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## Tamilnadu Samacheer Kalvi 9th Maths Solutions Chapter 2 Real Numbers Additional Questions

I. Multiple choice question

Question 1.

The decimal form of –\(\frac{3}{4}\) is ………

(a) – 0.75

(b) – 0.50

(c) – 0.25

(d) – 0.125

Solution:

(a) – 0.75

Question 2.

If a number has a non-terminating and non-recurring decimal expansion, then it is……….

(a) a rational number

(b) a natural number

(c) an irrational number

(d) an integer

Solution:

(c) an irrational number

Question 3.

Which one of the following has terminating decimal expansion?

(a) \(\frac{7}{9}\)

(b) \(\frac{8}{15}\)

(c) \(\frac{1}{12}\)

(d) \(\frac{5}{32}\)

Solution:

(d) \(\frac{5}{32}\)

Question 4.

Which of the following are irrational numbers?

(i) \(\sqrt{2+\sqrt3}\)

(ii) \(\sqrt{4+\sqrt25}\)

(iii) \(\sqrt[3]{5+\sqrt7}\)

(iv) \(\sqrt{8-\sqrt[3]8}\)

(a) (ii), (iii) and (iv)

(b) (i), (iii) and (iv)

(c) (i), (ii) and (iii)

(d) (i), (iii) and (iv)

Solution:

(d) (i), (iii) and (iv)

Question 5.

Irrational number has a

(a) terminating decimal

(b) no decimal part

(c) non-terminating and recurring decimal

(d) non-terminating and non-recurring decimal

Solution:

(d) non-terminating and non-recurring decimal

Question 6.

If \(\frac{1}{7}\) = 0.142857, then the value of \(\frac{3}{7}\) is……..

(a) 0.285741

(b) 0.428571

(c) 0.285714

(d) 0.574128

Solution:

(b) 0.428571

Question 7.

Which of the following are not rational numbers?

(a) 7√5

(b) \(\frac{7}{\sqrt{5}}\)

(c) \(\sqrt{36}\) – 9

(d) π + 2

Solution:

(c) \(\sqrt{36}\) – 9

Question 8.

The product of 2√5 and 6√5 is……….

(a) 12√5

(b) 60

(c) 40

(d) 8√5

Solution:

(b) 60

Question 9.

The rational number lying between \(\frac{1}{5}\) and \(\frac{1}{2}\)

(a) \(\frac{7}{20}\)

(b) \(\frac{2}{10}\)

(c) \(\frac{2}{7}\)

(d) \(\frac{3}{10}\)

Solution:

(a) \(\frac{7}{20}\)

Question 10.

The value of 0.03 + 0.03 is ……….

(a) 0.\(\overline { 09 }\)

(b) 0.\(\overline { 0303 }\)

(c) 0.\(\overline { 06 }\)

(d) 0

Solution:

(c) 0.06

Question 11.

The sum of \(\sqrt{343}\) + \(\sqrt{567}\) is

(a) 18√3

(b) 16√7

(c) 15√3

(d) 14√7

Solution:

(b) 16√7

Question 12.

If \(\sqrt{363}\) = x√3 then x = ………

(a) 8

(b) 9

(c) 10

(d) 11

Solution:

(d) 11

Question 13.

The rationalising factor of \(\frac{1}{\sqrt{7}}\) is ……….

(i) 7

(b) √7

(c) \(\frac{1}{7}\)

(d) \(\frac{1}{\sqrt{7}}\)

Solution:

(b) √7

Question 14.

The value of \((\frac{1}{3^5})^4\) is ……..

(a) 3^{20}

(b) 3^{-20}

(c) \(\frac{1}{3^{-20}}\)

(d) \(\frac{1}{3^{9}}\)

Solution:

(b) 3^{-20}

Question 15.

What is 3.976 × 10^{-4} written in decimal form?

(a) 0.003976

(b) 0.0003976

(c) 39760

(d) 0.03976

Solution:

(b) 0.0003976

II. Answer the following Questions.

Question 1.

Find any seven rational numbers between \(\frac{5}{8}\) and –\(\frac{5}{6}\)

Solution:

Let us convert the given rational numbers having the same denominators.

L.C.M of 8 and 6 is 24.

Now the rational numbers between

We can take any seven of them.

Question 2.

Find any three rational numbers between \(\frac{1}{2}\) and \(\frac{1}{5}\)

Solution:

Thus the three rational numbers are \(\frac{7}{20}\), \(\frac{17}{40}\) and \(\frac{37}{80}\)

Question 3.

Represent \(-\frac{2}{11}\), \(-\frac{5}{11}\) and \(-\frac{9}{11}\) on the number lines.

Solution:

To Represent \(-\frac{2}{11}\), \(-\frac{5}{11}\) and \(-\frac{9}{11}\) on the number line we make 11 markings each being equal distence \(\frac{1}{11}\) on the left of 0.

The point A represent \((-\frac{2}{11})\), the point B represents \((-\frac{5}{11})\) and the point C represents \((-\frac{9}{11})\)

Question 4.

Express the following in the form \(\frac{p}{q}\), where p and q are integers and q ≠ 0.

(i) 0.\(\overline { 47 }\)

Solution:

Let x = 0.474747…….. →(1)

100 x = 47.4747…….. →(2)

(2) – (1) ⇒ 100x – x = 47.4747……..

(-) 0.4747……..

99 x = 47.0000

x = \(\frac{47}{99}\)

∴ 0.\(\overline { 47 }\) = \(\frac{47}{99}\)

(ii) 0.\(\overline { 57 }\)

Solution:

Let x = 0.57777…….. →(1)

10 x = 5.77777…….. →(2)

100 x = 57.7777…….. →(3)

(3) – (2) ⇒ 100 x – 10 x = 57.7777……..

(-) 5.7777……..

99 x = 52.0000

x = \(\frac{52}{90}\) = \(\frac{26}{45}\)

∴ 0.\(\overline { 57 }\) = \(\frac{26}{45}\)

(iii) 0.\(\overline { 245 }\)

Solution:

Let x = 0.2454545…….. →(1)

10 x = 2.454545…….. →(2)

1000 x = 245.4545…….. →(3)

(3) – (2) ⇒ 1000 x – 10 x = 245.4545

(-) 2.4545………

990 x = 243.00000

x = \(\frac{243}{990}\) (or) \(\frac{27}{110}\)

∴ 0.\(\overline { 245 }\) = \(\frac{27}{110}\)

Question 5.

Without actual division classify the decimal expansion of the following numbers as terminating or non-terminating and recurring.

(i) \(\frac{7}{16}\)

(ii) \(\frac{13}{150}\)

(ii) –\(\frac{11}{75}\)

(iv) \(\frac{17}{200}\)

Solution:

(i) \(\frac{7}{16}\) = \(\frac{7}{2^4}\) = \(\frac{7}{2^{4} \times 5^{0}}\)

∴ \(\frac{7}{16}\) has a terminating decimal expansion.

(ii) \(\frac{13}{150}=\frac{13}{2 \times 3 \times 5^{2}}\)

Since it is not in the form of \(\frac{P}{2^{m} \times 5^{n}}\)

∴ \(\frac{13}{150}\) as non-terminating and recurring decimal expansion.

(iii) \(-\frac{11}{75}=-\frac{11}{3 \times 5^{2}}\)

Since it is not in the form of \(\frac{P}{2^{m} \times 5^{n}}\)

∴ –\(\frac{11}{75}\) as non-terminating and recurring decimal expansion.

(iv) \(\frac{17}{200}=\frac{17}{2^{3} \times 5^{2}}\)

∴ \(\frac{17}{200}\) has a terminating decimal expansion.

Question 6.

Find the value of \(\sqrt{27}\) + \(\sqrt{75}\) – \(\sqrt{108}\) + \(\sqrt{48}\)

Solution:

= 3√3 + 5√3 – 6√3 + 4√3

= 12√3 – 6√3

= 6√3

= 6 × 1.732

= 10.392

Question 7.

Evaluate \(\frac{\sqrt{2}+1}{\sqrt{2-1}}\)

Solution:

= 2√2 + 3

= 2 × 1.414 + 3

= 2.828 + 3

= 5.828

Question 8.

Solution:

= 69984 × 1021^{-21-20+9}

= 69984 × 10^{-32}

= 6.9984 × 10^{4} × 10^{-32}

= 6.9984 × 10^{-32+4}

= 6.9984 × 10^{-28}

Question 9.

Write

(a) 9.87 × 10^{9}

(b) 4.134 × 10^{-4} and

(c) 1.432 × 10^{-9} in decimal form.

Solution:

(a) 9.87 × 10^{9} = 9870000000

(b) 4.134 × 10^{-4 }= 0.0004134

(c) 1.432 × 10^{-9} = 0.000000001432