Q&A: practice questions with answers

Concept: Direct common tangent $=\sqrt{d^2-(r_1-r_2{)}^2}$; for circles touching externally $d=r_1+r_2$, so it becomes $2\sqrt{r_1{r}_2}$.

1. $r_1=18$, $r_2=8$, $d=AB=26\text{cm}$.
2. $CD=\sqrt{{26}^2-{10}^2}=\sqrt{676-100}=\sqrt{576}$.
3. $CD=24\text{cm}$ (same as $2\sqrt{18\times 8}=2\times 12$).

Trick: Touching circles: tangent $=2\sqrt{r_1{r}_2}=2\sqrt{144}=24$.

Answer: (C) $24\text{cm}$

Concept: $(r+1{)}^3=r^3+3r^2+3r+1$, so $r(r^2+3r+3)=(r+1{)}^3-1$.

1. $r=55\Rightarrow r+1=56$.
2. ${56}^3=175616$.
3. Value $=175616-1=175615$.

Trick: Spot the cube: $r^3+3r^2+3r$ is $(r+1{)}^3$ minus $1$. Option $175616$ is the trap.

Answer: (D) $175615$

Concept: Total work in man-days stays the same: $M_1{D}_1=M_2{D}_2$.

1. Let the original number be $M$: $M\times 20=(M-5)\times 40$.
2. $20M=40M-200\Rightarrow 20M=200$.
3. $M=10$.

Trick: Days doubled $\Rightarrow$ men halved; so the $5$ absent men are half the group: $M=10$.

Answer: (B) $10$

Concept: Add all 2020 (blue) bars, then add the 2021 Geography bar.

1. 2020 total $=75+100+60+125+75=435$.
2. Geography in 2021 $=75$.
3. Sum $=435+75=510$.

Trick: Option $435$ is only the 2020 total – remember to add Geography 2021.

Answer: (B) $510$

Concept: Divisible by both $2$ and $7$ means divisible by $\text{LCM}(2,7)=14$.

1. Count multiples of $14$ from $1$ to $1000$: $\lfloor \frac{1000}{14}\rfloor$.
2. $14\times 71=994\le 1000$, $14\times 72=1008>1000$.
3. Count $=71$.

Trick: Option $142$ counts multiples of $7$ only – "both" means LCM.

Answer: (B) $71$

Concept: Convert mixed fractions, then follow BODMAS (innermost bracket first).

1. $\frac{1}{2}-\frac{1}{7}=\frac{5}{14}$; $3\frac{1}{2}÷\frac{5}{14}=\frac{7}{2}\times \frac{14}{5}=\frac{49}{5}$.
2. $1\frac{2}{7}\times \frac{49}{5}=\frac{9}{7}\times \frac{49}{5}=\frac{63}{5}$.
3. $4\frac{1}{5}\times 1\frac{1}{2}=\frac{21}{5}\times \frac{3}{2}=\frac{63}{10}$.
4. $\frac{63}{5}÷\frac{63}{10}=2$.

Trick: Both parts reduce to $63$ over $5$ and $10$ – the ratio is $2$.

Answer: (C) $2$

Concept: Third proportional to $a$ and $b$ $=\frac{b^2}{a}$.

1. $a=0.024$, $b=0.12$.
2. $\frac{(0.12{)}^2}{0.024}=\frac{0.0144}{0.024}$.
3. $=0.6$, i.e. $0.600$.

Trick: $\frac{b}{a}=\frac{0.12}{0.024}=5$, so the third term $=b\times 5=0.6$ (a geometric sequence).

Answer: (C) $0.600$

Concept: For $x=a\cos \theta +b\sin \theta$ and $y=a\sin \theta -b\cos \theta$, squaring and adding removes $\theta$: $x^2+y^2=a^2+b^2$.

1. Directly, $x+y=(4-3)\cos \theta +(3+4)\sin \theta =\cos \theta +7\sin \theta$, which changes with $\theta$, so a fixed value comes only from $x^2+y^2$.
2. $x^2+y^2=16({\cos }^2\theta +{\sin }^2\theta )+9({\sin }^2\theta +{\cos }^2\theta )$ (the $24\sin \theta \cos \theta$ terms cancel).
3. $x^2+y^2=16+9=25$.

Trick: Same pair of coefficients $4,3$ in both expressions $\Rightarrow$ sum of squares $=4^2+3^2=25$.

Answer: (C) $25$

Concept: Solve numerator and denominator separately with BODMAS ("of" before ÷ and ×).

1. Numerator: $\frac{5}{8}-\{\frac{3}{8}-\frac{2}{8}\}=\frac{4}{8}=\frac{1}{2}$; $\frac{1}{2}$ of $8.8=4.4$; $4.4-1.2=3.2$.
2. Denominator: $\frac{1}{6}$ of $60=10$; $\frac{25}{6}÷2.5=\frac{5}{3}$; $\times 2=\frac{10}{3}$; $÷10=\frac{1}{3}$.
3. $\frac{3}{4}-\frac{3}{8}=\frac{3}{8}$; denominator $=\frac{1}{3}+\frac{3}{8}=\frac{17}{24}$.
4. Value $=3.2\times \frac{24}{17}=\frac{76.8}{17}=\frac{384}{85}=4\frac{44}{85}$.

Trick: $\frac{76.8}{17}\approx 4.52$; only $4\frac{44}{85}\approx 4.52$ matches.

Answer: (C) $4\frac{44}{85}$

Concept: Odd divisors use only the odd prime factors. Sum of divisors of $p^n$ $=1+p+\dots +p^n$.

1. $216=2^3\times 3^3$.
2. Odd divisors come from $3^3$ only: $1,3,9,27$.
3. Sum $=1+3+9+27=40$.

Trick: Ignore the power of $2$; sum of odd divisors $=\frac{3^4-1}{3-1}=\frac{80}{2}=40$. ($600$ is the sum of all divisors.)

Answer: (C) $40$

Concept: The idiom get on like a house on fire means two people become friendly very quickly and get on extremely well, the way fire spreads fast through a house.

1. Example: The new neighbours got on like a house on fire.
2. (A) only says 'like someone immediately' – incomplete; it misses the close, warm relationship.
3. (C) describes an unexplained fondness (a 'soft spot'); (D) describes making amends.

Trick: House on fire = spreads fast → friendship grows fast and strong.

Answer: (B) Get on extremely well with someone

Concept: A finite verb is needed in the main clause. 'Taking' is a participle and cannot stand alone as the verb. Since the trip is 'last' (completed), the simple past is required.

1. Subject: His last trip around the world – needs a main verb.
2. 'Last trip' shows a finished past event → simple past took.
3. His last trip around the world took a toll on his health.
4. (B) 'taken' needs has/had; (D) 'take' is present and wrong with a singular subject; (C) keeps the error.

Trick: '-ing' or V3 alone can never be the main verb – add a helper or use a finite form.

Answer: (A) took a toll

Concept: Against the clock means working in a great hurry to finish something before a deadline – as fast as possible.

1. Example: The workers raced against the clock to finish the bridge before the monsoon.
2. The idiom is figurative: no actual clock or running is involved, so (A) and (C) are literal and wrong.
3. (D) 'race with others' – the competition is with time, not with people.

Trick: Your rival is the clock (time), not other people.

Answer: (B) In a great hurry, as fast as possible

Concept: The verb receive is used in the active voice when the subject itself gets something: X received Y. 'Was received' is passive and would mean the knight himself was received (welcomed).

1. The knight is the one who got the compliments → active voice.
2. Correct: The knight received compliments from the king for his bravery.
3. 'was received compliments' wrongly mixes passive form with an object.
4. The other parts are correct.

Trick: If the subject is the receiver of an object, use active 'received', not 'was received'.

Answer: (D) The knight was received

Concept: Seldom is an adverb meaning 'rarely; not often'.

1. Example: He seldom visits his village.
2. Habitually, frequently and routinely all mean 'often/regularly' – the opposite meaning.

Yaad rakhein: Seldom = rarely; the other three are its antonyms.

Answer: (B) seldom

Concept: The correct spelling is serrated (with double 'r'), meaning 'having a jagged, saw-like edge'.

1. 'serated' misses one 'r' → incorrect.
2. prey, seizing and crushing are spelt correctly.
3. Correct sentence: Sharks often use their serrated or smooth teeth…

Yaad rakhein: Serrated comes from Latin serra = saw; keep the 'rr' of a saw.

Answer: (A) serated

Concept: Permit here means 'to allow / give permission'. The closest synonym among the options is grant (to allow or give).

1. Sentence: his purse did not allow (grant) that purchase.
2. 'Patent' is an exclusive right on an invention; 'warrant' means justify; 'tolerate' means put up with something unpleasant.
3. 'Grant' best keeps the sense of allowing.

Trick: Permit = allow = grant.

Answer: (C) grant

Concept: Utilitarian means 'designed to be useful or practical; necessary for use'. Its opposite is unnecessary.

1. Utilitarian ≈ useful, practical, functional.
2. (B) 'Necessary' is close in meaning, not opposite.
3. Evident (clear) and spacious (roomy) are unrelated.

Trick: Utilitarian = utility (use) → opposite = of no use = unnecessary.

Answer: (D) Unnecessary

Concept: Get a second wind means to get renewed energy to continue after feeling tired, like a runner who suddenly breathes easier.

1. Example: After lunch, the team got a second wind and finished the work.
2. 'Wind' here means breath/energy, not a chance – so (D) is wrong.
3. (A) and (C) are about repeating, unrelated.

Trick: Second wind = second breath = fresh energy.

Answer: (B) To have renewed energy

Concept: The correct spelling is Connoisseur (double 'n' and double 's'), meaning an expert judge of art, food or taste.

1. 'Connoiseur' has only one 's' → incorrect.
2. Diarrhoea, Pseudonym and Perseverance are correctly spelt (British spelling 'diarrhoea').

Yaad rakhein: Conno-iss-eur: two n's, two s's.

Answer: (C) Connoiseur