## Class 9 - Mathematics

#### Number System

MCQ
Q&A
Question:
Express each of the following numbers in the form p/q.
(i) 15.75 0 (ii) 8.0025Â (iii) -24.6875

(i) We have,
Â Â 15.75 = 1575/100
=> (1575 x 25)/(100 x 25)= 63/4

(ii)We have,
8.0025 = 80025/10000 = (80025 x 25)/(1000 x 25) = 3201/400

(iii) We have,
Â Â -25.6875 = -256875/10000 = (-256875 Ã· 625)/(1000 Ã·625) = -411/16

## Questions and Answers

Number System - Questions
1. If $\frac{{9}^{n}×{3}^{2}×{\left({3}^{-n/2}\right)}^{-2}×{\left(27\right)}^{n}}{33m×23}=\frac{1}{27}$ , Prove that m - n = 1.
2. Prove that: $\frac{{a}^{-1}}{{a}^{-1}+{b}^{-1}}+\frac{{a}^{-1}}{{a}^{-1}âˆ’{b}^{-1}}=\frac{{\mathrm{2b}}^{2}}{{b}^{2}âˆ’{a}^{2}}$
3. Find the value of x, if ${5}^{x-3}Ã—{3}^{2x-8}=8$
4. if x,y,z are positive real numbers show that :

$\sqrt{{x}^{-1}y}.\sqrt{{x}^{-1}z}.\sqrt{{z}^{-1}x}$
5. Show that :

$\frac{{x}^{a\left(b-c\right)}}{{x}^{b\left(a-c\right)}}÷{\left(\frac{{x}^{b}}{{x}^{a}}\right)}^{c}= 1\left(ii\right)\frac{{\left({x}^{a+b}\right)}^{2}{\left({x}^{b+c}\right)}^{2}{\left({x}^{c+a}\right)}^{2}}{\left(xaxbxc\right)4}= 1$
6. Simplify :
7. If a = 2 and b= 3, then find the values of each of the following :

(i) aa + bb (ii) ab + ba (iii) ab

(iv) (a/b)a (v) (1/a +1/b)a
8. Evalute each of the following:

(i) (2/11)4 x (11/3)2 x (3/2)3 (ii) (1/2)5 x (-2/3)4 x (3/5)-1

(iii) 255 x 260 - 297 x 218 (iv) (2/3)2 x (2/5)-3 x (3/5)2
9. Evalute each of the following:

(i) 52 x 54 (ii) 58 ÷ 53 (iii) (32)3 (iv) (11/12)3 (v) (3/4)-3
10. Simplify each of the following :

(i) (625)-1/4 (ii) (256/81)5/4 (iii) (243/32)-4/5 (iv) 5âˆš(32)-3
11. Simplify :

(i) (ii)
12. Assuming that x, y are positive real numbers, simplify each of the following:

(i) $\sqrt{{x}^{-2}{y}^{3}}$ (ii) ${\left({x}^{-2/3}{y}^{-1/2}\right)}^{2}$ (iii) ${\left(\sqrt{{x}^{-3}}\right)}^{5}$ (iv) ${\left(\sqrt{x}\right)}^{-2/3}\sqrt{{y}^{4}}÷\sqrt{{y}^{4}}$

(v) $3\sqrt{{\mathrm{xy}}^{2}÷{x}^{2}y}$ (vi) $4\sqrt{3\sqrt{{x}^{2}}}$
13. Identify âˆš45 as rational number or irrational number.
14. Prove that âˆš3 - âˆš2 is an irrational number.
15. Write three numbers whose decimal expansions are non-terminating non recurring.
16. Give examples of two irrational numbers, the product of which is:

(i) a rational number (ii) an irrational number.
17. Find a rational number between 2 and 1.
18. Express 0.999999... in the form p/q, where p and q are integers and q â‰ 0
19. Find two rational numbers between 0.23332333233332… and 0.2525525552555-52…
20. Simplify each of the following, removing radical signs and negative indices wherever they occur:

(i) (âˆš4)-3 (ii) (âˆš5)-3 (âˆš2)-3 (iii) 1/ 3âˆš4-5

(iv) (25)-1/3 x 3âˆš16 (v) (3âˆš8)-1/2 (vi) (âˆš4)-7 (âˆš2)-5
21. Express 0.001as a fraction in the simplest form.
22. Insert 100 rational numbers between -3/13 and 9/13.
23. Evaluate each of the following removing radical signs and negative indices wherever they occur:

(i) (64)1/3 (ii) (125)-1/3[NCERT] (iii) (27) - 2/3 (iv) (64/25)-3/2
24. Express each of the following decimals in the form p/q :
(i)Â 0.4Â Â (ii)Â 0.2 Â Â (iii)Â 0.3
(iv)0.4 Â Â (v)Â 0.5Â Â (vi)0.6
25. Represent 5/3 and -5/3 on the number line.
26. Examine whether the following numbers are rational or irrational:

(i) (âˆš2 + 2)2 (ii) (5 + âˆš5)(5 - âˆš5) (iii) 6/ 2âˆš3
27. Find one irrational number between the number a and b given below

a = 0.1111... = 0.1 and b = 0.1101
28. Find a rational number between - 2 and 6.
29. Express each of the following mixed recurring decimals in the form p/q. (i)4.32(ii)15.712
30. Find three different irrational numbers between the rational numbers 5/7 and 9/11.
31. Express each of the following numbers in the form p/q.
(i) 15.75 0 (ii) 8.0025Â (iii) -24.6875
32. Find two irrational numbers lying between âˆš2 and âˆš3.
33. Find three rational numbers between -2 and 5.
34. Represent 8/5 and -8/5 on the the number line.
35. Find two irrational numbers between 0.12 and 0.13
36. Express each of the following numbers in the form p/q.
(i)Â 0.15 Â Â Â (ii)Â 0.675Â (iii)Â 0.00026
37. Express each of the following decimals in the form p/q.
(i)Â 0.35Â Â Â (ii)Â Â 0.0.585
38. Insert 10 rational numbers betwveen -3/11 and 8/11.
39. Show that 0.2353535... = 0.235 can be expressed in the form p/q, where p and q are integers and q â‰ 0.
40. Insert a rational and irrational number between 2 and 3.
41. Prove that âˆšn is not a rational number, if n is not a perfect square.
42.

Show that 1.272727 = 1.27 can be expressed in the form p/q , where p and q are integers and q â‰ 0.

43. Find two irrational numbers between 2 and 2.5.
44. Convert the following decimal number in the form p/q.
(i) 5.2 (ii) 2343
45. Express the following decimals in the form p/q.
(i)0.32 (ii) 0.123 (iii) 0.00352
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