General Q&A for Class 12 · Page 122
5,215 free answers in General for Class 12 students, across CBSE, ICSE and State boards.
Question & Answer
Questions
- GeneralClass 12All
What is the product of a rational number and its multiplicative inverse?
The product is always one (1). Formula: (p/q) × (q/p) = 1 (where p ≠ 0) Examples: 3/4 × 4/3 = 12/12 = 1 7/5 × 5/7 = 35/35 = 1 (-2/9) × (-9/2) = 18/18 = 1 Explanation: This is the defining property of multiplicative inverse (reciprocal). The reciprocal is specifically the number that, when multiplied by the original, gives one. <strong
- GeneralClass 12All
What is the sum of a rational number and its additive inverse?
The sum is always zero (0). Formula: p/q + (-p/q) = 0 Examples: 3/7 + (-3/7) = 0 -5/9 + 5/9 = 0 2/3 + (-2/3) = 0 Explanation: This is the defining property of additive inverse. The additive inverse is specifically the number that, when added to the original, gives zero.
- GeneralClass 12All
Arrange in ascending order: 2/3, 5/6, 1/2
Step 1: Find LCM(3, 6, 2) = 6 Step 2: Convert all to denominator 6 2/3 = 4/6 5/6 = 5/6 1/2 = 3/6 Step 3: Compare numerators 3/6 < 4/6 < 5/6 Answer in ascending order: 1/2 < 2/3 < 5/6
- GeneralClass 12All
How do you compare two rational numbers?
There are three methods: Method 1: Cross Multiplication For a/b and c/d: Calculate a × d and b × c If a × d > b × c, then a/b > c/d If a × d < b × c, then a/b < c/d If a × d = b × c, then a/b = c/d Method 2: Same Denominator Convert both to the same denominator (LCM) Compare numerators Method 3: D
- GeneralClass 12All
Convert 0.454545... (0.4̄5̄) to a rational number.
Step 1: Let x = 0.454545... Step 2: Since 2 digits repeat, multiply by 100 100x = 45.454545... Step 3: Subtract 100x - x = 45.454545... - 0.454545... 99x = 45 Step 4: Solve x = 45/99 Step 5: Simplify (GCD = 9) 45/99 = 5/11 Answer: 0.4̄5̄ = 5/11
- GeneralClass 12All
Convert 0.666... (repeating) to a rational number.
Step 1: Let x = 0.666... Step 2: Multiply by 10 (one repeating digit) 10x = 6.666... Step 3: Subtract original from multiplied 10x - x = 6.666... - 0.666... 9x = 6 Step 4: Solve for x x = 6/9 = 2/3 Answer: 0.666... = 2/3 Verification: 2 ÷ 3 = 0.666... ✓
- GeneralClass 12All
How do you convert a terminating decimal to a rational number?
Count the decimal places and use that as the power of 10 in the denominator. Examples: (a) 0.5 1 decimal place 0.5 = 5/10 = 1/2 (b) 0.75 2 decimal places 0.75 = 75/100 = 3/4 (c) 2.125 3 decimal places 2.125 = 2125/1000 = 17/8 (d) 0.008 3 decimal places 0.008 = 8/1000
- GeneralClass 12All
How do you represent -2/5 on a number line?
Step 1: Identify position -2/5 is negative, so it lies to the LEFT of 0 It's between -1 and 0 Step 2: Divide the segment Divide the space between -1 and 0 into 5 equal parts Each part represents 1/5 Step 3: Mark the point Count 2 parts from 0 towards the left Mark this point as -2/5 Visual: -1 -4/5 -3/5 -
- GeneralClass 12All
Find five rational numbers between 1/3 and 1/2
Method 1: Common Denominator Convert to same denominator with enough space: 1/3 = 10/30 1/2 = 15/30 Five numbers between 10/30 and 15/30: 11/30, 12/30, 13/30, 14/30, 15/30 Simplified: 11/30, 2/5, 13/30, 7/15, 1/2 Wait, we need numbers BETWEEN 1/3 and 1/2, so: 11/30, 2/5, 13/30, 7/15 (not including 15/30 which equals 1/2) Better answer with more space:</st
- GeneralClass 12All
How many rational numbers lie between two rational numbers?
Infinitely many rational numbers exist between any two rational numbers. This is called the dense property of rational numbers. Example: Between 1/2 and 1: Using the mean method: (1/2 + 1)/2 = 3/4 (1/2 + 3/4)/2 = 5/8 (3/4 + 1)/2 = 7/8 So: 1/2 < 5/8 < 3/4 < 7/8 < 1 This process can continue forever, finding more numbers each time. General method:</stro
- GeneralClass 12All
What is the distributive property for rational numbers?
Multiplication distributes over addition. Formula: a × (b + c) = (a × b) + (a × c) Example: 2/3 × (1/4 + 1/5) Method 1 (add first): = 2/3 × (5/20 + 4/20) = 2/3 × 9/20 = 18/60 = 3/10 Method 2 (distribute): = (2/3 × 1/4) + (2/3 × 1/5) = 2/12 + 2/15 = 1/6 + 2/15 = 5/30 + 4/30 = 9/30 = 3/10 Both methods give 3/10 ✓
- GeneralClass 12All
Simplify: [(2/5) ÷ (3/7)] × (4/9)
Step 1: Solve division first (2/5) ÷ (3/7) = (2/5) × (7/3) = 14/15 Step 2: Multiply by 4/9 (14/15) × (4/9) = (14 × 4)/(15 × 9) = 56/135 Check for simplification: GCD(56, 135) = 1 Already in simplest form. Answer: 56/135
- GeneralClass 12All
Find the reciprocal of: (a) 7/9 (b) -3 (c) 1
(a) Reciprocal of 7/9: Flip the fraction: 9/7 Verification: 7/9 × 9/7 = 63/63 = 1 ✓ (b) Reciprocal of -3: Write as fraction: -3 = -3/1 Flip: -1/3 Verification: -3 × (-1/3) = 3/3 = 1 ✓ (c) Reciprocal of 1: 1 = 1/1 Flip: 1/1 = 1 Verification: 1 × 1 = 1 ✓
- GeneralClass 12All
What property is shown by: (2/3 × 4/5) × 1/2 = 2/3 × (4/5 × 1/2)?
This demonstrates the Associative Property of Multiplication. Definition: For any three rational numbers a, b, and c: (a × b) × c = a × (b × c) The grouping of multiplication doesn't matter; the result is the same. Verification: Left side: (2/3 × 4/5) × 1/2 = (8/15) × 1/2 = 8/30 = 4/15 Right side: 2/3 × (4/5 × 1/2) = 2/3 × (4/10) = 2/3 × 2/5
- GeneralClass 12All
Divide: (5/6) ÷ (10/9)
Step 1: Convert to multiplication by reciprocal (5/6) ÷ (10/9) = (5/6) × (9/10) Step 2: Multiply (5 × 9)/(6 × 10) = 45/60 Step 3: Simplify (GCD = 15) 45/60 = 3/4 Answer: 3/4
- GeneralClass 12All
Multiply: (2/3) × (9/4) × (1/6)
Method: Multiply all numerators and all denominators (2 × 9 × 1)/(3 × 4 × 6) = 18/72 Simplify: GCD(18, 72) = 18 18/72 = 1/4 Answer: 1/4 Smart approach: Cancel before multiplying: (2/3) × (9/4) × (1/6) = (2 × 9 × 1)/(3 × 4 × 6) = (1 × 3 × 1)/(3 × 2 × 3) [cancel
- GeneralClass 12All
What is the multiplicative identity for rational numbers?
The multiplicative identity is 1 (or 1/1). Property: For any rational number p/q: (p/q) × 1 = p/q Examples: 3/5 × 1 = 3/5 -7/9 × 1 = -7/9 1 × 4/11 = 4/11 One is called the multiplicative identity because multiplying any number by 1 gives back the same number.
- GeneralClass 12All
What is the multiplicative inverse (reciprocal) of a rational number?
The multiplicative inverse of p/q is q/p (provided p ≠ 0). Property: A number multiplied by its reciprocal equals 1. (p/q) × (q/p) = 1 Examples: Reciprocal of 3/4 is 4/3 Reciprocal of -5/7 is -7/5 Reciprocal of 2 (or 2/1) is 1/2 Reciprocal of -1 is -1 Note: Zero (0) has no reciprocal because 1/0 is undefined.
- GeneralClass 12All
How do you divide rational numbers?
Multiply by the reciprocal (multiplicative inverse) of the divisor. Formula: (a/b) ÷ (c/d) = (a/b) × (d/c) = (a × d)/(b × c) Example: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 Another example: (-5/6) ÷ (3/7) = (-5/6) × (7/3) = -35/18
- GeneralClass 12All
How do you multiply two rational numbers?
Multiply the numerators together and the denominators together. Formula: (a/b) × (c/d) = (a × c)/(b × d) Example: 2/3 × 5/7 = (2 × 5)/(3 × 7) = 10/21 With negatives: (-3/4) × (2/5) = (-3 × 2)/(4 × 5) = -6/20 = -3/10 Rule for signs: Positive × Positive = Positive Negative