RS Aggarwal Solutions: Exercise 1F - Number System

# RS Aggarwal Solutions: Exercise 1F - Number System | Extra Documents & Tests for Class 9 PDF Download

Q.1. Write the rationalising factor of the denominator in
Ans.

Here, the denominator i.e. 1 is a rational number. Thus, the rationalising factor of the denominator in

Q.2. Rationalise the denominator of each of the following.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
Ans.
(i)
On multiplying the numerator and denominator of the given number by √7, we get:

(ii)
On multiplying the numerator and denominator of the given number by √3, we get:

(iii)
On multiplying the numerator and denominator of the given number by 2 - √3, we get:

(iv)
On multiplying the numerator and denominator of the given number by √5 +2, we get:

(v)

On multiplying the numerator and denominator of the given number by 5-3√2, we get:

(vi)
Multiplying the numerator and denominator by √7+√6, we get

(vii)
Multiplying the numerator and denominator by √11+√7, we get

(viii)
Multiplying the numerator and denominator by 2+√2, we get

(ix)
Multiplying the numerator and denominator by 3-2√2, we get

Q.3. It being given that √2 = 1.414, √3 = 1.732, √5 = 2.236 and √10 = 3.162, find the value of three places of decimals, of each of the following.
(i)
(ii)
(iii)

Ans.
(i)

(ii)

(iii)

Q.4. Find rational numbers a and b such that

(i)
(ii)
(iii)
(iv)

Ans.

(i)

(ii)

(iii)

(iv)

Q.5. It being given  that √3 = 1.732, √5 = 2.236, √6 = 2.449  and √10 = 3.162, find to three places of decimal, the value of each of the following.

(i)
(ii)
(iii)
(iv)
(v)
(vi)
Ans.
(i)

= 0.213
(ii)

=3 × (2.236 − 1.732)
= 1.512
(iii)

(iv)

(v)

= 16.660
(vi)

= 4.441

Q.6. Simplify by rationalising the denominator.
(i)
(ii)
Ans.
(i)

(ii)

Q.7. Simplify
(i)
(ii)
(iii)
(iv)
Ans.
(i)

(ii)

= 0
(iii)

= 16 − √3
(iv)

= 0

Q.8. Prove that
(i)
(ii)
Ans.
(i)

= 2/2
= 1
(ii)

Q.9. Find the values of a and b if

Ans.

Comparing with the given expression, we get
a = 0 and b = 1
Thus, the values of a and b are 0 and 1, respectively.

Q.10. Simplify

Ans.

Q.11. If x = 3 + 2√2, check whether is rational or irrational.
Ans.

x = 3 + 2√2    .....(1)

Adding (1) and (2), we get
which is a rational number
Thus, is rational.

Q.12.
If x = 2 − √3, find value of
Ans.
x = 2 − √3    .....(1)

Subtracting (2) from (1), we get

Thus, the value of

Q.13. If x = 9 − 4√5, find the value of
Ans.
x = 9 − 4√5    .....(1)

Adding (1) and (2), we get

Squaring on both sides, we get

Thus, the value of x2 +  is 322.

Q.14. If x =  find the value of
Ans.

Adding (1) and (2), we get

Thus, the value of x +is 5.

Q.15. If a = 3 − 2√2, find the value of a-
Ans.
a = 3−2√2
⇒ a2 = (3−2√2)2
⇒ a2 = 9 + 8 − 12√2
⇒ a= 17 − 12√2    .....(1)

Subtracting (2) from (1), we get

Thus, the value of a2 -

Q.16. If x = √13 + 2√3, find the value of x −
Ans.

Subtracting (2) from (1), we get

Thus, the value of x

Q.17.

If x = 2 + √3, find the value of
Ans.

Adding (1) and (2), we get

Cubing both sides, we get

Thus, the value of

Q.18. If andshow that
Ans.
Disclaimer: The question is incorrect.

The question is incorrect. Kindly check the question.
The question should have been to show that x − y =

Q.19. If a = and b = show that 3a+ 4ab − 3b=
Ans.
According to question,

Now,
3a+ 4ab − 3b2
= 3(a− b2) + 4ab
= 3 (a + b)(a − b) + 4ab

Hence, 3a+ 4ab − 3b=

Q.20.
If a = and b =find the value of a2 + b2 – 5ab.
Ans.
According to question,

Now,

Hence, the value of a2 + b2 – 5ab is 93.

Q.21.
If p = and q = find the value of p2 + q2.

Ans.
According to question,

Now,
p2 + q2 = (p+q)− 2pq

Hence, the value of p2 + q2 is 47.

Q.22. Rationalise the denominator of each of the following.
(i)
(ii)
(iii)
Ans.
(i)

Hence, the rationalised form is
(ii)

Hence, the rationalised form is
(iii)

Hence, the rationalised form is

Q.23. Given, √2 = 1.414 and √6 = 2.449, find the value of correct to 3 places of decimal.
Ans.

Hence, the value of correct to 3 places of decimal is −1.465.

Q.24. If x = find the value of x3 – 2x2 – 7x + 5.
Ans.

Now,

Also,

Now,

Hence, the value of x– 2x2 – 7x + 5 is 3.

Q.25. Evaluate  it being given that √5 = 2.236 and √10 = 3.162.
Hint

Ans.

Hence,

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## FAQs on RS Aggarwal Solutions: Exercise 1F - Number System - Extra Documents & Tests for Class 9

 1. What are the different types of number systems?
Ans. The different types of number systems are: 1. Decimal number system: It is the most commonly used number system, which uses the base 10 and consists of digits from 0 to 9. 2. Binary number system: It is a base 2 number system, which uses only two digits, 0 and 1. It is widely used in computer systems. 3. Octal number system: It is a base 8 number system, which uses digits from 0 to 7. 4. Hexadecimal number system: It is a base 16 number system, which uses digits from 0 to 9 and alphabets A to F. It is often used in computer programming and digital electronics. 5. Roman numeral system: It is an ancient number system, which uses combinations of letters from the Latin alphabet to represent numbers.
 2. How to convert a decimal number to a binary number?
Ans. To convert a decimal number to a binary number, follow these steps: 1. Divide the decimal number by 2 and note down the remainder. 2. Divide the quotient obtained in step 1 by 2 again and note down the remainder. 3. Repeat step 2 until the quotient becomes 0. 4. The binary number is obtained by writing down the remainders in reverse order, starting from the last remainder obtained. For example, to convert decimal number 25 to binary: 25 ÷ 2 = 12 remainder 1 12 ÷ 2 = 6 remainder 0 6 ÷ 2 = 3 remainder 0 3 ÷ 2 = 1 remainder 1 1 ÷ 2 = 0 remainder 1 Therefore, the binary representation of 25 is 11001.
 3. What is the significance of the base in number systems?
Ans. The base of a number system determines the number of digits used in that system and the value each digit can take. It is also known as the radix. In the decimal number system, the base is 10, which means it has 10 digits from 0 to 9. Each digit's value is multiplied by powers of 10 based on its position in the number. In binary system, the base is 2, which means it has only two digits, 0 and 1. Each digit's value is multiplied by powers of 2 based on its position. The significance of the base is that it determines the counting system and the range of numbers that can be represented in that number system. Different bases are used in different applications, such as binary in computers and decimal in everyday life.
 4. How is the octal number system used in computer programming?
Ans. The octal number system is used in computer programming for several purposes, including: 1. Representation of file permissions: In Unix-like operating systems, file permissions are represented using octal digits. Each digit represents the permission for a specific user group, such as owner, group, and others. 2. Bitwise operations: Octal numbers are often used in bitwise operations, where each octal digit represents three binary digits. This makes it easier to perform operations on groups of bits. 3. Conversion between binary and hexadecimal: Octal numbers can be used as an intermediate step when converting between binary and hexadecimal numbers. Binary numbers can be grouped into groups of three bits, and each group can be represented by a single octal digit. 4. Addressing modes: In computer architecture, addressing modes are often represented using octal numbers. These modes define how data is accessed or stored in memory or registers.
 5. How is the Roman numeral system used in modern times?
Ans. Although the Roman numeral system is no longer widely used in everyday life, it still has some applications in modern times, such as: 1. Clock faces: Roman numerals are commonly used on clock faces to represent the hours. For example, the number 4 is represented as IV, and 9 is represented as IX. 2. Numbering of monarchs and popes: Roman numerals are often used to number monarchs and popes. For example, Queen Elizabeth II is the second monarch with that name. 3. Copyright dates: Roman numerals are sometimes used in copyright dates to give a traditional and timeless feel. For example, a book published in 2021 may have the copyright date MMXXI. 4. Book chapters and sections: Roman numerals are occasionally used to number chapters and sections in books or legal documents for a hierarchical structure. 5. Decorative purposes: Roman numerals are often used in decorative purposes, such as on buildings, statues, or awards, to give a classic and elegant appearance.

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