9th Class Mathematics - Chapter 1 - Review Exercise Solved

Review Exercise 1 Solved

Real and Complex Numbers

9th Class Mathematics - Chapter 1 - Solved Reference Guide

Question 1 Core MCQs

Choose the correct option.

\sqrt 7

is:

(a) Integer

(b) Rational number

(d) Natural number

ii.

π

and e are:

(a) Natural numbers

(b) Integers

(d) Irrational numbers ✔

iii. If n is not a perfect square then

\sqrt n

is:

(a) Rational number

(b) Natural number

(d) Irrational number ✔

iv.

\sqrt 3 + \sqrt 5

is:

(a) Whole number

(b) Integer

(d) Irrational number ✔

v. For all

x \in ℝ

x = x

is called:

(a) Reflexive property ✔

(b) Transitive property

(d) Trichotomy property

vi. Let

a, b, c \in ℝ

. If

a > b

and

b > c \Rightarrow a > c

is called:

(a) Trichotomy property

(b) Transitive property ✔

(d) Multiplicative property

vii. If

2 x \times 8 x = 64

then x =

(a) 32 ✔

(b) 34

(d) 23

viii. Let

a, b \in ℝ

. If

a = b \Rightarrow b = a

is called:

(a) Reflexive property

(b) Symmetric property ✔

(d) Additive property

ix.

\sqrt 75 + \sqrt 27

(a) √102

(b) 9√3

(d) 8√3 ✔

x. The product of

(3 + \sqrt 5)(3 - \sqrt 5)

is:

(a) Prime number

(b) Odd number

(d) Rational number ✔

iii

vii

viii

Additional MCQs Extended Practice

Comprehensive question bank from the history of numbers, real number sets, and surd properties.

Section A: History of Real Numbers

1. Which number system was used by the Sumerians?

(a) Decimal

(b) Hexadecimal

(d) Binary

2. The sexagesimal system is a number system with the base:

(a) 2

(b) 10

(d) 60 ✔

3. Which number system was used by the Egyptians?

(a) Decimal ✔

(b) Hexadecimal

(d) Binary

4. How many letters are used in Roman numeral system?

(a) 3

(b) 5

(d) 10

5. In Roman counting the letter "L" represents the number:

(a) 10

(b) 50 ✔

(d) 500

6. The invention of zero is attributed to:

(a) Arabs

(b) Egyptians

(d) Indians ✔

7. Which number system is known as Indo-Arabic numerals?

(a) Decimal ✔

(b) Hexadecimal

(d) Binary

8. Who did introduce the numerals (0-9) to Europe?

(a) Arabs ✔

(b) Egyptians

(d) Indians

9. Which of the following is one of the modern number systems?

(a) Roman Numerals

(b) Egyptian numerals

(d) Hexadecimal system ✔

Section B: Real Numbers

10.

Q \cup Q'

(a) Q'

(b) Q

(d) ∅

11.

Q \cap Q'

(a) Q'

(b) Q

(d) ∅ ✔

12.

Q

and

Q'

are ________ sets.

(a) disjoint ✔

(b) overlapping

(d) super

13. For each prime number P,

\sqrt P

is an:

(a) Irrational ✔

(b) Rational

(d) Whole

Section C: Properties of Real Numbers

14. Name the property of real numbers used in

π + (-π) = 0

(a) Additive inverse ✔

(b) Multiplicative inverse

(d) Multiplicative identity

15. Name the property of real numbers used in

12 \times 1 = 12

(a) Additive identity

(b) Additive inverse

(d) Multiplicative inverse

16. If

x < y

and

z < 0

then:

(a)

xz < yz

(b)

xz > yz

✔

(c)

xz = yz

(d)

x > y

17. If

a, b \in ℝ

then only one of

a = b

a < b

a > b

holds is called ________ property.

(a) Trichotomy ✔

(b) Transitive

(d) Multiplicative

Section D: Radical Expressions

18. In

\sqrt a

, the symbol

\sqrt

is called:

(a) radical sign ✔

(b) index

(d) base

19. In

n \sqrt a m

, 'n' is called:

(a) base

(b) radical sign

(d) radical

20.

(27 x) 2/3

(a) 3√x²9

(b) √x³9

(d) 93√x² ✔

21. Write

5 \sqrt x

in exponential form:

(a) x

(b)

x 5

(c)

x 1/5 ✔

(d)

x 5/2

22. Writing

m 2/3

with radical sign we get:

(a) 3√m² ✔

(b) √m³

(d) √m⁶

23. In

3 \sqrt 5

, the radicand is:

(a) 3

(b) 13

(d) 35

Section E: Surds

24. Which of the following is a Surd?

(a) √7 ✔

(b) √9

(d) √e

25. Which of the following is a binomial Surd?

(a) 4√3

(b) √16

(d) 2 - √3 ✔

26. A surd which contains a single term is called ________ surd.

(a) Monomial ✔

(b) Binomial

(d) None

27. Conjugate factor of the Surd

a + b \sqrt x

is:

(a)

a + b \sqrt x

(b)

a - b \sqrt x

✔

(c)

- a - b \sqrt x

(d)

- a + b \sqrt x

28.

(4 + \sqrt 2)(4 - \sqrt 2)

is equal to:

(a) 14 ✔

(b) -14

(d) 8

29.

(\sqrt x + \sqrt y)(\sqrt x - \sqrt y)(x + y)

(a)

(x + y)

(b)

(x - y)

(c)

(x 2 + y 2)

(d)

(x 2 - y 2)

✔

30.

1 2 - \sqrt 3

(a) 2 + √3 ✔

(b) 2 - √3

(d) -2 - √3

Question 2 Verification

a = 32

b = 53

, and

c = 75

, then verify that:
(i)

a (b + c) = ab + ac

(ii)

(a + b) c = ac + bc

Solution (i)

Left Hand Side (L.H.S):

L.H.S = a (b + c)

= 32 \times (53 + 75)

= 32 \times (25 + 21 15)

= 32 \times (4615)

Multiply numerator and denominator: cancel 3 and 15 (leaving 5 in denominator) and cancel 2 and 46 (leaving 23 in numerator):

= 1 \times 23 1 \times 5 = 235 — Eq. (1)

Right Hand Side (R.H.S):

R.H.S = ab + ac

= (32 \times 53) + (32 \times 75)

= 52 + 2110

Take LCM of denominators (2 and 10 is 10):

= 25 + 21 10

= 4610 = 235 — Eq. (2)

From Eq. (1) and Eq. (2), LHS = RHS. Hence verified.

Solution (ii)

Left Hand Side (L.H.S):

L.H.S = (a + b) c

= (32 + 53) \times 75

= (9 + 10 6) \times 75

= 196 \times 75

= 13330 — Eq. (1)

Right Hand Side (R.H.S):

R.H.S = ac + bc

= (32 \times 75) + (53 \times 75)

= 2110 + 3515

= 2110 + 73 [ ∵ 3515 = 73]

Take LCM of denominators (10 and 3 is 30):

= 21(3) + 7(10) 30

= 63 + 70 30 = 13330 — Eq. (2)

From Eq. (1) and Eq. (2), LHS = RHS. Hence verified.

Question 3 Associative Property

a = 43

b = 52

, and

c = 74

, then verify the associative property of real numbers w.r.t addition and multiplication.

Solution (i) w.r.t Addition

The associative property of real numbers with respect to addition states: $(a + b) + c = a + (b + c)$ .

Left Hand Side (L.H.S):

L.H.S = (a + b) + c

= (43 + 52) + 74

= (8 + 15 6) + 74

= 236 + 74

LCM of 6 and 4 is 12:

= 46 + 21 12 = 6712 — Eq. (1)

Right Hand Side (R.H.S):

R.H.S = a + (b + c)

= 43 + (52 + 74)

= 43 + (10 + 7 4)

= 43 + 174

LCM of 3 and 4 is 12:

= 16 + 51 12 = 6712 — Eq. (2)

From Eq. (1) and Eq. (2), LHS = RHS. Verified.

Solution (ii) w.r.t Multiplication

The associative property of real numbers with respect to multiplication states: $(a \times b) \times c = a \times (b \times c)$ .

Left Hand Side (L.H.S):

L.H.S = (a \times b) \times c

= (43 \times 52) \times 74

= 206 \times 74

= 103 \times 74 [ ∵ 206 = 103]

= 7012 = 356 — Eq. (1)

Right Hand Side (R.H.S):

R.H.S = a \times (b \times c)

= 43 \times (52 \times 74)

= 43 \times (358)

Cancel out common factor 4 and 8:

= 1 \times 35 3 \times 2 = 356 — Eq. (2)

From Eq. (1) and Eq. (2), LHS = RHS. Verified.

Question 4 Number Theory

Is 0 a rational number? Explain.

Solution

Yes, 0 is a rational number.

Explanation:

By definition, a rational number is any number that can be expressed in the fraction form $p q$ , where $p$ and $q$ are integers ( $p, q \in ℤ$ ) and the denominator $q$ is not equal to zero ( $q \neq 0$ ).

The number 0 can easily be written as fractions like:

01, 02, 03, 0 -5, etc.

Since the numerator (0) is an integer, and the denominators (1, 2, 3, etc.) are non-zero integers, 0 satisfies all conditions of a rational number.

Question 5 Definitions

State trichotomy property of real numbers.

Solution

The trichotomy property states that for any two real numbers a and b ( $a, b \in ℝ$ ), exactly one of the following three statements must be true:

(1) a > b (a is greater than b) (2) a = b (a is equal to b) (3) a < b (a is less than b)

Question 6 Rational Insertion

Find two rational numbers between 4 and 5.

Solution

Step 1: Find the first rational number by taking the average of 4 and 5:

Average r 1 = 4 + 5 2 = 92 (4.5)

Step 2: Find the second rational number by taking the average of 92 and 5:

Average r 2 = 92 + 5 2

= 9 + 10 2 2 = 19 2 \times 2 = 194 (4.75)

Conclusion: Two rational numbers between 4 and 5 are 92 and 194.

Question 7 Simplifications

Simplify the following:
(i)

5 \sqrt x 15 y 35 z 20

(ii)

3 \sqrt (27) 2x

(iii)

6(3) n+2 3 n+1 - 3 n

Solution (i)

Convert the 5th root to a fractional exponent:

= (x 15 y 35 z 20) 1/5

Distribute the exponent to each term: $(a m) n = a m \cdot n$ :

= x 15 \times 1/5 \cdot y 35 \times 1/5 z 20 \times 1/5

= x 3 y 7 z 4

Solution (ii)

Express 27 as base 3 ( $27 = 3 3$ ):

= 3 \sqrt (3 3) 2x

= 3 \sqrt 3 6x

= 3 \sqrt (3 2x) 3

Apply radical index rule: $3 \sqrt a 3 = a$ :

= 3 2x

Solution (iii)

Factor out base term $3 n$ :

= 6(3 n \times 3 2) 3 n \cdot 3 1 - 3 n \cdot 1

= 3 n (6 \times 9) 3 n (3 - 1)

Cancel out $3 n$ from both terms:

= 6 \times 9 2

= 542 = 27

Question 8 Word Problems

The sum of three consecutive odd integers is 51. Find the three integers.

Solution

Let the three consecutive odd integers be:

1 st odd integer = x 2 nd odd integer = x + 2 3 rd odd integer = x + 4

According to the given condition, their sum is 51:

x + (x + 2) + (x + 4) = 51

3 x + 6 = 51

3 x = 51 - 6

3 x = 45

x = 453 = 15

Calculate each odd integer:

1 st odd integer = x = 152 nd odd integer = 15 + 2 = 173 rd odd integer = 15 + 4 = 19

Conclusion: The three consecutive odd integers are 15, 17, and 19.

Question 9 Word Problems

Abdullah picked up 96 balls and placed them into two buckets. One bucket has twenty-eight more balls than the other bucket. How many balls were in each bucket?

Solution

Let the number of balls in the buckets be:

Balls in 1 st bucket = x Balls in 2 nd bucket = y

By 1^st condition (total balls is 96):

x + y = 96 — Eq. (1)

By 2^nd condition (one bucket has 28 more balls):

x = y + 28 \Rightarrow x - y = 28 — Eq. (2)

Add Eq. (1) and Eq. (2) to solve for x:

(x + y) + (x - y) = 96 + 28

2 x = 124

x = 1242 = 62

Substitute x = 62 in Eq. (1) to solve for y:

62 + y = 96

y = 96 - 62 = 34

Conclusion: The first bucket has 62 balls and the second bucket has 34 balls.

Question 10 Financial Math

Salma invested Rs. 350,000 in a bank, which paid simple profit at a rate

714 %

per annum. After 2 years, the rate was increased to 8% per annum. Find the amount she had at the end of 7 years.

Solution

Total time period = 7 years. We split the calculation into two terms:

Term 1: First 2 Years

Principal (P) = Rs. 350,000 Rate (R 1) = 7 14 % = 7.25% per annum Time (T 1) = 2 years

Profit 1 = P \times T 1 \times R 1

= 350,000 \times 2 \times 7.25%

= 700,000 \times 7.25 100

= 7,000 \times 7.25 = Rs. 50,750

Term 2: Remaining 5 Years

Principal (P) = Rs. 350,000 Rate (R 2) = 8% per annum Time (T 2) = 7 - 2 = 5 years

Profit 2 = P \times T 2 \times R 2

= 350,000 \times 5 \times 8%

= 1,750,000 \times 8100

= 17,500 \times 8 = Rs. 140,000

Total Calculation

Total Profit = Profit 1 + Profit 2 = 50,750 + 140,000 = Rs. 190,750

Total Accumulated Amount = Principal + Total Profit

= 350,000 + 190,750 = Rs. 540,750

Conclusion: Salma had an accumulated total amount of Rs. 540,750 at the end of 7 years.

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Mathematics 9

Real and Complex Numbers

Section A: History of Real Numbers

Section B: Real Numbers

Section C: Properties of Real Numbers

Section D: Radical Expressions

Section E: Surds

Term 1: First 2 Years

Term 2: Remaining 5 Years

Total Calculation

Accumulation Solution: