Question Bank — The World of Numbers

Section A — MCQs (10 questions, 1 mark each)

1. Which of the following sets correctly represents the natural numbers as described in the chapter?
   (a) {0, 1, 2, 3, …}
   (b) {1, 2, 3, 4, …}
   (c) {…, –2, –1, 0, 1, 2, …}
   (d) All fractions of the form p/q where q ≠ 0

2. The Ishango bone is historically significant because one of its columns groups notches into:
   (a) Even numbers only
   (b) Prime numbers between 10 and 20
   (c) Multiples of 10
   (d) Powers of 2

3. According to Brahmagupta’s rules stated in the chapter, the result of a × 0 is:
   (a) a
   (b) 0
   (c) –a
   (d) Undefined

4. The set of integers (Z) is formed by combining:
   (a) Natural numbers and zero only
   (b) Natural numbers, their negative counterparts, and zero
   (c) Rational numbers and irrational numbers
   (d) Positive fractions only

5. A rational number is defined in the chapter as any number that can be expressed in the form:
   (a) p/q where p and q are natural numbers
   (b) p/q where p and q are integers and q ≠ 0
   (c) p/q where p and q are positive integers
   (d) √n where n is a positive integer

6. Which property of rational numbers is illustrated by the fact that between any two rational numbers another rational number can always be found?
   (a) Closure
   (b) Density
   (c) Commutativity
   (d) Distributivity

7. The first formal proof of the irrationality of √2, as described in the chapter, was given by:
   (a) Brahmagupta
   (b) Āryabhaṭa
   (c) Hippasus using proof by contradiction
   (d) Mādhava

8. The decimal expansion of a rational number is always:
   (a) Non-terminating and non-repeating
   (b) Terminating or repeating
   (c) Non-terminating and repeating only
   (d) Terminating only

9. Assertion (A): The number 0.142857 (repeating) is a cyclic number.
   Reason (R): Multiplying 142857 by 1 through 6 produces cyclic permutations of the same digits.
   (a) Both A and R are true and R is the correct explanation of A.
   (b) Both A and R are true but R is not the correct explanation of A.
   (c) A is true but R is false.
   (d) A is false but R is true.

10. Assertion (A): Rational numbers are closed under addition, subtraction, multiplication and division (except division by zero).
    Reason (R): The sum, difference, product or quotient (when defined) of any two rational numbers is again a rational number.
    (a) Both A and R are true and R is the correct explanation of A.
    (b) Both A and R are true but R is not the correct explanation of A.
    (c) A is true but R is false.
    (d) A is false but R is true.

Section B — Very Short Answer (6 questions, 2 marks each)

1. State the definition of natural numbers as given in the chapter and give one example of one-to-one correspondence used by early humans.
2. According to Brahmagupta, what is the result of subtracting a number from itself? How did this lead to the concept of zero?
3. Using Brahmagupta’s laws, evaluate: (–8) × (–7) and explain the result in terms of debt.
4. Define a rational number. Why must q ≠ 0 in the form p/q?
5. What is the absolute value of a rational number? Give the absolute values of 5/3 and –5/3.
6. Distinguish between the decimal expansions of rational and irrational numbers with one example of each from the chapter.

Section C — Short Answer (5 questions, 3 marks each)

1. A merchant in Lothal receives 15 copper ingots for every 2 bags of spices. If he brings 12 bags, how many ingots does he receive? Show the calculation using one-to-one correspondence idea.
2. Find the sum: 2/5 + 3/10. Express the result in lowest terms and verify it equals a single rational number.
3. Represent –5/4 and 3/2 on a number line. Describe the steps to locate each point.
4. Without performing long division, determine whether 7/20 and 4/15 have terminating or repeating decimals. Justify using prime factors of the denominator.
5. State Brahmagupta’s rule for the product of two debts and verify with the example (–3) × (–4).

Section D — Long Answer (3 questions, 5 marks each)

1. Prove that √2 is irrational using the method of proof by contradiction as described in the chapter. Write all eight steps clearly.
2. A tailor has 15¾ metres of silk. Each kurta requires 2¼ metres. How many kurtas can he make exactly? Show all steps using operations on rational numbers.
3. Construct a line segment of length √2 on the number line following the three steps given in the chapter. Justify each step with the Pythagorean relation used.

Section E — Case/Source-Based (2 questions, 4 marks each)

Case 1: Temperature in Ladakh

The temperature in Ladakh is recorded as 4 °C at noon. By midnight it drops by 15 °C.

(a) Express the midnight temperature as an integer using Brahmagupta’s concept of debt.
(b) Write the calculation as an equation using integers.
(c) What does the result represent in terms of fortune and debt?
(d) If the temperature then rises by 7 °C by next noon, find the new temperature.

Case 2: Spice Trader’s Accounts

A spice trader takes a loan of ₹850. The next day he makes a profit of ₹1,200. The following week he incurs a loss of ₹450.

(a) Represent the three amounts as integers (debt and fortune).
(b) Write the sequence as a single equation using integers.
(c) Calculate his final financial standing.
(d) Explain the result using Brahmagupta’s rules for addition of fortunes and debts.