Class 9 Exam  >  Class 9 Notes  >  Mathematics (Maths) Class 9  >  Worksheet - 1: Number System

1: Number System Class 6 Worksheet Maths Chapter 1

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

1: Number System Class 6 Worksheet Maths Chapter 1
(b) √3
(c) 1/2
(d) 

1: Number System Class 6 Worksheet Maths Chapter 1
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 number1: Number System Class 6 Worksheet Maths Chapter 1in 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 =1: Number System Class 6 Worksheet Maths Chapter 1
multiply both side by 100,
100 x = 1: Number System Class 6 Worksheet Maths Chapter 1    ...(i)
multiply both side by 10
1000 x = 1: Number System Class 6 Worksheet Maths Chapter 1   ....(ii)
Subtract (ii) - (i)
1000 x - 100 x = 1: Number System Class 6 Worksheet Maths Chapter 1
900 x = 241
⇒ x = 241/900
Hence, option (c) is correct

Q.3. Every point on the number line represents, which of the following numbers?
(a) 
Natural numbers
(b) 
Irrational number
(c) 
Rational number
(d) 
Real number
Ans. 

Every point on the number line represents a: Real number
The number line includes all types of numbers, such as natural numbers, whole numbers, integers, rational numbers (including fractions), irrational numbers (like the square root of 2 or π), and even imaginary numbers. All these numbers together form the set of real numbers, which is represented on the number line. Therefore the correct option is (d).

Q.4. Which of the following is not a surd ?
(a) 
√6
(b) 
√7
(c) 
∛343
(d) 
√11
Ans. 

A surd is an irrational number that is expressed as the root of a non-perfect square or a non-perfect cube. An irrational number cannot be expressed as a fraction or a repeating decimal.

Let's go through each option again:

(a) √6: This is a surd because 6 is not a perfect square, and √6 is an irrational number.

(b) √7: This is a surd because 7 is not a perfect square, and √7 is an irrational number.

(c) ∛343: This is not a surd. 343 is a perfect cube because it can be expressed as 7^3, and ∛343 = 7, which is a rational number.

(d) √11: This is a surd because 11 is not a perfect square, and √11 is an irrational number.

So, the correct answer is: (c) ∛343

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.7. The conjugate pair of 2 + √3 is
(a) 2 - √3
(b) 2 + √3
(c) 2√3
(d) 4 - √3
Ans. 

To find the conjugate pair of the given number, 2 + √3, we need to change the sign of the square root term (√3). The conjugate pair is obtained by replacing the "+" with "-" or vice versa. So, the conjugate pair of 2 + √3 is 2 - √3.

Q.8. Simplify: 
1: Number System Class 6 Worksheet Maths Chapter 1
Ans.
1: Number System Class 6 Worksheet Maths Chapter 1
1: Number System Class 6 Worksheet Maths Chapter 1
1: Number System Class 6 Worksheet Maths Chapter 1

Q.9. Rationalise: 
1: Number System Class 6 Worksheet Maths Chapter 1
Ans.
1: Number System Class 6 Worksheet Maths Chapter 1
1: Number System Class 6 Worksheet Maths Chapter 1

Q.10. Find the value of 
1: Number System Class 6 Worksheet Maths Chapter 1
Ans.
1: Number System Class 6 Worksheet Maths Chapter 1
1: Number System Class 6 Worksheet Maths Chapter 1
1: Number System Class 6 Worksheet Maths Chapter 1
= 5+4 - 4√5 - 5 - 4 -  4√5  = -8√5

Q.11. If 1: Number System Class 6 Worksheet Maths Chapter 1
find the value of a & b.
Ans.
1: Number System Class 6 Worksheet Maths Chapter 1
Rationalising LHS
1: Number System Class 6 Worksheet Maths Chapter 1 
1: Number System Class 6 Worksheet Maths Chapter 1
1: Number System Class 6 Worksheet Maths Chapter 1
1: Number System Class 6 Worksheet Maths Chapter 1
1: Number System Class 6 Worksheet Maths Chapter 1
∴ a = 11/7 and b = 6/7

Q.12. Evaluate: 
1: Number System Class 6 Worksheet Maths Chapter 1
Ans.
1: Number System Class 6 Worksheet Maths Chapter 1 
1: Number System Class 6 Worksheet Maths Chapter 1

Q.13.  If 1: Number System Class 6 Worksheet Maths Chapter 1 , 
find the value of x3 - 2x2 - 7x+5
Ans.
1: Number System Class 6 Worksheet Maths Chapter 1
1: Number System Class 6 Worksheet Maths Chapter 1
x = 2 + √3
x3 - 2x2 - 7x + 5
= x(x- 2x - 7) + 5
= (2 + √3) [(2 + √3)2 - 2(2 + √3) - 7 ]+5
= (2 + √3) [4 + 3+4√3 - 4 + 2√3 - 7]+5
= (2 + √3)(-4 + 2√3) + 5
= (2 + √3) x 2(-2+√3)+5
= 2[(√3 + 2)(√3 - 2)] + 5
= 2[(√3)- 22 ]+5
= 2[3-4]+5
= 2(-1) + 5
= -2+5
= 3

Q.14. Express 1: Number System Class 6 Worksheet Maths Chapter 1 in p/q form.
Ans.
let x be the p/q form,
so, x = 1: Number System Class 6 Worksheet Maths Chapter 1 
10x = 1: Number System Class 6 Worksheet Maths Chapter 1
1000x = 1: Number System Class 6 Worksheet Maths Chapter 1
1000x - 10x = 1: Number System Class 6 Worksheet Maths Chapter 1-  1: Number System Class 6 Worksheet Maths Chapter 1
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
1: Number System Class 6 Worksheet Maths Chapter 1
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 27x = 9/3, find x
Ans.
27x = 9/3x
⇒ 33x = (3)2/(3)x
⇒ 33x = 32-x
⇒ 3x = 2-x
⇒ 4x = 2
⇒ x = 4/2
⇒ x = 0.5

Q.18. Write the value of 
1: Number System Class 6 Worksheet Maths Chapter 1
Ans.
1: Number System Class 6 Worksheet Maths Chapter 1
1: Number System Class 6 Worksheet Maths Chapter 1
= 15

The document 1: Number System Class 6 Worksheet Maths Chapter 1 is a part of the Class 9 Course Mathematics (Maths) Class 9.
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FAQs on 1: Number System Class 6 Worksheet Maths Chapter 1

1. What is the number system?
Ans. The number system is a system of representing numbers using a set of symbols or digits. It provides a way to express quantities and perform mathematical operations. In common usage, we use the decimal number system, which is based on the digits 0-9.
2. What are the different types of number systems?
Ans. There are several different types of number systems, including the decimal number system, binary number system, octal number system, and hexadecimal number system. Each system uses a different set of symbols or digits to represent numbers.
3. How does the binary number system work?
Ans. The binary number system is a base-2 system that uses only two digits, 0 and 1, to represent numbers. Each digit in a binary number is called a bit. The value of each bit depends on its position within the number. For example, the binary number 1010 represents the decimal number 10.
4. What is the significance of the decimal number system?
Ans. The decimal number system is widely used in everyday life and mathematics. It is a base-10 system that uses ten digits, 0-9, to represent numbers. The position of each digit in a decimal number determines its value. The decimal system is advantageous because it allows for easy representation of fractions and provides a familiar way to express quantities.
5. How is the hexadecimal number system used in computing?
Ans. The hexadecimal number system is a base-16 system that uses sixteen digits, 0-9 and A-F, to represent numbers. In computing, the hexadecimal system is commonly used to represent binary values in a more compact and human-readable form. Each hexadecimal digit corresponds to four binary digits (bits), making it easier to work with and understand binary numbers.
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