... (viii) 0.00011, .0001131331333 ... (ix) 1, 1.909009000 ... (x) 6.3753, 6.375414114111 ... 1 8 47 1 23 7. (i) (ii) (iii) (iv) (v) 5 9 9 999 90 133 8 40 (vi) (vii) (viii) 990 2475 99 7 6 8 34 3 9. (i) 5 (ii) (iii) 168 2 (iv) (v) 12 3 3 5 3 (vi) 5 − 2 6 (vii) 0 (viii) 2 (ix) 4 2 2 2 2+3 2 10. (i) 3 (ii) 30 (iii) (iv) 41 + 5 9 3 8 (v) 7 + 4 3 (vi) 3 2 − 2 3 (vii) 5 + 2 6 (viii) 9 + 2 15 9+4 6 (ix) 9 −5 11. (i) a = 11 (ii) a = (iii) b = (iv) a = 0, b = 1 11 6 12. 2 3 13. (i) 2.309 (ii) 2.449 (iii) 0.463 (iv) 0.414 (v) 0.318 PNPS 14. (i) 6 (ii) (iii) 9 (iv) 5 (v) 3– 3 (vi) –3 (vii) 16 ANSWERS 153

Q1Short answerExercise 1.3

3 (iv) ( 625)

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Rational numbers: (ii), (iii) Irrational numbers: (i), (iv)

Q1Short answerExercise 1.3

1 − 1 1 2 9 3 × 27 2 − (v) 1 2 (vi) 64 3 64 3 – 64 3 − 36 × 3 3

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Rational numbers: (ii), (iii) Irrational numbers: (i), (iv)

Q1Short answerExercise 1.3

1 8 3 × 16 3 (vii) − 32 3

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Rational numbers: (ii), (iii) Irrational numbers: (i), (iv)

Q1Long answerExercise 1.3

1 1 5−2 6 5−2 6 5−2 6 b= = = × = 2 2 = =5−2 6 a 5+2 6 5 + 2 6 5 − 2 6 5 − (2 6) 25 − 24 Therefore, a2 + b2 = (a + b)2 – 2ab Here, a + b = (5 + 2 6 ) + (5 – 2 6 ) = 10 ab = (5 + 2 6 ) (5 – 2 6 ) = 52 – ( 2 6 )2 = 25 – 24 = 1 Therefore, a2 + b2 = 102 – 2 × 1 = 100 – 2 = 98

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Rational numbers: (ii), (iii) Irrational numbers: (i), (iv)

Q1Short answerExercise 1.4

Express 0.6 + 0.7 + 0.47 in the form , where p and q are integers and q ≠ 0 . 7 3 2 5 3 2

Q2Short answerExercise 1.4

Simplify : – – . 10 + 3 6+ 5 15 + 3 2 4 3

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Q3Short answerExercise 1.4

If 2 = 1.414, 3 = 1.732 , then find the value of + . 3 3–2 2 3 3+2 2 3+ 5 2 1

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2.063

Q4Short answerExercise 1.4

If a = , then find the value of a + 2 . 2 a 3+ 2 3– 2

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Q5Short answerExercise 1.4

If x = and y = , then find the value of x2 + y2. 3– 2 3+ 2 ( ) −3 − 42

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Q6Short answerExercise 1.4

Simplify : ( 256 ) 4 1 2

Q7Short answerExercise 1.4

Find the value of 2 + 3 + 1 − − ( ) 216 3 ( ) 256 4 ( 243)− 5

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214