Q&A: practice questions with answers

Concept: $(x+y{)}^3-(x-y{)}^3=6x^{2}y+2y^3$.

1. Expression $=6x^{2}y+2y^3-6x^{2}y+6y^3$.
2. $=8y^3$.

Trick: Put $x=0,y=1$: $1-(-1)-6(-1)=8=8y^3$.

Answer: (A) $8y^3$

Concept: From $x^2+6x+1=0$: $x+6=-\frac{1}{x}$ and $x+\frac{1}{x}=-6$.

1. $(x+6{)}^3+\frac{1}{(x+6{)}^3}=-\frac{1}{x^3}-x^3=-(x^3+\frac{1}{x^3})$.
2. $x^3+\frac{1}{x^3}=(-6{)}^3-3(-6)=-216+18=-198$.
3. Required value $=198$.

Yaad rakhein: $t^3+\frac{1}{t^3}={(t+\frac{1}{t})}^3-3(t+\frac{1}{t})$; here $t=x$.

Answer: (D) 198

Concept: Rank by total marks of each student.

1. A $=60+70+80=210$; B $=50+45+55=150$.
2. C $=70+60+55=185$; D $=75+80+75=230$.
3. Highest: D.

Trick: Answer is the letter D, which is option (B).

Answer: (B) D

Concept: Apply BODMAS: $÷$ and $\times$ left to right before $+$ / $-$.

1. Numerator: $\frac{3}{7}\times \frac{9}{5}÷\frac{27}{7}=\frac{27}{35}\times \frac{7}{27}=\frac{1}{5}$; $\frac{5}{2}-\frac{1}{5}=\frac{23}{10}$.
2. Denominator: $\frac{7}{5}÷\frac{7}{2}\times \frac{5}{4}=\frac{2}{5}\times \frac{5}{4}=\frac{1}{2}$; $\frac{3}{2}+\frac{1}{2}=2$.
3. Value $=\frac{23}{10}÷2=\frac{23}{20}=1\frac{3}{20}$.

Trick: Convert mixed numbers first: $1\frac{4}{5}=\frac{9}{5}$, $3\frac{6}{7}=\frac{27}{7}$.

Answer: (C) $1\frac{3}{20}$

Concept: Mean proportional of $a,b$ is $\sqrt{ab}$, and $ab=\text{HCF}\times \text{LCM}$.

1. $ab=15\times 60=900$.
2. Mean proportional $=\sqrt{900}=30$.

Yaad rakhein: Product of two numbers $=\text{HCF}\times \text{LCM}$.

Answer: (A) 30

Concept: Area of sector $=\frac{1}{2}\times l\times r$.

1. $128=\frac{1}{2}\times 64\times r$.
2. $r=\frac{128}{32}=4\text{cm}$.

Yaad rakhein: Sector area $=\frac{1}{2}lr$, like a triangle with base $l$ and height $r$.

Answer: (A) 4 cm

Concept: Add each player’s centuries over the 5 years.

1. A $=1+0+2+1+1=5$; B $=1+2+0+0+1=4$; C $=0+0+1+1+0=2$.
2. D $=0+1+1+0+1=3$; E $=1+0+1+0+1=3$.
3. D and E are equal ($3$ each).

Trick: Only one pair has equal totals.

Answer: (A) D and E

Concept: $\text{HCF}=\frac{\text{HCF of numerators}}{\text{LCM of denominators}}$.

1. HCF$(11,9,16,10)=1$.
2. LCM$(25,20,15,33)=2^2\times 3\times 5^2\times 11=3300$.
3. HCF $=\frac{1}{3300}$.

Yaad rakhein: HCF of fractions: H/L.

Answer: (A) $\frac{1}{3300}$

Concept: Take ages as $3x$ and $2x$ and use the product.

1. $3x\times 2x=486\Rightarrow x^2=81\Rightarrow x=9$.
2. Ages: $27$ and $18$.
3. After 5 years: $32:23$.

Trick: $486=27\times 18$ — spot it directly.

Answer: (D) 32 : 23

Concept: Solve brackets first; division left to right.

1. $16÷8÷2=1$; first bracket $=1+7+1=9$.
2. $6\times 2^2+6=30$; $\frac{2}{\sqrt{36}}=\frac{1}{3}$; second bracket $=10$.
3. Total $=9+10=19$.

Trick: $16÷8÷2$ is $(16÷8)÷2=1$, not $16÷4$.

Answer: (C) 19

Concept: The blank needs an idiom for a quiet, peaceful moment just before trouble or a tense event begins.

1. The family meeting on a property dispute is likely to be tense and noisy; the speaker is enjoying coffee before it starts.
2. The calm before the storm = a quiet period just before a difficult or turbulent event — fits exactly.
3. The better of is incomplete (get the better of = defeat); the corridors of power = places where important decisions are made; the bottom line = the most important fact/final result.

Trick: Coming trouble + present peace = calm before the storm.

Answer: (C) the calm before the storm

Concept: Phrasal verbs have fixed meanings: come by = to obtain/get something (often by chance); come about = to happen.

1. The sentence asks how the person got his purse, so the meaning needed is 'obtain'.
2. Come about means 'happen', which does not fit with an object (his purse).
3. Come into = inherit; come off = become detached/succeed; come out = be revealed. Only come by = obtain.

Trick: How did you come by it? = How did you get it?

Answer: (D) come by

Concept: A is used before a consonant sound, an before a vowel sound.

1. person, hobby, job begin with consonant sounds → a person, a hobby, a job.
2. activity begins with a vowel sound → an activity.
3. Option B wrongly has 'an job', C has 'an person', D has 'a activity'. Only A uses all articles correctly.

Yaad rakhein: Judge by the first sound, not the spelling: a university, an hour.

Answer: (A) the sentence using 'a person', 'an activity', 'a hobby', 'a job'

Concept: Pester means to trouble or annoy someone with repeated requests or interruptions.

1. Pester ≈ annoy, bother, harass (e.g. Children pester parents for toys).
2. Abuse = treat badly/insult (much stronger); Adore = love deeply; Amuse = entertain.
3. So the closest synonym is Annoy.

Trick: Pest (कीड़ा) keeps bothering you → pester = annoy.

Answer: (C) Annoy

Concept: Burn one's bridges = to do something that makes it impossible to go back to an earlier situation.

1. The underlined part describes an irreversible step — exactly the meaning of burning your bridges.
2. Painting the town red = celebrating wildly; knocking your socks off = greatly impressing; a red-letter day = a memorable, happy day.
3. Sometimes burning your bridges strengthens you, because there is no way back.

Trick: Burn the bridge behind you → no return path.

Answer: (C) Burning your bridges

Concept: A clause like I have ever come across compares one person with all others, so the superlative degree is required (the most …).

1. 'She is the very ambitious woman I have ever come across' uses the positive degree very ambitious — wrong.
2. Correct: 'She is the most ambitious woman I have ever come across.'
3. The other segments are grammatically fine, so the error lies in 'the very ambitious woman'.

Trick: '… ever + present perfect' after a noun → use the superlative.

Answer: (C) the very ambitious woman

Concept: After to (infinitive marker), the verb must be in its base form (V1): to speak, to go, to write.

1. 'to spoke' (V2) and 'to spoken' (V3) are wrong; 'to speaking' is also wrong here because afraid to takes the infinitive.
2. The sentence is in the present (I am …), so no need to change it to was.
3. Correct: 'I am not afraid to speak the truth.'

Yaad rakhein: to + V1 always: to speak, never to spoke.

Answer: (B) am not afraid to speak

Concept: Wild goose chase = a long, hopeless search for something that cannot be found; a futile pursuit.

1. A wild goose is almost impossible to catch, so chasing it wastes time and effort.
2. Example: Looking for the lost ring on the beach was a wild goose chase.
3. 'Working very hard' misses the idea of uselessness; 'crazy' and 'betray' are unrelated.

Trick: Chasing a wild goose = getting nothing → futile.

Answer: (D) Futile search

Concept: Have an axe to grind = to have a private or selfish motive behind what one does.

1. Saira cannot keep friends because she always has selfish motives — she wants to serve her own interests.
2. 'To use weapons carefully' is a literal trap; 'work for others' and 'selflessness' are the opposite meaning.
3. Hence: to have one's own interests to serve.

Trick: Grinding (sharpening) your own axe = selfish motive.

Answer: (B) To have one’s own interests to serve

Concept: Vacant = empty, not filled (a vacant post is one with nobody in it).

1. The opposite of a vacant post is a post that is filled, i.e. occupied.
2. Empty is a synonym, not an antonym.
3. Indistinct (unclear) and Lucid (clear) are unrelated.

Yaad rakhein: Vacant seat ↔ occupied seat.

Answer: (C) Occupied