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Maths MCQ

Cubes and Cube Roots · Maharashtra Board STD 8 (MSBSHSE - English Medium). 16 questions.

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16 questions · auto-graded multiple-choice test.

MCQ 11 Mark
Tick $(\checkmark)$ the correct answer: By what least number should $648$ be multiplied to get a perfect cube?
  • A
    $3$
  • B
    $6$
  • ✓
    $9$
  • D
    $8$
Answer
Correct option: C.
$9$
Factorising $648,$
We get,
$\begin{array}{c|c}2&648\\\hline2&324\\\hline2&162\\\hline3&81\\\hline3&27\\\hline3&9\\\hline3&3\\\hline&1\end{array}$
$648=2\times2\times2\times3\times3\times3\times3$
$=2^3\times3^3\times3$
$\therefore$ In order to get a perfect cube, we have to multiply by $3 \times 3$ to complete the triplet.
Required number $= 9$
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MCQ 21 Mark
Tick $(\checkmark)$ the correct answer:
$(0.8)^3=$ ?
  • A
    $51.2$
  • B
    $5.12$
  • ✓
    $0.512$
  • D
    None of these.
Answer
Correct option: C.
$0.512$
$(0.8)^3$
$(0.8)^3$
$=0.8 \times 0.8 \times 0.8$
$=0.512$
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MCQ 31 Mark
Mark $(\checkmark)$ against the correct answer:
Which of the following numbers is a perfect cube?
  • A
    121
  • B
    169
  • C
    196
  • ✓
    216
Answer
Correct option: D.
216
216

Solution:

$121=11 \times 11$

$169=13 \times 13$

$196=7 \times 7 \times 2 \times 2$

$216=2 \times 2 \times 2 \times 3 \times 3 \times 3$

$=(2)^3 \times(3)^3$

$=(6)^3$

$216=(6)^3$

Hence, 216 is a perfect cube.
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MCQ 41 Mark
Mark $(\checkmark)$ against the correct answer: $\sqrt[3]{216\times64}=\ ?$
  • A
    $64$
  • B
    $32$
  • ✓
    $24$
  • D
    $36$
Answer
Correct option: C.
$24$
$\sqrt[3]{216\times64}$
$=\sqrt[3]{216}\times\sqrt[3]{64}$
$=\sqrt[3]{2\times2\times2\times3\times3\times3}\times\\\sqrt[3]{2\times2\times2\times2\times2\times2}$
$=\sqrt[3]{(2)^3\times(3)^3}\times\sqrt[3]{(2)^3\times(2)^3}$
$=\sqrt[3]{(6)^3}\times\sqrt[3]{(4)^3}$
$=6\times4$
$\sqrt[3]{216\times64}=24$
$\therefore\sqrt[3]{216\times64}=24$
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MCQ 51 Mark
Mark $(\checkmark)$ against the correct answer: By what least number should $324$ be multiplied to get a perfect cube?
  • A
    $12$
  • B
    $14$
  • C
    $16$
  • ✓
    $18$
Answer
Correct option: D.
$18$
$\begin{array}{c|c}2&324\\\hline2&162\\\hline3&81\\\hline3&27\\\hline3&9\\\hline3&3\\\hline&1\end{array}$
$324=2\times2\times3\times3\times3\times3$
$=2\times2\times3\times(3)^3$
Therefore, to show that the given number is the product of three triplets,
we need to multiply $324$ by $(2 \times 3 \times 3).$
In other words, we need to multiply $324$ by $18$ to make it a perfect cube
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MCQ 61 Mark
Tick $(\checkmark)$ the correct answer: $\sqrt[3]{125\times64}=\ ?$
  • A
    $100$
  • B
    $40$
  • ✓
    $20$
  • D
    $30$
Answer
Correct option: C.
$20$
$\sqrt[3]{125\times64}$
$=\sqrt[3]{5\times5\times5\times4\times4\times4}$
$=\sqrt[3]{5^3\times4^3}$
$=5\times4$
$=20$
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MCQ 71 Mark
Tick $(\checkmark)$ the correct answer: $\sqrt[3]{\frac{64}{343}}=\ ?$
  • A
    $\frac{4}{9}$
  • ✓
    $\frac{4}{7}$
  • C
    $\frac{4}{9}$
  • D
    $\frac{4}{9}$
Answer
Correct option: B.
$\frac{4}{7}$
$\sqrt[3]{\frac{64}{343}}$
$=\sqrt[3]{\frac{4\times4\times4}{7\times7\times7}}$
$=\sqrt[3]{\frac{4^3}{7^3}}$
$=\frac{4}{7}$
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MCQ 81 Mark
Tick $(\checkmark)$ the correct answer: $\sqrt[3]{\frac{-512}{729}}=\ ?$
  • A
    $\frac{-7}{9}$
  • ✓
    $\frac{-8}{9}$
  • C
    $\frac{7}{9}$
  • D
    $\frac{8}{9}$
Answer
Correct option: B.
$\frac{-8}{9}$
$\sqrt[3]{\frac{-512}{729}}$
$\begin{array}{c|c}8&512\\\hline8&64\\\hline8&8\\\hline&1\end{array}$
$\begin{array}{c|c}9&729\\\hline9&81\\\hline9&9\\\hline&1\end{array}$
$=\sqrt[3]{\frac{(-8)\times(-8)\times(-8)}{9\times9\times9}}$
$=\sqrt[3]{\frac{(-8)^3}{(9)^3}}$
$=\frac{-8}{9}$
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MCQ 91 Mark
Tick $(\checkmark)$ the correct answer: Which of the following numbers is a perfect cube?
  • A
    $141$
  • B
    $294$
  • ✓
    $216$
  • D
    $496$
Answer
Correct option: C.
$216$
 
$1.141$
$=3 \times 47$
$\begin{array}{c|c}3&141\\\hline47&47\\\hline&1\end{array}$
$2.294$
$=2 \times 7 \times 7 \times 3$ 
$\begin{array}{c|c}2&294\\\hline7&147\\\hline7&21\\\hline3&3\\\hline&1\end{array}$
$3.216$
$=2 \times 2 \times 2 \times 3 \times 3 \times 3$
$=2^3 \times 3^3$
$\begin{array}{c|c}2&216\\\hline2&108\\\hline2&54\\\hline3&27\\\hline3&9\\\hline3&3\\\hline&1\end{array}$
$4.496$
​​​​​​​$=2 \times 2 \times 2 \times 2 \times 31$
$\begin{array}{c|c}2&496\\\hline2&248\\\hline2&124\\\hline2&62\\\hline31&31\\\hline&1\end{array}$​​​​​​​
We see that $216$ is a perfect cube.
 
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MCQ 101 Mark
Mark $(\checkmark)$ against the correct answer: $\Big(1\frac{3}{4}\Big)^3=\ ?$
  • A
    $1\frac{27}{64}$
  • B
    $2\frac{27}{64}$
  • ✓
    $5\frac{23}{64}$
  • D
    None of these.
Answer
Correct option: C.
$5\frac{23}{64}$
$\Big(1\frac{3}{4}\Big)^3=\Big(\frac{7}{4}\Big)^3$
$=\frac{(7)^3}{(4)^3}$
$=\frac{7\times7\times7}{4\times4\times4}$
$=\frac{343}{64}$
$\Big(1\frac{3}{4}\Big)^3=\frac{343}{64}=5\frac{23}{64}$
$\therefore\Big(1\frac{3}{4}\Big)^3=5\frac{23}{64}$
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MCQ 111 Mark
Mark $(\checkmark)$ against the correct answer: Which of the following is a cube of an odd number?
  • A
    $216$
  • B
    $512$
  • ✓
    $343$
  • D
    $1000$
Answer
Correct option: C.
$343$
The cube of an odd number will always be an odd number.
Therefore, $343$ is the cube of an odd number.
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MCQ 121 Mark
Tick $(\checkmark)$ the correct answer: By what least number should 1536 be divided to get a perfect cube?
  • A
    $3$
  • B
    $4$
  • ✓
    $6$
  • D
    $8$
Answer
Correct option: C.
$6$
Factorising $1536,$
We get,
$\begin{array}{c|c}2&1536\\\hline2&768\\\hline2&384\\\hline2&192\\\hline2&96\\\hline2&48\\\hline2&24\\\hline2&12\\\hline2&6\\\hline3&3\\\hline&1\end{array}$
$1536=2\times2\times2\times2\times2\times2\times2\times2\times2\times3$
$=2^3\times2^3\times2^3\times3$
We see that $3$ is left
$\therefore$ In order to get a perfect cube, we should divide it by $3.$
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MCQ 131 Mark
Mark $(\checkmark)$ against the correct answer: $\sqrt[3]{\frac{-343}{729}}=\ ?$
  • A
    $\frac{7}{9}$
  • ✓
    $\frac{-7}{9}$
  • C
    $\frac{-9}{7}$
  • D
    $\frac{9}{7}$
Answer
Correct option: B.
$\frac{-7}{9}$
By prime factorisation method
$\sqrt[3]{\frac{-343}{729}}$
$=\frac{\sqrt[3]{-343}}{\sqrt[3]{729}}$
$=\frac{\sqrt[3]{(-7)\times(-7)\times(-7)}}{\sqrt[3]{3\times3\times3\times3\times3\times3}}$
$=\frac{\sqrt[3]{(-7)^3}}{\sqrt[3]{(3)^3\times(3)^3}}$
$=\frac{\sqrt[3]{(-7)^3}}{\sqrt[3]{(9)^3}}$
$=\frac{-7}{9}$
$\therefore\sqrt[3]{\frac{-343}{729}}=\frac{-7}{9}$
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MCQ 141 Mark
Tick $(\checkmark)$ the correct answer: $\Big(1\frac{3}{10}\Big)^3=\ ?$
  • A
    $1\frac{27}{1000}$
  • B
    $2\frac{27}{1000}$
  • ✓
    $2\frac{197}{1000}$
  • D
    None of these.
Answer
Correct option: C.
$2\frac{197}{1000}$
$\Big(1\frac{3}{10}\Big)^3$
$=\Big(\frac{13}{10}\Big)^3$
$=\frac{13\times13\times13}{10\times10\times10}$
$=\frac{2197}{1000}$
$=2\frac{197}{1000}$
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MCQ 151 Mark
Mark $(\checkmark)$ against the correct answer: $\frac{\sqrt[3]{128}}{\sqrt[3]{250}}=\ ?$
  • A
    $\frac{3}{5}$
  • ✓
    $\frac{4}{5}$
  • C
    $\frac{2}{5}$
  • D
    None of these.
Answer
Correct option: B.
$\frac{4}{5}$
Resolving the numerator and the denominator into prime factors:
$\frac{\sqrt[3]{128}}{\sqrt[3]{250}}$
$=\sqrt[3]{\frac{128}{250}}$
$=\sqrt[3]{\frac{2\times8\times8}{2\times5\times5\times5}}$
$=\sqrt[3]{\frac{\not{2}\times8\times8}{\not{2}\times5\times5\times5}}$
$=\sqrt[3]{\frac{8\times8}{5\times5\times5}}$
$=\sqrt[3]{\frac{(2)^3\times(2)^3}{(5)^3}}$
$=\frac{2\times2}{5}$
$=\frac{4}{5}$
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MCQ 161 Mark
Tick $(\checkmark)$ the correct answer: Which of the following numbers is a perfect cube?
  • A
    $1152$
  • ✓
    $1331$
  • C
    $2016$
  • D
    $739$
Answer
Correct option: B.
$1331$
$1.1152$
$=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3$
$=2^3 \times 2^3 \times 3^2$
$\begin{array}{c|c}2&1152\\\hline2&576\\\hline2&288\\\hline2&144\\\hline2&72\\\hline2&36\\\hline2&18\\\hline3&9\\\hline3&3\\\hline&1\end{array}$
$2.1131$
$=11 \times 11 \times 11$
$=(11)^3$
$\begin{array}{c|c}11&1331\\\hline11&121\\\hline11&11\\\hline&1\end{array}$
$3.2016$
$=2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7$
$=2^3 \times 2 \times 2 \times 3 \times 3 \times 7$
$\begin{array}{c|c}2&2016\\\hline2&1008\\\hline2&504\\\hline2&252\\\hline2&126\\\hline3&63\\\hline3&21\\\hline7&7\\\hline&1\end{array}$
$4.739$
$=1 \times 739$
We see that $1331$ is a perfect cube.
 
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