MCQs for Real Numbers: Introduction & Explanation

Answer: (c) Non-terminating and recurring

Explanation: Terminating decimal refers to a number with a finite number of digits after the decimal. A recurring decimal means a number with infinitely repeating digits.

22/7 = 3.14285714286

When we divide 22 by 7 we get a non-terminating and recurring decimal value. So, the correct option is (c) Non-terminating and recurring.

Answer: (c) 2n + 1

Explanation: As given, n is an integer, it can either be an even integer or an odd integer. So we can multiply it by 2 to get an even integer. Since 2n represents the even numbers, so, 2n + 1 will always represent the odd numbers.

For example, if n = 3, then 2n = 6 and 2n + 1 = 7.

Answer: (b) 13

Explanation: HCF is the Highest Common Factor. It can be calculated by multiplying all the factors that appear in the factorisation of both numbers. The prime factorisation of 26 and 91 are;

26 = 2 x 13

91 = 7 x 13

13 is the only factor appearing in the factorization of both 26 and 91. Hence, HCF (26, 91) = 13

Answer: (c) (3 + √7) (3 – √7)

Explanation: Irrational numbers are the numbers that cannot be expressed as ratio of two integers.

If we solve, (3 + √7) (3 – √7), we get;

(3 + √7) (3 – √7)

= 32 – (√7)2 [By a2 – b2 = (a – b) (a + b)]

= 9 – 7

= 2

Answer: (b) Irrational number

Explanation: The sum of a rational number and an irrational number is always irrational. By definition, an irrational number is a decimal form that goes on forever without repeating (a non-recurring, non-terminating decimal). By definition, a rational number is a decimal form that either terminates or repeats. So, if we add a rational number to an irrational number it will still be a non-recurring, non-terminating decimal.

Answer: (c) Maybe rational or irrational

Explanation: The product of two irrational numbers can be either rational or irrational depending upon what the number is. For example, √17 x √17 = 17, so it is a rational number. Whereas √17 x √7 = 10.908712114… so it is an irrational number.

Answer: (b) Rational number

Explanation: By definition, a rational number is any number that can be represented as a ratio of two integers. So, as given in the question, p and q are integers and are represented in the form of p/q, then it is a rational number.

Answer: (c) 13

Explanation: 72 – 7 = 65 and 128 – 11 = 117

By definition HCF (65, 117) is the largest number that divides 72 and 128 and leaves the remainder 7 and 11.

Factorising 65 and 117,

65 = 13 x 5

117 = 13 x 3 x 3

Therefore, HCF (65, 117) = 13

Answer: (b) 60

Explanation: The least number divisible by 1, 2, 3, 4, and 5 will be LCM of 1, 2, 3, 4, 5.

Hence, LCM (1, 2, 3, 4, 5) = 2 x 3 x 2 x 5 = 60

Answer: (c) rational or irrational

Explanation: When two irrational numbers are added the result can be either rational or irrational. Although in most cases the sum is irrational, in cases like √3 + (-√3) the sum is zero which is a rational number.

Answer: (c) 0 ≤ r < b

Explanation: As we already know Euclid’s division lemma states - for any two positive integers a and b, there exist two unique positive integers q and r such that a = bq + r, where r must satisfy 0 ≤ r < b.

Answer: (c) 33

Explanation: 396 > 231

Using Euclid’s division algorithm,

396 = 231 × 1 + 165

231 = 165 × 1 + 66

165 = 66 × 2 + 33

66 = 33 × 2 + 0

Therefore, the HCF (396, 231) is 33.

Answer: (c) an odd integer

Explanation:

We know that an odd number in the form (2N + 1) where N is a natural number ,

so, p² -1 = (2N + 1)² -1

= 4N² + 4N + 1 -1

= 4N² + 4N

Substituting N = 1, 2,…

When N = 1,

4N² + 4N = 4(1)² + 4(1) = 4 + 4 = 8 , it is divisible by 8.

When N = 2,

4N² + 4N = 4(2)² + 4(2) =16 + 8 = 24, it is also divisible by 8 .

When N = 3,

4N² + 4N = 4(3)² + 4(3) = 36 + 12 = 48 , divisible by 8

It is concluded that 4Q² + 4Q is divisible by 8 for all natural numbers.

Hence, p² -1 is divisible by 8 for all odd values of p.

Answer: (d) a × b

Explanation: For any two positive integers a and b, HCF (a, b) × LCM (a, b) = a × b.

For example, take two numbers 9 and 12.

First, we need to find the highest common factor (HCF) of 9 and 12.

9 = 3×3

12 = 2 × 2× 3

∴ HCF of 9 and 12 = 3

Lowest common multiple (LCM) of 12 and 18 = 2 x 2 x 3 x 3 = 108.

HCF × LCM = 3 × 36 = 108

Also 9 × 12 = 108

Therefore, the product of HCF and LCM of 9 and 12 = product of 9 and 12.

Answer: (b) 0, 1, 2

Explanation: According to Euclid’s division lemma, a positive integer a can be represented as,

a = 3q + r, where 0 ≤ r < 3 and r is an integer.

Therefore, the possible values of r are 0, 1, or 2. So, the correct option is (b) 0, 1, 2.

Answer: (c) 22338

Explanation: It is given that the HCF of the two numbers 306,657 is 9. We have to find its LCM.

We know that, LCM×HCF = Product of the two given numbers.

Therefore, LCM×HCF = 306×657

⇒LCM = 306×657/HCF

⇒LCM=306×657/HCF

⇒LCM=306×657 / 9

⇒LCM=306×657 / 9

∴ LCM=22338

Answer: (c) 1

Explanation: The lowest possible value of n = 1.

If we put n =1

(24)n = 24 which is divisible by 8.

So, the correct option is (c) 1.