Q&A: practice questions with answers

Concept: In a 100% stacked bar, each model's share $=$ upper edge $-$ lower edge of its segment.

1. 2020, model A: $0\%$ to $40\%$ ⇒ $40\%$ of $65$ lakh $=26$ lakh.
2. 2019, model E: $75\%$ to $90\%$ ⇒ $15\%$ of $40$ lakh $=6$ lakh.
3. Total $=26+6=32$ lakh $=32,00,000$.

Yaad rakhein: Read segment heights, not the top edge of the segment.

Answer: (B) $32,00,000$

Concept: Add the four cities for each year and pick the smallest total.

1. 2016: $18+25+18+20=81$; 2017: $25+23+18.5+18=84.5$.
2. 2018: $23+24+18.4+17.6=83$; 2019: $24.4+19.5+19+16=78.9$.
3. 2020: $22.5+21+20+17.5=81$. Least $=78.9$ in 2019.

Yaad rakhein: 2019 has the lowest Kolkata bar ($16$) and no year makes it up elsewhere.

Answer: (C) 2019

Concept: Net effect of $+a\%$ then $-b\%$: $a-b-\frac{ab}{100}$.

1. $26-12-\frac{26\times 12}{100}=14-3.12=10.88\%$.
2. Check: CP $100$ → MP $126$ → SP $126\times 0.88=110.88$.

Trick: Successive change formula saves time.

Answer: (A) $10.88\%$

Concept: ${(x^2+\frac{1}{x^2})}^2=x^4+\frac{1}{x^4}+2$ and ${(x+\frac{1}{x})}^2=x^2+\frac{1}{x^2}+2$.

1. $x^2+\frac{1}{x^2}=\sqrt{47+2}=7$.
2. $x+\frac{1}{x}=\sqrt{7+2}=3$ (as $x>0$).
3. $x+\frac{1}{x}-2=3-2=1$.

Trick: Add $2$ and take the root: $47\to 7\to 3$.

Answer: (A) $1$

Concept: $\frac{\sec \theta -1}{\sec \theta +1}=\frac{1-\cos \theta }{1+\cos \theta }={\left(\frac{1-\cos \theta }{\sin \theta }\right)}^2=(\csc \theta -\cot \theta {)}^2$.

1. Multiply numerator and denominator by $1-\cos \theta$: $\frac{(1-\cos \theta {)}^2}{{\sin }^2\theta }$.
2. So ${\left(\frac{\sec \theta -1}{\sec \theta +1}\right)}^n=(\csc \theta -\cot \theta {)}^{2n}$.
3. $2n=1\Rightarrow n=\frac{1}{2}=0.5$.

Trick: $\theta ={90}^{\circ }$: LHS $=1$ for any $n$, so try $\theta ={60}^{\circ }$: ${\left(\frac{1}{3}\right)}^n=\frac{1}{\sqrt{3}}\Rightarrow n=0.5$.

Answer: (B) $0.5$

Concept: If $DE\parallel BC$, corresponding angles are equal: $\angle AED=\angle ACB$.

1. $\angle AED=\angle ACB={40}^{\circ }$.
2. In $△ADE$: $\angle DAE+\angle ADE={180}^{\circ }-{40}^{\circ }={140}^{\circ }$.

Trick: It equals $\angle A+\angle B$ of the big triangle $={180}^{\circ }-\angle C$.

Answer: (C) ${140}^{\circ }$

Concept: Surface area of a cube $=6a^2$, volume $=a^3$.

1. $6a^2=54\Rightarrow a^2=9\Rightarrow a=3\text{cm}$.
2. Volume $=3^3=27{\text{cm}}^3$.

Trick: $V={\left(\frac{S}{6}\right)}^{3/2}=9^{3/2}=27$.

Answer: (C) $27{\text{cm}}^3$

Concept: ${\sec }^{2}A-{\tan }^{2}A=1$.

1. $16{\sec }^{2}A-16{\tan }^{2}A=16({\sec }^{2}A-{\tan }^{2}A)$.
2. $=16\times 1=16$.

Yaad rakhein: Take the common factor out first.

Answer: (C) $16$

Concept: Price of mixture $=\frac{q_1{p}_1+q_2{p}_2}{q_1+q_2}$.

1. $\frac{3\times 300+2\times 375}{5}=\frac{900+750}{5}$.
2. $=\frac{1650}{5}=330$ per kg.

Trick: $300+(375-300)\times \frac{2}{5}=300+30=330$.

Answer: (B) ₹$330$

Concept: $\frac{1+\tan \theta }{1-\tan \theta }+\frac{1-\tan \theta }{1+\tan \theta }=\frac{2(1+{\tan }^2\theta )}{1-{\tan }^2\theta }=\frac{2}{\cos 2\theta }$, and $2{\cos }^2\theta -1=\cos 2\theta$.

1. $\frac{(1+t{)}^2+(1-t{)}^2}{1-t^2}=\frac{2(1+t^2)}{1-t^2}$ with $t=\tan \theta$.
2. $\frac{1+{\tan }^2\theta }{1-{\tan }^2\theta }=\frac{1}{\cos 2\theta }$.
3. Value $=\cos 2\theta \times \frac{2}{\cos 2\theta }=2$.

Trick: Put $\theta =0^{\circ }$: $(2-1)(1+1)=2$.

Answer: (A) $2$

Concept: Spot the word whose letters are wrongly arranged.

1. Vociferous (loud and noisy) is spelt with -ferous, not "-firous".
2. Affliction, Episcopal and Innocuous are all spelt correctly.

Yaad rakhein: Many adjectives end in -ferous (carrying/producing): vociferous, coniferous, carboniferous.

Answer: (A) Vocifirous

Concept: Find the misspelt word among words ending in -ulate.

1. The correct spelling is Stipulate (to demand as part of an agreement) — only one p.
2. Manipulate, Postulate and Calculate are correct.

Yaad rakhein: sti-pu-late: single p, just like po-stu-late.

Answer: (C) Stippulate

Concept: The word to be replaced is ruin (to spoil or destroy).

1. Despoil = to spoil, ruin or rob something of its value — same meaning as ruin.
2. Fix, revamp (renovate) and recover are all positive — the opposite idea.

Trick: despoil = de + spoil → spoil completely = ruin.

Answer: (A) despoil

Concept: Clever, sharp and sound in judgement = Shrewd.

1. Shrewd: having sharp powers of judgement, e.g. a shrewd businessman.
2. Shriek = a loud scream; Shrift = confession (short shrift); Shrike = a bird.

Yaad rakhein: Shrewd = smart + sensible judgement.

Answer: (C) Shrewd

Concept: Emulate = to copy or imitate someone you admire, trying to equal them.

1. To emulate someone you pay close attention to and follow them.
2. Its opposite is Neglect (ignore, pay no attention).
3. Respect is close in spirit; spoilt and bash are unrelated.

Trick: Emulate = follow a role model; Neglect = ignore it.

Answer: (B) Neglect

Concept: A person who does something for pleasure, not as a paid job = Amateur.

1. Amateur comes from Latin amator = lover — one who does it out of love.
2. Professional is the exact opposite; Performer is too general; Ascetic = one who renounces pleasures.

Trick: Amateur = "ama" (love) → does it for love, not money.

Answer: (B) Amateur

Concept: Resume = to begin again after a pause.

1. Students will resume (restart/continue) their studies.
2. Opposite: cease = to stop or bring to an end.
3. Usher = guide in; presume = suppose; announce = declare — none mean stop.

Trick: Resume ↔ Cease (restart ↔ stop).

Answer: (A) cease

Concept: Check subject–verb agreement, the phrasal verb and article use in each part.

1. "The investors were" — plural subject with plural verb: correct.
2. "put off by" — phrasal verb put off = discouraged; passive "were put off by" is correct.
3. "the dirty visage of the workshop" — visage = face/outward appearance; adjective + noun + of-phrase is correct.

Trick: Plural subject + were + past participle (put off) = correct passive.

Answer: (C) No error

Concept: Check the double consonants and "sc/ss" patterns.

1. The correct word is Abscission (natural shedding of leaves/fruit in plants): ab-sci-ssion.
2. Abstruse (hard to understand), Incision (a cut) and Abyssal (very deep) are correct.

Yaad rakhein: Abscission = abs + sciss (cut, as in scissors) + ion.

Answer: (A) Abcission

Concept: Dull (of light/colour) = lacking brightness, dim.

1. The most direct opposite of dull is Bright.
2. Clever opposes dull only in the sense of intelligence; coloured and youthful are not opposites.
3. Bright covers both senses (shiny and intelligent), so it is the best answer.

Trick: Dull ↔ Bright is the standard antonym pair.

Answer: (B) Bright