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The correct answer is (b) 2^m×6^n

To elaborate: Let’s, take a NUMBER where q is of the FORM 2^m×5^n, say 2^50×5^10and P can be any integer

\(\FRAC {p}{2^{50}\times 5^{10}} = \frac {p \times 5^{40}}{10^{50}}\)

The number \(\frac {p \times 5^{40}}{10^{50}}\) will terminate after 50 decimal places.

Hence, if q is of the form 2^m×5^n, it will terminate after some decimal places.

Right answer is (a) 0

Explanation: LET the number be N.

If the number is divided by 3 it leaves 5 as remainder.

By EUCLID’s division lemma,

n = 3q + 5 where q is quotient

3N = 3(3q+5)

3n = 9Q + 15

3n + 3 = 9q + 18 = 3 × 3q + 3 × 6 = 3 (3q + 6)

Hence, the remainder is 0.

Correct option is (b) False

To explain I would say: Irrational numbers cannot be written in the FORM of \(\FRAC {p}{q}\).

For EXAMPLE, ∛4 cannot be written in a fraction form as it has non-terminating and non-repeating decimals.

The correct option is (b) True

To EXPLAIN: Let us CONSIDER a composite number, say 25

25 can be FACTORIZED as 5 × 5 × 1. This FACTORIZATION is unique for 25 and no other number can be represented in the same manner.

The CORRECT choice is (a) True

To explain: According to Euclid’s Division lemma, any two numbers can be written in the form of a = bq + rwhere a and B are integers and q and r whole numbers.

For Example: 28 when divided by 7 give 4 as quotient and 0 as remainder.

So, according to Euclid’s Division Lemma,

28 = 7 × 4 + 0

Right answer is (c) 1

The best I can explain: The required NUMBER divides (100-3) i.e. 97 and (25-3) i.e. 22 exactly.

Now, 97 = 97 × 1 and 22 = 2 × 11

HCF of 97 and 22 is 1.

Hence, the required number is 1.

The correct option is (b) 6

Easy explanation: We have,

\(\frac {57}{2^4 5^6} = \frac {57 \TIMES 2^2}{2^6 5^6} = \frac {228}{10^6}\) = 0.000228

The number \(\frac {57}{2^4 5^6}\) will TERMINATE after 6 decimal places.

Correct answer is (c) 2^m×5^n

To explain: Let’s, take a number where q is of the form 2^m×5^n, say 2^50×5^10and p can be any integer

\(\frac {p}{2^{50}\times 5^{10}} = \frac {p \times 5^{40}}{10^{50}}\)

The number \(\frac {p \times 5^{40}}{10^{50}}\)will TERMINATE after 50 decimal PLACES.

Hence, if q is of the form 2^m×5^n, it will terminate after some decimal places.

The correct ANSWER is (b) True

Easiest explanation: Consider TWO rational numbers, say \(\frac {8}{9}, \frac {3}{5}\)

SUM of these NUMBER = \(\frac {8}{9} + \frac {3}{5} = \frac {67}{45}\), which is rational number.

Hence, the sum of two rational numbers is a rational number.

Correct option is (d) 3m, 3m + 1

Easiest explanation: LET a be any arbitrary number.

Then, by Euclid’s division LEMMA,

a = 3q + r where 0 ≤ r ≤ 3

a^2 = (3q + r)^2 = 9q^2 + r^2 + 6qr

When r = 0,

Then, a^2 = 9q^2 + 0^2 + 6q(0)

= 9q^2 = 3(3q^2) = 3m where m = 3q^2

Now, r = 1

a^2 = (3q+r)^2

= 9q^2 + (1)^2 + 6q(1)

= 9q^2 + 1 + 6q

= 3(3q^2 + 2Q) + 1 = 3m + 1 where m = 3q^2 + 2q

When r = 2,

a^2 = (3q+r)^2

= 9q^2 + (2)^2 + 6q(2)

= 9q^2 + 12q + 4

= 3(3q^2) + 3(4)q + 3 × 1 + 1

= 3[(3q^2) + (4)q + 1] + 1

= 3m + 1 where m = (3q^2) + (4)q + 1

Hence, square of number n is of the form 3m or 3m + 1

Right option is (a) 0

The best I can explain: If 6^n ends with 0, then it should have 5 as a factor.

In CASE of 6 only 3 and 2 are factors of 6.

Also, from the FUNDAMENTAL theorem of arithmetic, prime factorisation of each number is unique.

Hence, 6^n can never end with 0.

Correct option is (c) 2181

To ELABORATE: According to EUCLID’s Division LEMMA,

Number = (quotient × divisor) + remainder = 60 × 35 + 81 = 2181

The correct choice is (a) False

The explanation is: CONSIDER two irrational numbers, SAY, √2and √5

Sum of these number = √2+ √3 = 3.14626… which is an irrational number.

Hence, the sum of two irrational numbers is an irrational number.

The correct choice is (B) False

Easiest explanation: CONSIDER an irrational number, say √10

√10 × √10=10

10 is a rational number. HENCE, the product of two irrational NUMBERS is not always irrational.

Correct answer is (b) False

The EXPLANATION: TAKE a RATIONAL and an irrational number, say 2 and 3√3

PRODUCT of 2 × 3√3 = 6√3.

6√3 is an irrational number

Hence, the product of a rational and an irrational number is a irrational number.

Right choice is (b) False

Easiest explanation: \(\FRAC {33}{2} \times \frac {5}{4} = \frac {165}{8}\)

\(\frac {165}{8}\) is a rational number

The correct option is (b) \(\frac {22}{7}\)

For explanation: An irrational NUMBER is expressible in the decimal FORM as non-terminating and non-REPEATING decimals.

From the GIVEN OPTIONS,

π, 1.5353353335…, 2.7878878887… are non-terminating and non-repeating decimal.

Whereas, \(\frac {22}{7}\) is non-terminating but repeating decimal.

Right choice is (a) 108

For EXPLANATION: The two buckets contain 546 and 764 liters of water.

546 can be FACTORIZED as 2 × 2 × 3 × 3 × 3 × 5 and 764 as 2 × 2 × 3 × 3 × 3 × 7.

To find the maximum capacity of the container which can measure the water of either buckets exact NUMBER of times, we have to find the HCF of the two numbers.

HCF of 546 and 764 = 2^2 × 3^3=108

Hence, the maximum capacity of the container is 108 liters.

The correct ANSWER is (a) 31

Best explanation: A prime number has two FACTORS the number itself and 1. In case of 31 there are two factors i.e. 31 and 1. HENCE, it is a prime number.

Right choice is (c) 9

To explain: A PRIME number has two factors the number itself and 1. In CASE of 9 there are three factors i.e. 3 × 3 × 1. Hence, it is not a prime number.

Right answer is (c) 1

To explain I would say: 80 = 1×2×2×2×2×5

567 = 1×3×3×3×3×7. The largest common FACTOR between the TWO numbers is 1.

Right choice is (a) π

To explain: A rational number has terminating or non-terminating but repeating decimals.

In case of π, it has a non-terminating as well as non-repeating DECIMAL.

The other THREE numbers have terminating or non-terminating but repeating decimal, THEREFORE, they are rational numbers.

Hence, it is an IRRATIONAL number.

Correct CHOICE is (a) 3

Explanation: 21 can be written as 3 × 7 × 1 and 90 can be written as 3 × 3 × 2 × 5.

HCF is the product of smallest POWER of each prime factor involved in the numbers.

Therefore, HCF = 3 × 1 = 3

Right CHOICE is (d) 35

To EXPLAIN I would say: For two numbers a and B, we know that

(a × b) = HCF of (a, b) × LCM of (a, b)

Here a = 97, HCF is 1 and LCM is 3395

97 × b = 1 × 3395

B = \(\frac {3395}{97}\) = 35

Right answer is (d) 3780

Easiest explanation: The THREE BOXES contain 60, 84 and 108 number of cookies.

60 can be factorized as 2 × 2 × 3 × 5, 84 as 2 × 2 × 3 × 7 and 108 as 2 × 2 × 3 × 3 × 3

To FIND the least number of cookies that can be FILLED in the container, we have to find the LCM of the three numbers

LCM of 60, 84 and 108 = 2 × 2 × 3 × 3 × 3 × 5 × 7 = 3780

Hence, the least number of cookies that can be filled in the container is 3780.

Correct ANSWER is (b) 1

The EXPLANATION: For two numbers a and b, we know that

(a × b) = HCF of (a, b) × LCM of (a, b)

Here a = 61 and b = 131, and LCM is 7991

61 × 131 = HCF × 7991

HCF = \(\frac {7991}{7991}\) = 1

The correct CHOICE is (c) Non-terminating and non-repeating decimal

To EXPLAIN I would SAY: An IRRATIONAL number has both non-terminating as well as non-repeating DECIMALS.

For example, the number 1.353353335… has non-terminating as well as non-repeating decimals.