Practice set 2.1 Page no: 21...An irrational number is a number that cannot be expressed as a...

$: Practice set 2.1 Page no: 21...An irrational number is a number that cannot be expressed as a fraction p/q for any integers p and q and q ≠ 0. Since √5 cannot be written as p/q$
Maharashtra Board Solutions For Class 9 Maths Part 1

Chapter 2 – Real Numbers

Practice set 2.1 Page no: 21 1. Classify the decimal form of the given rational numbers into terminating and non-terminating recurring type. i. 13/5 ii. 2/11 iii. 29/16 iv. 17/125 v. 11/6 Solution:

i. ∵ The division is exact ∴ it is a terminating decimal.

ii. ∵ The division never ends and the digits ‘18’ is repeated endlessly ∴ it is a non-terminating recurring type decimal.

iii. ∵ The division is exact ∴ it is a terminating decimal.

iv. ∵ The division is exact ∴ it is a terminating decimal.

v. ∵ The division never ends and the digit ‘3’ is repeated endlessly ∴ it is a non-terminating recurring type decimal. 2. Write the following rational numbers in decimal form.

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i. 127/200 ii. 25/99 iii. 23/7 iv. 4/5 v. 17/8 Solution:

i.

ii.

iii.

iv.

v. 3. Write the following rational numbers in form.

i.

ii.

iii.

iv.

v. Solution:





Practice set 2.2 Page no: 25 1. Show that is 4√2 an irrational number. Solution:

2. Prove that 3 + √5 is an irrational number. Solution:





3. Represent the numbers √5 and √10 on a number line. Solution: Given √5

Given that √10

4. Write any three rational numbers between the two numbers given below. (i) 0.3 and -0.5





Solution:





(ii) -2.3 and -2.33 Solution:





(iii) 5.2 and 5.3 Solution:





(iv) -4.5 and 4.6 Solution:





Practice set 2.3 Page no: 30 1. State the order of the surds given below.

Solution:





2. State which of the following are surds. Justify.

Solution:

iv. √256 = √162 = 16 Surds are numbers left in root form (√) to express its exact value. It has an infinite number of non-recurring decimals. Therefore, surds are irrational numbers. It is not a surd ∵ it is a rational number.





3. Classify the given pair of surds into like surds and unlike surds. i. √52, 5√13 ii. √68, 5√3 iii. 4√18, 7√2 iv. 19√12, 6√3 v. 5√22, 7√33 vi. 5√5, √75 Solution: Two or more surds are said to be similar or like surds if they have the same surd-factor. Two or more surds are said to be dissimilar or unlike when they are not similar. Therefore, i. Given √52, 5√13 given surd can be written as √52 = √ (2×2×13) = 2√13 5√13 ∵ both surds have same surd-factor that is √13. ∴ they are like surds. ii. Given √68, 5√3 Given surd can be written as √68 = √ (2×2×17) = 2√17





5√3 ∵ both surds have different surd-factors √17 and √3. ∴ they are unlike surds. iii. Given 4√18, 7√2 Given surd can be written as 4√18 = 4 √ (2×3×3) = 4×3√2 = 12√2 7√2 ∵ both surds have same surd-factor i.e., √2. ∴ they are like surds. iv. Given 19√12, 6√3 Given surd can be written as 19√12 = 19√ (2×2×3) = 19×2√3 = 38√3 6√3 ∵ both surds have same surd-factor i.e., √3. ∴ they are like surds. v. Given 5√22, 7√33 ∵ both surds have different surd-factors √22 and √33. ∴ they are unlike surds. vi. Given 5√5, √75 5√5 Given surd can be written as √75 = √ (5×5×3) = 5√3 ∵ both surds have different surd-factors √5 and √3. ∴ they are unlike surds. 4. Simplify the following surds. i. √27 ii. √50 iii. √250 iv. √112 v. √168 Solution:





Practice set 2.4 Page no: 32 1. Multiply i. √3(√7 – √3) ii. (√5 – √7) √2 iii. (3√2 – √3) (4√3 – √2) Solution: i. Given √3 (√7 – √3) =√3 × √7 – √3 × √3 [∵ √a (√b – √c) = √a × √b – √a × √c] =√21 – 3 ii. Given (√5 – √7) √2 =√5 × √2 – √7 × √2 [∵√a (√b – √c) = √a × √b – √a × √c] = √10 – √14 iii. Given (3√2 – √3) (4√3 – √2) =3√2 (4√3 – √2) – √3 (4√3 – √2) [∵ √a (√b – √c) = √a × √b – √a × √c] = 3√2 × 4√3 – 3√2 × √2 – √3 × 4√3 + √3 × √2 = 12√6 – 3 × 2 – 4 × 3 + √6 = 12√6 – 6 – 12 + √6 = 13√6 – 18 2. Rationalize the denominator.





iv. Solution:





Practice set 2.5 Page no: 33 1. Find the value. (i) |15 - 2| (ii) |4 - 9| (iii) |7| × |-4| Solution: i. Given |15 - 2| Absolute value describes the distance of a number on the number line from 0 without considering which direction from zero the number lies. The absolute value of a number is never negative. Therefore, |15 - 2| = |13| = 13 ii. Given |4 - 9| Absolute value describes the distance of a number on the number line from 0 without considering which direction from zero the number lies. The absolute value of a number is never negative. Therefore, |4 - 9| = |-5| = 5 iii. Given |7| × |-4| Absolute value describes the distance of a number on the number line from 0 without considering which direction from zero the number lies. The absolute value of a number is never negative. Therefore, |7| × |-4| = 7 × 4 = 28 2. Solve. i. |3x - 5| = 1 ii. |7 – 2x| = 5

iii.





iv. Solution:





Problem set 2 Page no: 34 1. Choose the correct alternative answer for the questions given below. i. Which one of the following is an irrational number? A. √16/25 B. √5 C. 3/9 D. √196 Solution: B. √5 Explanation: An irrational number is a number that cannot be expressed as a fraction p/q for any integers p and q and q ≠ 0. Since √5 cannot be written as p/q it is an irrational number Therefore √5 is an irrational number. ii. Which of the following is an irrational number? A. 0.17

B.

C. D. 0.101001000.... Solution: D. 0.101001000.... Explanation: An irrational number is a number that cannot be expressed as a fraction p/q for any integers p and q and q ≠ 0. 0.101001000.... is an irrational number because it is a non-terminating and non-`repeating decimal. Therefore, 0.101001000.... is an irrational number. iii. Decimal expansion of which of the following is non-terminating recurring?





A. 2/5 B. 3/16 C. 3/11 D. 137/25 Solution: C. 3/11 Explanation: A non-terminating recurring decimal representation means that the number will have an infinite number of digits to the right of the decimal point and those digits will repeat themselves.

∵ it has an infinite number of digits to the right of the decimal point which are repeating themselves ∴ it is a non-terminating recurring decimal. iv. Every point on the number line represent, which of the following numbers? A. Natural numbers B. Irrational numbers C. Rational numbers D. Real numbers. Solution: D. Real numbers. Explanation: Every point of a number line is assumed to correspond to a real number, and every real number to a point. Therefore, every point on the number line represent a real number. v. The number 0.4 in p/q form is …………. A. 4/9 B. 40/9 C. 3.6/9 D. 36/9 Solution:





C. 3.6/9 Explanation:

vi. What is √n, if n is not a perfect square number? A. Natural number B. Rational number C. Irrational number D. Options A, B, C all are correct. Solution: C. Irrational number Explanation: If n is not a perfect square number, then √n cannot be expressed as ratio of a and b where a and b are integers and b ≠ 0 Therefore, √n is an Irrational number vii. Which of the following is not a surd? A. √7 B. 3√17 C. 3√64 D. √193 Solution: C. 3√64 Explanation:





viii. What is the order of the surd ? A. 3 B. 2 C. 6 D. 5 Solution: C. 6 Explanation:

ix. Which one is the conjugate pair of 2√5 + √3? A. -2√5 + √3 B. -2√5 - √3 C. 2√3 + √5 D. √3 + 2√5 Solution: A. -2√5 + √3 Explanation: A math conjugate is formed by changing the sign between two terms in a binomial. For





instance, the conjugate of x + y is x - y. Now, 2√5 + √3 = √3 + 2√5 Its conjugate pair = √3 - 2√5 = -2√5 + √3 ∴ The conjugate pair of 2√5 + √3 = -2√5 + √3 x. The value of |12 – (13 + 7) × 4| is ........... A. -68 B. 68 C. -32 D. 32 Solution: B. 68 Explanation: |12 – (13 + 7) × 4| = |12 – 20 × 4| (Solving it according to BODMAS) ⇒ |12 – (13 + 7) × 4| = |12 – 80| ⇒ |12 – (13 + 7) × 4| = |-68| ⇒ |12 – (13 + 7) × 4| = 68



Practice set 2.1 Page no: 21...An irrational number is a number that cannot be expressed as a...

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