Number Systems — Class 9 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 9 Mathematics chapter "Number Systems" — 8 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 9 Mathematics chapter "Number Systems" — 8 important questions with detailed answers for CBSE board exam p…
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NCERT Solutions for Class 9 Mathematics — Number Systems. Step-by-step answers to all 8 textbook questions from this chapter, written for CBSE board preparation and free to use.
Number Systems — All 8 Questions Solved
Q1. Classify √2 as rational or irrational. Justify your answer.
√2 is irrational.
Justification (Proof by contradiction):
Suppose √2 is rational. Then √2 = p/q where p and q are coprime integers (gcd(p,q) = 1).
Squaring both sides: 2 = p²/q²
Therefore: p² = 2q²
This means p² is even, so p must be even. Let p = 2m.
Substituting: (2m)² = 2q²
4m² = 2q²
2m² = q²
This means q² is even, so q is even.
But if both p and q are even, they have a common factor of 2, contradicting our assumption that gcd(p,q) = 1.
Therefore, √2 cannot be rational.
Conclusion: √2 is irrational.
Q2. Express 2.3̄ (2.333...) as a fraction in lowest terms.
Let x = 2.3̄ = 2.333...
Step 1: Write x = 2 + 0.333...
Let y = 0.333...
Step 2: Multiply y by 10
10y = 3.333...
Step 3: Subtract to eliminate the repeating part
10y - y = 3.333... - 0.333...
9y = 3
y = 3/9 = 1/3
Step 4: Find x
x = 2 + 1/3 = 6/3 + 1/3 = 7/3
Answer: 2.3̄ = 7/3
Q3. Rationalize the denominator: 1/(√5 - √3)
Expression: 1/(√5 - √3)
Step 1: Identify the conjugate
Conjugate of (√5 - √3) is (√5 + √3)
Step 2: Multiply numerator and denominator by the conjugate
= 1/(√5 - √3) × (√5 + √3)/(√5 + √3)
= (√5 + √3)/[(√5 - √3)(√5 + √3)]
Step 3: Apply difference of squares formula (a - b)(a + b) = a² - b²
= (√5 + √3)/[(√5)² - (√3)²]
= (√5 + √3)/(5 - 3)
= (√5 + √3)/2
Answer: (√5 + √3)/2
Q4. If a = 1/(2 - √3), find the value of a² - 2a.
Step 1: Rationalize a
a = 1/(2 - √3) × (2 + √3)/(2 + √3)
= (2 + √3)/[(2 - √3)(2 + √3)]
= (2 + √3)/(4 - 3)
= 2 + √3
Step 2: Calculate a²
a² = (2 + √3)²
= 4 + 4√3 + 3
= 7 + 4√3
Step 3: Calculate 2a
2a = 2(2 + √3) = 4 + 2√3
Step 4: Find a² - 2a
a² - 2a = (7 + 4√3) - (4 + 2√3)
= 7 + 4√3 - 4 - 2√3
= 3 + 2√3
Answer: a² - 2a = 3 + 2√3
Q5. Locate √5 on the number line using the Pythagorean theorem.
Step 1: Express √5 using Pythagorean theorem
√5 = √(1² + 2²)
This is the hypotenuse of a right triangle with legs 1 and 2.
Step 2: Construction on number line
- Mark point 0 (origin) and point 2 on the number line.
- At point 2, draw a perpendicular line of length 1 unit.
- Using compass, measure the hypotenuse from 0 to the endpoint of the perpendicular.
- This distance equals √5.
Step 3: Locate √5
- With center at 0 and radius equal to the hypotenuse length, draw an arc on the number line.
- The arc intersects the number line at point √5 ≈ 2.236.
Conclusion: √5 lies between 2 and 3 on the number line, closer to 2.
Q6. Simplify: (5 + 2√3)/(7 + 4√3) by rationalizing the denominator.
Expression: (5 + 2√3)/(7 + 4√3)
Step 1: Multiply by conjugate of denominator
Conjugate of (7 + 4√3) is (7 - 4√3)
= (5 + 2√3)/(7 + 4√3) × (7 - 4√3)/(7 - 4√3)
Step 2: Calculate denominator using (a + b)(a - b) = a² - b²
= (5 + 2√3)(7 - 4√3)/[(7)² - (4√3)²]
= (5 + 2√3)(7 - 4√3)/(49 - 48)
= (5 + 2√3)(7 - 4√3)/1
Step 3: Expand numerator
= 5(7) + 5(-4√3) + 2√3(7) + 2√3(-4√3)
= 35 - 20√3 + 14√3 - 8(3)
= 35 - 20√3 + 14√3 - 24
= 11 - 6√3
Answer: (5 + 2√3)/(7 + 4√3) = 11 - 6√3
Q7. Prove that 0.142857̄ = 1/7 (where the bar indicates repeating decimal).
Let x = 0.142857142857... = 0.1̄4̄2̄8̄5̄7̄ (6 digits repeat)
Step 1: The block 142857 repeats every 6 decimal places
Multiply x by 10⁶ = 1,000,000
1,000,000x = 142857.142857...
Step 2: Subtract the original equation from this
1,000,000x - x = 142857.142857... - 0.142857...
999,999x = 142857
x = 142857/999,999
Step 3: Simplify the fraction
Note: 999,999 = 7 × 142,857
Verify: 7 × 142,857 = 999,999 ✓
Therefore: x = 142,857/(7 × 142,857) = 1/7
Conclusion: 0.142857̄ = 1/7
Q8. If x = 3 + 2√2, find the value of (x + 1/x).
Given: x = 3 + 2√2
Step 1: Find 1/x
1/x = 1/(3 + 2√2)
Rationalize by multiplying with (3 - 2√2)/(3 - 2√2)
= (3 - 2√2)/[(3 + 2√2)(3 - 2√2)]
= (3 - 2√2)/(9 - 8)
= 3 - 2√2
Step 2: Calculate x + 1/x
x + 1/x = (3 + 2√2) + (3 - 2√2)
= 3 + 2√2 + 3 - 2√2
= 6
Answer: x + 1/x = 6
Number Systems — Practice MCQs with Answers
Attempt these 20 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. Which of the following is an irrational number?
- A. 2/3
- B. √4
- C. √7
- D. -5
Answer: C. √7 — √7 is an irrational number because it cannot be expressed in the form p/q, where p and q are integers and q≠0, and its decimal expansion is non-terminating and non-repeating.
MCQ 2. The decimal expansion of 3/8 is:
- A. 0.375
- B. 0.38
- C. 0.333...
- D. 0.83
Answer: A. 0.375 — Dividing 3 by 8 results in a terminating decimal 0.375, which is characteristic of a rational number whose denominator (in simplest form) has only prime factors 2 or 5.
MCQ 3. Express 0.6 (with a bar over 6) in the form p/q.
- A. 3/5
- B. 2/3
- C. 6/10
- D. 1/6
Answer: B. 2/3 — By setting the repeating decimal to x, multiplying by a power of 10 to shift the decimal, and subtracting the original equation, we can isolate x as a fraction.
MCQ 4. A rational number between 3 and 4 is:
- A. 7/2
- B. 3/2
- C. 4/3
- D. 1
Answer: A. 7/2 — 7/2 simplifies to 3.5, which lies exactly in the middle of 3 and 4.
MCQ 5. Simplify: 3√2 + 2√2 - 5√2
- A. 0
- B. √2
- C. 10√2
- D. 5√2
Answer: A. 0 — Treat √2 as a common variable; combine the coefficients just like combining algebraic terms (3x + 2x - 5x = 0x).
MCQ 6. Simplify: (√5 + √2)(√5 - √2)
- A. 7
- B. 3
- C. √10
- D. 2√5
Answer: B. 3 — This expression is in the form (a+b)(a-b), which simplifies to a² - b².
MCQ 7. Rationalise the denominator of 1/(√7 - √6).
- A. √7 + √6
- B. √7 - √6
- C. 1/(√7 + √6)
- D. 1/(√7 - √6)
Answer: A. √7 + √6 — To rationalise the denominator, multiply both numerator and denominator by the conjugate of the denominator, which is √7 + √6.
MCQ 8. Simplify: (2^(2/3)) * (2^(1/3))
- A. 2^(1/3)
- B. 2^(2/9)
- C. 2
- D. 4^(2/9)
Answer: C. 2 — When multiplying powers with the same base, add their exponents according to the law a^m * a^n = a^(m+n).
MCQ 9. Find the value of 125^(1/3).
- A. 5
- B. 25
- C. 1/5
- D. 3
Answer: A. 5 — The exponent 1/3 represents the cube root of the number.
MCQ 10. Every rational number is a:
- A. Natural number
- B. Whole number
- C. Integer
- D. Real number
Answer: D. Real number — Real numbers are the set of all rational and irrational numbers. Since every rational number can be plotted on the number line, it falls under the category of real numbers.
MCQ 11. Which of the following is NOT an irrational number?
- A. √10
- B. √25
- C. √7
- D. π
Answer: B. √25 — √25 simplifies to 5, which is an integer and thus a rational number. The other options are irrational numbers.
MCQ 12. Simplify: (64)^( -1/2 )
- A. 8
- B. -8
- C. 1/8
- D. -1/8
Answer: C. 1/8 — A negative exponent means to take the reciprocal of the base raised to the positive exponent. The exponent 1/2 means to take the square root.
MCQ 13. Between any two distinct rational numbers, there exist:
- A. No rational number
- B. Exactly one rational number
- C. Infinitely many rational numbers
- D. Exactly one irrational number
Answer: C. Infinitely many rational numbers — The property of density for rational numbers states that between any two distinct rational numbers, there exists an infinite number of other rational numbers.
MCQ 14. Simplify: (2√3 - 3√2)²
- A. 6 - 12√6
- B. 30 - 12√6
- C. 12 - 18
- D. 30 + 12√6
Answer: B. 30 - 12√6 — Expand the square using the algebraic identity (a-b)² = a² - 2ab + b².
MCQ 15. If x^(1/2) = 7, what is x?
- A. √7
- B. 49
- C. 1/7
- D. 14
Answer: B. 49 — The exponent 1/2 denotes a square root. To find x, square both sides of the equation.
MCQ 16. The decimal expansion of an irrational number is:
- A. Terminating
- B. Non-terminating recurring
- C. Non-terminating non-recurring
- D. Always an integer
Answer: C. Non-terminating non-recurring — Irrational numbers are characterized by decimal expansions that neither terminate nor repeat.
MCQ 17. Which of the following statements is TRUE?
- A. Every integer is a whole number.
- B. Every natural number is a whole number.
- C. Every rational number is an integer.
- D. Every real number is irrational.
Answer: B. Every natural number is a whole number. — Natural numbers (1, 2, 3...) are a subset of whole numbers (0, 1, 2, 3...).
MCQ 18. Evaluate: (81)^(0.25)
- A. 3
- B. 9
- C. 27
- D. 1/3
Answer: A. 3 — Convert the decimal exponent to a fraction (0.25 = 1/4) and then find the fourth root of 81.
MCQ 19. Rationalise the denominator of 5/√7.
- A. 5√7/7
- B. 5√7
- C. √7/5
- D. 35/7
Answer: A. 5√7/7 — To rationalize the denominator with a single square root term, multiply both the numerator and the denominator by that square root.
MCQ 20. If a = 2 + √3, then what is 1/a?
- A. 2 - √3
- B. 1/(2 + √3)
- C. 2 + √3
- D. 1 - √3
Answer: A. 2 - √3 — To find 1/a, substitute 'a' and then rationalise the denominator by multiplying by its conjugate.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.