Case Based Questions: Number System | Mathematics (Maths) Class 9 PDF Download

Case Study 1

During revision hours, two students Vimal and Sunil were discussing with each other about the topic of rationalising the denominator.

Vimal explains that simplification of √3 / (√5 + √3) by rationalising the denominator is multiplying numerator and denominator by (√5 - √3).

And Sunil explains the simplification of (√5 + √3)(√5 - √3) by using the identity (a + b)(a - b) = a² - b².

On the basis of the above information, solve the following questions:

Q 1. The rationalising factor of √5 + √3 is:

a. 1 / (√5 + √3)

b. (√5 - √3)

c. -√5 - √3

d. (√5 + √3)

Q 2. According to Vimal explanation, the simplification of √3 / (√5 + √3) is:

a. (5 - √15)

b. (5 + √15) / 2

c. (√15 - 3) / 2

d. 5 + √15

Q 3. According to Sunil explanation, the simplification of (√3 + √2)(√3 - √2) is:

a. 1

b. -1

c. 2

d. 3

Q 4. Addition of two irrational numbers:

a. is always rational

b. is always irrational

c. may be rational or irrational

d. is always integer

Q 5. The square root of natural number is a/an:

a. rational

b. irrational

c. rational or irrational

d. None of these

Solutions

1. (b) The rationalising factor of (√5 + √3) is (√5 - √3).

So, option (b) is correct.

2. (c) We have, √3 / (√5 + √3)
By rationalisation the denominator

√3 / (√5 + √3) = √3 / (√5 + √3) × (√5 - √3) / (√5 - √3)

= (√3(√5 - √3)) / ((√5)² - (√3)²)

= (√3√5 - √3√3) / (5 - 3)

= (√15 - 3) / 2
So, option (c) is correct.

3. (a) Using identity (a + b)(a - b) = a² - b²

Therefore (√3 + √2)(√3 - √2) = (√3)² - (√2)²

= 3 - 2 = 1
So, option (a) is correct.

4. (c) Addition of two irrational numbers may be rational or irrational.

e.g. (2 + √5) + (1 - √5) = 3, which is rational.

(ii) (2 + √5) + √3 = 2 + √5 + √3, which is irrational.

So, option (c) is correct.

5. (c) The square root of a natural number is a rational or irrational number.

So, option (c) is correct.

Case Study 2

One day a math teacher taught students about the number system. She drew a number line on the black board and represented different types of numbers such as natural numbers, integers, rational numbers, etc. A number of the form p/q is said to be a rational number, if q ≠ 0 and p and q are integers.On the basis of the above information, solve the following questions:

Q 1. A rational number between 1/3 and 1/7 is:

a. 21/5

b. 17/21

c. 5/21

d. 5/21

Q 2. An irrational number between √3 and √5 is:

a. 2.1

b. √3.5

c. 3.5

d. √7

Q 3. Decimal number 1.5 in the form of p/q is:

a. 14/9

b. 11/9

c. 14/9

d. 11/9

Q 4. The sum of two rational numbers is always:

a. integers

b. naturals

c. rational

d. irrational

Q 5. A terminating or repeating decimal number is a/an:

a. rational

b. irrational

c. rational or irrational

d. whole number
Solutions:

1. (d) A rational number between 1/3 and 1/7 is

1/3 + 1/7 = (7 + 3) / (2 × 21) = 10 / 42 = 5 / 21

So, option (d) is correct.

2. (c) Since, √3 = 1.732 and √5 = 2.236

But √3.5 = 1.870, which lies in the given interval.

Hence, irrational number √3.5 lies between √3 and √5.

So, option (c) is correct.

3. (d) Let x = 1.5

⇒ x = 1.555... ...(1)

Multiplying both sides by 10, we get

10x = 15.55... ...(2)

Subtracting eq. (1) from eq. (2), we get

9x = 14 ⇒ x = 14 / 9

So, option (d) is correct.

4. (c) The sum of two rational numbers is always a rational number.

So, option (c) is correct.

5. (a) A terminating or repeating decimal number is a rational number.

So, option (a) is correct.