Case Study

Ms Mehta teaches maths in a school. One day after teaching the lesson of number system, she wanted to check the understanding of the students of her class. So, she wrote two numbers, $\dfrac{3}{11}\text{ and } 0.\overline{52}$ on the blackboard and asked few questions based on them. You please try to answer the following questions asked by Ms Mehta.

1. The decimal expansion of $\dfrac{3}{11}$ is:
   (a) terminating
   (b) non-terminating
   (c) non-terminating non-repeating
   (d) non-terminating repeating

2. $0.\overline{52}$ is:
   (a) non-terminating non-repeating
   (b) non-terminating repeating
   (c) non-terminating
   (d) terminating

3. The decimal form of $\dfrac{3}{11}$:
   (a) 0.27
   (b) 0.2727
   (c) $0.\overline{27}$
   (d) 0.3

4. $0.\overline{52}$ as vulgar fraction becomes:
   
   (a) $\dfrac{52}{99}$
   
   (b) $\dfrac{52}{100}$
   
   (c) $\dfrac{26}{25}$
   
   (d) $\dfrac{13}{25}$

5. The sum of $0.\overline{52} \text{ and } \dfrac{3}{11}$ is :
   (a) $\dfrac{79}{99}$
   
   (b) $\dfrac{70}{99}$
   
   (c) $\dfrac{52}{99}$
   
   (d) $\dfrac{40}{99}$

Assertion (A): The number obtained on rationalizing the denominator of $\dfrac{1}{\sqrt{5} - 2}$ is $2 + \sqrt{5}$.

Reason (R): If the product of two irrational numbers is rational, then each one is called the rationalizing factor of the other.

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false.

The sum of all rational numbers between 0 and 0.1 is :

1. finite

2. infinite

3. can't say anything

4. none of these

Four rational numbers p, q, r and s are such that q is the reciprocal of p and s is the reciprocal of r. The value of the expression $\Big([p + \dfrac{1}{q}] ÷ [r + \dfrac{1}{s}]\Big) ÷ \Big([s + \dfrac{1}{r}] ÷ [q + \dfrac{1}{p}]\Big)$ is equal to:

1. 1

2. 0

3. pr

4. $\dfrac{s}{q}$