Real Numbers and Their Properties: MCQ test, Set 7

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

How many integers are there between $-11$ and $-4$ ?
A $7$ B $5$ C $8$ D $6$

Answer and solution

Answer: (D) $6$

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

The integers between $-11$ and $-4$ are $-10, -9, -8, -7, -6, -5$ , that is $6$ numbers.
The integers between $1$ and $7$ are
A $2, 3, 4, 5, 6$ B $1, 2, 3, 4, 5, 6, 7$ C $3, 4, 5, 6$ D $2, 3, 4, 5, 6, 7$

Answer and solution

Answer: (A) $2, 3, 4, 5, 6$

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

So the integers between $1$ and $7$ are $2, 3, 4, 5, 6$ .
Which of the following belongs to $Q^*$ , the set of irrational numbers?
A $\sqrt{61}$ B $1.938$ C $2.\overline{2}$ D $\displaystyle -\frac{19}{7}$

Answer and solution

Answer: (A) $\sqrt{61}$

$\sqrt{61} = 7.8102\ldots$ is a non terminating non recurring decimal, so it is irrational.

The other numbers can be written in the form $\displaystyle \frac{p}{q}$ , so they are rational.
$5.\overline{1}$ is
A an integer B a rational number C a natural number D an irrational number

Answer and solution

Answer: (B) a rational number

$5.\overline{1}$ is a non terminating recurring decimal, so it is rational.

It is not a whole number, so it is not an integer or a natural number.
$-10$ belongs to which of these sets?
A $Q$ (rational numbers) B $N$ (natural numbers) C $W$ (whole numbers) D $Q^*$ (irrational numbers)

Answer and solution

Answer: (A) $Q$ (rational numbers)

So $-10$ belongs to $Q$ ; it does not belong to $W$ , $Q^*$ or $N$ .
$\pi$ belongs to which of these sets?
A $W$ (whole numbers) B $N$ (natural numbers) C $Q^*$ (irrational numbers) D $Z$ (integers)

Answer and solution

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

$\pi = 3.14159\ldots$ is a non terminating non recurring decimal, so it is irrational.

So $\pi$ belongs to $Q^*$ ; it does not belong to $Z$ , $W$ or $N$ .
Add $-15\sqrt{21}$ and $6\sqrt{21}$ .
A $-9\sqrt{21}$ B $-9\sqrt{42}$ C $-21\sqrt{21}$ D $9\sqrt{21}$

Answer and solution

Answer: (A) $-9\sqrt{21}$

$-15\sqrt{21} + 6\sqrt{21} = (-15 + 6)\sqrt{21}$

$= -9\sqrt{21}$
Add $3 + 11\sqrt{3}$ and $-6 - \sqrt{3}$ .
A $9 + 10\sqrt{3}$ B $-3 + 12\sqrt{3}$ C $7\sqrt{3}$ D $-3 + 10\sqrt{3}$

Answer and solution

Answer: (D) $-3 + 10\sqrt{3}$

$3 + 11\sqrt{3} - 6 - \sqrt{3} = (3 - 6) + (11 - 1)\sqrt{3}$

$= -3 + 10\sqrt{3}$

Only like terms are added: numbers with numbers, $\sqrt{3}$ terms with $\sqrt{3}$ terms.
Subtract $-9\sqrt{22}$ from $-7\sqrt{22}$ .
A $-2\sqrt{22}$ B $16\sqrt{22}$ C $2\sqrt{22}$ D $-16\sqrt{22}$

Answer and solution

Answer: (C) $2\sqrt{22}$

$-7\sqrt{22} - (-9\sqrt{22}) = -7\sqrt{22} + 9\sqrt{22}$

$= 2\sqrt{22}$
Subtract $(9 + 4\sqrt{10})$ from $(-10 - 5\sqrt{10})$ .
A $-1 - 9\sqrt{10}$ B $19 + 9\sqrt{10}$ C $-19 - 9\sqrt{10}$ D $-19 - \sqrt{10}$

Answer and solution

Answer: (C) $-19 - 9\sqrt{10}$

$(-10 - 5\sqrt{10}) - (9 + 4\sqrt{10}) = -10 - 5\sqrt{10} - 9 - 4\sqrt{10}$

$= -19 - 9\sqrt{10}$

The sign of every term of the bracket being subtracted changes.
$2\sqrt{7} \times 9\sqrt{19} =$
A $11\sqrt{133}$ B $18\sqrt{26}$ C $-18\sqrt{133}$ D $18\sqrt{133}$

Answer and solution

Answer: (D) $18\sqrt{133}$

$2\sqrt{7} \times 9\sqrt{19} = 2 \times 9 \times \sqrt{7 \times 19}$

$= 18\sqrt{133}$
$\sqrt{5} \times \sqrt{21} =$
A $5\sqrt{21}$ B $\sqrt{26}$ C $\sqrt{105}$ D $105$

Answer and solution

Answer: (C) $\sqrt{105}$

$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$

$\sqrt{5} \times \sqrt{21} = \sqrt{105}$
$-20\sqrt{22} - 11\sqrt{22} + 25\sqrt{22} =$
A $-5\sqrt{22}$ B $-56\sqrt{22}$ C $6\sqrt{22}$ D $-6\sqrt{22}$

Answer and solution

Answer: (D) $-6\sqrt{22}$

$-20\sqrt{22} - 11\sqrt{22} + 25\sqrt{22} = (-20 - 11 + 25)\sqrt{22}$

$= -6\sqrt{22}$
$-10048\sqrt{15} + 90570\sqrt{15} - 25275\sqrt{15} + 49930\sqrt{15} =$
A $175823\sqrt{15}$ B $105177\sqrt{15}$ C $105187\sqrt{15}$ D $-105177\sqrt{15}$

Answer and solution

Answer: (B) $105177\sqrt{15}$

Positive terms: $90570\sqrt{15} + 49930\sqrt{15} = 140500\sqrt{15}$

Negative terms: $10048\sqrt{15} + 25275\sqrt{15} = 35323\sqrt{15}$

$-10048\sqrt{15} + 90570\sqrt{15} - 25275\sqrt{15} + 49930\sqrt{15} = 140500\sqrt{15} - 35323\sqrt{15}$

$= (140500 - 35323)\sqrt{15} = 105177\sqrt{15}$
$3\sqrt{32} \times (-3\sqrt{8}) =$
A $-144$ B $7$ C $-2304$ D $144$

Answer and solution

Answer: (A) $-144$

$3\sqrt{32} \times (-3\sqrt{8}) = 3 \times (-3) \times \sqrt{32 \times 8}$

$= -9 \times \sqrt{256}$

$\sqrt{256} = 16$

$= -9 \times 16 = -144$
$-6\sqrt{72} \times (-4\sqrt{2}) =$
A $-120$ B $288$ C $3456$ D $-288$

Answer and solution

Answer: (B) $288$

$-6\sqrt{72} \times (-4\sqrt{2}) = -6 \times (-4) \times \sqrt{72 \times 2}$

$= 24 \times \sqrt{144}$

$\sqrt{144} = 12$

$= 24 \times 12 = 288$
$\displaystyle \frac{\sqrt{42}}{\sqrt{7}} =$
A $6$ B $\sqrt{35}$ C $\sqrt{6}$ D $\sqrt{294}$

Answer and solution

Answer: (C) $\sqrt{6}$

$\displaystyle \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

$\displaystyle \frac{\sqrt{42}}{\sqrt{7}} = \sqrt{6}$
$\displaystyle \frac{8}{\sqrt{8}} =$
A $8\sqrt{8}$ B $\sqrt{8}$ C $8$ D $1$

Answer and solution

Answer: (B) $\sqrt{8}$

$\displaystyle \frac{a}{\sqrt{a}} = \frac{\sqrt{a} \times \sqrt{a}}{\sqrt{a}} = \sqrt{a}$

$\displaystyle \frac{8}{\sqrt{8}} = \sqrt{8}$
$\sqrt{252}$ in the simplest form is
A $6\sqrt{42}$ B $6\sqrt{7}$ C $7\sqrt{6}$ D $36\sqrt{7}$

Answer and solution

Answer: (B) $6\sqrt{7}$

$\sqrt{252} = \sqrt{36 \times 7} = \sqrt{36} \times \sqrt{7} = 6\sqrt{7}$
$\sqrt{245} - \sqrt{320} =$
A $\sqrt{5}$ B $-\sqrt{10}$ C $-15\sqrt{5}$ D $-\sqrt{5}$

Answer and solution

Answer: (D) $-\sqrt{5}$

$\sqrt{245} = \sqrt{49 \times 5} = \sqrt{49} \times \sqrt{5} = 7\sqrt{5}$

$\sqrt{320} = \sqrt{64 \times 5} = \sqrt{64} \times \sqrt{5} = 8\sqrt{5}$

$7\sqrt{5} - 8\sqrt{5} = -\sqrt{5}$
If $a = 11\sqrt{13}$ and $b = -7\sqrt{13}$ , then $a + b$ is
A $4$ B $4\sqrt{13}$ C $4\sqrt{26}$ D $18\sqrt{13}$

Answer and solution

Answer: (B) $4\sqrt{13}$

$a + b = 4\sqrt{13} \in R$
If $41 - 8 = 33$ , then $8 - 41 =$
A $-33$ B $33$ C $-49$ D $49$

Answer and solution

Answer: (A) $-33$

$8 - 41 = -(41 - 8) = -33$

Subtraction is not commutative: $33 \ne -33$ .
$13 - (39 - 15) =$
A $67$ B $-41$ C $37$ D $-11$

Answer and solution

Answer: (D) $-11$

$13 - (39 - 15) = 13 - 24 = -11$

Note $(13 - 39) - 15 = -41$ : subtraction is not associative.
$\displaystyle -\frac{23}{10} \times \underline{\quad\quad} = -\frac{23}{10}$ . The number in the blank is
A $\displaystyle -\frac{10}{23}$ B $\displaystyle -\frac{23}{10}$ C $1$ D $0$

Answer and solution

Answer: (C) $1$

$a \times 1 = a$ : $1$ is the multiplicative identity.
The multiplicative inverse of $6\sqrt{11}$ is
A $\displaystyle \frac{1}{6\sqrt{11}}$ B $\displaystyle -\frac{1}{6\sqrt{11}}$ C $1$ D $-6\sqrt{11}$

Answer and solution

Answer: (A) $\displaystyle \frac{1}{6\sqrt{11}}$

Multiplicative inverse of $a$ is $\displaystyle \frac{1}{a}$ , since $\displaystyle a \times \frac{1}{a} = 1$ .

$\displaystyle 6\sqrt{11} \times \frac{1}{6\sqrt{11}} = 1$

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MCQ test, Set 7