Number Systems Exemplar — Class 9 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 9 Mathematics chapter "Number Systems Exemplar" — 7 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 Exemplar" — 7 important questions with detailed answers for CBSE boa…
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
NCERT Solutions for Class 9 Mathematics — Number Systems Exemplar. Step-by-step answers to all 7 textbook questions from this chapter, written for CBSE board preparation and free to use.
Number Systems Exemplar — All 7 Questions Solved
Q1. Show that 5 - 2√3 is irrational.
Assume 5 - 2√3 is rational. Then 5 - 2√3 = p/q where p, q are integers and q ≠ 0. Rearranging: 2√3 = 5 - p/q = (5q - p)/q. So √3 = (5q - p)/(2q). Since p, q are integers, (5q - p)/(2q) is rational. But this means √3 is rational, which contradicts the fact that √3 is irrational. Therefore, 5 - 2√3 must be irrational.
Q2. Find the value of (81)^(3/4).
(81)^(3/4) = (81^(1/4))^3. Now 81 = 3^4, so 81^(1/4) = (3^4)^(1/4) = 3. Therefore (81)^(3/4) = 3^3 = 27.
Q3. Simplify: (√5 + √3)/(√5 - √3).
Multiply numerator and denominator by (√5 + √3): [(√5 + √3)/(√5 - √3)] × [(√5 + √3)/(√5 + √3)] = (√5 + √3)^2 / [(√5)^2 - (√3)^2] = (5 + 2√15 + 3) / (5 - 3) = (8 + 2√15) / 2 = 4 + √15.
Q4. If x = 3 + 2√2, find x + 1/x.
First find 1/x = 1/(3 + 2√2). Rationalize: 1/(3 + 2√2) × (3 - 2√2)/(3 - 2√2) = (3 - 2√2)/(9 - 8) = 3 - 2√2. Therefore x + 1/x = (3 + 2√2) + (3 - 2√2) = 6.
Q5. Express 0.3456̄ (where 3456 repeats) as a fraction.
Let x = 0.34563456... Then 10000x = 3456.34563456... Subtracting: 10000x - x = 3456. So 9999x = 3456, giving x = 3456/9999. Simplifying by GCD(3456, 9999) = 3: x = 1152/3333 = 384/1111.
Q6. Prove that √2 is irrational.
Assume √2 is rational. Then √2 = p/q in lowest terms (GCD(p,q) = 1). Squaring: 2 = p^2/q^2, so p^2 = 2q^2. This means p^2 is even, so p is even. Let p = 2m. Then (2m)^2 = 2q^2, so 4m^2 = 2q^2, giving 2m^2 = q^2. Thus q^2 is even, so q is even. But if both p and q are even, GCD(p,q) ≥ 2, contradicting our assumption. Therefore √2 is irrational.
Q7. Find the decimal expansion of 1/7.
Divide 1 by 7: 1 ÷ 7 = 0.142857142857... The remainders are 1, 3, 2, 6, 4, 5, 1 (repeats). So 1/7 = 0.1̄4̄2̄8̄5̄7̄ (period of 6 digits).
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.