MCQ test, Set 5

Answer: (C) $\displaystyle \frac{-12}{1}$

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

$\displaystyle -12 = \frac{-12}{1}$

Answer: (D) $4$

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

The integers between $0$ and $5$ are $1, 2, 3, 4$, that is $4$ numbers.

Answer: (B) $\sqrt{94}$

$\sqrt{94} = 9.6953\ldots$ 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.

Answer: (C) a rational number

$2.\overline{4}$ 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.

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

$\displaystyle -\frac{29}{2}$ is in the form $\displaystyle \frac{p}{q}$ with $q \ne 0$, so it is rational.

So $\displaystyle -\frac{29}{2}$ belongs to $Q$; it does not belong to $W$, $N$ or $Q^*$.

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

$4.\overline{1}$ is a non terminating recurring decimal, so it is rational.

So $4.\overline{1}$ belongs to $Q$; it does not belong to $W$, $N$ or $Z$.

Answer: (B) $27\sqrt{22}$

$11\sqrt{22} + 16\sqrt{22} = (11 + 16)\sqrt{22}$

$= 27\sqrt{22}$

Answer: (A) $-7 + 2\sqrt{22}$

$8 + 5\sqrt{22} - 15 - 3\sqrt{22} = (8 - 15) + (5 - 3)\sqrt{22}$

$= -7 + 2\sqrt{22}$

Only like terms are added: numbers with numbers, $\sqrt{22}$ terms with $\sqrt{22}$ terms.

Answer: (C) $30\sqrt{11}$

$7\sqrt{11} - (-23\sqrt{11}) = 7\sqrt{11} + 23\sqrt{11}$

$= 30\sqrt{11}$

Answer: (A) $13 + 6\sqrt{10}$

$(4 + 11\sqrt{10}) - (-9 + 5\sqrt{10}) = 4 + 11\sqrt{10} + 9 - 5\sqrt{10}$

$= 13 + 6\sqrt{10}$

The sign of every term of the bracket being subtracted changes.

Answer: (D) $108\sqrt{65}$

$9\sqrt{5} \times 12\sqrt{13} = 9 \times 12 \times \sqrt{5 \times 13}$

$= 108\sqrt{65}$

Answer: (A) $\sqrt{102}$

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

$\sqrt{6} \times \sqrt{17} = \sqrt{102}$

Answer: (A) $-43\sqrt{6}$

$14\sqrt{6} - 28\sqrt{6} - 29\sqrt{6} = (14 - 28 - 29)\sqrt{6}$

$= -43\sqrt{6}$

Answer: (D) $-23937\sqrt{14}$

Positive terms: $5856\sqrt{14} + 3207\sqrt{14} = 9063\sqrt{14}$

Negative terms: $33000\sqrt{14}$

$5856\sqrt{14} + 3207\sqrt{14} - 33000\sqrt{14} = 9063\sqrt{14} - 33000\sqrt{14}$

$= -(33000 - 9063)\sqrt{14} = -23937\sqrt{14}$

Answer: (C) $360$

$-2\sqrt{54} \times (-5\sqrt{24}) = -2 \times (-5) \times \sqrt{54 \times 24}$

$= 10 \times \sqrt{1296}$

Long division: $3 \times 3 = 9$, remainder $3$; bring down $96$ to get $396$; $66 \times 6 = 396$. So $\sqrt{1296} = 36$

$= 10 \times 36 = 360$

Answer: (C) $-64$

$-2\sqrt{2} \times 2\sqrt{128} = -2 \times 2 \times \sqrt{2 \times 128}$

$= -4 \times \sqrt{256}$

$\sqrt{256} = 16$

$= -4 \times 16 = -64$

Answer: (B) $\sqrt{5}$

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

$\displaystyle \frac{\sqrt{30}}{\sqrt{6}} = \sqrt{5}$

Answer: (C) $\sqrt{14}$

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

$\displaystyle \frac{14}{\sqrt{14}} = \sqrt{14}$

Answer: (D) $7\sqrt{5}$

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

Answer: (A) $2\sqrt{15}$

$\sqrt{60} = \sqrt{4 \times 15} = \sqrt{4} \times \sqrt{15} = 2\sqrt{15}$

Answer: (B) $-399$

$a \times b = -21 \times 19 = -399 \in R$

Answer: (D) $\displaystyle \frac{1}{7}$

$\displaystyle 10 \div 70 = \frac{10}{70} = \frac{1}{7}$

Division is not commutative.

Answer: (B) $\displaystyle \frac{3}{5}$

$\displaystyle (45 \div 15) \div 5 = 3 \div 5 = \frac{3}{5}$

Do the bracket first. Division is not associative, so $45 \div (15 \div 5) = 15$ is different.

Answer: (A) $1$

$a \times 1 = a$: $1$ is the multiplicative identity.

Answer: (B) $\displaystyle -\frac{1}{28}$

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

$\displaystyle -28 \times (-\frac{1}{28}) = 1$