uestion 1. Classify the following numbers as rational or irrational: (i) 2 − √5 (ii) (3 + √23) − √23 (iii) 2√7 / 7√7 (iv) 1/√2 (v) 2π
Solution.
(i) 2 − √5 — difference of a rational and an irrational — irrational.
(ii) (3 + √23) − √23 = 3 — rational.
(iii) 2√7/7√7 = 2/7 — rational.
(iv) 1/√2 — quotient of a rational and irrational — irrational.
(v) 2π — product of a rational and irrational — irrational.Ans.
Question 2. Simplify each of the following expressions: (i) (3+√3)(2+√2) (ii) (3+√3)(3−√3) (iii) (√5+√2)² (iv) (√5−√2)(√5+√2)
Solution.
(i) (3+√3)(2+√2) = 6 + 3√2 + 2√3 + √6
(ii) (3+√3)(3−√3) = 9 − 3 = 6
(iii) (√5+√2)² = 5 + 2√10 + 2 = 7 + 2√10
(iv) (√5−√2)(√5+√2) = 5 − 2 = 3Ans.
Question 3. π is defined as the ratio of the circumference (c) of a circle to its diameter (d), i.e., π = c/d. This seems to contradict the fact that π is irrational. How will you resolve this?
Solution. There is no contradiction. When we measure a length with a scale or other device, we only get an approximate rational value — so either c or d is actually irrational, even though it appears rational when measured. Hence π is irrational.Ans.
Question 4. Represent √9.3 on the number line.
Solution. Mark a distance of 9.3 units from point A to point B. From B, mark a further 1 unit to point C. Find the midpoint O of AC and draw a semicircle with centre O, radius OC. Draw a perpendicular to AC at B, meeting the semicircle at D; then BD = √9.3. Treating BC as the number line (B = 0, C = 1), draw an arc with centre B and radius BD to meet the number line at E — point E represents √9.3.Ans.
Question 5. Rationalise the denominators of the following: (i) 1/√7 (ii) 1/(√7−√6) (iii) 1/(√5+√2) (iv) 1/(√7−2)
Solution.
(i) 1/√7 = √7/7
(ii) 1/(√7−√6) = (√7+√6)/(7−6) = √7 + √6
(iii) 1/(√5+√2) = (√5−√2)/(5−2) = (√5−√2)/3
(iv) 1/(√7−2) = (√7+2)/(7−4) = (√7+2)/3