CBSE Class 10 Mathematics Real Numbers Assignment Set 09

Fill in the Blanks

Question 1. 144 as a product of its prime factors is ________.
Answer:
\( 144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = \) \( 2^4 \times 3^2 \)
In simple words: Split 144 into primes and write the repeated primes as powers.

Teacher's Note:
1. Divide by 2 till the number turns odd, then by 3.
2. Write the answer with powers, in ascending order of the primes.

Question 2. The least value of r in the relation a = bq + r is ________.
Answer:
In Euclid's division lemma 0 ≤ r < b, so the least value of r is 0 (zero).
In simple words: The remainder can never be negative, and the smallest possible remainder is zero.

Teacher's Note:
1. Recall the condition 0 ≤ r < b.
2. When r = 0, b divides a exactly.

Question 3. If every positive even integer is of the form 2q, then every positive odd integer is of the form ________, where q is some integer.
Answer:
Dividing by 2 leaves remainder 0 or 1. Remainder 0 gives the even form 2q, so remainder 1 gives the odd form 2q + 1.
In simple words: Even numbers are 2q, odd numbers are one more, 2q + 1.

Teacher's Note:
1. Use Euclid's lemma with b = 2.
2. The same idea gives 3q, 3q + 1, 3q + 2 for divisor 3.

Question 4. In the prime factorisation of 72, the power of 2 is ________.
Answer:
\( 72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2 \), so the power of 2 is 3.
In simple words: 72 contains three factors of 2.

Teacher's Note:
1. Keep dividing by 2: 72, 36, 18, 9.
2. Count how many times 2 was used.

Question 5. For finding the H.C.F. of two positive integers, an algorithm which is used is ________.
Answer:
Euclid's division algorithm.
In simple words: Repeated division with remainders gives the HCF.

Teacher's Note:
1. The algorithm stops when the remainder is 0.
2. The last divisor is the HCF.

Question 6. A lemma is a proven statement used for ________.
Answer:
A lemma is a proven statement used for proving another statement.
In simple words: A lemma is a helper result used to prove a bigger result.

Teacher's Note:
1. Euclid's division lemma is used to build the division algorithm.
2. A theorem is a main result; a lemma supports it.

Question 7. The product of two numbers is equal to the ________ of their HCF and LCM.
Answer:
Product of two numbers = HCF × LCM, so the blank is product.
In simple words: HCF × LCM gives back the product of the two numbers.

Teacher's Note:
1. The relation holds for two numbers only.
2. For three numbers it does not hold in this simple form.

Question 8. \( \frac{37}{50} \) is a ________ decimal expansion.
Answer:
\( 50 = 2 \times 5^2 \) is of the form \( 2^m 5^n \), so the expansion is terminating (\( \frac{37}{50} = 0.74 \)).
In simple words: The denominator has only 2 and 5 as prime factors, so the decimal ends.

Teacher's Note:
1. Factorise the denominator first.
2. Only 2 and 5 in the denominator means terminating.

Question 9. An ________ is a series of well-defined steps which gives a procedure for solving a type of problem.
Answer:
An algorithm.
In simple words: An algorithm is a step by step method.

Teacher's Note:
1. Euclid's division algorithm is a standard example.
2. The word comes from the name of a mathematician, Al-Khwarizmi.

Question 10. \( \sqrt{2} \) is a/an ________ number.
Answer:
\( \sqrt{2} \) is an irrational number.
In simple words: \( \sqrt{2} \) cannot be written as a fraction of integers.

Teacher's Note:
1. The proof uses contradiction and the idea of coprime integers.
2. Its decimal expansion never ends and never repeats.

True and False Statements

Question 11. For given positive integers a and b, there exist whole numbers q and r which satisfy the equation a = bq + r, where 0 ≤ r < b.
Answer:
True. This is Euclid's division lemma.
In simple words: Division always gives a quotient and a smaller remainder.

Teacher's Note:
1. It is the statement of the lemma itself.
2. Both q and r are whole numbers.

Question 12. Every composite number can be expressed in the form of a product of its prime factors, and this decomposition is unique, apart from the order in which the prime factors occur.
Answer:
True. This is the Fundamental Theorem of Arithmetic.
In simple words: Prime factorisation is unique except for order.

Teacher's Note:
1. Remember it as the unique factorisation theorem.
2. It is used to find HCF and LCM.

Question 13. The sum of two prime numbers is always an even number.
Answer:
False. For example, 2 + 3 = 5, which is odd. The sum is even only when both primes are odd.
In simple words: Because 2 is a prime, the sum can be odd.

Teacher's Note:
1. A single counter-example is enough to prove a statement false.
2. 2 is the only even prime.

Question 14. \( \sqrt{2} \) and \( \sqrt{3} \) are rational numbers.
Answer:
False. Both are irrational.
In simple words: Both square roots are non-repeating, non-ending decimals.

Teacher's Note:
1. The square root of a prime is irrational.
2. The proof of irrationality uses contradiction.

Question 15. Every point on the number line is of the form \( \sqrt{m} \).
Answer:
False. The number −3 is a point on the number line, but \( \sqrt{m} \) is never negative. So not every point is of the form \( \sqrt{m} \).
In simple words: The root sign never gives a negative number.

Teacher's Note:
1. One counter-example is enough to show a statement is false.
2. Here m is taken as a positive number, so its root is not negative.

Question 16. The sum of two rational numbers is a rational number.
Answer:
True. Rational numbers are closed under addition: \( \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \).
In simple words: Adding fractions gives a fraction.

Teacher's Note:
1. The denominator bd is non-zero.
2. The numerator ad + bc is an integer.

Question 17. The quotient of two integers is always a rational number.
Answer:
False. The quotient of two integers is rational only if the divisor is not zero; \( \frac{1}{0} \) is not defined.
In simple words: Division by zero is not allowed.

Teacher's Note:
1. A rational number needs a non-zero denominator.
2. This is why q ≠ 0 is always stated.

Question 18. \( \frac{1}{0} \) is not a rational number.
Answer:
True. The denominator of a rational number cannot be zero.
In simple words: A fraction with zero in the denominator has no meaning.

Teacher's Note:
1. Rational numbers are of the form p/q with q ≠ 0.
2. Division by zero is undefined.

Question 19. If \( x = \frac{p}{q} \) is a rational number such that the prime factorisation of q is of the form \( 2^m 5^n \), where m and n are non-negative integers, then x has a decimal expansion which terminates.
Answer:
True. Multiplying numerator and denominator suitably turns the denominator into a power of 10.
In simple words: Only 2s and 5s in the denominator means the decimal stops.

Teacher's Note:
1. This is Theorem 5 of the chapter.
2. Check the denominator in lowest terms.

Question 20. The number of irrational numbers between 15 and 18 is finite.
Answer:
False. Between any two real numbers there are infinitely many irrational numbers (for example \( \sqrt{226}, \sqrt{227}, \sqrt{228}, \ldots \)).
In simple words: There is no end to irrational numbers between two numbers.

Teacher's Note:
1. Between two numbers there are infinitely many rational and irrational numbers.
2. Squares of 15 and 18 are 225 and 324, so square roots of 226 to 323 all lie between them.

Match the Following

Question 21. Match Column I with Column II.
(1) A rational number is always    (A) irrational number
(2) An irrational number is always    (B) rational number
(3) \( \frac{4}{7} \) is a    (C) non-terminating and non-repeating
(4) \( \sqrt{5} \) is an    (D) terminating or non-terminating recurring
(a) 1-D, 2-C, 3-B, 4-A
(b) 1-A, 2-B, 3-C, 4-D
(c) 1-C, 2-D, 3-A, 4-C
(d) 1-B, 2-C, 3-A, 4-D
Answer:
Answer: (a) 1-D, 2-C, 3-B, 4-A
A rational number has a terminating or non-terminating recurring decimal (D). An irrational number has a non-terminating, non-repeating decimal (C). \( \frac{4}{7} \) is a rational number (B). \( \sqrt{5} \) is an irrational number (A).
In simple words: Rational decimals end or repeat; irrational decimals never end or repeat.

Teacher's Note:
1. Match the nature of the decimal first.
2. Then match the named numbers.

Question 22. Match the following. Column II: (A) 2, (B) 5, (C) 9.
(1) H.C.F. of 306 and 657
(2) H.C.F. of the smallest composite number and the smallest prime number
(3) H.C.F. of 475 and 495
(a) 1-A, 2-B, 3-C
(b) 1-C, 2-B, 3-A
(c) 1-C, 2-A, 3-B
(d) 1-B, 2-A, 3-C
Answer:
Answer: (c) 1-C, 2-A, 3-B
(1) 657 = 306 × 2 + 45, 306 = 45 × 6 + 36, 45 = 36 × 1 + 9, 36 = 9 × 4 + 0, so HCF = 9 (C).
(2) The smallest composite number is 4 and the smallest prime is 2, so HCF (4, 2) = 2 (A).
(3) 495 = 475 × 1 + 20, 475 = 20 × 23 + 15, 20 = 15 × 1 + 5, 15 = 5 × 3 + 0, so HCF = 5 (B).
In simple words: Use Euclid's algorithm for each pair.

Teacher's Note:
1. The smallest composite number is 4, and 1 is not composite.
2. Verify each HCF by prime factors if time permits.