Question:
Find two rational numbers between 0.23332333233332… and 0.2525525552555-52…
Answer:
Let a = 0.232332333233332… and b = 0.25255255525552… The numbers a and b are both irrational numbers. The decimal representations of a and b are non-terminating and non-repeating. We observe that in the first decimal place a and b have the same digit 2 but the digits in the second place of their decimal representatio are distinct. Number a has a 3 while number b has 5. So, a
Consider the numbers c = 0.25 and d = 0.2525
Clearly, c and d both are rational numbers such that a < c < d < b.
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.
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 35.
Express each of the following decimals in the form p/q.
(i)Â 0.
35Â Â Â (ii) 0.
0.585
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