Q&A: practice questions with answers

Concept: $\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}$, $s=\frac{a+b+c}{2}$.

1. $s=\frac{8+12+16}{2}=18$
2. Area $=\sqrt{18\times 10\times 6\times 2}=\sqrt{2160}$
3. $=\sqrt{144\times 15}=12\sqrt{15}{\text{cm}}^2$

Trick: Sides $4(2,3,4)$: area of the $2,3,4$ triangle is $\frac{3\sqrt{15}}{4}$; multiply by $4^2=16$ to get $12\sqrt{15}$.

Answer: (D) $12\sqrt{15}$

Concept: Let $\frac{1}{d}=a$, $\frac{1}{u}=b$ (d, u = downstream, upstream speeds); still-water speed $=\frac{d+u}{2}$.

1. $40a+25b=7.5$ and $48a+36b=10$
2. Multiply: $240a+150b=45$, $240a+180b=50\Rightarrow 30b=5,b=\frac{1}{6}$
3. $40a=7.5-\frac{25}{6}=\frac{10}{3}\Rightarrow a=\frac{1}{12}$; so $d=12$, $u=6$
4. Boat speed $=\frac{12+6}{2}=9\text{km/h}$

Trick: Check: $\frac{40}{12}+\frac{25}{6}=3\frac{1}{3}+4\frac{1}{6}=7\frac{1}{2}$ h ✓.

Answer: (C) $9$

Concept: The largest number dividing all given numbers is their HCF.

1. $24=2^3\times 3$, $56=2^3\times 7$, $96=2^5\times 3$
2. Common: $2^3=8$

Trick: HCF divides differences: $56-24=32$, $96-56=40$; $\text{HCF}(32,40)=8$.

Answer: (C) $8$

Concept: TSA of solid hemisphere $=3\pi r^2$; volume is conserved: $\frac{2}{3}\pi r^3=\frac{1}{3}\pi r^{2}h$.

1. $3\times \frac{22}{7}\times r^2=462\Rightarrow r^2=49,r=7\text{cm}$
2. $\frac{2}{3}\pi r^3=\frac{1}{3}\pi r^{2}h\Rightarrow h=2r$
3. $h=14\text{cm}$

Yaad rakhein: Hemisphere $\to$ cone with the same radius always gives $h=2r$.

Answer: (A) $14\text{cm}$

Concept: Read each option's value from the table and pick the maximum.

1. A, 2011 $=700000$; D, 2012 $=700000$; D, 2014 $=700000$
2. B, 2014 $=710000$ — the highest value in the whole table

Trick: Scan for the single largest entry: 710000 occurs only once (B, 2014).

Answer: (B) B, 2014

Concept: Majority $=$ (winner% $-$ loser%) of valid votes.

1. Let voters $=V$. Cast $=0.9V$; valid $=0.9\times 0.9V=0.81V$
2. Margin $=(54\%-46\%)=8\%$ of valid $=0.08\times 0.81V=0.0648V$
3. $0.0648V=1620\Rightarrow V=25000$

Trick: Valid votes $=\frac{1620}{8\%}=20250$; $V=\frac{20250}{0.81}=25000$.

Answer: (A) $25000$

Concept: Apply the discounts one after another, checking the condition after the first.

1. List price $=12\times 750=9000$
2. After 8%: $9000\times 0.92=8280$ ($>5000$, so extra 20% applies)
3. Final $=8280\times 0.8=6624$

Trick: Net factor $=0.92\times 0.8=0.736$; $9000\times 0.736=6624$.

Answer: (B) ₹6,624

Concept: Selling at a fraction of SP gives a known loss; use it to relate SP and CP.

1. $\frac{3}{8}\text{SP}=0.55\text{CP}$ (45% loss)
2. $\frac{\text{SP}}{\text{CP}}=\frac{0.55\times 8}{3}=\frac{4.4}{3}\approx 1.4667$
3. Profit $\approx 46.67\%\approx 47\%$

Trick: Take CP $=300$: $\frac{3}{8}\text{SP}=165\Rightarrow \text{SP}=440$, profit $=\frac{140}{300}\times 100\approx 47\%$.

Answer: (C) 47 Percent

Concept: Rationalise: $\sqrt{\frac{1-\sin \theta }{1+\sin \theta }}=\frac{1-\sin \theta }{\cos \theta }=\sec \theta -\tan \theta$.

1. Multiply inside by $\frac{1-\sin \theta }{1-\sin \theta }$: $\sqrt{\frac{(1-\sin \theta {)}^2}{{\cos }^2\theta }}$
2. $=\frac{1-\sin \theta }{\cos \theta }=\sec \theta -\tan \theta$
3. With $\theta ={45}^{\circ }$: $\sec {45}^{\circ }-\tan {45}^{\circ }$

Trick: Value check: $\sec {45}^{\circ }-\tan {45}^{\circ }=\sqrt{2}-1\approx 0.414>0$; option (B) is negative, so it cannot be a square root.

Answer: (D) $\sec {45}^{\circ }-\tan {45}^{\circ }$

Concept: Simplify each part, then solve the resulting quadratic.

1. $(1÷x)\times (x^3-2x^2)=x^2-2x$
2. $(4^3÷2^5)\times x=\frac{64}{32}x=2x$
3. Sum $=x^2-2x+2x=x^2$; $\frac{x^2}{25}=1\Rightarrow x^2=25$
4. $x=5$ or $x=-5$ (both non-zero, so valid)

Trick: $x^2=25$ always has two roots $\pm 5$ — don't drop the negative one.

Answer: (D) Both 5 and $-5$

Concept: A one-word substitute replaces a whole phrase with a single precise word.

1. Zoology (zoo = animal + -logy = study) is the scientific study of animals – their behaviour, structure, physiology, classification and distribution.
2. Archaeology = study of ancient human remains; Petrology = study of rocks; Ecology = study of organisms and their environment.

Trick: Greek zōion = animal, so zo- words (zoology, zoo) relate to animals.

Answer: (A) Zoology

Concept: A synonym has the same or nearly the same meaning.

1. Peril = serious and immediate danger (e.g. 'The ship was in peril').
2. Danger matches exactly; Doubt = uncertainty; Benefit and Safety are opposite in sense.

Trick: 'Perilous journey' = dangerous journey.

Answer: (B) Danger

Concept: Spot the word whose letters are wrong.

1. The correct spelling is Perspicuous (clearly expressed, easy to understand) – there is no extra 'i' before 'ous'.
2. Purvey (to supply), Peruse (to read carefully) and Pervade (to spread through) are spelt correctly.

Trick: perspicu-ous: like 'conspicuous', the ending is just -uous.

Answer: (A) Perspicuious

Concept: A synonym has the same meaning as the given word.

1. Erudite = having or showing great knowledge gained from study; learned.
2. Scholarly means the same. Excited, Impressive and Lazy are unrelated.

Trick: Erudite ≈ 'erudition' = deep learning → scholar.

Answer: (A) Scholarly

Concept: Find the single word for 'honour given for some achievement'.

1. Laurel (often 'laurels') = honour or praise earned for an achievement; in ancient Greece winners wore laurel wreaths.
2. Placard = a poster/sign; Party = a celebration or group; Gift = a present, not necessarily for achievement.

Trick: 'Rest on one's laurels' = be content with past honours.

Answer: (B) Laurel

Concept: Choose the word with the same meaning.

1. Impasse = a situation in which no progress is possible because people cannot agree.
2. Deadlock means the same. Resolution, Peaceful and Harmony suggest the opposite (settlement/agreement).

Trick: im-passe = 'no passing' → stuck → deadlock.

Answer: (A) Deadlock

Concept: An antonym has the opposite meaning.

1. Novice = a beginner, a person new to an activity.
2. Its opposite is Expert (highly skilled person).
3. Apprentice is a synonym (learner); Mature and Stylish do not oppose 'beginner' in skill.

Trick: Novice ↔ Expert, Amateur ↔ Professional.

Answer: (B) Expert

Concept: Pick the correct spelling of the underlined word.

1. The word is spelt c-a-l-e-n-d-a-r: '-ar' at the end, 'a' after c and 'e' after l.
2. 'calender' (a machine for pressing cloth) is a different word; celender and calunder are wrong.

Trick: Two a's and an e: calendar.

Answer: (B) calendar

Concept: Some nouns take fixed prepositions.

1. We 'take/cast a glance at' something, not 'on' it.
2. Correct: He took a hurried glance at the newspaper and swiftly left for work.
3. So the error is in 'on the newspaper'.

Trick: glance at, look at, stare at – the eye-verbs take 'at'.

Answer: (C) on the newspaper

Concept: Simple past active → was/were + V3; object becomes subject, subject goes after 'by'.

1. Active: The officers (subject) kept (past) us (object) waiting …
2. Object 'us' → subject 'We'; verb 'kept' → 'were kept'.
3. Passive: We were kept waiting for one hour by the officers.
4. Other options change the tense or meaning and drop 'by the officers'.

Trick: Past simple passive = was/were + V3.

Answer: (A) We were kept waiting for one hour by the officers.