Q1. Find a rational number between 1 and 2.
Sol: Express 1 and 2 as rational numbers with the same denominator:
1 = 1010, 2 = 2010
Now, the rational numbers between 1010 and 2010 are:
1110, 1210, 1310, 1410, 1510, 1610, 1710, 1810, 1910
Out of these, select any five:1110, 1210, 1310, 1410, 1510
Answer: The five rational numbers between 1 and 2 are:
1110, 1210, 1310, 1410, 1510
Q2. Find two rational numbers between 0.1 and 0.3.
Sol: Express and as rational numbers with the same denominator:
Now, the rational numbers between and can be written with a larger denominator to find numbers in between.
Let us express them with a denominator of 100:The rational numbers between and are:
any two:
Q3. Express 3 18 in the form of a decimal.
We have, 3 18 = 258
Now, dividing 25 by 8,![]()
Since, the remainder is 0.
∴ The process of division terminates.∴ 258 = 3.125
Thus, 3 18 = 3.125
Q4. Express 0.33 as a rational number.
Let x = 0.33 = 0.3333... (1)
Multiplying (1) by 100, we have 100x = 100 x 0.3333…
⇒ 100x = 33.3333 …(2)
Subtracting (2) from (1), we have
100x – x = 33.3333… – 0.3333…
⇒ 99x = 33⇒ x = 3399 = 13
Thus, 0.33 = 13
Q5. Simplify: (4+ √3) (4 −√3)
∵ (a + b)(a – b) = a2 – b2
(4 +√3) (4 −√3) = (4)2 – ( √3)2 = 16 – 3 = 13
Thus, (4 +√3) (4 −√3) = 13
Q6. Simplify: (√3 +√2)2
∵ (a + b)2 = a2 + 2ab + b2
(√3 +√2)2 = (√3)2 + √2 ( √3 ×2) + (√2)2 = 3 + 2 √6 + 2 = 5 + 2 √6
Thus, (√3 +√2)2 = 5 + 2√6
Q7. Rationalise the denominator of 1√6 - √5
Multiply and divide the given number by √6 + √5
1√6 - √5
= √6 + √5(√6 - √5) (√6 + √5)
= √6 + √56 - 5
= √6 + √5
Q8. Find (64)-1/3
As, 64 = 4 x 4 x 4 = 43
∴ (64)-1/3 = 1(64)1/3 = 1(43)1/3
= 143×1/3 = 14
Thus, (64)-1/3 = 14
Q9. Find a rational number lying between 15 and 12
Rational numbers between 12 and 15 are infinite. Some of them are 310 , 410 , 45100 , 35100 .Step-by-step explanation:As per the question, We need to find drational numbers lying between 15 and 12 As we know,Hence, Rational numbers between 15 and 12 are infinite. Some of them are 310 , 410 , 45100 , 35100
- Rational Numbers are numbers that can be expressed in the form of p/q where q is not equal to zero.
- Now, we know that 15 = 0.2 and 12 = 0.5
- So, numbers between 0.2 and 0.5 are infinite. Some of them are 0.3,0.4,0.45,,0.35 etc.
- And these may be written as 310 , 410 , 45100 , 35100 etc.
Q10. Express 0.245 as a fraction in the simplest form.
Let X =0.245 = 0.24545 ... (1)
Then, multiplying (1) by 10,
We have 10X = 10 x 0.24545...
⇒ 10x = 2.4545 ... (2)
Again multiplying (1) by 1000,
we get 1000 x X = 0.24545... x 1000
⇒ 1000X = 245.4545 ... (3)
Subtracting (2) from (3),
1000X – 10X = 245.4545... – 2.4545...⇒ 990x = 243
⇒ x = 243900 = 27110
Thus, 0.245 = 27110
45 videos|409 docs|53 tests
|
1. What are the different types of number systems? | ![]() |
2. How do you convert a binary number to a decimal number? | ![]() |
3. What is the significance of the hexadecimal number system in computing? | ![]() |
4. Can you explain how to perform addition in the binary number system? | ![]() |
5. What are the applications of the octal number system? | ![]() |