M l aggarwal understanding icse mathematics class 9 Solutions for Chapter 1 rational and irrational numbers

Insert a rational number between and 2/9 and 3/8 arrange in descending order.

Insert a rational number between and 2/9 and 3/8 arrange in descending order.

Insert two rational numbers between 1/3 and 1/4 and arrange in ascending order.

Insert two rational numbers between 1/3 and 1/4 and arrange in ascending order.

Insert two rational numbers between – 1/3 and – 1/2 and arrange in ascending order.

Insert two rational numbers between – 1/3 and – 1/2 and arrange in ascending order.

Insert three rational numbers between 1/3 and 4/5, and arrange in descending order.

Insert three rational numbers between 1/3 and 4/5, and arrange in descending order.

Insert three rational numbers between 4 and 4.5.

Insert three rational numbers between 4 and 4.5.

Find six rational numbers between 3 and 4.

Find six rational numbers between 3 and 4.

Find five rational numbers between 3/5 and 4/5.

Find five rational numbers between 3/5 and 4/5.

Find ten rational numbers between -2/5 and 1/7.

Find ten rational numbers between -2/5 and 1/7.

Find six rational numbers between 1/2 and 2/3.

Find six rational numbers between 1/2 and 2/3.

Prove that, √5 is an irrational number.

Prove that, √5 is an irrational number.

Prove that, √7 is an irrational number.

Prove that, √7 is an irrational number.

Prove that √6 is an irrational number.

Prove that 1/√11 is an irrational number.

Prove that 1/√11 is an irrational number.

Prove that √2 is an irrational number. Hence show that 3 — √2 is irrational.

Prove that √2 is an irrational number. Hence show that 3 — √2 is an irrational.

Prove that, √3 is an irrational number. Hence, show that 2/5×√3 is an irrational

number.

Prove that, √3 is an irrational number. Hence, show that 2/5×√3 is an irrational number.

Prove that √5 is an irrational number. Hence, show that -3 + 2√5 is an irrational number.

Prove that √5 is an irrational number. Hence, show that -3 + 2√5 is an irrational number.

Prove that √5 is an irrational number. Hence, show that -3 + 2√5 is an irrational number.

Prove that the following numbers are irrational:

(i) 5 +√2

(ii) 3 - 5√3

(iii) 2√3 - 7

(iv) √2 +√5

Locate √10 and √17 on the amber line.

Write the decimal expansion of each of the following numbers and say what kind

of decimal expansion each has:

(i) 36/100

(ii) 4 1/8

(iii) 2/9

(iv) 2/11

(v) 3/13

(vi) 329/400

Without actually performing the king division, State whether the following

rational numbers will have a terminating decimal expansion or a non-terminating

repeating decimal expansion:

(i) 13/3125

(ii) 17/8

(iii) 23/75

(iv) 6/15

(v) 1258/625

(vi) 77/210

Without actually performing the long division, find if 987/10500 will have terminating or non-terminating repeating decimal expansion.

Give reasons for your answer.

Write the denominator of the rational number 257/5000 in form 2^{m} \times 5^{r} where

m, n is non-negative integers. Hence, write its decimal expansion on without actual

division.

Write the decimal expansions of the following numbers which have terminating

decimal expansions:

(i) 17/8

(ii) 13/3125

(iii) 7/80

(iv) 6/15

(v) 2^{2} \times 7 / 5^{4}

(vi) 237/1500

Write the decimal expansions of the following numbers which have terminating

decimal expansions:

(i) 17/8

(ii) 13/3125

(iii) 7/80

(iv) 6/15

(v) 2^{2} \times 7 / 5^{4}

(vi) 237/1500

Write the decimal expansion of 1/7. Hence, write the decimal expression of?

2/7,3/7, 4/7, 5/7, and 6/7.

Express the following numbers in the form p/q’. Where p and q are both integers

and q≠0;

.. Classify the following numbers as rational or irrational:

(i) √23

(ii) √225

(iii) 0.3796

(iv) 7.478478

(v) 1.101001000100001…

10. Insert… following.

(i) One irrational number between 1/3 and ½

(ii) One irrational number between -2/5 and ½

(iii) One irrational number between 0 and 0.1

Insert two irrational numbers between 2 and 3.

Write two irrational numbers between 4/9 and 7/11.

Find one rational number between √2 and √3.

Find two rational numbers between √12 and √15.

Insert irrational numbers between √5 and √7.

Insert two irrational numbers between √3 and √7.

Simplify the following:

(i) √45 - 3√20 + 4√5

(ii) 3√3 + 2√27 + 7/√3

(iii) 6√5 × 2√5

(iv) 8√15 ÷ 2√3

(v) √24/8 + √54/9

(vi) 3/√8 + 1/√2

1. Simplify the following:

(i) (5 + √7) (2 + √5)

(ii) (5 + √5) (5 - √5)

(v) (√2 + √3) (√5 + √7)

(vi) (4 + √5) (√3 - √7)

If √2 = 1.414, then find the value of

(i) √8 + √50 + √72 + √98

(ii) 3√32 - 2√50 + 4√128 - 20√18

If √3 = 1.732, then find the value of

(i) √27 + √75 + √108 - √243

(ii) 5√12 - 3√48 + 6√75 + 7√108

State which of the following are rational or irrational decimals.

(i) √(4/9), -3/70, √(7/25), √(16/5)

(ii) -√(2/49), 3/200, √(25/3), -√(49/16)

State which of the following are rational or irrational decimals.

(i) -3√2

(ii) √(256/81)

(iii) √(27×16)

(iv) √(5/36)

State which of the following are irrational numbers.

(i) 3 - √(7/25)

(ii) -2/3 + ∛2

(iii) 3/√3

(iv) -2/7 ∛5

(v) (2 - √3) (2 + √3)

Prove the following are irrational numbers.

(i) ∛2

(ii) ∛3

(iii) ∜5

Find the greatest and the smallest real numbers.

(i) 2√3, 3/√2, -√7, √15

(ii) -3√2, 9/√5, -4, 4/3 √5, 3/2√3

Write in ascending order.

(i) 3√2, 2√3, √15, 4

(ii) 3√2, 2√8, 4, √50, 4√3

. Write in descending order.

(i) 9/√2, 3/2 √5, 4√3, 3√(6/5)

(ii) 5/√3, 7/3 √2, -√3, 3√5, 2√7

Arrange in ascending order.

Rationalize the following:

(i) 3/4√5

(ii) 5√7 / √3

(iii) 3/(4 - √7)

(iv) 17/(3√2 + 1)

(v) 16/ (√41 - 5)

(vi) 1/ (√7 - √6)

(vii) 1/ (√5 + √2)

(viii) (√2 + √3) / (√2 - √3)

. Simplify:

(i) (7 + 3√5) / (7 - 3√5)

(ii) (3 - 2√2) / (3 + 2√2)

(iii) (5 - 3√14) / (7 + 2√14)

Simplify:

[7√3 / (√10 + √3)] – [2√5 / (√6 + √5)] – [3√2 / (√15 + 3√2)]

Simplify:

[1/(√4 + √5)] + [1/(√5 + √6)] + [1/(√6 + √7)] + [1/(√7 + √8)] + [1/(√8 + √9)]

Given, find the value of a and b, if

(i) [3 - √5] / [3 + 2√5] = -19/11 + a√5

(ii) [√2 + √3] / [3√2 - 2√3] = a - b√6

(iii) {[7 + √5]/[7 - √5]} – {[7 - √5]/[7 + √5]} = a + 7/11 b√5

Simplify:

{[7 + 3√5] / [3 + √5]} – {[7 - 3√5] / [3 - √5]} = p + q√5

If √2 = 1.414, √3 = 1.732, find

(i) √2/(2 + √2)

(ii) 1/(√3 + √2)

If a = 2 + √3, find 1/a, (a – 1/a)

Solve:

If x = 5 - 2√6, find 1/x, \left(x^{2}-1 / x^{2}\right)

If p = (2-√5)/(2+√5) and q = (2+√5)/(2-√5), find the values of

(i) p + q

(ii) p – q

If x = (√2 - 1)/( √2 + 1) and y = (√2 + 1)/( √2 - 1), find

(i) x + y

(ii) xy