MCQ test, Set 6

Answer: (C) $z \times z$

$a^2 = a \times a$

So $z^2 = z \times z$ ($z$ multiplied by itself, not by $2$).

Answer: (D) $676$

$676 = 26 \times 26 = 26^2$

The other numbers are not in the table of squares.

Answer: (C) $841$

In the table, $24^2$ and $26^2$ differ by $100$, $23^2$ and $27^2$ by $200$, $22^2$ and $28^2$ by $300$, $21^2$ and $29^2$ by $400$.

$29^2 = 441 + 400 = 841$

Answer: (B) $1600$

$50^2 - 30^2 = 2500 - 900 = 1600$

Answer: (C) $72$

Rule: multiply the number before $5$ by the next number, then write $25$.

$8 \times 9 = 72$, so $85^2 = 7225$

Answer: (B) $153$

$a^2 = b \Rightarrow \sqrt{b} = a$

$153^2 = 23409 \Rightarrow \sqrt{23409} = 153$

(The square root is not half the number.)

Answer: (A) $16$

$16^2 = 16 \times 16 = 256$

$\therefore \sqrt{256} = 16$

Answer: (D) $169$

$\sqrt{b} = a \Rightarrow b = a^2$

$13^2 = 13 \times 13 = 169$

Answer: (A) $25$

Take one factor from each pair: $5 \times 5 = 25$

Answer: (D) $55$

$\sqrt{5 \times 5 \times 11 \times 11} = \sqrt{(5 \times 5) \times (11 \times 11)}$

$= 5 \times 11 = 55$

Answer: (C) $62$

Pairs: $\overline{38}\;\overline{44}$.

$6 \times 6 = 36 \le 38$: first digit $6$, and $38 - 36 = 2$

Bring down $44$: we get $244$. New divisor: $6 \times 2 = 12$; $122 \times 2 = 244$ ($123 \times 3 = 369$ is too big); $244 - 244 = 0$

Remainder $0$. $\therefore \sqrt{3844} = 62$

Answer: (B) $16$

Double the root found so far: $8 \times 2 = 16$

Then $161 \times 1 = 161$ ($162 \times 2 = 324$ is too big), so $\sqrt{6561} = 81$

Answer: (C) $274$

Pair the digits from the right: $\overline{7}\;\overline{50}\;\overline{76}$

$2 \times 2 = 4 \le 7$: first digit $2$, and $7 - 4 = 3$

Bring down $50$: we get $350$. New divisor: $2 \times 2 = 4$; $47 \times 7 = 329$ ($48 \times 8 = 384$ is too big); $350 - 329 = 21$

Bring down $76$: we get $2176$. New divisor: $27 \times 2 = 54$; $544 \times 4 = 2176$ ($545 \times 5 = 2725$ is too big); $2176 - 2176 = 0$

Remainder $0$. $\therefore \sqrt{75076} = 274$

Answer: (B) $15$

$\sqrt{3} \times \sqrt{75} = \sqrt{3 \times 75} = \sqrt{225} = 15$

Answer: (A) $26$

$(\sqrt{a})^2 = \sqrt{a} \times \sqrt{a} = a$

$\therefore (\sqrt{26})^2 = 26$

Answer: (D) $5$

$\displaystyle \frac{\sqrt{175}}{\sqrt{7}} = \sqrt{\frac{175}{7}} = \sqrt{25} = 5$

Answer: (D) $\sqrt{34}$

$\displaystyle \frac{34}{\sqrt{34}} = \frac{\sqrt{34} \times \sqrt{34}}{\sqrt{34}} = \sqrt{34}$

Answer: (B) $34$

$\sqrt{100} + \sqrt{576} = 10 + 24 = 34$

$\sqrt{a} + \sqrt{b} \ne \sqrt{a + b}$, so do not add first.

Answer: (B) $3$

$\sqrt{225} - \sqrt{144} = 15 - 12 = 3$

$\sqrt{a} - \sqrt{b} \ne \sqrt{a - b}$, so do not subtract first.

Answer: (A) $41$

$\sqrt{9^2 + 40^2} = \sqrt{81 + 1600} = \sqrt{1681} = 41$

$\sqrt{a^2 + b^2} \ne a + b$

Answer: (D) $58$

$a^2 + b^2 = 3^2 + 7^2 = 9 + 49 = 58$

Note: $a^2 + b^2 \ne (a + b)^2$ (here $(a + b)^2 = 100$).

Answer: (A) $0.09$

$\displaystyle \sqrt{0.0081} = \sqrt{\frac{81}{10000}}$

$\displaystyle = \frac{\sqrt{81}}{\sqrt{10000}}$

$\displaystyle = \frac{9}{100}$

$= 0.09$

Answer: (D) $\displaystyle \frac{18}{100}$

$\displaystyle \sqrt{0.0324} = \sqrt{\frac{324}{10000}} = \frac{\sqrt{324}}{\sqrt{10000}} = \frac{18}{100}$

Answer: (C) $1.9$

$\displaystyle \sqrt{3.61} = \sqrt{\frac{361}{100}} = \frac{\sqrt{361}}{\sqrt{100}} = \frac{19}{10} = 1.9$

Answer: (A) $56.25$

$\displaystyle 7.5 = \frac{75}{10}$

$\displaystyle \left(\frac{75}{10}\right)^2 = \frac{5625}{100} = 56.25$

Check: $\displaystyle \sqrt{56.25} = \frac{\sqrt{5625}}{\sqrt{100}} = 7.5$