Click Here To Download Ex : 1.1 Ex : 1.3 MCQs & Shorts Review Exercise Exercise 1.2 – Complete Explanation Guide Question 1: Rationalize the Denominator Concept Explanation: Rationalize denominator ka matlab hai ke denominator se root (√) ko hatana hai. Iske liye hum conjugate use karte hain. Conjugate Rule: (a + b)(a – b) = a² – b² Agar denominator (a + √b) hai toh conjugate (a – √b) se multiply karein Agar denominator (a – √b) hai toh conjugate (a + √b) se multiply karein (i) 13 / (4 + √3) Step 1: Denominator mein (4 + √3) hai Step 2: Conjugate se multiply karein: (4 – √3) text = 13/(4+√3) × (4-√3)/(4-√3) = 13(4-√3) / (4+√3)(4-√3) = 13(4-√3) / (16 – 3) = 13(4-√3) / 13 = 4 – √3 ✅ Answer: 4 – √3 (ii) (√2 + √5) / √3 Step 1: Denominator mein √3 hai Step 2: √3 se multiply karein (numerator aur denominator dono ko) text = (√2 + √5)/√3 × √3/√3 = √3(√2 + √5) / 3 = (√6 + √15) / 3 ✅ Answer: (√6 + √15) / 3 (iii) (√2 – 1) / √5 Step 1: Denominator mein √5 hai Step 2: √5 se multiply karein text = (√2 – 1)/√5 × √5/√5 = √5(√2 – 1) / 5 = (√10 – √5) / 5 ✅ Answer: (√10 – √5) / 5 (iv) (6 – 4√2) / (6 + 4√2) Step 1: Denominator mein (6 + 4√2) hai Step 2: Conjugate (6 – 4√2) se multiply karein text = (6-4√2)/(6+4√2) × (6-4√2)/(6-4√2) = (6-4√2)² / (6+4√2)(6-4√2) = (36 – 48√2 + 32) / (36 – 32) = (68 – 48√2) / 4 = 17 – 12√2 ✅ Answer: 17 – 12√2 (v) (√3 – √2) / (√3 + √2) Step 1: Denominator mein (√3 + √2) hai Step 2: Conjugate (√3 – √2) se multiply karein text = (√3-√2)/(√3+√2) × (√3-√2)/(√3-√2) = (√3-√2)² / (√3+√2)(√3-√2) = (3 – 2√6 + 2) / (3 – 2) = (5 – 2√6) / 1 = 5 – 2√6 ✅ Answer: 5 – 2√6 (vi) 4√3 / (√7 + √5) Step 1: Denominator mein (√7 + √5) hai Step 2: Conjugate (√7 – √5) se multiply karein text = 4√3/(√7+√5) × (√7-√5)/(√7-√5) = 4√3(√7-√5) / (√7+√5)(√7-√5) = 4√3(√7-√5) / (7 – 5) = 4√3(√7-√5) / 2 = 2√3(√7-√5) = 2√21 – 2√15 ✅ Answer: 2√21 – 2√15 Question 2: Simplify the Following Concept Explanation: (a/b)⁻ⁿ = (b/a)ⁿ aᵐ × aⁿ = aᵐ⁺ⁿ aᵐ ÷ aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ a⁰ = 1 (i) (81/16)⁻³/⁴ text = (16/81)³/⁴ = (2⁴ / 3⁴)³/⁴ = (2/3)⁴ × ³/⁴ = (2/3)³ = 8/27 ✅ Answer: 8/27 (ii) (3/4)⁻² ÷ (4/9)³ × 16/27 Step 1: (3/4)⁻² = (4/3)² = 16/9 Step 2: (4/9)³ = 64/729 Step 3: 16/9 ÷ 64/729 × 16/27 text = 16/9 × 729/64 × 16/27 = (16 × 729 × 16) / (9 × 64 × 27) = (16 × 9 × 16) / (9 × 64) [729/27 = 27, 27/9 = 3, 729/27 = 27] = (16 × 16) / 64 = 256/64 = 4 ✅ Answer: 4 (iii) (0.027)⁻¹/³ text 0.027 = 27/1000 = 3³/10³ = (3/10)³ (0.027)⁻¹/³ = (3/10)³ × ⁻¹/³ = (3/10)⁻¹ = 10/3 ✅ Answer: 10/3 (iv) √(x³⁴y²¹z³⁵ / y³⁴z⁷) text = √(x³⁴ × y²¹⁻³⁴ × z³⁵⁻⁷) = √(x³⁴ × y⁻¹³ × z²⁸) = (x³⁴ × y⁻¹³ × z²⁸)¹/² = x¹⁷ × y⁻¹³/² × z¹⁴ = x¹⁷ z¹⁴ / y¹³/² = x¹⁷ z¹⁴ / √(y¹³) = x¹⁷ z¹⁴ / (y¹³/²) ✅ Answer: x¹⁷ z¹⁴ / y¹³/² (v) [5(25)ⁿ⁺¹ – 25(5)²ⁿ] / [5(5)²ⁿ⁺² – (25)ⁿ⁺¹] Step 1: 25 = 5² text = [5(5²)ⁿ⁺¹ – 25(5)²ⁿ] / [5(5)²ⁿ⁺² – (5²)ⁿ⁺¹] = [5(5)²ⁿ⁺² – 5²(5)²ⁿ] / [5(5)²ⁿ⁺² – (5)²ⁿ⁺²] = [5(5)²ⁿ⁺² – 25(5)²ⁿ] / [5(5)²ⁿ⁺² – 5²ⁿ⁺²] = [5(5)²ⁿ⁺² – 25(5)²ⁿ] / [5(5)²ⁿ⁺² – (5)²ⁿ⁺²] Step 2: Let 5²ⁿ = a, 5² = 25 text = [5 × 25 × 5²ⁿ – 25 × 5²ⁿ] / [5 × 25 × 5²ⁿ – 25 × 5²ⁿ] = [125a – 25a] / [125a – 25a] = 100a / 100a = 1 ✅ Answer: 1 (vi) [(16)ⁿ⁺¹ + 20(4²ⁿ)] / [2⁵ × 8ⁿ⁺²] Step 1: 16 = 2⁴, 4 = 2², 8 = 2³ text = [(2⁴)ⁿ⁺¹ + 20(2²)²ⁿ] / [2⁵ × (2³)ⁿ⁺²] = [2⁴ⁿ⁺⁴ + 20(2⁴ⁿ)] / [2⁵ × 2³ⁿ⁺⁶] = [2⁴ⁿ(2⁴ + 20)] / [2³ⁿ⁺¹¹] = [2⁴ⁿ(16 + 20)] / [2³ⁿ⁺¹¹] = [2⁴ⁿ(36)] / [2³ⁿ⁺¹¹] = 36 × 2⁴ⁿ⁻³ⁿ⁻¹¹ = 36 × 2ⁿ⁻¹¹ ✅ Answer: 36 × 2ⁿ⁻¹¹ (vii) (64)²/³ ÷ (9)³/² text (64)²/³ = (4³)²/³ = 4² = 16 (9)³/² = (3²)³/² = 3³ = 27 16 ÷ 27 = 16/27 ✅ Answer: 16/27 (viii) (3ⁿ × 9ⁿ⁺¹) / (3ⁿ⁻¹ × 9ⁿ⁻¹) text = [3ⁿ × (3²)ⁿ⁺¹] / [3ⁿ⁻¹ × (3²)ⁿ⁻¹] = [3ⁿ × 3²ⁿ⁺²] / [3ⁿ⁻¹ × 3²ⁿ⁻²] = 3³ⁿ⁺² / 3³ⁿ⁻³ = 3⁵ = 243 ✅ Answer: 243 (ix) (5ⁿ⁺³ – 6.5ⁿ⁺¹) / (9 × 5ⁿ – 2³ × 5ⁿ) text = [5ⁿ⁺¹(5² – 6)] / [5ⁿ(9 – 8)] = [5ⁿ⁺¹(25 – 6)] / [5ⁿ(1)] = [5ⁿ⁺¹(19)] / 5ⁿ = 5 × 19 = 95 ✅ Answer: 95 Question 3: If x = 3 + √8 Concept Explanation: x + 1/x ka formula use karein (a + b)² = a² + 2ab + b² (a – b)² = a² – 2ab + b² Given: x = 3 + √8 = 3 + 2√2 Find 1/x: Conjugate use karein text 1/x = 1/(3+2√2) = (3-2√2)/(9-8) = 3 – 2√2 (i) x + 1/x text = (3+2√2) + (3-2√2) = 6 ✅ Answer: 6 (ii) x – 1/x text = (3+2√2) – (3-2√2) = 4√2 ✅ Answer: 4√2 (iii) x² + 1/x² Formula: x² + 1/x² = (x + 1/x)² – 2 text = (6)² – 2 = 36 – 2 = 34 ✅ Answer: 34 (iv) x² – 1/x² Formula: x² – 1/x² = (x – 1/x)(x + 1/x) text = (4√2)(6) = 24√2 ✅ Answer: 24√2 (v) x⁴ + 1/x⁴ Formula: x⁴ + 1/x⁴ = (x² + 1/x²)² – 2 text = (34)² – 2 = 1156 – 2 = 1154 ✅ Answer: 1154 (vi) (x – 1/x)² text = (4√2)² = 32 ✅ Answer: 32 Question 4: Find p and q Given: (8 – 3√2) / (4 + 3√2) = p + q√2 Step 1: Rationalize denominator text = (8-3√2)/(4+3√2) × (4-3√2)/(4-3√2) = (8-3√2)(4-3√2) / (16 – 18) = (32 – 24√2 – 12√2 + 18)
9th Class Ch # 1: Real Numbers ( Ex : 1.1 Notes – New Book 2026-27)
Click Here To Download Ex : 1.2 Ex : 1.3 Imp. MCQs and Shorts Review Exercise Exercise 1.1 – Complete Explanation Guide Question 1: Identify Rational or Irrational Numbers Concept Explanation: Rational Numbers: Numbers that can be written as p/q where q ≠ 0 Terminating decimals (e.g., 0.5, 2.75) Repeating decimals (e.g., 0.333…, 1.252525…) All integers and fractions Irrational Numbers: Numbers that cannot be written as p/q Non-terminating, non-repeating decimals Square roots of non-perfect squares π (pi), e (Euler’s number) (i) 2.353535 Type: ✅ Rational Number Explanation: Yeh number repeating decimal hai (353535 repeat ho raha hai) Jo bhi number repeat hota hai, wo rational hota hai Isay p/q ki form mein likha ja sakta hai Iski rational form: 233/99 (approx) (ii) 0.6 Type: ✅ Rational Number Explanation: Yeh terminating decimal hai (khatam hone wala decimal) 0.6 = 6/10 = 3/5 Jo bhi terminating decimal hai, wo rational hota hai p/q form mein likha ja sakta hai (q ≠ 0) (iii) 2.236067… Type: ❌ Irrational Number Explanation: Yeh non-terminating, non-repeating decimal hai Ye dots (…) show kar rahe hain ke ye number kabhi khatam nahi hota Yeh asal mein √5 hai √5 irrational hai kyunke 5 perfect square nahi hai Isay p/q form mein nahi likha ja sakta (iv) √7 Type: ❌ Irrational Number Explanation: √7 means “7 ka square root” 7 perfect square nahi hai (1²=1, 2²=4, 3²=9) Jo number perfect square nahi hai, uska square root irrational hota hai Iski decimal value: 2.64575131… (kabhi khatam nahi hoti) (v) e Type: ❌ Irrational Number Explanation: e (Euler’s number) ≈ 2.718281828… Yeh ek famous irrational number hai Non-terminating, non-repeating decimal hai Mathematics mein natural logarithms mein use hota hai (vi) π (pi) Type: ❌ Irrational Number Explanation: π (pi) ≈ 3.1415926535… Yeh sab se famous irrational number hai Circle ke perimeter aur diameter ka ratio hai Iski decimal value kabhi khatam nahi hoti aur koi pattern nahi repeat karta (vii) 5 + √1 Type: ✅ Rational Number Explanation: √1 = 1 (kyunke 1 × 1 = 1) 5 + 1 = 6 6 ek integer hai aur sab integers rational hote hain Isay 6/1 likha ja sakta hai (viii) √3 + √13 Type: ❌ Irrational Number Explanation: √3 irrational hai (3 perfect square nahi) √13 irrational hai (13 perfect square nahi) Do irrational numbers ka sum bhi irrational hota hai Isay p/q form mein nahi likha ja sakta (ix) (2 – √2)(2 + √2) Type: ✅ Rational Number Explanation: Yeh (a – b)(a + b) = a² – b² ka formula use kar raha hai (2 – √2)(2 + √2) = 2² – (√2)² = 4 – 2 = 2 2 ek integer hai, jo rational hai Isay 2/1 likha ja sakta hai (x) 15/4 Type: ✅ Rational Number Explanation: Yeh already p/q ki form mein hai p = 15, q = 4 (q ≠ 0) Isliye yeh rational number hai Question 2: Represent Numbers on Number Line Concept Explanation: Number line ek straight line hoti hai jis par numbers ko show kiya jata hai. Zero center mein hota hai, right side positive numbers, left side negative numbers. (i) √2 (Square root of 2) How to represent: Step 1: Ek number line banayein aur 0 aur 1 mark karein Step 2: Point 1 se perpendicular line 1 unit upward draw karein Step 3: 0 se perpendicular line ke end tak line draw karein (ye √2 hai) Step 4: Compass se 0 se √2 tak ka distance measure karein Step 5: Number line par 0 se right side ye distance mark karein Explanation: √2 ≈ 1.414 (1 aur 2 ke darmiyan) Pythagoras theorem use hoti hai (1² + 1² = 2) (ii) √3 (Square root of 3) How to represent: Step 1: Pehle √2 locate karein (upar wali method se) Step 2: Point √2 se perpendicular line 1 unit upward draw karein Step 3: 0 se perpendicular line ke end tak line draw karein (ye √3 hai) Step 4: Compass se 0 se √3 tak ka distance measure karein Step 5: Number line par 0 se right side ye distance mark karein Explanation: √3 ≈ 1.732 (1 aur 2 ke darmiyan) √2 aur √3 dono irrational hain (iii) 100/5 How to represent: Step 1: Pehle simplify karein: 100 ÷ 5 = 20 Step 2: Number line par 0 aur 20 ke darmiyan distance measure karein Step 3: 0 se right side 20 mark karein Explanation: 100/5 = 20 (integer hai) Sab integers number line par show ho sakte hain (iv) -2 (Negative 2) How to represent: Step 1: Number line par 0 mark karein Step 2: 0 se left side 2 units count karein Step 3: Us point ko -2 mark karein Explanation: Negative numbers 0 ke left side show hote hain -2 ka matlab hai 0 se 2 units left (v) √8 (Square root of 8) How to represent: Step 1: Simplify karein: √8 = √(4 × 2) = 2√2 Step 2: Pehle √2 locate karein Step 3: 0 se √2 tak ka distance measure karein aur 2 times karein Step 4: Number line par 0 se right side ye distance mark karein Explanation: √8 ≈ 2.828 (2 aur 3 ke darmiyan) Isay 2√2 likha ja sakta hai Question 3: Express as Rational Number (p/q) Concept Explanation: Kisi bhi number ko p/q ki form mein likhna hai, jahan q ≠ 0. (i) 0.4 Solution: 0.4 = 4/10 Simplify: 4/10 ÷ 2/2 = 2/5 ✅ p = 2, q = 5 (ii) 0.37 Solution: 0.37 = 37/100 Already simplified (37 prime hai) ✅ p = 37, q = 100 (iii) 0.21 Solution: 0.21 = 21/100 Already simplified (21 aur 100 ka common factor 1 hai) ✅ p = 21, q = 100 Question 4: Name the Property Used Concept Explanation: Properties of real numbers ko pehchanana hai. (i) (a + 4) + b = a + (4 + b) Answer: ✅ Associative Property of Addition Explanation: Numbers ko different groups mein add karne se result same aata hai Formula: (a + b) + c = a + (b + c) (ii) √2 + √3 = √3 + √2 Answer: ✅ Commutative Property of Addition Explanation: Numbers ka order change karne se sum same aata hai Formula: a + b = b + a (iii) x – x = 0 Answer: ✅ Additive Inverse Property Explanation: Kisi bhi number mein uski negative value add karne se 0 aata hai Formula: a + (-a) = 0 (iv) a(b + c) = ab + ac Answer: ✅ Distributive Property Explanation: Multiplication addition par distribute hoti hai Formula: a × (b + c) =