Q&A: practice questions with answers

Concept: Volume is conserved on recasting. Frustum volume $=\frac{\pi h}{3}(R^2+r^2+Rr)$.

1. $V=\frac{\pi \times 45}{3}({28}^2+7^2+28\times 7)=15\pi \times 1029=15435\pi$
2. Cylinder $r=3k,h=5k$: $\pi \times 9k^2\times 5k=45\pi k^3$
3. $45k^3=15435\Rightarrow k^3=343\Rightarrow k=7$; $r=21,h=35$
4. CSA $=2\pi rh=2\times \frac{22}{7}\times 21\times 35=4620{\text{cm}}^2$

Trick: ${28}^2+7^2+28\times 7=49(16+1+4)=49\times 21$ — factor out $7^2$.

Answer: (D) 4620

Concept: Average speed $=\frac{\text{Total distance}}{\text{Total time}}$; first express each segment's time.

1. $t_1=\frac{10}{40}=\frac{1}{4}\text{h}$
2. $t_2=\frac{t_1}{2}=\frac{1}{8}$, $t_4=t_1=\frac{1}{4}$, $t_3=\frac{t_4}{2}=\frac{1}{8}$
3. Total time $=\frac{1}{4}+\frac{1}{8}+\frac{1}{8}+\frac{1}{4}=\frac{3}{4}\text{h}$
4. Average $=\frac{40}{3/4}=\frac{160}{3}\text{km/h}$

Trick: Times in ratio $2:1:1:2$ (total $6$ parts, first part $=\frac{1}{4}$ h $=2$ parts).

Answer: (C) $\frac{160}{3}\text{km/h}$

Concept: $1+{\tan }^2\theta ={\sec }^2\theta$.

1. ${\tan }^2\theta +{\tan }^4\theta ={\tan }^2\theta (1+{\tan }^2\theta )$
2. $={\tan }^2\theta {\sec }^2\theta =({\sec }^2\theta -1){\sec }^2\theta$
3. $={\sec }^4\theta -{\sec }^2\theta$

Trick: Put $\theta ={45}^{\circ }$: $1+1=2$ and ${\sec }^4-{\sec }^2=4-2=2$ ✓.

Answer: (B) ${\sec }^4\theta -{\sec }^2\theta$

Concept: $(a-b{)}^2=2(a^2+b^2)-(a+b{)}^2$ with $a^2+b^2={\text{hyp}}^2$.

1. $a+b=41$, $a^2+b^2={29}^2=841$
2. $2ab={41}^2-841=1681-841=840$
3. $(a-b{)}^2=841-840=1\Rightarrow a-b=1\text{cm}$

Trick: $20$–$21$–$29$ is a Pythagorean triplet: $20+21=41$, difference $1$.

Answer: (B) $1\text{cm}$

Concept: HCF $\times$ LCM $=$ product of the numbers; take numbers $3k$ and $4k$.

1. $3k\times 4k=12k^2=2700\Rightarrow k^2=225\Rightarrow k=15$
2. Numbers $45$ and $60$
3. Sum $=105$

Trick: Sum $=7k$, so the answer must be a multiple of $7$ — only $105$ is.

Answer: (B) 105

Concept: $\text{SI}=\frac{P\times R\times T}{100}$; split the sum as $x$ and $10000-x$.

1. In 2 years: A gives $16\%$ of $x$, B gives $20\%$ of $(10000-x)$
2. $0.16x+0.2(10000-x)=1680$
3. $2000-0.04x=1680\Rightarrow x=8000$

Trick: Alligation: average rate $=\frac{1680}{10000\times 2}=8.4\%$; ratio A:B $=(10-8.4):(8.4-8)=1.6:0.4=4:1$ → A $=$ ₹$8000$.

Answer: (A) ₹8,000

Concept: In $a:b::c:d$, product of extremes $=$ product of means: $ad=bc$.

1. $42:36::35:d$
2. $d=\frac{36\times 35}{42}$
3. $=30$

Trick: $\frac{36}{42}=\frac{6}{7}$, so $d=35\times \frac{6}{7}=30$.

Answer: (C) 30

Concept: Final price $=\text{CP}\times (1+\text{markup})\times (1-d_1)(1-d_2)(1-d_3)$.

1. Take CP $=100$; MP $=170$
2. After discounts: $170\times 0.95\times 0.8\times 0.8$
3. $=170\times 0.608=103.36$
4. Gain $=3.36\%$

Trick: $0.95\times 0.8\times 0.8=0.608$; $1.7\times 0.608=1.0336$.

Answer: (D) $3.36\%$

Concept: $A=P{(1+\frac{r}{100})}^n$; the amount owed now is the principal grown for the years already passed.

1. $P\times (1.1{)}^5=161051\Rightarrow P=\frac{161051}{1.61051}=100000$
2. After 3 years: $100000\times (1.1{)}^3=100000\times 1.331$
3. $=133100$

Trick: Or discount back 2 years: $\frac{161051}{1.21}=133100$.

Answer: (C) ₹1,33,100

Concept: If $a^2+b^2=c^2$, the triangle is right-angled and area $=\frac{1}{2}ab$.

1. $5^2+{12}^2=25+144=169={13}^2$
2. Right angle between sides $5$ and $12$
3. Area $=\frac{1}{2}\times 5\times 12=30{\text{cm}}^2$

Trick: Spot the $5$–$12$–$13$ triplet; no need for Heron's formula.

Answer: (A) $30{\text{cm}}^2$

Concept: Words ending in '-ate' keep the root spelling; 'assassinate' has a double 's' twice (as-sas-sin-ate).

1. Interrogate, Alienate and Deserve are spelt correctly.
2. 'Assasinate' misses one 's'; the correct form is Assassinate (to murder an important person).
3. So option (D) is the incorrect spelling.

Yaad rakhein: Assassinate = ASS + ASS + IN + ATE – two pairs of 'ss'.

Answer: (D) Assasinate

Concept: The idiom 'bend over backwards' means to try very hard, often more than is required, to help or please someone.

1. Bending backwards is physically difficult – it shows great effort to please.
2. Option (A) 'working hard to try to impress' matches this meaning.
3. 'Working slowly' is the opposite; 'going backward' is the literal reading; 'wrongful means' is unrelated.

Yaad rakhein: Bend over backwards = go out of your way to please/help someone.

Answer: (A) Working hard to try to impress

Concept: Adjectives ending in '-ant/-ent' must be learnt: 'blatant' ends in '-ant'.

1. 'Blatant' means open and shameless (a blatant disregard of rules).
2. The sentence writes 'blatent', which is wrong; correct is blatant.
3. 'Advisory', 'disregard' and 'violation' are spelt correctly.

Yaad rakhein: Blatant – both vowels are 'a': bl-A-t-A-nt.

Answer: (B) blatent

Concept: A verb is modified by an adverb, not an adjective. 'Swayed' is a verb, so it needs the adverb 'daintily'.

1. 'Dainty' is an adjective (delicate, pretty).
2. It describes how they swayed, so the adverb is required: 'swayed daintily'.
3. Correct sentence: The dancers swayed daintily on the stage last night.

Yaad rakhein: Verb + how? → adverb (-ly): swayed daintily, spoke softly.

Answer: (D) swayed dainty

Concept: A published notice of someone's death, usually with a short life sketch, is an obituary.

1. Obituary – death notice with biography in a newspaper (matches).
2. Oblivion – state of being forgotten; Elegy – a sad poem for the dead.
3. Advertisement – a public promotion; not specific to death.

Yaad rakhein: Obituary → 'obit' = death; Elegy = poem, Obituary = newspaper notice.

Answer: (A) Obituary

Concept: 'Aggregate' (total, whole) is built from 'ag- + greg-' and has a double 'g'.

1. Adjacent, Accomplish and Allure are spelt correctly.
2. 'Agregate' misses one 'g'; correct spelling is Aggregate.
3. Hence option (C) is incorrectly spelt.

Yaad rakhein: aGGregate – like 'aGGressive', it starts with double g.

Answer: (C) Agregate

Concept: 'Birds of a (the same) feather' refers to people who have similar interests, character or tastes.

1. Nitin and Uday share the same interests – so they are 'birds of the same feather'.
2. 'Under the weather' = unwell; 'blowing smoke' = talking without substance.
3. 'Different kettles of fish' means things are entirely different – the opposite idea.

Yaad rakhein: Birds of a feather flock together – similar people stay together.

Answer: (B) birds of the same feather

Concept: 'Moving' means in motion or changing. Its opposite is something fixed, constant or unchanging – 'steady'.

1. In the sentence, 'steady tick-tock' means a regular, unchanging beat.
2. 'Steady' (firm, stable, not moving) is opposite to 'moving'.
3. 'Marking', 'Chimed' and 'Passage' (passing) all relate to action or movement, not its opposite.

Yaad rakhein: Moving ≠ Steady (stable, fixed).

Answer: (A) Steady

Concept: 'Than' is used for comparison; 'then' is an adverb of time (next, after that).

1. 'Better' is a comparative adjective, so it must be followed by 'than'.
2. Correct: Her paintings are way better than her sketches.
3. 'Off' and 'more' make no sense here ('more better' is a double comparative).

Yaad rakhein: Comparative (-er/better/more) + THAN; time sequence → THEN.

Answer: (B) than

Concept: 'Get off on the wrong foot' means to start a relationship or activity badly, making a poor first impression.

1. The speaker does not want a bad beginning.
2. Option (C) – to start a relationship or activity badly – matches.
3. Option (D) is a literal reading; (A) and (B) give positive or neutral starts.

Yaad rakhein: Wrong foot = bad start; right foot = good start.

Answer: (C) To start a relationship or activity badly