Question:
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
Answer:
(i) Let x = 0. 1 Then,
   x=0.11111…             ...(i)
=>    10x = 1.11111…         ...(ii)
On substracting (i) from (ii), we get
   9x=1
=>Â Â Â x= 1/9
=>Â Â Â 0. 1 = 1/9 i.e., 0.11111... = 1/9
Let x = 0. 2 Then,
   x = 0.222222...         ...(i)
=>Â Â Â 10x = 2.222222... Â Â Â Â Â Â Â Â Â Â Â Â ...(ii)
On substracting (i) from (ii), we get
   9x = 2
=>Â Â Â x=2/9
=>Â Â Â 0. 2 = 2/9 i.e., 0.22222... = 2/9
(iii)Let x=0. 3 Then,
   x= 0.33333...         ...(i)
=>Â Â Â 10x = 3.33333... Â Â Â Â Â Â Â Â ...(ii)
On substracting (i) from (ii), we get
   9x=3
=>Â Â Â x = 3/9
=> Â Â Â 0. 3 = 3/9 i.e., 0.33333... = 3/9
Proceeding as above, we obtain
   0. 4 = 4/9 i.e., 0.44444... 4/9
   0. 5 =5/9 i.e., 0.55555... = 5/9
   0. 6 = 6/9 i.e., 0.666666... = 6/9
Also, we have
   0. 7 =7/9 i.e., 0.77777 = ... 7/9
   0. 8 =8/9 i.e., 0.88888 = ... 8/9
and ,   0. 9 =9/9 i.e., 0.99999 = ...9/9
Number System
Q 1.
Show that :
Q 2.
Prove that:
Q 3.
if x,y,z are positive real numbers show that :
Q 4.
If a = 2 and b= 3, then find the values of each of the following :
(i) a
a + b
b (ii) a
b + b
a (iii) a
b (iv) (a/b)
a (v) (1/a +1/b)
a
Q 5.
If
, Prove that m - n = 1.
Q 6.
Find the value of x, if
Q 7.
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) 2
55 x 2
60 - 2
97 x 2
18 (iv) (2/3)
2 x (2/5)
-3 x (3/5)
2
Q 8.
Simplify :
(i)
(ii)
Q 9.
Evalute each of the following:
(i) 5
2 x 5
4 (ii) 5
8 ÷ 5
3 (iii) (3
2)
3 (iv) (11/12)
3 (v) (3/4)
-3
Q 10.
Prove that √
3 - √
2 is an irrational number.
Q 11.
Simplify :
Q 12.
Express 0.999999... in the form p/q, where p and q are integers and q ≠0
Q 13.
Find a rational number between 2 and 1.
Q 14.
Simplify each of the following :
(i) (625)
-1/4 (ii) (256/81)
5/4 (iii) (243/32)
-4/5 (iv) 5√
(32)-3
Q 15.
Find two rational numbers between 0.23332333233332… and 0.2525525552555-52…
Q 16.
Give examples of two irrational numbers, the product of which is:
(i) a rational number (ii) an irrational number.
Q 17.
Write three numbers whose decimal expansions are non-terminating non recurring.
Q 18.
Identify √
45 as rational number or irrational number.
Q 19.
Represent 5/3 and -5/3 on the number line.
Q 20.
Express 0.
001as a fraction in the simplest form.
Q 21.
Find three different irrational numbers between the rational numbers 5/7 and 9/11.
Q 22.
Assuming that x, y are positive real numbers, simplify each of the following:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Q 23.
Find two irrational numbers lying between √
2 and √
3.
Q 24.
Find a rational number between - 2 and 6.
Q 25.
Insert 100 rational numbers between -3/13 and 9/13.
Q 26.
Represent 8/5 and -8/5 on the the number line.
Q 27.
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
Q 28.
Insert 10 rational numbers betwveen -3/11 and 8/11.
Q 29.
Find three rational numbers between -2 and 5.
Q 30.
Find one irrational number between the number a and b given below
a = 0.1111... = 0.1 and b = 0.1101
Q 31.
Insert a rational and irrational number between 2 and 3.
Q 32.
Prove that √
n is not a rational number, if n is not a perfect square.
Q 33.
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
Q 34.
Express each of the following decimals in the form p/q.
(i)Â 0.
35Â Â Â (ii) 0.
0.585
Q 35.
Show that 0.2353535... = 0.2
35 can be expressed in the form p/q, where p and q are integers and q ≠0.
Q 36.
Find two irrational numbers between 0.12 and 0.13
Q 37.
Examine whether the following numbers are rational or irrational:
(i) (√
2 + 2)
2 (ii) (5 + √
5)(5 - √
5) (iii) 6/ 2√
3
Q 38.
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
Q 39.
Find two irrational numbers between 2 and 2.5.
Q 40.
Simplify each of the following, removing radical signs and negative indices wherever they occur:
(i) (√
4)
-3 (ii) (√
5)
-3 (√
2)
-3 (iii) 1/ 3√
4-5(iv) (25)
-1/3 x 3√
16 (v) (3√
8)
-1/2 (vi) (√
4)
-7 (√
2)
-5
Q 41.
Express each of the following mixed recurring decimals in the form p/q.
(i)4.3
2(ii)15.
712
Q 42.
Convert the following decimal number in the form p/q.
(i) 5.
2 (ii) 23
43
Q 43.
Show that 1.272727 = 1.27 can be expressed in the form p/q , where p and q are integers and q ≠0.
Q 44.
Express each of the following numbers in the form p/q.
(i) 15.75 0 (ii) 8.0025Â (iii) -24.6875
Q 45.
Express the following decimals in the form p/q.
(i)0.
32 (ii) 0.12
3 (iii) 0.003
52