NCERT MCQ Solutions for Class 9 Maths Chapter 1 Number Systems

To find a rational number between 3 and 4, we need to check which of the given options lies in the interval 3 and 4.
Checking each option:
(a). 3/2 =1.5, 1.5 is less than 3, so it is not between 3 and 4.
(b). 4/3 ≈ 1.33, 1.33 is also less than 3, so it is not between 3 and 4.
(c). 7/2 = 3.5, 3.5 is between 3 and 4, so this is a valid option. 
(d). 7/4 = 1.75, 1.75 is less than 3, so it is not between 3 and 4.
Conclusion:
The only rational number in the given options that lies between 3 and 4 is 7/2.

We are given the repeating decimal 0.6666… and need to express it in the form p/q.
Let x = 0.6666…
Multiply by 10 to shift the decimal point 
10x = 6.666…
Subtract the two equations 
10x – x = 6.6666… – 0.6666..
9x = 6
Solve for x 
x = 6/9
Simplify the fraction 6/9 = 2/3

Let’s analyze each statement carefully:
Option (a): Every whole number is a natural number
Natural numbers (N): {1,2, 3, 4,…}
Whole numbers (W): {0,1,2,3,4,…}
Since whole number include 0, but natural number do not, this statement is false.
Option (b): Every integer is a rational number
Integers (Z): {…, -3, -2, -1, 0, 1,2,3,…)
Rotational number (Q): Numbers that can be expressed as p/q, where p and q are integers, and q ≠ 0.
Every integer x can be written as x/1, which is in the form of a rotational number.
Thus this statement is true.
Option (c): Every rational number is an integer.
Example 1/2 is a rational number but not an integer.
So, this statement is false.
Option (d): Every integer is a whole number 
Whole number do not include negative numbers, but integers do.
Example: -1 is an integer but not a whole number.
Thus, this statement is false.
Final Answer: The correct answer is Every integer is a rotational number.

We are given the rational number: 3/40
We need to convert it into decimal form and determine the correct answer from the given options.
Convert the Fraction to Decimal
To convert 3/40 into a decimal, we divide 3 by 40: 3 ÷ 40
Performing the division:
3.000 (Adding decimal point and zeros to divide properly)
40 goes into 30 0 times. (So, we consider 300)
40 goes into 300 7 times (since 40 × 7 = 280)Remainder: 300 – 280 = 20
Bring down a 0 (making it 200)
40 goes into 200 5 times (since 40 × 5 = 200)Remainder: 0 (Division is complete)
Thus, 3/40 = 0.075
Step 2: Compare with Given Options
The given options are:
(a) 0.75
(b) 0.12
(c)
(d)
Since we found that 3/40 = 0.075, the correct answer is 0.075.

We are given the expression: (3 + √3) (3 – √3)
Apply the identity
This follows the difference of squares identity:
(a + b) (a – b) = a² – b²
where a = 3 and b = √3.
Substitute and solve
(3 + √3)(3 – √3) = 3² – (√3)²
= 9 – 3
= 6
The correct answer is 6.

We  are given that m and n are two natural numbers, and mⁿ = 32
We need to find the value of nᵐⁿ.
Find Possible Value of m and n
Since 32 and 2⁵, we look for natural numbers m and n such that: mⁿ = 32
Possible values are:
m = 2, n = 5( since 2⁵ = 32)
Compute nᵐⁿ 
We substitute n = 5 and mⁿ = 32: 
nᵐⁿ = 5³²
Identify the correct option
None of the options match 5³² directly.
Please verify if there was a misinterpretation in the problem statement or options. If you meant nᵐ*ⁿ instead of nᵐⁿ , let’s check:
nᵐ*ⁿ = 5²×⁵ = 5¹⁰.

We need to find a rational number between 3/5 and 4/5 from the given options. 
Convert to Decimal Form
3/5 = 0.6
4/5 = 0.8
Now, we check which given options lie in the range (0.6, 0.8).
Convert the Options to Decimal Form 
7/5 = 1.4 (Not between 0.6 and 0.8)
7/10 = 0.7 (Between 0.6 and 0.8)
3/10 = 0.3 (Less than 0.6)
4/10 = 0.4 ( Less than 0.6)
The only rational number between 3/5 and 4/5 is 7/10.

We are given the expression: 1/√2
To rationalize the denominator, we multiply both the numerator and denominator by √2 to eliminate the square root in the denominator 
Multiply by √2/√2
1/√2 × √2/√2 = √2/2.

A rational number is any number that can be expressed as p/q, where p and q are integers, and q ≠ 0.
Now, let’s check each option:
√3 The square root of 3 is irrational because it cannot be written as a fraction of two integers.
√2 The square root of 2 is also irrational because it has a non-repeating , non-terminating decimal expansion.
0 Zero can be written as 0/1, which is in the form p/q where p = 0 and q = 1. Since it satisfies the definition of a rational number, 0 is rational.
√5 The square root of 5 is irrational for the same reasons as √2 and √3. 
Conclusion: The only rational number in the given option is 0.

A rational number p/q has a terminating decimal expansion if and only if the prime factorization of its denominator q (after simplify the fraction) contains only the prime factorization of its denominator q (after simplifying the fraction) contains only the prime factor 2 and/or 5.
2 or 5 
If the denominator has only 2, and 5, or both(e.g., 10, 20, 25, etc.), the decimal expansion terminates.
3 or 5
The presence of 3 in the denominator(e.g., 1/3 = 0.3333…) causes a non-terminating repeating decimal.
9 or 11
9 = 3³ and 11 are not factors that produce a terminating decimal. 
3 or 7 
Both 3 and 7 cause non-terminating repeating decimals.