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### Real NumbersClass 9th Mathematics Part I MHB Solution

Problem Set 2

Question 1.

Choose the correct alternative answer for the questions given below.

i. Which one of the following is an irrational number?
A. √16/25

B. √5

C. 3/9

D. √196

An irrational number is a number that cannot be expressed as a fraction for any integers p and q and q ≠ 0.

since it can be written as  , it is a rational number.

since it can be written as  , it is a rational number.

since it can be written as  , it is a rational number.

Since √5 cannot be written as  it is an irrational number

Therefore √5 is an irrational number.

Question 2.

Which of the following is an irrational number?
A. 0.17

B.

C.

D. 0.101001000....

An irrational number is a number that cannot be expressed as a fraction for any integers p and q and q ≠ 0.

.

Since it can be written as  ,

it is a rational number.

is a rational number because it is a non-terminating but repeating decimal.

is a rational number because it is a non-terminating but repeating decimal.

0.101001000.... is an irrational number because it is a non-terminating and non-`repeating decimal.

Therefore, 0.101001000.... is an irrational number.

Question 3.

Decimal expansion of which of the following is non-terminating recurring?
A. 2/5

B. 3/16

C. 3/11

D. 137/25

A non-terminating recurring decimal representation means that the number will have an infinite number of digits to the right of the decimal point and those digits will repeat themselves.

∵ it does not have an infinite number of digits to the right of the decimal point ∴ it is not a non-terminating recurring decimal.

∵ it does not have an infinite number of digits to the right of the decimal point ∴ it is not a non-terminating recurring decimal.

∵ it has an infinite number of digits to the right of the decimal point which are repeating themselves ∴ it is a non-terminating recurring decimal.

∵ it does not have an infinite number of digits to the right of the decimal point ∴ it is not a non-terminating recurring decimal.

Therefore,  is a non-terminating recurring decimal.

Question 4.

Every point on the number line represent, which of the following numbers?
A. Natural numbers

B. Irrational numbers

C. Rational numbers

D. Real numbers.

Every point of a number line is assumed to correspond to a real number, and every real number to a point. Therefore, Every point on the number line represent a real number.

Question 5.

The number 0.4 in p/q form is ………….
A. 4/9

B. 40/9

C. 3.6/9

D. 36/9

∵ the denominator of all the above options is 9 ∴ we multiply both numerator and denominator by 0.9 as 10 × 0.9 = 9

Question 6.

What is √n, if n is not a perfect square number?
A. Natural number

B. Rational number

C. Irrational number

D. Options A, B, C all are correct.

If n is not a perfect square number, then √n cannot be expressed as ratio of a and b where a and b are integers and b ≠ 0

Therefore, √n is an Irrational number

Question 7.

Which of the following is not a surd?
A. √7

B. 3√17

C. 3√64

D. √193

Which is a rational number

Therefore,  is not a surd.

Question 8.

What is the order of the surd ?
A. 3

B. 2

C. 6

D. 5

Therefore, the order of the surd  is 6.

Question 9.

Which one is the conjugate pair of 2√5 + √3?
A. -2√5 + √3

B. -2√5 - √3

C. 2√3 + √5

D. √3 + 2√5

A math conjugate is formed by changing the sign between two terms in a binomial. For instance, the conjugate of x + y is x - y.

Now,

2√5 + √3 = √3 + 2√5

Its conjugate pair = √3 - 2√5 = -2√5 + √3

∴ The conjugate pair of 2√5 + √3 = -2√5 + √3

Question 10.

The value of |12 – (13 + 7) × 4| is ...........
A. -68

B. 68

C. -32

D. 32

|12 – (13 + 7) × 4| = |12 – 20 × 4| (Solving it according to BODMAS)

⇒ |12 – (13 + 7) × 4| = |12 – 80|

⇒ |12 – (13 + 7) × 4| = |-68|

⇒ |12 – (13 + 7) × 4| = 68

Question 11.

Write the following numbers in p/q form.

i. 0.555 ii.

iii. 9.315 315 ... iv. 357.417417...

v.

i.

ii.

Let

⇒ 1000x = 29568.568568......

Now,

1000x - x = 29568.568568 – 29.568568

⇒999x = 29539.0

iii.

Let x = 9.315315…

⇒ 1000x = 9315.315315......

Now,

1000x - x = 9315.315315 – 9.315315

⇒999x = 9306.0

iv.

Let x = 357.417417…

⇒ 1000x = 357417.417417…

Now,

1000x - x = 357417.417417 – 357.417417

⇒999x = 357060.0

v.

Let

⇒ 1000x = 30219.219219…

Now,

1000x - x = 30219.219219 – 30.219219

⇒999x = 30189.0

Question 12.

Write the following numbers in its decimal form.

i. -5/7 ii. 9/11

iii. √5 iv. 121/13

v. 29/8

i.

ii.

iii.

√5 = 2.236067977…….

iv.

v.

Question 13.

Show that 5 + √7 is an irrational number.

Let us assume that 5 + √7 is a rational number

where, b≠0 and a, b are integers

∵ a, b are integers ∴ a – 5b and b are also integers

is rational which cannot be possible ∵ which is an irrational number

∵ it is contradicting our assumption ∴ the assumption was wrong

Hence, 5 + √7 is an irrational number

Question 14.

Write the following surds in simplest form.

i.  ii.

i.

ii.

Question 15.

Write the simplest form of rationalizing factor for the given surds.

i. √32 ii. √50

iii. √27 iv. 3/5√10

v. 3√72 vi. 4√11

i. √32

∴ Its rationalizing factor = √2

ii. √50

∴ Its rationalizing factor = √2

iii. √27

∴ Its rationalizing factor = √3

∵ √10 cannot be further simplified

∴ Its rationalizing factor = √10

v. 3√72

∴ Its rationalizing factor = √2

vi. 4√11

∵ √11 cannot be further simplified

∴ Its rationalizing factor = √11

Question 16.

Simplify.

i.

ii.

iii.

iv.

v.

i.

= 4√3 + 3√3 – √3

= 7√3 – √3

= 6√3

ii.

iii.

iv.

v.

Question 17.

Rationalize the denominator.

i.  ii.

iii.  iv.

v.

i.

ii.

iii.

iv.

v.