Class 9 Maths - Number System CBSE Worksheets- 1

Q.1. Which of the following is an irrational number?
(a) 

(b) √3
(c) 1/2
(d) 

Ans.

An irrational number is a number that cannot be expressed as a fraction of two integers and has an infinite non-repeating decimal representation. Let's evaluate the options:

(a) √49/64: This is a rational number because both the numerator and denominator are perfect squares, and their square root can be expressed as a fraction of integers. √49/64 = 7/8.

(b) √3: This is an irrational number because the square root of 3 cannot be expressed as a fraction of integers, and its decimal representation goes on infinitely without repeating.

(c) 1/2: This is a rational number because it can be expressed as a fraction of integers.

(d) -√1/4: This is a rational number because √1/4 = 1/2, and the negative sign only changes the sign of the rational number.

So, the irrational number among the given options is: (b) √3

Q.2. The numberin p/q form is
(a) 267/1000
(b) 26/10
(c) 241/900
(d) 241/999
Ans. (c)
Solution:
let x be the p/q form, x =
multiply both side by 100,
100 x =  ...(i)
multiply both side by 10
1000 x =  ....(ii)
Subtract (ii) - (i)
1000 x - 100 x =
900 x = 241
⇒ x = 241/900
Hence, option (c) is correct

Q.4. The decimal representation of a rational number is either:

(a) Terminating or repeating

(b) Non-terminating and non-repeating

(c) Only terminating

(d) Only repeating

Ans: (a) Terminating or repeating

•  A rational number is any number that can be expressed as a fraction $\frac{p}{q}\backslash frac\{p\}\{q\}$, where p and q are integers and $q\ne 0q\; \backslash neq\; 0$

• A terminating decimal is one that has a finite number of digits after the decimal point. 

• A repeating decimal is one where a block of digits repeats infinitely.

Q.5. Show that 3√5 is an irrational number.
Ans. 
Let us assume 3√5 is a rational number,
3√5 = p/q, where p and q are co-primes,
√5 = p/3q
Clearly, √5 is irrational, while number on right q ≠ 0 are rational
∴ Irrational = Rational
But above deduced can't be right. Therefore our supposition is wrong making 3√5 an irrational number.

Q.6. What is the decimal form of the following no's.
(a) 18/11
(b) 3/26
(c) 1/17
(d) 2/13
Ans.
(a) 18/11 = 1.63636363...
(b) 3/26 = 0.11538461538
(c) 1/17 = 0.05882352941
(d) 2/13 = 0.15384615384

Q.8. Simplify: 

Ans.




Q.9. Rationalise: 

Ans.



Q.10. Find the value of 

Ans.



= 5+4 - 4√5 - 5 - 4 -  4√5  = -8√5

Q.11. If ,
find the value of a & b.
Ans.

Rationalising LHS
 




∴ a = 11/7 and b = 6/7

Q.12. Evaluate: 

Ans.
 


Q.13. Write the value of 

Ans.


= 15

Q.14. Express  in p/q form.
Ans.
let x be the p/q form,
so, x =  
10x =
1000x =
1000x - 10x = -  
990x = 15555
x= 15555/990
= 1037/66

Q.15. Insert five rational no's between 3/5 and 4/5.
Ans.
3/5 and 4/5

30/50 and 40/50
∴ pick any five number between 30 and 40
31/50,  32/50,  36/50,  37/50,  39/50

Q.16. Insert 3 irrational number between 2.6 and 3.8
Ans.
2.6 and 3.8
irrational numbers are non repeating non - terminating
2.61010010001.....
2.802002000200002......
3.604004000400004.......

Q.17. If 27^x = 9/3^x , find x
Ans.
27^x = 9/3^x
⇒ 3^3x = (3)²/(3)^x
⇒ 3^3x = 3^2-x
⇒ 3x = 2-x
⇒ 4x = 2
⇒ x = 4/2
⇒ x = 0.5