Q&A: practice questions with answers

Concept: The number must be a multiple of the LCM and a perfect square.

1. $\text{LCM}(10,12,15,18,20)=180=2^2\times 3^2\times 5$.
2. Make every power even: multiply by 5 → $900={30}^2$.

Trick: Check options: 400 and 625 are not divisible by 18; 180 is not a square.

Answer: (C) $900$

Concept: Solve innermost brackets first; “of” before ÷.

1. $(4\text{ of }2÷4)=8÷4=2$; $(4÷4\text{ of }2)=4÷8=\frac{1}{2}$.
2. $\{4+2+\frac{1}{2}\}=\frac{13}{2}$; $5+1÷2=\frac{11}{2}$.
3. Bracket $=\frac{11}{2}-\frac{13}{2}=-1$; $-7÷(-1)=7$.

Trick: Negative ÷ negative is positive – that rules out options (A) and (B).

Answer: (D) $7$

Concept: For a fixed distance, time $\propto \frac{1}{\text{speed}}$.

1. Speed ratio $5:9$ ⇒ time ratio $9:5$.
2. Saving $=4$ parts $=40$ min ⇒ 1 part $=10$ min.
3. Usual time $=9\times 10=90$ min.

Trick: Usual time $=\frac{\text{saving}}{1-\frac{5}{9}}=\frac{40}{4/9}=90$.

Answer: (C) $90$

Concept: Use $(a+b{)}^2=a^2+b^2+2ab$ with $a^2+b^2={26}^2$.

1. $a+b=60-26=34$.
2. $2ab={34}^2-{26}^2=1156-676=480\Rightarrow ab=240$.
3. Area $=\frac{1}{2}ab=120{\text{cm}}^2$.

Trick: Triplet 10-24-26 (= 2 × 5-12-13): area $=\frac{1}{2}\times 10\times 24=120$.

Answer: (D) $120$

Concept: % more $=\frac{\text{difference}}{\text{base}}\times 100$, base = students with 400 or more.

1. Less than 350: $30+45+60+35=170$.
2. 400 or more: $35$.
3. More by $\frac{170-35}{35}\times 100=\frac{135}{35}\times 100\approx 385.7\%$.

Trick: $\frac{170}{35}=4.857$ times → $385.7\%$ more (subtract 100%).

Answer: (C) 385.7%

Concept: $6^2+8^2={10}^2$, so $\angle A={90}^{\circ }$; the median to the hypotenuse equals half the hypotenuse.

1. $36+64=100$ ⇒ right angle at A, BC is the hypotenuse.
2. $AD=\frac{BC}{2}=5$ cm.

Trick: Apollonius check: $A{D}^2=\frac{2(36+64)-100}{4}=25$.

Answer: (B) $5$

Concept: Numbers $=\text{HCF}\times$ each co-prime factor.

1. Numbers $=29\times 13=377$ and $29\times 15=435$.
2. Larger $=435$.

Trick: Larger number = HCF × larger factor.

Answer: (A) $435$

Concept: Area of a path around a circle $=\pi (R^2-r^2)=\pi (R+r)(R-r)$, where $R=r+\text{width}$.

1. $r=\frac{10}{2}=5\text{m}$, width $=64\text{cm}=0.64\text{m}$, so $R=5.64\text{m}$.
2. Area $=\frac{22}{7}\times (5.64+5)\times (5.64-5)=\frac{22}{7}\times 10.64\times 0.64$.
3. $=\frac{22}{7}\times 6.8096\approx 21.4{\text{m}}^2$.
4. None of the options 10, 11, 15, 9 is close to $21.4$. (A value near 10 would need a path only $32\text{cm}$ wide: $\frac{22}{7}\times 10.32\times 0.32\approx 10.4$.)

Trick: Use $(R+r)(R-r)$ instead of squaring decimals: $10.64\times 0.64\approx 6.81$, then $\times \frac{22}{7}\approx 21.4$.

Answer: The area of the path as printed is about $21.4{\text{m}}^2$, so no option fits; this question was treated as a bonus question.

Concept: Build the number backwards from the last quotient $k$.

1. Last step: $7k+5$; previous: $4(7k+5)+3=28k+23$.
2. Number $=3(28k+23)+2=84k+71$.
3. Divided by $84$ ($=3\times 4\times 7$) → remainder $71$.

Trick: Remainder $=r_1+d_1{r}_2+d_1{d}_2{r}_3=2+3\times 3+12\times 5=71$.

Answer: (A) $71$

Concept: From the midpoint M both feet are at the same distance $d$, so height $=d\tan \theta$.

1. $h_A=d\tan {30}^{\circ }=\frac{d}{\sqrt{3}}$; $h_B=d\tan {60}^{\circ }=\sqrt{3}d$.
2. $\frac{h_B}{h_A}=\frac{\sqrt{3}}{1/\sqrt{3}}=3$ → $3:1$.

Trick: $\frac{\tan {60}^{\circ }}{\tan {30}^{\circ }}=3$ always.

Answer: (B) 3 : 1

Concept: An action that started in the past and is still continuing, with a period of time (for two hours), takes the Present Perfect Continuous: has/have + been + V-ing.

1. Subject She is singular → has been.
2. She is the one who is waiting (active), so the verb must be waiting, not passive being waited.
3. Correct: She has been waiting for you for two hours.
4. had wait is a wrong form; have been waited is passive and plural.

Trick: for + period / since + point of time with an ongoing action → has/have been + V-ing.

Answer: (B) has been waiting

Concept: Suave means smooth, polished and charming in manner (especially a man).

1. Opposite of polished, gracious behaviour = rude (impolite, ill-mannered).
2. charming and polite are close to synonyms of suave.
3. brisk means quick and active – unrelated.

Yaad rakhein: Suave ≈ sophisticated, urbane; its antonyms: rude, crude, boorish.

Answer: (B) rude

Concept: Coveted means greatly wanted or longed for (e.g. a coveted award).

1. Closest meaning = desired.
2. despised (hated) and renounced (given up) are opposite in sense.
3. covered is only a look-alike word (trap).

Trick: Covet = to desire strongly what someone else has – so coveted = much desired.

Answer: (A) desired

Concept: Before a noun (an accident) we need a prepositional phrase of cause such as owing to / due to / because of. In order to shows purpose and takes a verb.

1. an accident is a noun phrase, not a clause.
2. since and because are conjunctions – they need a full clause (subject + verb) after them.
3. So only owing to fits: The train was delayed owing to an accident.

Trick: Cause + noun → owing to / because of; cause + clause → because / since; purpose + verb → in order to.

Answer: (B) owing to

Concept: Bear the brunt of = to receive the worst or main part of an attack, anger or difficulty.

1. Brunt = the chief force or shock of something unpleasant.
2. Example: The coastal villages bore the brunt of the cyclone.
3. So the meaning is ‘bear the maximum fury’; the other options do not match.

Yaad rakhein: Brunt = main impact; bear the brunt = suffer the heaviest blow.

Answer: (D) bear the maximum fury

Concept: After a preposition (of) a verb takes the -ing (gerund) form. Been is a past participle and cannot follow of directly.

1. of been stationary ✗ → of being stationary ✓.
2. Correct: The two trains gave the illusion of being stationary.
3. Other segments are correct.

Trick: Preposition + verb → always V-ing (of being, without going, before leaving).

Answer: (B) of been stationary

Concept: Shove means to push roughly or suddenly.

1. Among the options, the closest is poke (to push/prod sharply).
2. pull is the opposite direction of force; halt = stop; shout = cry loudly.

Yaad rakhein: Shove ≈ push, thrust, jostle, poke.

Answer: (B) poke

Concept: Lofty means very tall / of great height (also: noble, elevated).

1. Closest meaning = towering (very high).
2. humble and plain are nearly opposite; silent is unrelated.

Trick: Loft = upper storey → lofty = high up.

Answer: (D) towering

Concept: The sword of Damocles = a danger that may strike at any moment; an impending threat.

1. From a Greek tale: Damocles sat on a throne with a sword hanging above him by a single hair.
2. It shows that power and good fortune carry constant danger.
3. So it means ‘an impending threat’, not a real or rare sword.

Yaad rakhein: Sword hanging by a hair over the head = ever-present danger.

Answer: (C) an impending threat

Concept: A person or thing well known for something bad is notorious.

1. famous and renowned mean well known for good reasons.
2. icon = a person admired as a symbol.
3. Evil reputation → notorious (e.g. a notorious criminal).

Trick: Famous = good fame; notorious = bad fame (infamous).

Answer: (B) notorious