Real Numbers
Class 9th Mathematics Part I MHB Solution
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.
