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RATIONAL AND IRRATIONAL NUMBERS - Questions & Answers

EXERCISE 1 (A)

1. Multiple Choice Type : Choose the correct answer from the options given below.
(a) Let zero = p/q, where p and q are integers. What additional condition will make 0 = p/q a rational number :
(i) q = 0     (ii) p ≠ 0     (iii) q ≠ 0     (iv) p ≠ 0 and q ≠ 0
Answer: (iii) q ≠ 0
Step: The denominator of a rational number can never be zero.

(b) Every non-terminating decimal number is a :
(i) recurring decimal     (ii) real number     (iii) non-recurring decimal     (iv) circulating decimal
Answer: (ii) real number
Step: Non-terminating decimals can be recurring (rational) or non-recurring (irrational), both of which fall under real numbers.

(c) 7.478478..... is a :
(i) terminating rational     (ii) recurring     (iii) neither rational non-terminating     (iv) not real
Answer: (ii) recurring
Step: The block of digits '478' is repeating continuously, making it a recurring decimal.

(d) 71 / 75 is :
(i) terminating     (ii) non-terminating     (iii) periodic decimal     (iv) not a rational number
Answer: (ii) non-terminating
Step: The prime factorization of the denominator 75 is 3 × 52. Since it has a prime factor other than 2 or 5 (which is 3), it is a non-terminating decimal.

(e) Which of the following is terminating :
(i) 13 / 85     (ii) 9 / 524     (iii) 51 / 405     (iv) none of these
Answer: (iv) none of these
Step 1: For (i), 85 = 5 × 17 (contains 17, so non-terminating).
Step 2: For (ii), 524 = 4 × 131 = 22 × 131 (contains 131, so non-terminating).
Step 3: For (iii), 51 / 405 = 17 / 135. 135 = 33 × 5 (contains 3, so non-terminating).

2. Are the following statements true or false ? Give reasons for your answers.
(i) Every whole number is a natural number.
Answer: False.
Reason: Zero (0) is a whole number, but it is not a natural number (natural numbers start from 1).

(ii) Every whole number is a rational number.
Answer: True.
Reason: Any whole number 'n' can be written in the form of p/q as 'n/1', where the denominator is not zero.

(iii) Every integer is a rational number.
Answer: True.
Reason: Any integer 'm' can be expressed as 'm/1', satisfying the definition of a rational number.

(iv) Every rational number is a whole number.
Answer: False.
Reason: Rational numbers include fractions like 1/2 or 3/4, which are not whole numbers.

3. Arrange -5/9, 7/12, -2/3 and 11/18 in the ascending order of their magnitudes. Also, find the difference between the largest and the smallest of these rational numbers. Express this difference as a decimal fraction correct to one decimal place.
Step 1: The given rational numbers are -5/9, 7/12, -2/3, 11/18.
Step 2: Find the LCM of the denominators 9, 12, 3, and 18, which is 36.
Step 3: Convert each to equivalent fractions with denominator 36.
-5/9 = (-5 × 4) / (9 × 4) = -20/36
7/12 = (7 × 3) / (12 × 3) = 21/36
-2/3 = (-2 × 12) / (3 × 12) = -24/36
11/18 = (11 × 2) / (18 × 2) = 22/36
Step 4: Arrange numerators in ascending order: -24 < -20 < 21 < 22.
Step 5: Ascending order of fractions is: -2/3, -5/9, 7/12, 11/18.
Step 6: The largest fraction is 11/18 and the smallest is -2/3.
Step 7: Difference = Largest - Smallest = 11/18 - (-2/3) = 11/18 + 12/18 = 23/18.
Step 8: Decimal value of 23/18 = 1.277...
Answer: Difference correct to one decimal place = 1.3

4. Arrange 5/8, -3/16, -1/4 and 17/32 in the descending order of their magnitudes. Also, find the sum of the lowest and the largest of these rational numbers. Express the result obtained as a decimal fraction correct to two decimal places.
Step 1: The given rational numbers are 5/8, -3/16, -1/4, 17/32.
Step 2: Find the LCM of denominators 8, 16, 4, 32, which is 32.
Step 3: Convert each to equivalent fractions with denominator 32.
5/8 = 20/32
-3/16 = -6/32
-1/4 = -8/32
17/32 = 17/32
Step 4: Arrange numerators in descending order: 20 > 17 > -6 > -8.
Step 5: Descending order of fractions is: 5/8, 17/32, -3/16, -1/4.
Step 6: The largest is 5/8 and the lowest is -1/4.
Step 7: Sum = Largest + Lowest = 5/8 + (-1/4) = 5/8 - 2/8 = 3/8.
Step 8: Decimal value of 3/8 = 0.375.
Answer: Result correct to two decimal places = 0.38

5. Without doing any actual division, find which of the following rational numbers have terminating decimal representation :
(i) 7/16 (ii) 23/125 (iii) 9/14 (iv) 32/45 (v) 43/50
Step 1: Rule - If prime factors of the denominator consist of only 2, only 5, or both 2 and 5, it is a terminating decimal.
(i) 7/16: Denominator is 16 = 24. Factors are only 2. Answer: Terminating.
(ii) 23/125: Denominator is 125 = 53. Factors are only 5. Answer: Terminating.
(iii) 9/14: Denominator is 14 = 2 × 7. Contains 7. Answer: Non-terminating.
(iv) 32/45: Denominator is 45 = 32 × 5. Contains 3. Answer: Non-terminating.
(v) 43/50: Denominator is 50 = 2 × 52. Factors are only 2 and 5. Answer: Terminating.

EXERCISE 1 (B)

1. Multiple Choice Type : Choose the correct answer from the options given below.
(a) The negative of an irrational number is :
(i) a rational number     (ii) an irrational number     (iii) a rational number and an irrational number     (iv) a whole number
Answer: (ii) an irrational number

(b) √8(√8 - 1) is always :
(i) rational     (ii) irrational     (iii) whole number     (iv) natural number
Answer: (ii) irrational
Step: √8 × √8 - √8 = 8 - √8. Since we are subtracting an irrational number from a rational number, the result is irrational.

(c) For the given figure length of OA is :
(i) √5     (ii) √3
Answer: (i) √5
Step: Based on the coordinate plane in the figure, the base is 1 and the perpendicular height is 2. Using Pythagoras theorem, OA = √(12 + 22) = √(1 + 4) = √5.

(d) 2√3 × 3√8 is :
(i) rational     (ii) irrational     (iii) neither rational nor irrational     (iv) 12×√5
Answer: (ii) irrational
Step: 2√3 × 3√8 = (2 × 3) √(3 × 8) = 6√24 = 6 × 2√6 = 12√6, which is an irrational number.

(e) Two irrational numbers between 8 and 11 are :
(i) √65 and √120     (ii) √69 and 10.5     (iii) 8.2 and √125     (iv) 3 and √110
Answer: (i) √65 and √120
Step: Express 8 and 11 as square roots: 8 = √64 and 11 = √121. Any non-perfect square root between √64 and √121 is irrational. √65 and √120 both fit this condition perfectly.

2. State, whether the following numbers are rational or not :
(i) (2 + √2)2
Step: Expand using (a + b)2 = a2 + 2ab + b2
= 4 + 4√2 + 2 = 6 + 4√2
Answer: Not rational (Irrational).

(ii) (3 - √3)2
Step: Expand using (a - b)2 = a2 - 2ab + b2
= 9 - 6√3 + 3 = 12 - 6√3
Answer: Not rational (Irrational).

(iii) (5 + √5)(5 - √5)
Step: Use identity (a + b)(a - b) = a2 - b2
= 25 - 5 = 20
Answer: Yes, it is rational.

(iv) (√3 - √2)2
Step: Expand using (a - b)2
= 3 - 2√6 + 2 = 5 - 2√6
Answer: Not rational (Irrational).

3. Find the square of :
(i) 3√5 / 5
Step: (3√5 / 5)2 = (32 × 5) / 52
= (9 × 5) / 25 = 45 / 25 = 9/5 = 1 4/5
Answer: 9/5

(ii) √3 + √2
Step: (√3 + √2)2 = 3 + 2 + 2√6
Answer: 5 + 2√6

(iii) √5 - 2
Step: (√5 - 2)2 = 5 + 4 - 4√5
Answer: 9 - 4√5

(iv) 3 + 2√5
Step: (3 + 2√5)2 = 9 + (2√5)2 + 2(3)(2√5) = 9 + 20 + 12√5
Answer: 29 + 12√5

4. State, in each case, whether true or false :
(i) √2 + √3 = √5
Answer: False
(ii) 2√4 + 2 = 6
Step: 2(2) + 2 = 4 + 2 = 6.
Answer: True
(iii) 3√7 - 2√7 = √7
Step: (3 - 2)√7 = 1√7.
Answer: True
(iv) 2/7 is an irrational number.
Answer: False
(v) 5/11 is a rational number.
Answer: True
(vi) All rational numbers are real numbers.
Answer: True
(vii) All real numbers are rational numbers.
Answer: False
(viii) Some real numbers are rational numbers.
Answer: True

5. Given universal set = { -6, -5 3/4, -√4, -3/5, -3/8, 0, 4/5, 1, 1 2/3, √8, 3.01, π, 8.47 }
From the given set, find :
(i) set of rational numbers
Step: Identify all numbers that can be written as p/q. Note that -√4 = -2.
Answer: {-6, -5 3/4, -√4, -3/5, -3/8, 0, 4/5, 1, 1 2/3, 3.01, 8.47}
(ii) set of irrational numbers
Step: Identify numbers that cannot be written as simple fractions.
Answer: {√8, π}
(iii) set of integers
Step: Identify whole positive and negative numbers including zero.
Answer: {-6, -√4, 0, 1}
(iv) set of non-negative integers
Step: Filter the integers for positive values and zero.
Answer: {0, 1}

6. Prove that each of the following numbers is irrational :
(i) √3 + √2
Step 1: Let us assume that x = √3 + √2 is a rational number.
Step 2: Squaring both sides, x2 = (√3 + √2)2
Step 3: x2 = 3 + 2 + 2√6 = 5 + 2√6
Step 4: Rearranging, we get √6 = (x2 - 5) / 2
Step 5: Since x is rational, (x2 - 5)/2 must also be a rational number.
Step 6: But this means √6 is rational, which contradicts the fact that √6 is irrational.
Answer: Hence, our assumption is wrong, and √3 + √2 is irrational.

(ii) 3 - √2
Step 1: Let us assume y = 3 - √2 is a rational number.
Step 2: Squaring both sides, y2 = (3 - √2)2
Step 3: y2 = 9 + 2 - 6√2 = 11 - 6√2
Step 4: Rearranging, we get √2 = (11 - y2) / 6
Step 5: Since y is rational, (11 - y2)/6 must be rational.
Step 6: This contradicts the fact that √2 is irrational.
Answer: Hence, 3 - √2 is irrational.

7. Write a pair of irrational numbers whose sum is irrational.
Step: Take √2 and √3. Their sum is √2 + √3, which remains irrational.
Answer: √2 and √3

8. Write a pair of irrational numbers whose sum is rational.
Step: Take a conjugate pair where the irrational parts cancel out when added.
Answer: (3 + √2) and (3 - √2). Sum = 6.

9. Write a pair of irrational numbers whose difference is irrational.
Step: Take two random irrational numbers with different surds.
Answer: (5 + √3) and (2 + √2). Difference = 3 + √3 - √2.

10. Write a pair of irrational numbers whose difference is rational.
Step: Take two irrational numbers with the same surd part so it cancels during subtraction.
Answer: (4 + √5) and (2 + √5). Difference = 2.

11. Write a pair of irrational numbers whose product is irrational.
Step: Take two distinct basic surds.
Answer: √2 and √3. Product = √6.

12. Write a pair of irrational numbers whose product is rational.
Step: Multiply conjugate pairs.
Answer: (3 + √2) and (3 - √2). Product = 32 - (√2)2 = 9 - 2 = 7.

EXERCISE 1 (C)

1. Multiple Choice Type : Choose the correct answer from the options given below.
(a) If x = √5 - 2, x + 1/x is equal to :
(i) 2√5     (ii) 4     (iii) 4√5     (iv) -4
Step 1: First find 1/x = 1/(√5 - 2). Rationalize to get (√5 + 2)/(5 - 4) = √5 + 2.
Step 2: Add x and 1/x: (√5 - 2) + (√5 + 2) = 2√5.
Answer: (i) 2√5

(b) If x = 1 + √2, then (x + 1/x)2 is :
(i) 2√2     (ii) 8     (iii) 4     (iv) 4√2
Step 1: 1/x = 1/(√2 + 1). Rationalize to get √2 - 1.
Step 2: x + 1/x = (√2 + 1) + (√2 - 1) = 2√2.
Step 3: Square it: (2√2)2 = 4 × 2 = 8.
Answer: (ii) 8

(c) (2√27 + 3√12) / 4√3 is equal to :
(i) 2√3     (ii) 3√2     (iii) 3     (iv) √3 + √2
Step 1: Simplify roots: √27 = 3√3 and √12 = 2√3.
Step 2: Numerator = 2(3√3) + 3(2√3) = 6√3 + 6√3 = 12√3.
Step 3: Divide by denominator: 12√3 / 4√3 = 3.
Answer: (iii) 3

(d) (√5 - √3)2 is :
(i) 8 + 2√15     (ii) 8 + √15     (iii) 8 - √15     (iv) 8 - 2√15
Step: (√5)2 + (√3)2 - 2√5√3 = 5 + 3 - 2√15 = 8 - 2√15.
Answer: (iv) 8 - 2√15

(e) 3 / (4 + √7) is equal to :
(i) 1/3 (4 - √7)     (ii) 3(4 - √7)     (iii) 1/3 (4 + √7)     (iv) 3(4 + √7)
Step 1: Rationalize the denominator by multiplying top and bottom by (4 - √7).
Step 2: Denominator becomes 42 - (√7)2 = 16 - 7 = 9.
Step 3: 3(4 - √7) / 9 = (4 - √7) / 3 = 1/3 (4 - √7).
Answer: (i) 1/3 (4 - √7)

(f) 1 / (7 - √5) is equal to :
(i) 4(7 + √5)     (ii) 1/44 (7 + √5)     (iii) 1/44 (7 - √5)     (iv) 4(7 - √5)
Step 1: Multiply numerator and denominator by (7 + √5).
Step 2: Denominator = 72 - 5 = 49 - 5 = 44.
Step 3: Expression becomes (7 + √5) / 44.
Answer: (ii) 1/44 (7 + √5)

(g) If x = √2 - 1, then (x - 1/x)2 is :
(i) 2√2     (ii) 8     (iii) 4     (iv) 2 - √2
Step 1: 1/x = 1/(√2 - 1). Rationalize to get √2 + 1.
Step 2: x - 1/x = (√2 - 1) - (√2 + 1) = -2.
Step 3: Square it: (-2)2 = 4.
Answer: (iii) 4

(h) (5 - √7) / (5 + √7) - (5 + √7) / (5 - √7) is equal to :
(i) 10√7 / 9     (ii) 1/9 √7     (iii) 10√7 / 9     (iv) -10√7 / 9
Step 1: Find the common denominator: (5 + √7)(5 - √7) = 25 - 7 = 18.
Step 2: Numerator becomes (5 - √7)2 - (5 + √7)2
Step 3: Expand: (25 + 7 - 10√7) - (25 + 7 + 10√7) = -20√7.
Step 4: Fraction is -20√7 / 18 = -10√7 / 9.
Answer: (iv) -10√7 / 9

2. State, with reason, which of the following are surds and which are not :
(i) √180
Step: √180 = √(36 × 5) = 6√5. Since it leaves an irrational root, it is a surd.
Answer: Surd

(ii) 3√27
Step: 3√27 = 3. Since the result is a rational number, it is not a surd.
Answer: Not a surd

(iii) 5√128
Step: 128 = 27. The 5th root is irrational. Therefore, it is a surd.
Answer: Surd

(iv) 3√64
Step: 3√64 = 4, which is a rational number.
Answer: Not a surd

(v) 3√25 × 3√40
Step: Combine them: 3√(25 × 40) = 3√1000 = 10 (Rational).
Answer: Not a surd

(vi) 3√-125
Step: 3√-125 = -5 (Rational).
Answer: Not a surd

(vii) √π
Step: A surd must be the root of a rational number. π is irrational.
Answer: Not a surd

(viii) √(3 + √2)
Step: The number inside the root (3 + √2) is irrational, so by definition, it is not a surd.
Answer: Not a surd

3. Write the lowest rationalizing factor of :
(i) 5√2
Answer: √2
(ii) √24
Step: √24 = 2√6.
Answer: √6
(iii) √5 - 3
Answer: √5 + 3
(iv) 7 - √7
Answer: 7 + √7
(v) √18 - √50
Step: 3√2 - 5√2 = -2√2. We need to rationalize the surd √2.
Answer: √2
(vi) √5 - √2
Answer: √5 + √2
(vii) √13 + 3
Answer: √13 - 3

4. Rationalize the denominators of :
(i) 2√3 / √5
Step: Multiply numerator and denominator by √5.
Answer: 2√15 / 5

(ii) (√6 - √5) / (√6 + √5)
Step 1: Multiply by (√6 - √5).
Step 2: Numerator = (√6 - √5)2 = 6 + 5 - 2√30 = 11 - 2√30.
Step 3: Denominator = 6 - 5 = 1.
Answer: 11 - 2√30

5. Find the values of 'a' and 'b' in each of the following :
(i) (2 + √3) / (2 - √3) = a + b√3
Step 1: Rationalize left side by multiplying by (2 + √3).
Step 2: (2 + √3)2 / (4 - 3) = 4 + 3 + 4√3 = 7 + 4√3.
Step 3: Compare with a + b√3.
Answer: a = 7, b = 4

(ii) (√7 - 2) / (√7 + 2) = a√7 + b
Step 1: Rationalize by multiplying by (√7 - 2).
Step 2: (√7 - 2)2 / (7 - 4) = (7 + 4 - 4√7) / 3 = (11 - 4√7) / 3.
Step 3: Split the fraction: 11/3 - (4/3)√7.
Step 4: Compare with b + a√7 (note the positions of a and b).
Answer: a = -4/3, b = 11/3

(iii) 3 / (√3 - √2) = a√3 - b√2
Step 1: Rationalize by multiplying by (√3 + √2).
Step 2: 3(√3 + √2) / (3 - 2) = 3√3 + 3√2.
Step 3: Compare with a√3 - b√2.
Answer: a = 3, b = -3

6. Simplify :
(i) 22 / (2√3 + 1) + 17 / (2√3 - 1)
Step 1: Take LCM of denominators: (2√3 + 1)(2√3 - 1) = 12 - 1 = 11.
Step 2: Numerator = 22(2√3 - 1) + 17(2√3 + 1) = 44√3 - 22 + 34√3 + 17.
Step 3: Simplify numerator: 78√3 - 5.
Answer: (78√3 - 5) / 11

(ii) √2 / (√6 - √2) - √3 / (√6 + √2)
Step 1: Rationalize the first term: [√2(√6 + √2)] / 4 = (√12 + 2) / 4 = (2√3 + 2) / 4 = (√3 + 1) / 2.
Step 2: Rationalize the second term: [√3(√6 - √2)] / 4 = (3√2 - √6) / 4.
Step 3: Subtract: [2(√3 + 1) - (3√2 - √6)] / 4 = (2√3 + 2 - 3√2 + √6) / 4.
Answer: (2 + 2√3 - 3√2 + √6) / 4

7. If x = (√5 - 2) / (√5 + 2) and y = (√5 + 2) / (√5 - 2) ; find :
(i) x2
Step: Simplify x by rationalizing: (√5 - 2)2 / 1 = 9 - 4√5. Then x2 = (9 - 4√5)2 = 81 + 80 - 72√5 = 161 - 72√5.
Answer: 161 - 72√5

(ii) y2
Step: Simplify y by rationalizing: (√5 + 2)2 / 1 = 9 + 4√5. Then y2 = (9 + 4√5)2 = 81 + 80 + 72√5 = 161 + 72√5.
Answer: 161 + 72√5

(iii) xy
Step: Notice that x and y are reciprocals. So their product is 1.
Answer: 1

(iv) x2 + y2 + xy
Step: Add values from previous steps: (161 - 72√5) + (161 + 72√5) + 1 = 322 + 1.
Answer: 323

8. If m = 1 / (3 - 2√2) and n = 1 / (3 + 2√2), find :
(i) m2
Step: Rationalize m: 3 + 2√2. m2 = (3 + 2√2)2 = 9 + 8 + 12√2 = 17 + 12√2.
Answer: 17 + 12√2

(ii) n2
Step: Rationalize n: 3 - 2√2. n2 = (3 - 2√2)2 = 9 + 8 - 12√2 = 17 - 12√2.
Answer: 17 - 12√2

(iii) mn
Step: They are reciprocals, so their product is 1.
Answer: 1

9. If x = 2√3 + 2√2, find :
(i) 1/x
Step: 1 / (2√3 + 2√2). Rationalize: (2√3 - 2√2) / (12 - 8) = 2(√3 - √2) / 4 = (√3 - √2) / 2.
Answer: (√3 - √2) / 2

(ii) x + 1/x
Step: 2√3 + 2√2 + (√3 - √2)/2 = (4√3 + 4√2 + √3 - √2) / 2.
Answer: (5√3 + 3√2) / 2

(iii) (x + 1/x)2
Step: [(5√3 + 3√2) / 2]2 = (75 + 18 + 30√6) / 4.
Answer: (93 + 30√6) / 4

10. If x = 1 - √2, find the value of (x - 1/x)3.
Step 1: 1/x = 1/(1 - √2) = (1 + √2) / (1 - 2) = -1 - √2.
Step 2: x - 1/x = (1 - √2) - (-1 - √2) = 1 - √2 + 1 + √2 = 2.
Step 3: (2)3 = 8.
Answer: 8

11. If x = 5 - 2√6, find : x2 + 1/x2
Step 1: 1/x = 1/(5 - 2√6) = 5 + 2√6.
Step 2: x + 1/x = 5 - 2√6 + 5 + 2√6 = 10.
Step 3: (x + 1/x)2 = 100 ⇒ x2 + 1/x2 + 2 = 100.
Answer: 98

12. If √2 = 1.4 and √3 = 1.7, find the value of each of the following, correct to one decimal place :
(i) 1 / (√3 - √2)
Step: Rationalize to get √3 + √2. Substitute values: 1.7 + 1.4 = 3.1.
Answer: 3.1

(ii) 1 / (3 + 2√2)
Step: Rationalize to get 3 - 2√2. Substitute value: 3 - 2(1.4) = 3 - 2.8 = 0.2.
Answer: 0.2

(iii) (2 - √3) / √3
Step: Rationalize by multiplying with √3 to get (2√3 - 3) / 3. Substitute value: (2(1.7) - 3) / 3 = (3.4 - 3) / 3 = 0.4 / 3 = 0.133...
Answer: 0.1

13. Evaluate : (4 - √5) / (4 + √5) + (4 + √5) / (4 - √5)
Step 1: Use common denominator (4 + √5)(4 - √5) = 16 - 5 = 11.
Step 2: Numerator is (4 - √5)2 + (4 + √5)2 = (16 + 5 - 8√5) + (16 + 5 + 8√5) = 42.
Answer: 42 / 11 (or 3 9/11)

14. If (2 + √5) / (2 - √5) = x and (2 - √5) / (2 + √5) = y ; find the value of x2 - y2.
Step 1: Notice x2 - y2 = (x + y)(x - y).
Step 2: x + y = [(2 + √5)2 + (2 - √5)2] / (4 - 5) = (9 + 4√5 + 9 - 4√5) / -1 = 18 / -1 = -18.
Step 3: x - y = [(2 + √5)2 - (2 - √5)2] / (4 - 5) = (9 + 4√5 - 9 + 4√5) / -1 = 8√5 / -1 = -8√5.
Step 4: Product = -18 × -8√5 = 144√5.
Answer: 144√5

TEST YOURSELF

1. Multiple Choice Type : Choose the correct answer from the options given below.
(a) Since, 90 = 2 × 3 × 3 × 5, 23/90 is not a terminating decimal :
(i) true     (ii) false     (iii) none of these
Answer: (i) true
Step: Since the prime factorization of the denominator contains 3, the decimal will be non-terminating.

(b) √27 is irrational and √3 is also irrational, then which of the following is rational :
(i) √27 - √3     (ii) √27 + √3     (iii) √27 × √3     (iv) none of these
Answer: (iii) √27 × √3
Step: √27 × √3 = √81 = 9, which is a rational number.

(c) If x = √6 - √5, then x - 1/x is equal to :
(i) 1     (ii) 11     (iii) 2√6     (iv) -2√5
Answer: (iv) -2√5
Step: 1/x = √6 + √5. Then x - 1/x = (√6 - √5) - (√6 + √5) = -2√5.

(d) (2 + √3) / (2 - √3) + (2 - √3) / (2 + √3) is equal to :
(i) 14     (ii) 2√3     (iii) 1     (iv) 8√3
Answer: (i) 14
Step: Take LCM (which is 4 - 3 = 1). Numerator is (2 + √3)2 + (2 - √3)2 = (7 + 4√3) + (7 - 4√3) = 14.

(e) Statement (1) : If a = 3√3 and b = 5 / √12, then a × b is irrational.
Answer: False. a × b = (3√3) × (5 / 2√3) = 15 / 2, which is a rational number.

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Quick Review Flashcards - Click to flip and test your knowledge!
Question
Into which two main categories is the complete number system divided?
Answer
Imaginary numbers and Real numbers.
Question
How is a square root of a negative number (e.g. $\sqrt{-4}$) classified?
Answer
As an imaginary number.
Question
What is the mathematical definition of a rational number?
Answer
A number that can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.
Question
In the rational number $\frac{a}{b}$, what condition must $a$ and $b$ meet regarding their common factors for it to be in simplest form?
Answer
They must have no common factor other than $1$ (they must be co-primes).
Question
Which symbol is used to denote the set of all rational numbers?
Answer
The letter $Q$.
Question
According to the source, what is the usual sign requirement for the denominator $b$ in a rational number?
Answer
It is usually positive.
Question
Under what condition are two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ considered equal?
Answer
If and only if $a \times d = b \times c$.
Question
If $\frac{a}{b} > \frac{c}{d}$ for positive denominators, what must be true of the products $a \times d$ and $b \times c$?
Answer
$a \times d > b \times c$.
Question
What formula can be used to find a single rational number that lies exactly between any two rational numbers $a$ and $b$?
Answer
$\frac{a+b}{2}$
Question
Beyond the mean method, what alternative fraction using numerators $a, c$ and denominators $b, d$ always lies between $\frac{a}{b}$ and $\frac{c}{d}$?
Answer
$\frac{a+c}{b+d}$
Question
When inserting $n$ rational numbers between $x$ and $y$ ($x < y$), what is the formula for the common difference $d$?
Answer
$d = \frac{y - x}{n + 1}$
Question
In the method for finding a large number of rationals, what are the first three terms inserted between $x$ and $y$?
Answer
$x + d$, $x + 2d$, and $x + 3d$.
Question
To find 4 rational numbers between two fractions using the 'alternative method', by what factor should you multiply the numerators and denominators after finding a common denominator?
Answer
Multiply by $n + 1$, which is $5$.
Question
What property of rational numbers states that the sum, difference, and product of any two rational numbers is always a rational number?
Answer
Closure property.
Question
Under what specific condition is the division of one rational number by another guaranteed to be a rational number?
Answer
When the divisor is a non-zero rational number.
Question
What name is given to decimals where the division ends and no remainder is left?
Answer
Terminating decimals.
Question
How are decimals described when a digit or a set of digits repeats continually in the decimal part?
Answer
Non-terminating recurring decimals (or periodic/circulating decimals).
Question
What term refers to the specific repeating digit or set of digits in a recurring decimal?
Answer
The period of the recurring decimal.
Question
In recurring decimal notation, where is a dot or a bar placed to indicate repetition?
Answer
Above the repeating digit or over the entire set of repeating digits.
Question
What is the period of the decimal representation of $\frac{4}{7} = 0.\overline{571428}$?
Answer
571428
Question
What distinguishes a 'mixed recurring decimal' from a 'pure recurring decimal'?
Answer
In a mixed recurring decimal, at least one digit in the decimal part is not repeating.
Question
What is the shortcut formula for the numerator when converting a recurring decimal to a fraction?
Answer
(All digits in the decimal part) minus (all non-recurring digits in the decimal part).
Question
What is the shortcut rule for determining the denominator when converting a recurring decimal to a fraction?
Answer
A number of nines equal to the repeating digits, followed by a number of zeros equal to the non-repeating digits.
Question
Without performing division, how can you identify if a rational number is convertible into a terminating decimal?
Answer
The prime factors of the denominator must only be $2$ and/or $5$ (expressed as $2^m \times 5^n$).
Question
If the prime factors of the denominator of a rational number in simplest form include a factor other than $2$ or $5$, what type of decimal is produced?
Answer
A non-terminating recurring decimal.
Question
What defines an irrational number in terms of its decimal representation?
Answer
A non-terminating and non-recurring decimal.
Question
Why is $\pi$ classified as an irrational number despite often being used as $\frac{22}{7}$ in calculations?
Answer
Because its decimal representation is non-terminating and non-recurring; $\frac{22}{7}$ is only an approximate value.
Question
Under what condition is the square root of a natural number $m$ ($\sqrt{m}$) considered an irrational number?
Answer
If $m$ is not a perfect square.
Question
How can you find one irrational number between two positive rational numbers $a$ and $b$ if $ab$ is not a perfect square?
Answer
$\sqrt{ab}$
Question
What is the result of the operation: (a rational number) + (an irrational number)?
Answer
An irrational number.
Question
Is the sum of two irrational numbers always an irrational number?
Answer
No, it may or may not be irrational (e.g. $(3 + \sqrt{5}) + (6 - \sqrt{5}) = 9$).
Question
What is the result of multiplying a non-zero rational number by an irrational number?
Answer
An irrational number.
Question
How do you compare two irrational numbers with different indices, such as $\sqrt[3]{4}$ and $\sqrt{3}$?
Answer
Convert them to like surds by finding the L.C.M. of their indices to make the indices the same.
Question
How is the set of Real Numbers ($R$) defined in relation to rational and irrational numbers?
Answer
The union of the set of rational numbers ($Q$) and the set of irrational numbers ($\bar{Q}$).
Question
What is a 'surd' or 'radical'?
Answer
An irrational root of a positive rational number.
Question
In the expression $\sqrt[n]{x}$, what is the term '$n$' called?
Answer
The order of the surd.
Question
Is every irrational number a surd?
Answer
No; for example, $\pi$ is irrational but not a surd.
Question
Is every surd an irrational number?
Answer
Yes, by definition a surd must be an irrational root.
Question
What are 'rationalising factors'?
Answer
Two surds whose product results in a rational number.
Question
What is the least rationalising factor of $\sqrt{27}$?
Answer
$\sqrt{3}$ (since $\sqrt{27} = 3\sqrt{3}$ and $3\sqrt{3} \times \sqrt{3} = 9$).
Question
What is the rationalising factor for a denominator of the form $a + \sqrt{b}$?
Answer
$a - \sqrt{b}$
Question
What is the rationalising factor for a denominator of the form $\sqrt{x} + \sqrt{y}$?
Answer
$\sqrt{x} - \sqrt{y}$
Question
In the context of the number system tree, what sub-categories make up the set of Integers?
Answer
Negative Integers, Zero, and Positive Integers (Natural Numbers).
Question
What set of numbers is formed by combining Zero and Positive Integers?
Answer
Whole Numbers ($W$).
Question
If $x$ and $y$ are rational and $\sqrt{z}$ is irrational, what does $x + \sqrt{z} = y + \sqrt{z}$ imply?
Answer
$x = y$
Question
If $a + b\sqrt{x} = c + d\sqrt{x}$ where $a, b, c, d$ are rational and $\sqrt{x}$ is irrational, what are the values of $a$ and $b$?
Answer
$a = c$ and $b = d$.
Question
What is the order of the surd $\sqrt[3]{10}$?
Answer
Order 3.
Question
Determine the rationalising factor of $2\sqrt{125}$.
Answer
$\sqrt{5}$
Question
Identify the type of number: $\frac{22}{7}$.
Answer
Rational number.
Question
Identify the type of number: $3.01\dots$ (non-terminating, non-recurring).
Answer
Irrational number.
Question
Is the difference of two irrational numbers always irrational?
Answer
No, it may be rational (e.g. $(8 - \sqrt{10}) - (3 - \sqrt{10}) = 5$).
Question
What is the decimal representation of $\frac{1}{11}$?
Answer
$0.\overline{09}$
Question
Define the term 'pure arithmetic' as presented in the unit heading.
Answer
The study of the properties and relations of numbers, specifically rational and irrational numbers in this context.
Question
Why is $\sqrt[3]{64}$ not a surd?
Answer
Because its value is $4$, which is a rational number.
Question
If $x = 3$ and $y = 5$, what is the first rational number inserted between them using $d = \frac{y-x}{n+1}$ for $n=3$?
Answer
$3.5$ (or $3\frac{1}{2}$).
Question
What characterises 'non-integral rationals'?
Answer
Rational numbers that are not integers, such as fractions like $\frac{5}{8}$ or $-\frac{3}{7}$.
Question
How is the number 0 classified within the real number system tree?
Answer
As an integer, a whole number, and a rational number.