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Test: Real Numbers (Hard) - Class 10 MCQ


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20 Questions MCQ Test - Test: Real Numbers (Hard)

Test: Real Numbers (Hard) for Class 10 2024 is part of Class 10 preparation. The Test: Real Numbers (Hard) questions and answers have been prepared according to the Class 10 exam syllabus.The Test: Real Numbers (Hard) MCQs are made for Class 10 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Real Numbers (Hard) below.
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Test: Real Numbers (Hard) - Question 1
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Show that x is rational if: x2 = 0.0004

Detailed Solution for Test: Real Numbers (Hard) - Question 1
x2 = 0.0004

= 2 / 100 = 0.02, which is rational

Test: Real Numbers (Hard) - Question 2
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Without doing any actual division, find which of the following rational numbers have terminating decimal representation:

(i) 7 / 16

(ii) 23 / 125

(iii) 9 / 14

(iv) 32 / 45

(v) 43 / 50

(vi) 17 / 40

(vii) 61 / 75

(viii) 123 / 250

Detailed Solution for Test: Real Numbers (Hard) - Question 2
The rational not having denominator as multiple of 2m x 5n will be non terminating.

(i) 7 / 16 = 7 / 2 x 2 x 2 x 2 denominator multiple of 2m x 5n hence terminating.

(ii) 23 / 125 = 23 / 5 x 5 x 5 denominator multiple of 2m x 5n hence terminating.

(iii) 9 / 14 = 9 / 7 x 2 denominator is not multiple of 2m x 5n hence non-terminating.

(iv) 32 / 45 = 32 / 3 x 3 x 5 denominator is not multiple of 2m x 5n hence non-terminating.

(v) 43 / 50 = 43 / 5 x 5 x 2 denominator multiple of 2m x 5n hence terminating.

(vi) 17 / 40 = 17 / 2 x 2 x 2 x 5 denominator multiple of 2m x 5n hence terminating.

(vii) 61 / 75 = denominator is not multiple of 2m x 5n hence non-terminating.

(viii) 123 / 250 = 123 / 5 x 5 x 5 x 2 denominator multiple of 2m x 5n hence terminating.

(i), (ii), (v), (vi) and (viii) will have terminating decimal.

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Test: Real Numbers (Hard) - Question 3
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Prove that 3 + √5 is an irrational number.

Detailed Solution for Test: Real Numbers (Hard) - Question 3
Let us assume that √5 is a rational number which can be expressed in the form of p / q, where p and q are integers, q ≠ 0 and p and q are co prime that is HCF (p, q) = 1.
Test: Real Numbers (Hard) - Question 4
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Find the greatest number that will divide 93, 111 and 129, leaving remainder 3 in each case.
Detailed Solution for Test: Real Numbers (Hard) - Question 4
Since Remainder is 3 in each case numbers are

93 - 3 = 90

111 - 3 = 108

129 - 3 = 126

Required number will be H.C.F of 90,108 and 126

H.C.F of 90 and 108

∴ The greatest number is 18.

Test: Real Numbers (Hard) - Question 5
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If the product of two numbers is 540 and their HCF is 30, find their LCM.
Detailed Solution for Test: Real Numbers (Hard) - Question 5
The product of two numbers = 540

HCF x LCM = 540

30 x LCM = 540

LCM = 540 / 30 = 18

Test: Real Numbers (Hard) - Question 6
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π is an irrational number.
Detailed Solution for Test: Real Numbers (Hard) - Question 6
π is an irrational number, which means that it is a real number that cannot be expressed by a simple fraction. That's because pi is what mathematicians call an "infinite decimal" — after the decimal point, the digits go on forever and ever. ... (These rational expressions are only accurate to a couple of decimal places.)
Test: Real Numbers (Hard) - Question 7
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The sum of two irrational numbers is an irrational number.
Detailed Solution for Test: Real Numbers (Hard) - Question 7
An irrational number is a number that cannot be expressed as a fraction for any integers and. Irrational numbers have decimal expansions that neither terminate nor become periodic.
Test: Real Numbers (Hard) - Question 8
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If the product of two co-primes is 553, then their LCM = _____
Detailed Solution for Test: Real Numbers (Hard) - Question 8
The HCF of co-prime no.s is 1

∴ Product of the two numbers = HCF × LCM

Thus LCM = 553

Test: Real Numbers (Hard) - Question 9
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The value of x in following factor tree is ____
Detailed Solution for Test: Real Numbers (Hard) - Question 9
1 × 42 = 42

2 × 21 = 42

3 × 14 = 42

6 × 7 = 42

Therefore, the factors of 42 in pairs are (1, 42), (2, 21), (3, 14) and (6, 7).

Test: Real Numbers (Hard) - Question 10
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Product of any irrational number with a rational number 'x' is always rational. Then, x is ____.
Detailed Solution for Test: Real Numbers (Hard) - Question 10
Product of any rational no. with irrational is always irrational only when irrational no. is not a perfect square and is not having 0 , 1 as roots.
Test: Real Numbers (Hard) - Question 11
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The prime factorisation of 320 is____.
Detailed Solution for Test: Real Numbers (Hard) - Question 11
320 = 2 x 2 x 2 x 2 x 2 x 2 x 5

320 = 26 x 51

Test: Real Numbers (Hard) - Question 12
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Any number ending with '0' must have ___ and ___ as its prime factors.
Detailed Solution for Test: Real Numbers (Hard) - Question 12
For example: 10, 20, 30, .....

The factors of 10 = 1, 2, 5 and

The prime factors of 10 = 2 and 5

The factors of 20 = 1, 2, 4, 5, 10 and 20

The prime factors of 20 = 2 and 5

∴ Any number ending with '0' must have '2' and '5' as its prime factors.

Test: Real Numbers (Hard) - Question 13
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HCF of two numbers is always a factor of their LCM.
Detailed Solution for Test: Real Numbers (Hard) - Question 13
This statement HCF of two numbers is always a factor of their LCM is TRUE.

Because HCF is a factor of both the numbers which are factors of their LCM.

Hence, HCF of two numbers is always a factor of their LCM.

Test: Real Numbers (Hard) - Question 14
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Every even integer is of the form 2m, where m is an integer.
Detailed Solution for Test: Real Numbers (Hard) - Question 14
This statement every even integer is of the form 2m, where m is an integer is TRUE.

For Example : Let the value of m be -2, 4 & 0

Then the values for 2m = 2 × -2 , 2 × 4 & 2 × 0

= - 4 , 8 & 0

Test: Real Numbers (Hard) - Question 15
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If p and q are relatively prime numbers, what is their LCM?
Detailed Solution for Test: Real Numbers (Hard) - Question 15
If p and q are prime numbers, they do not have any common factor.

p = 1 × p

q = 1 × q

HCF of a and b = 1.

Their LCM = 1 × p × q

LCM of p and q = pq

Test: Real Numbers (Hard) - Question 16
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For what value of n, 2n x 5n ends in 5.
Detailed Solution for Test: Real Numbers (Hard) - Question 16
2m × 5n can also be written as

2n × 5n = (2 × 5)n = (10)n

which always ends in zero, as 102 = 100, 103 = 1000,.....

Thus, there is no value of n for which (2n × 5n ) ends in 5.

Test: Real Numbers (Hard) - Question 17
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LCM of two prime numbers is 1.
Detailed Solution for Test: Real Numbers (Hard) - Question 17
A prime number is a natural number greater than 1 which has no divisor other than 1 and itself. The least common multiple (LCM) of two numbers is the smallest non zero number which is a multiple of both numbers. considering the L.C.M of two primes : 2 and 3 , here the L.C.M = 2 × 3 = 6.
Test: Real Numbers (Hard) - Question 18
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LCM of two or more ____ is equal to their product.
Detailed Solution for Test: Real Numbers (Hard) - Question 18
Prime numbers are numbers that have only 2 factors: 1 and themselves. For example, the first 5 prime numbers are 2, 3, 5, 7, and 11. By contrast, numbers with more than 2 factors are called composite numbers.
Test: Real Numbers (Hard) - Question 19
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If a and b are two odd prime numbers, then a2 - b2 is ____(prime/composite/neither prime nor composite).
Detailed Solution for Test: Real Numbers (Hard) - Question 19
Composite numbers can be defined as the whole numbers that have more than two factors. Whole numbers that are not prime are composite numbers, because they are divisible by more than two numbers.
Test: Real Numbers (Hard) - Question 20
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HCF of 870 and 225 is:
Detailed Solution for Test: Real Numbers (Hard) - Question 20
Factors of 870 = 5 × 2 × 3 × 29

Factors of 225 = 5 × 5 × 3 × 3

Common factors = 5 × 3 = 15

H.C.F = 15

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