Mathematics · Textbook solutions

Real Numbers

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 145 questions

Properties of Rational Numbers and Decimal Expansion

2 q

Solved Examples

Worked · 2
  1. 2.1 SolvedEx.1
    Express the recurring decimal 0.777....0.777.... in pq\frac{p}{q} form.
  2. 2.1 SolvedEx.2
    Express the recurring decimal 7.529529529...7.529529529... in pq\frac{p}{q} form.

Practice Set 2.1

15 q
  1. Classify the decimal form of the given rational numbers into terminating and non-terminating recurring type.
    Ex 2.1 Q1(i)
    135\frac{13}{5}
  2. Ex 2.1 Q1(ii)
    211\frac{2}{11}
  3. Ex 2.1 Q1(iii)
    2916\frac{29}{16}
  4. Ex 2.1 Q1(iv)
    17125\frac{17}{125}
  5. Ex 2.1 Q1(v)
    116\frac{11}{6}
  6. Write the following rational numbers in decimal form.
    Ex 2.1 Q2(i)
    127200\frac{127}{200}
  7. Ex 2.1 Q2(ii)
    2599\frac{25}{99}
  8. Ex 2.1 Q2(iii)
    237\frac{23}{7}
  9. Ex 2.1 Q2(iv)
    45\frac{4}{5}
  10. Ex 2.1 Q2(v)
    178\frac{17}{8}
  11. Write the following rational numbers in pq\frac{p}{q} form.
    Ex 2.1 Q3(i)
    0.6‾0.\overline{6}
  12. Ex 2.1 Q3(ii)
    0.37‾0.\overline{37}
  13. Ex 2.1 Q3(iii)
    3.17‾3.\overline{17}
  14. Ex 2.1 Q3(iv)
    15.89‾15.\overline{89}
  15. Ex 2.1 Q3(v)
    2.514‾2.\overline{514}

Irrational Numbers and the Number Line

3 q

Solved Examples

Worked · 3
  1. 2.2 SolvedEx.1
    Prove that 2\sqrt{2} is an irrational number, using indirect proof.
  2. 2.2 SolvedEx.2
    Find the square root of 2 and 3 using the division method, and hence state what kind of decimal expansion 2\sqrt{2} and 3\sqrt{3} have.
  3. 2.2 SolvedEx.3
    We know that 3\sqrt{3} is an irrational number because its decimal expansion is non-terminating, non-recurring. Show whether 2+32+\sqrt{3} is irrational or not.

Practice Set 2.2

7 q
  1. Ex 2.2 Q1
    Show that 424\sqrt{2} is an irrational number.
  2. Ex 2.2 Q2
    Prove that 3+53+\sqrt{5} is an irrational number.
  3. Ex 2.2 Q3
    Represent the numbers 5\sqrt{5} and 10\sqrt{10} on a number line.
  4. Write any three rational numbers between the two numbers given below.
    Ex 2.2 Q4(i)
    0.30.3 and −0.5-0.5
  5. Ex 2.2 Q4(ii)
    −2.3-2.3 and −2.33-2.33
  6. Ex 2.2 Q4(iii)
    5.25.2 and 5.35.3
  7. Ex 2.2 Q4(iv)
    −4.5-4.5 and −4.6-4.6

Surds — Order, Comparison and Operations

12 q

Solved Examples

Worked · 12
  1. 2.3 SolvedEx.1
    Simplify : 73+2937\sqrt{3} + 29\sqrt{3}
  2. 2.3 SolvedEx.2
    Simplify : 73−2937\sqrt{3} - 29\sqrt{3}
  3. 2.3 SolvedEx.3
    Simplify : 138+128−5813\sqrt{8} + \frac{1}{2}\sqrt{8} - 5\sqrt{8}
  4. 2.3 SolvedEx.4
    Simplify : 85+20−1258\sqrt{5} + \sqrt{20} - \sqrt{125}
  5. 2.3 SolvedEx.5
    Multiply the surds 7×42\sqrt{7} \times \sqrt{42}.
  6. 2.3 SolvedEx.6
    Divide the surds : 125÷5\sqrt{125} \div \sqrt{5}
  7. 2.3 SolvedEx.7
    Simplify : 50×18\sqrt{50} \times \sqrt{18}
  8. 2.3 SolvedEx.8
    If the surd 2\sqrt{2} is multiplied by 2\sqrt{2}, what do we get? Hence state a rationalizing factor of 2\sqrt{2}.
  9. 2.3 SolvedEx.9
    Multiply 2×8\sqrt{2} \times \sqrt{8}.
  10. 2.3 SolvedEx.10
    Find the rationalizing factor of 27\sqrt{27}.
  11. 2.3 SolvedEx.11
    Rationalize the denominator of 15\frac{1}{\sqrt{5}}.
  12. 2.3 SolvedEx.12
    Rationalize the denominator of 327\frac{3}{2\sqrt{7}}.

Practice Set 2.3

46 q
  1. State the order of the surds given below.
    Ex 2.3 Q1(i)
    73\sqrt[3]{7}
  2. Ex 2.3 Q1(ii)
    5125\sqrt{12}
  3. Ex 2.3 Q1(iii)
    104\sqrt[4]{10}
  4. Ex 2.3 Q1(iv)
    39\sqrt{39}
  5. Ex 2.3 Q1(v)
    183\sqrt[3]{18}
  6. State which of the following are surds. Justify.
    Ex 2.3 Q2(i)
    513\sqrt[3]{51}
  7. Ex 2.3 Q2(ii)
    164\sqrt[4]{16}
  8. Ex 2.3 Q2(iii)
    815\sqrt[5]{81}
  9. Ex 2.3 Q2(iv)
    256\sqrt{256}
  10. Ex 2.3 Q2(v)
    643\sqrt[3]{64}
  11. Ex 2.3 Q2(vi)
    227\sqrt{\frac{22}{7}}
  12. Classify the given pair of surds into like surds and unlike surds.
    Ex 2.3 Q3(i)
    52\sqrt{52}, 5135\sqrt{13}
  13. Ex 2.3 Q3(ii)
    68\sqrt{68}, 535\sqrt{3}
  14. Ex 2.3 Q3(iii)
    4184\sqrt{18}, 727\sqrt{2}
  15. Ex 2.3 Q3(iv)
    191219\sqrt{12}, 636\sqrt{3}
  16. Ex 2.3 Q3(v)
    5225\sqrt{22}, 7337\sqrt{33}
  17. Ex 2.3 Q3(vi)
    555\sqrt{5}, 75\sqrt{75}
  18. Simplify the following surds.
    Ex 2.3 Q4(i)
    27\sqrt{27}
  19. Ex 2.3 Q4(ii)
    50\sqrt{50}
  20. Ex 2.3 Q4(iii)
    250\sqrt{250}
  21. Ex 2.3 Q4(iv)
    112\sqrt{112}
  22. Ex 2.3 Q4(v)
    168\sqrt{168}
  23. Compare the following pair of surds.
    Ex 2.3 Q5(i)
    727\sqrt{2}, 535\sqrt{3}
  24. Ex 2.3 Q5(ii)
    247\sqrt{247}, 274\sqrt{274}
  25. Ex 2.3 Q5(iii)
    272\sqrt{7}, 28\sqrt{28}
  26. Ex 2.3 Q5(iv)
    555\sqrt{5}, 727\sqrt{2}
  27. Ex 2.3 Q5(v)
    4424\sqrt{42}, 929\sqrt{2}
  28. Ex 2.3 Q5(vi)
    535\sqrt{3}, 99
  29. Ex 2.3 Q5(vii)
    77, 252\sqrt{5}
  30. Simplify.
    Ex 2.3 Q6(i)
    53+835\sqrt{3} + 8\sqrt{3}
  31. Ex 2.3 Q6(ii)
    95−45+1259\sqrt{5} - 4\sqrt{5} + \sqrt{125}
  32. Ex 2.3 Q6(iii)
    748−27−37\sqrt{48} - \sqrt{27} - \sqrt{3}
  33. Ex 2.3 Q6(iv)
    7−357+27\sqrt{7} - \frac{3}{5}\sqrt{7} + 2\sqrt{7}
  34. Multiply and write the answer in the simplest form.
    Ex 2.3 Q7(i)
    312×183\sqrt{12} \times \sqrt{18}
  35. Ex 2.3 Q7(ii)
    312×7153\sqrt{12} \times 7\sqrt{15}
  36. Ex 2.3 Q7(iii)
    38×53\sqrt{8} \times \sqrt{5}
  37. Ex 2.3 Q7(iv)
    58×285\sqrt{8} \times 2\sqrt{8}
  38. Divide, and write the answer in simplest form.
    Ex 2.3 Q8(i)
    98÷2\sqrt{98} \div \sqrt{2}
  39. Ex 2.3 Q8(ii)
    125÷50\sqrt{125} \div \sqrt{50}
  40. Ex 2.3 Q8(iii)
    54÷27\sqrt{54} \div \sqrt{27}
  41. Ex 2.3 Q8(iv)
    310÷5\sqrt{310} \div \sqrt{5}
  42. Rationalize the denominator.
    Ex 2.3 Q9(i)
    35\frac{3}{\sqrt{5}}
  43. Ex 2.3 Q9(ii)
    114\frac{1}{\sqrt{14}}
  44. Ex 2.3 Q9(iii)
    57\frac{5}{\sqrt{7}}
  45. Ex 2.3 Q9(iv)
    693\frac{6}{9\sqrt{3}}
  46. Ex 2.3 Q9(v)
    113\frac{11}{\sqrt{3}}

Binomial Surds and Rationalization of the Denominator

4 q

Solved Examples

Worked · 4
  1. 2.4 SolvedEx.1
    Multiply. 2(8+18)\sqrt{2}\left(\sqrt{8}+\sqrt{18}\right)
  2. 2.4 SolvedEx.2
    Multiply. (3−2)(23−32)\left(\sqrt{3}-\sqrt{2}\right)\left(2\sqrt{3}-3\sqrt{2}\right)
  3. 2.4 SolvedEx.3
    Rationalize the denominator 15−3\dfrac{1}{\sqrt{5}-\sqrt{3}}.
  4. 2.4 SolvedEx.4
    Rationalize the denominator 832+5\dfrac{8}{3\sqrt{2}+\sqrt{5}}.

Practice Set 2.4

7 q
  1. Multiply.
    Ex 2.4 Q1(i)
    3(7−3)\sqrt{3}\left(\sqrt{7}-\sqrt{3}\right)
  2. Ex 2.4 Q1(ii)
    (5−7)2\left(\sqrt{5}-\sqrt{7}\right)\sqrt{2}
  3. Ex 2.4 Q1(iii)
    (32−3)(43−2)\left(3\sqrt{2}-\sqrt{3}\right)\left(4\sqrt{3}-\sqrt{2}\right)
  4. Rationalize the denominator.
    Ex 2.4 Q2(i)
    17+2\dfrac{1}{\sqrt{7}+\sqrt{2}}
  5. Ex 2.4 Q2(ii)
    325−32\dfrac{3}{2\sqrt{5}-3\sqrt{2}}
  6. Ex 2.4 Q2(iii)
    47+43\dfrac{4}{7+4\sqrt{3}}
  7. Ex 2.4 Q2(iv)
    5−35+3\dfrac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}

Absolute Value

3 q

Solved Examples

Worked · 3
  1. 2.5 SolvedEx.1
    Find the values ∣3∣|3|, ∣−3∣|-3| and ∣0∣|0|.
  2. 2.5 SolvedEx.2
    Find the value. (i) ∣9−5∣|9-5| (ii) ∣8−13∣|8-13| (iii) ∣8∣−∣−3∣|8|-|-3| (iv) ∣8∣×∣4∣|8| \times |4|
  3. 2.5 SolvedEx.3
    Solve ∣x−5∣=2|x-5|=2.

Practice Set 2.5

7 q
  1. Find the value.
    Ex 2.5 Q1(i)
    ∣15−2∣|15-2|
  2. Ex 2.5 Q1(ii)
    ∣4−9∣|4-9|
  3. Ex 2.5 Q1(iii)
    ∣7∣×∣−4∣|7| \times |-4|
  4. Solve.
    Ex 2.5 Q2(i)
    ∣3x−5∣=1|3x-5|=1
  5. Ex 2.5 Q2(ii)
    ∣7−2x∣=5|7-2x|=5
  6. Ex 2.5 Q2(iii)
    ∣8−x2∣=5\left|\dfrac{8-x}{2}\right|=5
  7. Ex 2.5 Q2(iv)
    ∣5+x4∣=5\left|5+\dfrac{x}{4}\right|=5

Problem Set 2

39 q
  1. Choose the correct alternative answer for the questions given below.
    Prob Q1(i)
    Which one of the following is an irrational number ?
    1. A.
      1625\sqrt{\frac{16}{25}}
    2. B.
      5\sqrt{5}
    3. C.
      39\frac{3}{9}
    4. D.
      196\sqrt{196}
  2. Prob Q1(ii)
    Which of the following is an irrational number?
    1. A.
      0.170.17
    2. B.
      1.513‾1.\overline{513}
    3. C.
      0.2746‾0.27\overline{46}
    4. D.
      0.101001000…0.101001000\ldots
  3. Prob Q1(iii)
    Decimal expansion of which of the following is non-terminating recurring ?
    1. A.
      25\frac{2}{5}
    2. B.
      316\frac{3}{16}
    3. C.
      311\frac{3}{11}
    4. D.
      13725\frac{137}{25}
  4. Prob Q1(iv)
    Every point on the number line represent, which of the following numbers?
    1. A.
      Natural numbers
    2. B.
      Irrational numbers
    3. C.
      Rational numbers
    4. D.
      Real numbers
  5. Prob Q1(v)
    The number 0.4˙0.\dot{4} in pq\frac{p}{q} form is .....
    1. A.
      49\frac{4}{9}
    2. B.
      409\frac{40}{9}
    3. C.
      3.69\frac{3.6}{9}
    4. D.
      369\frac{36}{9}
  6. Prob Q1(vi)
    What is n\sqrt{n}, if nn is not a perfect square number ?
    1. A.
      Natural number
    2. B.
      Rational number
    3. C.
      Irrational number
    4. D.
      Options A, B, C all are correct.
  7. Prob Q1(vii)
    Which of the following is not a surd ?
    1. A.
      7\sqrt{7}
    2. B.
      173\sqrt[3]{17}
    3. C.
      643\sqrt[3]{64}
    4. D.
      193\sqrt{193}
  8. Prob Q1(viii)
    What is the order of the surd 53\sqrt[3]{\sqrt{5}} ?
    1. A.
      33
    2. B.
      22
    3. C.
      66
    4. D.
      55
  9. Prob Q1(ix)
    Which one is the conjugate pair of 25+32\sqrt{5}+\sqrt{3} ?
    1. A.
      −25+3-2\sqrt{5}+\sqrt{3}
    2. B.
      −25−3-2\sqrt{5}-\sqrt{3}
    3. C.
      23−52\sqrt{3}-\sqrt{5}
    4. D.
      3+25\sqrt{3}+2\sqrt{5}
  10. Prob Q1(x)
    The value of ∣12−(13+7)×4∣|12-(13+7)\times 4| is ...............
    1. A.
      −68-68
    2. B.
      6868
    3. C.
      −32-32
    4. D.
      3232
  11. Write the following numbers in pq\frac{p}{q} form.
    Prob Q2(i)
    0.5550.555
  12. Prob Q2(ii)
    29.568‾29.\overline{568}
  13. Prob Q2(iii)
    9.315 315…9.315\,315\ldots
  14. Prob Q2(iv)
    357.417417…357.417417\ldots
  15. Prob Q2(v)
    30.219‾30.\overline{219}
  16. Write the following numbers in its decimal form.
    Prob Q3(i)
    −57\frac{-5}{7}
  17. Prob Q3(ii)
    911\frac{9}{11}
  18. Prob Q3(iii)
    5\sqrt{5}
  19. Prob Q3(iv)
    12113\frac{121}{13}
  20. Prob Q3(v)
    298\frac{29}{8}
  21. Prob Q4
    Show that 5+75+\sqrt{7} is an irrational number.
  22. Write the following surds in simplest form.
    Prob Q5(i)
    348\frac{3}{4}\sqrt{8}
  23. Prob Q5(ii)
    −5945-\frac{5}{9}\sqrt{45}
  24. Write the simplest form of rationalising factor for the given surds.
    Prob Q6(i)
    32\sqrt{32}
  25. Prob Q6(ii)
    50\sqrt{50}
  26. Prob Q6(iii)
    27\sqrt{27}
  27. Prob Q6(iv)
    3510\frac{3}{5}\sqrt{10}
  28. Prob Q6(v)
    3723\sqrt{72}
  29. Prob Q6(vi)
    4114\sqrt{11}
  30. Simplify.
    Prob Q7(i)
    47147+38192−1575\frac{4}{7}\sqrt{147}+\frac{3}{8}\sqrt{192}-\frac{1}{5}\sqrt{75}
  31. Prob Q7(ii)
    53+227+135\sqrt{3}+2\sqrt{27}+\frac{1}{\sqrt{3}}
  32. Prob Q7(iii)
    216−56+294−36\sqrt{216}-5\sqrt{6}+\sqrt{294}-\frac{3}{\sqrt{6}}
  33. Prob Q7(iv)
    412−75−7484\sqrt{12}-\sqrt{75}-7\sqrt{48}
  34. Prob Q7(v*)
    248−75−132\sqrt{48}-\sqrt{75}-\frac{1}{\sqrt{3}}
  35. Rationalize the denominator.
    Prob Q8(i)
    15\frac{1}{\sqrt{5}}
  36. Prob Q8(ii)
    237\frac{2}{3\sqrt{7}}
  37. Prob Q8(iii)
    13−2\frac{1}{\sqrt{3}-\sqrt{2}}
  38. Prob Q8(iv)
    135+22\frac{1}{3\sqrt{5}+2\sqrt{2}}
  39. Prob Q8(v)
    1243−2\frac{12}{4\sqrt{3}-\sqrt{2}}