Problem Set 2 Real Numbers Class 9th Mathematics Part I MHB Solution

Real Numbers
Class 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


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.




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