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Real Numbers and Their Properties

MCQ test, Set 2

  • Set 2 of 10
  • 25 questions
  • 25 marks
  • 30 minutes

Download the PDF (with key sheet and solutions, 89.3 KB)

Choose one answer for each question, then check your score at the end. Each correct answer carries 1 mark; there is no negative marking.

  1. The pq\displaystyle \frac{p}{q} form of −29-29 is
    Answer and solution

    Answer: (D) −291\displaystyle \frac{-29}{1}

    Every integer can be written with q=1q = 1 (q≠0q \ne 0).

    −29=−291\displaystyle -29 = \frac{-29}{1}

  2. How many integers are there between −2-2 and 44?
    Answer and solution

    Answer: (B) 55

    Z={…,−2,−1,0,1,2,3,…}Z = \{\ldots, -2, -1, 0, 1, 2, 3, \ldots\}

    The integers between −2-2 and 44 are −1,0,1,2,3-1, 0, 1, 2, 3, that is 55 numbers.

  3. Which of the following is a rational number?
    Answer and solution

    Answer: (A) 25\sqrt{25}

    25=5=51\displaystyle \sqrt{25} = 5 = \frac{5}{1}, so it is rational.

    The other numbers are non terminating non recurring decimals, so they are irrational.

  4. 33=5.7445…\sqrt{33} = 5.7445\ldots is
    Answer and solution

    Answer: (D) a non terminating non recurring decimal, so irrational

    33=5.7445…\sqrt{33} = 5.7445\ldots is a non terminating non recurring decimal, so it is irrational.

  5. 35\sqrt{35} belongs to which of these sets?
    Answer and solution

    Answer: (C) Q∗Q^* (irrational numbers)

    35=5.9160…\sqrt{35} = 5.9160\ldots is a non terminating non recurring decimal, so it is irrational.

    So 35\sqrt{35} belongs to Q∗Q^*; it does not belong to NN, QQ or ZZ.

  6. −25-\sqrt{25} belongs to which of these sets?
    Answer and solution

    Answer: (C) QQ (rational numbers)

    −25=−5-\sqrt{25} = -5

    So −25-\sqrt{25} belongs to QQ; it does not belong to WW, NN or Q∗Q^*.

  7. Add 4104\sqrt{10} and −1410-14\sqrt{10}.
    Answer and solution

    Answer: (A) −1010-10\sqrt{10}

    410−1410=(4−14)104\sqrt{10} - 14\sqrt{10} = (4 - 14)\sqrt{10}

    =−1010= -10\sqrt{10}

  8. Add 1113−61511\sqrt{13} - 6\sqrt{15} and 13+515\sqrt{13} + 5\sqrt{15}.
    Answer and solution

    Answer: (A) 1213−1512\sqrt{13} - \sqrt{15}

    1113−615+13+515=(11+1)13+(−6+5)1511\sqrt{13} - 6\sqrt{15} + \sqrt{13} + 5\sqrt{15} = (11 + 1)\sqrt{13} + (-6 + 5)\sqrt{15}

    =1213−15= 12\sqrt{13} - \sqrt{15}

  9. Subtract (−6−1022)(-6 - 10\sqrt{22}) from (−6+1022)(-6 + 10\sqrt{22}).
    Answer and solution

    Answer: (B) 202220\sqrt{22}

    (−6+1022)−(−6−1022)=−6+1022+6+1022(-6 + 10\sqrt{22}) - (-6 - 10\sqrt{22}) = -6 + 10\sqrt{22} + 6 + 10\sqrt{22}

    =2022= 20\sqrt{22}

  10. Subtract −42-4\sqrt{2} from 929\sqrt{2}.
    Answer and solution

    Answer: (D) 13213\sqrt{2}

    92−(−42)=92+429\sqrt{2} - (-4\sqrt{2}) = 9\sqrt{2} + 4\sqrt{2}

    =132= 13\sqrt{2}

  11. 123×(−73)=12\sqrt{3} \times (-7\sqrt{3}) =
    Answer and solution

    Answer: (C) −252-252

    123×(−73)=12×(−7)×3×312\sqrt{3} \times (-7\sqrt{3}) = 12 \times (-7) \times \sqrt{3} \times \sqrt{3}

    =−84×3=−252= -84 \times 3 = -252 (since a×a=a\sqrt{a} \times \sqrt{a} = a)

  12. 823×33=8\sqrt{23} \times 3\sqrt{3} =
    Answer and solution

    Answer: (D) 246924\sqrt{69}

    823×33=8×3×23×38\sqrt{23} \times 3\sqrt{3} = 8 \times 3 \times \sqrt{23 \times 3}

    =2469= 24\sqrt{69}

  13. 36+116−256=3\sqrt{6} + 11\sqrt{6} - 25\sqrt{6} =
    Answer and solution

    Answer: (B) −116-11\sqrt{6}

    36+116−256=(3+11−25)63\sqrt{6} + 11\sqrt{6} - 25\sqrt{6} = (3 + 11 - 25)\sqrt{6}

    =−116= -11\sqrt{6}

  14. −45492+69732+51302−170502=-4549\sqrt{2} + 6973\sqrt{2} + 5130\sqrt{2} - 17050\sqrt{2} =
    Answer and solution

    Answer: (C) −94962-9496\sqrt{2}

    Positive terms: 69732+51302=1210326973\sqrt{2} + 5130\sqrt{2} = 12103\sqrt{2}

    Negative terms: 45492+170502=2159924549\sqrt{2} + 17050\sqrt{2} = 21599\sqrt{2}

    −45492+69732+51302−170502=121032−215992-4549\sqrt{2} + 6973\sqrt{2} + 5130\sqrt{2} - 17050\sqrt{2} = 12103\sqrt{2} - 21599\sqrt{2}

    =−(21599−12103)2=−94962= -(21599 - 12103)\sqrt{2} = -9496\sqrt{2}

  15. −628×2112=-6\sqrt{28} \times 2\sqrt{112} =
    Answer and solution

    Answer: (B) −672-672

    −628×2112=−6×2×28×112-6\sqrt{28} \times 2\sqrt{112} = -6 \times 2 \times \sqrt{28 \times 112}

    =−12×3136= -12 \times \sqrt{3136}

    Long division: 5×5=255 \times 5 = 25, remainder 66; bring down 3636 to get 636636; 106×6=636106 \times 6 = 636. So 3136=56\sqrt{3136} = 56

    =−12×56=−672= -12 \times 56 = -672

  16. 728×(−37)=7\sqrt{28} \times (-3\sqrt{7}) =
    Answer and solution

    Answer: (C) −294-294

    728×(−37)=7×(−3)×28×77\sqrt{28} \times (-3\sqrt{7}) = 7 \times (-3) \times \sqrt{28 \times 7}

    =−21×196= -21 \times \sqrt{196}

    196=14\sqrt{196} = 14

    =−21×14=−294= -21 \times 14 = -294

  17. 20052=\displaystyle \frac{\sqrt{200}}{5\sqrt{2}} =
    Answer and solution

    Answer: (D) 22

    200=100×2=102\sqrt{200} = \sqrt{100} \times \sqrt{2} = 10\sqrt{2}

    10252=2\displaystyle \frac{10\sqrt{2}}{5\sqrt{2}} = 2

  18. 4×255=\displaystyle \frac{4 \times \sqrt{25}}{\sqrt{5}} =
    Answer and solution

    Answer: (D) 454\sqrt{5}

    255=255=5\displaystyle \frac{\sqrt{25}}{\sqrt{5}} = \sqrt{\frac{25}{5}} = \sqrt{5}

    ∴4×5=45\therefore 4 \times \sqrt{5} = 4\sqrt{5}

  19. 243\sqrt{243} in the simplest form is
    Answer and solution

    Answer: (C) 939\sqrt{3}

    243=81×3=81×3=93\sqrt{243} = \sqrt{81 \times 3} = \sqrt{81} \times \sqrt{3} = 9\sqrt{3}

  20. 486=\sqrt{486} =
    Answer and solution

    Answer: (A) 969\sqrt{6}

    486=81×6=81×6=96\sqrt{486} = \sqrt{81 \times 6} = \sqrt{81} \times \sqrt{6} = 9\sqrt{6}

  21. If a=67a = 6\sqrt{7} and b=77b = 7\sqrt{7}, then a+ba + b is
    Answer and solution

    Answer: (A) 13713\sqrt{7}

    a+b=137∈Ra + b = 13\sqrt{7} \in R

  22. If 62−12=5062 - 12 = 50, then 12−62=12 - 62 =
    Answer and solution

    Answer: (B) −50-50

    12−62=−(62−12)=−5012 - 62 = -(62 - 12) = -50

    Subtraction is not commutative: 50≠−5050 \ne -50.

  23. For a=−36a = -36, b=12b = 12 and c=6c = 6, which of the following is true?
    Answer and solution

    Answer: (A) a+(b+c)=(a+b)+ca + (b + c) = (a + b) + c

    −36+(12+6)=−18=(−36+12)+6-36 + (12 + 6) = -18 = (-36 + 12) + 6

    Real numbers are associative under addition and multiplication only.

  24. In 8+0=0+8=88 + 0 = 0 + 8 = 8, 00 is called the
    Answer and solution

    Answer: (B) additive identity

    Adding 00 or multiplying by 11 leaves the number unchanged: these are the identities.

  25. The additive inverse of 77 is
    Answer and solution

    Answer: (C) −7-7

    Additive inverse of aa is −a-a, since a+(−a)=0a + (-a) = 0.

    7+(−7)=07 + (-7) = 0