Q.1: Find five rational numbers between 1 and 2.
Solution:
We have to find five rational numbers between 1 and 2.
So, let us write the numbers with denominator 5 + 1 = 6
Thus, 6/6 = 1, 12/6 = 2
From this, we can write the five rational numbers between 6/6 and 12/6 as:
7/6, 8/6, 9/6, 10/6, 11/6
Q.2: Find five rational numbers between 3/5 and 4/5.
Solution:
We have to find five rational numbers between 3/5 and 4/5.
So, let us write the given numbers by multiplying with 6/6, (here 6 = 5 + 1)
Now,
3/5 = (3/5) × (6/6) = 18/30
4/5 = (4/5) × (6/6) = 24/30
Thus, the required five rational numbers will be: 19/30, 20/30, 21/30, 22/30, 23/30
Q.4: Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.
Solution:
No, since the square root of a positive integer 16 is equal to 4. Here, 4 is a rational number.
Q.6: Show that
Solution:
Let x = 0.3333….
Multiply with 10,
10x = 3.3333…
Now, 3.3333… = 3 + x (as we assumed x = 0.3333…)
Thus, 10x = 3 + x
10x – x = 3
9x = 3
x = 1/3
Therefore, 0.3333… = 1/3. Here, 1/3 is in the form of p/q and q ≠ 0.
Q.8: Find three different irrational numbers between the rational numbers 5/7 and 9/11.
Solution:
The given two rational numbers are 5/7 and 9/11.
5/7 = 0.714285714…..
9/11 = 0.81818181……
Hence, the three irrational numbers between 5/7 and 9/11 can be:
0.720720072000…
0.730730073000…
0.808008000…
Q.10: Add 2√2+ 5√3 and √2 – 3√3.
Solution:
(2√2 + 5√3) + (√2 – 3√3)
= 2√2 + 5√3 + √2 – 3√3
= (2 + 1)√2 + (5 – 3)√3
= 3√2 + 2√3
Q.11: Simplify: (√3+√7) (√3-√7).
Solution:
(√3 + √7)(√3 – √7)
Using the identity (a + b)(a – b) = a2 – b2,
(√3 + √7)(√3 – √7) = (√3)2 – (√7)2
= 3 – 7
= -4
Q.12: Rationalise the denominator of 1/[7+3√3].
Solution:
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
Q.14: Simplify:
(i) 72/3.71/5
(ii) 101/2/101/4
Solution:
(i) 72/3.71/5
Bases are equal, so add the powers.
7(2/3 + 1/5)
= 7(10 + 3)/15
= 713/15
(ii) 101/2/101/4
Bases are equal, so subtract the powers.
= 10 (1/2 – 1/4)
= 101/4
Q.15: What is the product of a rational and an irrational number?
a) Always an integer
b) Always a rational number
c) Always an irrational number
d) Sometimes rational and sometimes irrational
Correct answer: Option (c)
Explanation:
The product of a rational and an irrational number is always an irrational number.
For example, 2 is a rational number and √3 is irrational. Thus, 2√3 is always an irrational number.
Q.16: What is the value of (256)0.16 X (256)0.09?
a) 4
b) 16
c) 64
d) 256.25
Correct answer: Option (a)
(256)0.16 x (256)0.09 = (256)(0.16 + 0.09)
= (256)0.25
= (256)(25/100)
= (256)(1/4)
= (44)(1/4)
= 44(1/4)
= 4
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