Q&A: practice questions with answers

Concept: Number of glasses $=\frac{\text{bowl volume}}{\text{glass volume}}={\left(\frac{R}{r}\right)}^3$ (both hemispheres).

1. $2\pi R=88$ ⇒ $R=14\text{cm}$
2. Glass radius $r=3.5\text{cm}$
3. ${\left(\frac{14}{3.5}\right)}^3=4^3=64$ persons

Trick: Similar solids: volume ratio $=$ (radius ratio)${}^3$ — no need to compute $\frac{2}{3}\pi r^3$.

Answer: (B) $64$

Concept: $x^3+y^3+z^3-3xyz=(x+y+z)[(x+y+z{)}^2-3(xy+yz+zx)]$.

1. RHS $=1\times [1-3(-1)]=4$
2. $x^3+y^3+z^3=4+3xyz=4+3(-1)=1$

Trick: Here $x,y,z$ are roots of $t^3-t^2-t+1=0$, i.e. $1,1,-1$: $1+1-1=1$ ✓.

Answer: (A) $1$

Concept: Shares held $=$ % share $\times$ total shares.

1. Pandey $=5\%$ of $8,00,000=40,000$
2. After buying $10,000$ from Varma: $40,000+10,000=50,000$

Trick: $1\%$ of 8 lakh $=8000$ ⇒ $5\%=40,000$.

Answer: (B) $50000$

Concept: If $\sec X=\text{cosec}Y$ (acute), then $X+Y={90}^{\circ }$.

1. $6A+(A-{29}^{\circ })={90}^{\circ }$
2. $7A={119}^{\circ }$ ⇒ $A={17}^{\circ }$

Yaad rakhein: $\sec \theta =\text{cosec}({90}^{\circ }-\theta )$ — just set the two angles to add up to ${90}^{\circ }$.

Answer: (C) $17$

Concept: $AR$ and $QS$ are heights on the same base $PC$, so $\frac{AR}{QS}=\frac{\text{ar}(APC)}{\text{ar}(QPC)}$.

1. Let $\text{ar}(ABC)=49$. $\text{ar}(APC)=\frac{AP}{AB}\times 49=\frac{4}{7}\times 49=28$; $\text{ar}(PBC)=21$
2. $PQ\parallel AC$ ⇒ $\frac{BQ}{BC}=\frac{BP}{BA}=\frac{3}{7}$ ⇒ $\frac{QC}{BC}=\frac{4}{7}$
3. $\text{ar}(QPC)=\frac{4}{7}\times 21=12$
4. $\frac{AR}{9}=\frac{28}{12}$ ⇒ $AR=21\text{cm}$

Trick: Perpendiculars to a common line compare like areas on that line: ratio $=\frac{28}{12}=\frac{7}{3}$, so $AR=9\times \frac{7}{3}$.

Answer: (C) $21$

Concept: Fourth proportional: $a:b=c:d$ ⇒ $ad=bc$.

1. $4:a=16a:81$
2. $4\times 81=16a^2$ ⇒ $a^2=\frac{81}{4}$
3. $a=\frac{9}{2}$

Yaad rakhein: Product of extremes $=$ product of means.

Answer: (C) $\frac{9}{2}$

Concept: Valid votes $=$ total $-$ invalid; the loser gets the remaining share of valid votes.

1. Valid $=75\%$ of $15000=11250$
2. Other candidate $=40\%$ of $11250=4500$

Trick: Directly: $15000\times 0.75\times 0.4=4500$.

Answer: (B) $4,500$

Concept: Internal bisector divides the opposite side in the ratio of the adjacent sides: $\frac{AB}{AC}=\frac{BD}{DC}$.

1. $DC=7.5-5=2.5\text{cm}$
2. $\frac{AB}{AC}=\frac{5}{2.5}=\frac{2}{1}$

Yaad rakhein: The longer segment ($BD$) lies next to the longer side ($AB$).

Answer: (A) $2:1$

Concept: Let numbers be $3k$ and $7k$.

1. $\frac{3k-11}{7k-11}=\frac{7}{20}$
2. $60k-220=49k-77$ ⇒ $11k=143$ ⇒ $k=13$
3. Smaller $=3\times 13=39$

Trick: Option check: $39\to 91$; $\frac{28}{80}=\frac{7}{20}$ ✓.

Answer: (A) $39$

Concept: Average speed $=\frac{\text{total distance}}{\text{total time}}$; the car moves only for three quarters.

1. Distance $=20+25+15=60\text{km}$
2. Time $=3\times \frac{1}{4}=\frac{3}{4}$ h
3. Speed $=60÷\frac{3}{4}=80\text{km/h}$

Trick: Equal time intervals ⇒ average of the interval speeds: $\frac{80+100+60}{3}=80$.

Answer: (A) $80$

Concept: Uniformity means sameness – everything being alike. Its opposite is difference or diversity.

1. In the sentence, variety means a number of different kinds (of birds).
2. Variety = diversity, the exact opposite of uniformity.
3. Great, found and sanctuaries have no opposite relation with uniformity.

Trick: Uniform = one form; variety = many forms.

Answer: (B) Variety

Concept: Hit the sack is an informal idiom: 'sack' refers to a bed (old straw-filled mattress).

1. To hit the sack = to go to bed / go to sleep.
2. Example: I was so tired that I hit the sack at 9 p.m.
3. Running, cracking exams and making money have nothing to do with bed.

Yaad rakhein: Hit the sack = hit the hay = go to sleep.

Answer: (C) Going to sleep

Concept: Check each word letter by letter for the standard spelling.

1. Mountan is wrong – correct spelling is Mountain (–ain, like 'captain').
2. Weekend, Dessert (sweet dish) and Journey are spelt correctly.

Yaad rakhein: Mountain, captain, certain – all end in –ain.

Answer: (A) Mountan

Concept: Doubling rule: a final consonant is doubled before –ed when the last syllable is stressed (oc-CUR).

1. occur + ed → occurred (double r).
2. So Occured is misspelt.
3. Equally, Unwilling and Argument (argue drops 'e') are correct.

Trick: Stress on last syllable → double the consonant: occurred, preferred, referred.

Answer: (D) Occured

Concept: A person or thing trying to appear more important or talented than it really is is called pretentious.

1. Pretentious = showy, claiming undeserved importance – exact match.
2. Imperious = arrogant and domineering (giving orders).
3. Airy = light/spacious or casual; Glorious = deserving great honour.

Trick: Pretentious comes from 'pretend' – pretending to be greater.

Answer: (A) Pretentious

Concept: The French-origin word for rules of polite behaviour is spelt etiquette.

1. Correct: e-t-i-q-u-e-t-t-e (one t at the start, double t at the end).
2. Given 'ettiquete' reverses this, so it is misspelt.
3. Rationalisation, elopement and occasion are correct.

Yaad rakhein: Etiquette – single t first, 'tt' + e at the end.

Answer: (A) ettiquete

Concept: Stative verbs (understand, know, believe, like, love) express a state of mind and are not used in continuous tenses.

1. 'I am understanding' is wrong.
2. Correct: I understand (that) you like her because she has a generous nature.
3. The other segments are correct.

Trick: Mental-state verbs → no –ing: know, understand, believe, prefer.

Answer: (B) I am understanding;

Concept: Monotonous = dull because of being the same all the time (boring).

1. Its opposite is something that holds attention – interesting.
2. Pious = religious; Precarious = unstable/risky; Threatening = menacing – none is the opposite of boring.

Trick: Mono-tone = one tone = boring ↔ interesting.

Answer: (B) Interesting

Concept: The adverb from 'scarce' is formed by adding –ly, keeping the 'c-e'.

1. scarce + ly → scarcely (= hardly).
2. 'scarely' drops the 'c', so segment (C) has the spelling error.
3. Correct sentence: Joe was so tired that he could scarcely stand.

Yaad rakhein: Scarcely = scarce + ly.

Answer: (C) scarely stand.

Concept: Marriage terms: poly = many, andr = man/husband, gyny = woman.

1. Polyandry = a woman having more than one husband at the same time.
2. Monogamy = one spouse; Endogamy = marrying within one's own group; Polymathy = wide learning in many subjects.

Trick: Andro = male → polyandry (many husbands); polygyny (many wives).

Answer: (C) Polyandry