Q1. Write four rational numbers equivalent to 5/7.
We have,
Four rational numbers equivalent to 5/7 are 10/14, 15/21, 20/28, 25/35.
Q2. Find nine rational numbers between 0.1 and 0.2.
Let x = 0.1, y = 0.2 and n = 9.
The nine rational numbers between x and y are:
(x + d), (x + 2d), (x + 3d), (x + 4d), (x + 5d), (x + 6d), (x + 7d), (x + 8d) and (x + 9d).
Nine rational numbers between 0.1 and 0.2 are:
(0.1 + 0.01); (0.1 + 0.02); (0.1 + 0.03); (0.1 + 0.04); (0.1 + 0.05); (0.1 + 0.06); (0.1 + 0.07); (0.1 + 0.08) and (0.1 + 0.09)
The nine rational numbers between 0.1 and 0.2 are 0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17, 0.18 and 0.19.
Q3. Represent √2 on the number line (or on the real line).
 Draw XOX' and mark O as 0.
 Take OA = 1 unit.
 Draw AB ⊥ OX and cut off AB = 1 unit.
 Now, OAB is a right triangle.
∵ OA^{2} + AB^{2} = OB^{2}
⇒ 1^{2} + 1^{2} = OB^{2}
⇒ 2 = OB^{2}
⇒ OB =√2 With centre O and OB as radius, draw an arc intersecting OX at C.
 Thus, OC = √2 on the real line XOX'.
Q4. Represent √3 on the real line.
 Draw XOX ' and mark O as 0 on it.
 Take OA = 1 unit.
 Draw AB ⊥ OX and cut off AB = 1 unit.
 Join OB such that OAB is a right triangle.
∵ OA^{2} + AB^{2} = OB^{2}
⇒ 1^{2} + 1^{2} = OB^{2}
⇒ OB =√2 Again, draw BC ⊥ OB and cut off BC = 1 unit. Join OC.
 Now, OBC is a right triangle.
∵ OB^{2} + BC^{2} = OC^{2}
⇒ √2^{2} + 1^{2} = OC^{2}
⇒ 2 + 1 = OB^{2}
⇒ OC =√3 With centre as O and radius as OC draw an arc to intersect OX at D.
 Thus, OD =√3 on the real line XOX'.
Q5. Express as a rational number in simplest form.
⇒ 100x = 100 x (0.3838…)
⇒ 100x = 38.3838...(2)
Subtracting (1) from (2),
We have, 100x – x = (38.3838…) – (0.3838…)
Q6. Express in the form of .
∴ 10x = 10 x (0.5353…)
or 10x = 5.333... ...(1)
Also, 100x = 53.333... ...(2)
Subtracting (1) from (2),
⇒ 100x – 10x = (53.333…) – (5.333…)
⇒ 90x = 48
Q7. Express is the form of p/q in the simplest form.
∴ 1000x = 1000 x (0.003003…)
or 1000x = 3.003003... …(2)
Subtracting (1) from (2),
We have 1000x – x = (3.003003…) – (0.003003…)
⇒ 999x = 3
⇒ x = 3/999 = 1/333
Thus,
Q8. Find the sum of (3√3 +7√2) and (√3  5√2)
We have (3√3 +7√2) + (√3  5√2)
⇒ √3 3+7√2 + √3  5√2
⇒ (3√3+√3) + 7√2  5√2)
⇒ √3(3+1) + √2(72)
⇒ √3(4) + √2(5) = (4√3 + 2√2)
Q9. Divide 15 √12 by 3√3.
Q10. Rationalize the denominator of
⇒ (Prime factorize the numbers under the root in the denominator)
⇒
Q11. Rationalize the denominator of
Q12. If ‘a’ and ‘b’ are rational numbers and find the values of ‘a’ and ‘b’.
Comparing
Q13. If , what is the value of x^{3}  5x^{2} + 8x  4?
(x  2)^{3} = x^{3}  6x^{2} + 12x  8 ...(1)
(x  2)^{2} = x^{2}  4x + 4 ... (2)
(1) + (2) = x^{3}  5x^{2} + 8x  4
Q14. Find the value of when
∵
Q15. Rationalise the denominator of 1/[7+3√3].
1/(7 + 3√3)
By rationalizing the denominator,
= [1/(7 + 3√3)] [(7 – 3√3)/(7 – 3√3)]
= (7 – 3√3)/[(7)^{2} – (3√3)^{2}]
= (7 – 3√3)/(49 – 27)
= (7 – 3√3)/22
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