### Real Numbers

Class 9th Mathematics Part I MHB Solution

**Problem Set 2**

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

Answer:

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....

Answer:

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

Answer:

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.

Answer:

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

Answer:

∵ 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.

Answer:

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

Answer:

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

Answer:

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

Answer:

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

Answer:

|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.

Answer:

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

Answer:

i.

ii.

iii.

√5 = 2.236067977…….

iv.

v.

Question 13.

Show that 5 + √7 is an irrational number.

Answer:

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.

Answer:

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

Answer:

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.

Answer:

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.

Answer:

i.

ii.

iii.

iv.

v.