Review Exercise - Complete Explanation Guide
Question 1: Multiple Choice Questions
(i) √7 is:
Answer: (c) irrational number
Explanation:
√7 = 2.64575131… (non-terminating, non-repeating decimal)
7 is not a perfect square (1²=1, 2²=4, 3²=9)
Numbers that are not perfect squares have irrational square roots
(ii) π and e are:
Answer: (d) irrational numbers
Explanation:
π = 3.1415926535… (never ends, never repeats)
e = 2.718281828… (never ends, never repeats)
Both are famous irrational numbers
(iii) If n is not a perfect square, then √n is:
Answer: (d) irrational number
Explanation:
Perfect squares: 1, 4, 9, 16, 25, 36…
If n is not in this list, √n is irrational
Example: √2, √3, √5, √6, √7, √8, √10…
(iv) √3 + √5 is:
Answer: (d) irrational number
Explanation:
√3 is irrational, √5 is irrational
Sum of two irrational numbers is irrational
√3 + √5 = 1.732… + 2.236… = 3.968… (irrational)
(v) For all x ∈ R, x = x is called:
Answer: (a) reflexive property
Explanation:
Reflexive property: a = a (every number is equal to itself)
This is the most basic property of equality
(vi) Let a, b, c ∈ R, then a > b and b > c ⇒ a > c is called:
Answer: (b) transitive
Explanation:
Transitive property: If a > b and b > c, then a > c
This property applies to inequalities
(vii) 2ˣ × 8ˣ = 64 then x =
Solution:
2ˣ × 8ˣ = 64 2ˣ × (2³)ˣ = 2⁶ 2ˣ × 2³ˣ = 2⁶ 2⁴ˣ = 2⁶ 4x = 6 x = 6/4 x = 3/2
Answer: (a) 3/2
(viii) Let a, b ∈ R, then a = b and b = a is called:
Answer: (b) symmetric
Explanation:
Symmetric property: If a = b, then b = a
Equality works both ways
(ix) √75 + √27 =
Solution:
√75 = √(25 × 3) = 5√3 √27 = √(9 × 3) = 3√3 √75 + √27 = 5√3 + 3√3 = 8√3
Answer: (d) 8√3
(x) The product of (3+√5)(3-√5) is:
Solution:
(3+√5)(3-√5) = 3² - (√5)² = 9 - 5 = 4
4 is an integer, which is a rational number.
Answer: (d) rational number
Question 2: Verify Distributive Property
Given: a = 3/2, b = 5/3, c = 7/5
(i) a(b + c) = ab + ac
LHS:
b + c = 5/3 + 7/5 = (25 + 21)/15 = 46/15 a(b + c) = 3/2 × 46/15 = (3 × 46)/(2 × 15) = 138/30 = 23/5
RHS:
ab = 3/2 × 5/3 = 15/6 = 5/2 ac = 3/2 × 7/5 = 21/10 ab + ac = 5/2 + 21/10 = 25/10 + 21/10 = 46/10 = 23/5
✅ Verified: LHS = RHS = 23/5
(ii) (a + b)c = ac + bc
LHS:
a + b = 3/2 + 5/3 = (9 + 10)/6 = 19/6 (a + b)c = 19/6 × 7/5 = (19 × 7)/(6 × 5) = 133/30
RHS:
ac = 3/2 × 7/5 = 21/10 = 63/30 bc = 5/3 × 7/5 = 35/15 = 7/3 = 70/30 ac + bc = 63/30 + 70/30 = 133/30
✅ Verified: LHS = RHS = 133/30
Question 3: Verify Associative Property
Given: a = 4/3, b = 5/2, c = 7/4
Associative Property of Addition: (a + b) + c = a + (b + c)
LHS:
a + b = 4/3 + 5/2 = (8 + 15)/6 = 23/6 (a + b) + c = 23/6 + 7/4 = (46 + 21)/12 = 67/12
RHS:
b + c = 5/2 + 7/4 = (10 + 7)/4 = 17/4 a + (b + c) = 4/3 + 17/4 = (16 + 51)/12 = 67/12
✅ Verified: (a + b) + c = a + (b + c) = 67/12
Associative Property of Multiplication: (a × b) × c = a × (b × c)
LHS:
a × b = 4/3 × 5/2 = 20/6 = 10/3 (a × b) × c = 10/3 × 7/4 = 70/12 = 35/6
RHS:
b × c = 5/2 × 7/4 = 35/8 a × (b × c) = 4/3 × 35/8 = 140/24 = 35/6
✅ Verified: (a × b) × c = a × (b × c) = 35/6
Question 4: Is 0 a Rational Number?
Answer: Yes, 0 is a rational number.
Explanation:
A rational number is any number that can be expressed in the form p/q where q ≠ 0.
0 = 0/1 = 0/2 = 0/3 = ...
Here p = 0, q = 1 (q ≠ 0)
Therefore, 0 is a rational number.
Question 5: State Trichotomy Property
Answer:
The Trichotomy Property states that for any two real numbers a and b, exactly one of the following is true:
a > b (a is greater than b)
a = b (a is equal to b)
a < b (a is less than b)
Explanation:
There is no fourth possibility
Only one can be true at a time
Example: For 3 and 5, only 3 < 5 is true
Question 6: Find Two Rational Numbers Between 4 and 5
Method 1: Using Fractions
4 = 4/1 = 8/2 = 12/3 = 16/4 = 20/5 5 = 5/1 = 10/2 = 15/3 = 20/4 = 25/5
Between 4 and 5:
9/2 = 4.5
14/3 ≈ 4.67
✅ Answer: 9/2 and 14/3 (or 4.5 and 4.67)
Method 2: Using Decimals
4.1 = 41/10 4.5 = 9/2 4.7 = 47/10
✅ Answer: 4.1 and 4.5
Question 7: Simplify the Following
(i) √(x¹⁵ y³⁵ / z³⁰)
Solution:
= √(x¹⁵) × √(y³⁵) / √(z³⁰) = x¹⁵/² × y³⁵/² / z¹⁵ = x⁷·⁵ × y¹⁷·⁵ / z¹⁵ = x⁷ × √x × y¹⁷ × √y / z¹⁵ = x⁷ y¹⁷ √(xy) / z¹⁵
✅ Answer: x⁷ y¹⁷ √(xy) / z¹⁵
(ii) ³√(27)²ˣ
Solution:
= ³√(3³)²ˣ = ³√(3⁶ˣ) = (3⁶ˣ)¹/³ = 3²ˣ
✅ Answer: 3²ˣ
(iii) 6(3)ˣ⁺² / (3ˣ⁺¹ – 3ˣ)
Solution:
= 6(3)ˣ⁺² / [3ˣ(3¹ - 1)] = 6(3)ˣ⁺² / [3ˣ(3 - 1)] = 6(3)ˣ⁺² / [3ˣ(2)] = 6(3)ˣ⁺² / (2 × 3ˣ) = 3(3)ˣ⁺² / 3ˣ = 3 × 3² = 3 × 9 = 27
✅ Answer: 27
Question 8: Sum of Three Consecutive Odd Integers
Given: Sum = 51
Concept: Consecutive odd integers differ by 2
Solution:
Let the three consecutive odd integers be:
First = x
Second = x + 2
Third = x + 4
x + (x + 2) + (x + 4) = 51 3x + 6 = 51 3x = 45 x = 15
Therefore:
First = 15
Second = 17
Third = 19
✅ Answer: 15, 17, 19
Verification: 15 + 17 + 19 = 51 ✅
Question 9: Balls in Two Buckets
Given:
Total balls = 96
One bucket has 28 more than the other
Solution:
Let the smaller bucket have = x balls
Then the larger bucket has = x + 28 balls
x + (x + 28) = 96 2x + 28 = 96 2x = 68 x = 34
Therefore:
Smaller bucket = 34 balls
Larger bucket = 34 + 28 = 62 balls
✅ Answer: 34 and 62 balls
Verification: 34 + 62 = 96 ✅
Question 10: Simple Profit Calculation
Given:
Principal = Rs. 3,50,000
Rate 1 = 7¼% = 29/4% = 7.25% for 2 years
Rate 2 = 8% for remaining 5 years (7 – 2 = 5 years)
Concept: Simple Interest = P × R × T / 100
Solution:
Step 1: Interest for first 2 years
I₁ = 3,50,000 × 7.25 × 2 / 100 I₁ = 3,50,000 × 14.5 / 100 I₁ = 3,50,000 × 0.145 I₁ = 50,750
Step 2: Interest for next 5 years
I₂ = 3,50,000 × 8 × 5 / 100 I₂ = 3,50,000 × 40 / 100 I₂ = 3,50,000 × 0.4 I₂ = 1,40,000
Step 3: Total Interest
Total Interest = I₁ + I₂ = 50,750 + 1,40,000 = 1,90,750
Step 4: Total Amount
Amount = Principal + Total Interest = 3,50,000 + 1,90,750 = 5,40,750
✅ Answer: Rs. 5,40,750