Q&A: practice questions with answers

Concept: $SP=MP\times \frac{100-\text{discount}\%}{100}$.

1. SP = $4,500\times \frac{81}{100}$.
2. $=45\times 81=$ ₹3,645.

Trick: 19% of 4,500 = $900-45=855$ (20% minus 1%); $4,500-855=3,645$.

Answer: (B) ₹3,645

Concept: In three terms of an AP, the middle term is the mean of the outer two.

1. Middle radius = $(5+15)÷2$.
2. $=20÷2=10$ cm.

Trick: For any three AP terms a, b, c: $2b=a+c$.

Answer: (C) 10 cm

Concept: Volume of a cylinder = $\pi r^{2}h$.

1. $h=$ diameter $=2\times 6=12$ cm.
2. $V=\pi \times 6^2\times 12=432\pi {\text{cm}}^3$.

Trick: When $h=2r$, $V=2\pi r^3=2\pi \times 216=432\pi$.

Answer: (A) 432π cm³

Concept: Volume of any prism = base area × height (length).

1. $V=35\times 8$.
2. $=280{\text{cm}}^3$.

Trick: Same rule works for all prisms and cylinders: area of cross-section × length.

Answer: (C) 280 cm³

Concept: $\cot A$ $=\frac{\text{base}}{\text{perpendicular}}$; $\cos A$ $=\frac{\text{base}}{\text{hypotenuse}}$.

1. $\cot A=\frac{3}{1}$ ⇒ base = 3, perpendicular = 1.
2. Hypotenuse = $\sqrt{3^2+1^2}=\sqrt{10}$.
3. $\cos A=\frac{3}{\sqrt{10}}$.

Trick: Since $\cot A$ exceeds 1, A is below ${45}^{\circ }$, so $\cos A$ must exceed $\frac{1}{\sqrt{2}}\approx 0.71$; $\frac{3}{\sqrt{10}}\approx 0.95$ fits.

Answer: (D) $\frac{3}{\sqrt{10}}$

Concept: Degrees = radians $\times \frac{180}{\pi }$.

1. $5\times \frac{180}{\pi }=\frac{900}{\pi }$.
2. With $\pi =3.14$: $900÷3.14\approx {286.62}^{\circ }$.

Trick: 1 radian ≈ 57.3°, so 5 rad ≈ 286.5° — only 286.62° is close.

Answer: (C) 286.62°

Concept: cot θ = tan(90° − θ).

1. tan P = cot Q = tan(90° − Q).
2. Both acute ⇒ P = 90° − Q ⇒ P + Q = 90°.

Trick: Any 'co-' pair (sin–cos, tan–cot, sec–cosec) being equal means the angles add to 90°.

Answer: (B) 90°

Concept: Sum of interior angles of an n-sided polygon = $(n-2)\times {180}^{\circ }$.

1. Pentagon: $3\times {180}^{\circ }={540}^{\circ }$.
2. $5y+50=540\Rightarrow y={98}^{\circ }$.

Trick: Angles are in AP, so the middle one $(y+10)=\frac{540}{5}={108}^{\circ }$; smallest = $108-10={98}^{\circ }$.

Answer: (B) 98°

Concept: An altitude is perpendicular to the opposite side; for perpendicular lines $m_1\times m_2=-1$.

1. Slope of QR = $\frac{3-1}{5-1}=\frac{2}{4}=\frac{1}{2}$.
2. Slope of PS = $-1÷\left(\frac{1}{2}\right)=-2$.

Trick: Perpendicular slope = 'flip and change sign': $\frac{1}{2}\to -\frac{2}{1}$. P's coordinates are not even needed.

Answer: (B) −2

Concept: Find all three side lengths of each triangle and compare.

1. △ABC: $AB=5$, $AC=12$, $BC=\sqrt{25+144}=13$.
2. △XYZ: $XY=12$, $XZ=5$, $YZ=\sqrt{144+25}=13$.
3. Sides 5, 12, 13 match ⇒ congruent by SSS.

Trick: Both are right triangles with legs 5 and 12 (the 5-12-13 triplet) — one look at the axis-parallel sides is enough.

Answer: (A) Yes, by SSS

Concept: Write each root as (number) $\times \sqrt{5}$ by taking out perfect squares.

1. $\sqrt{45}=3\sqrt{5}$, $\sqrt{20}=2\sqrt{5}$ $\Rightarrow$ bracket $=\sqrt{5}$.
2. $2\sqrt{5}\times \sqrt{5}=2\times 5=10$.

Trick: Pull out the square factor first ($45=9\times 5$, $20=4\times 5$) — the surd then cancels neatly.

Answer: (D) 10

Concept: Angle at the centre = 2 × angle at the circumference (same arc).

1. Central angle = 100°; the inscribed angle stands on the same arc.
2. Angle at circumference = 100° ÷ 2 = 50°.

Trick: On the opposite (minor-arc) side the angle would be 180° − 50° = 130° — read 'same side' carefully.

Answer: (A) 50°

Concept: Segment area = sector area − area of the triangle formed by the two radii.

1. Sector = $\frac{90}{360}\times \pi \times 4^2=4\pi$.
2. Triangle (right-angled) = $\frac{1}{2}\times 4\times 4=8$.
3. Segment = $(4\pi -8){\text{cm}}^2$.

Trick: For a ${90}^{\circ }$ angle: segment = $r^2(\frac{\pi }{4}-\frac{1}{2})$; with $r^2=16\to 4\pi -8$.

Answer: (A) 4π − 8

Concept: ${\sin }^{2}A+{\cos }^{2}A-2\sin A\cos A=(\sin A-\cos A{)}^2$.

1. $\cos A=\frac{4}{5}\Rightarrow \sin A=\frac{3}{5}$ (3-4-5 triangle).
2. ${(\frac{3}{5}-\frac{4}{5})}^2={(-\frac{1}{5})}^2=\frac{1}{25}$.

Trick: Alternatively $1-2\sin A\cos A=1-2\times \frac{12}{25}=\frac{1}{25}$.

Answer: (C) $\frac{1}{25}$

Concept: Together time = $\frac{a\times b}{a+b}$.

1. One-day work: $\frac{1}{30}+\frac{1}{60}=\frac{3}{60}=\frac{1}{20}$.
2. Time = $\frac{30\times 60}{30+60}=\frac{1800}{90}=20$ days.

Trick: When one takes double the other's time (30 and 60), together time = $\frac{2}{3}$ of the faster one: $\frac{2}{3}\times 30=20$.

Answer: (D) 20 days

Concept: Correct spelling: N-A-R-R-A-T-I-V-E (two r's, one t, ends in -ive).

1. 'Narrative' = a story or an account of events; it fits 'the ___ of the novel'.
2. 'narative' drops one r; 'nerritive' and 'naritivve' change vowels and letters.

Trick: Link it to 'narrate' (double r) + -ive → narrative.

Answer: (A) narrative

Concept: Acquisitive = excessively eager to get and own things (money, property).

1. Greedy = wanting more than one needs — same sense.
2. Generous, Selfless, Benevolent all mean giving to others — opposite sense.

Trick: Acquisitive comes from 'acquire' (to get) — the one who always wants to get is greedy.

Answer: (A) Greedy

Concept: Apocryphal = (of a story) widely told but probably not true; of doubtful origin.

1. Opposite: Authentic = genuine, proven true.
2. Spurious, Dubious, Questionable are close in meaning to apocryphal — they are synonyms, not antonyms.

Trick: When three options mean the same thing, they are synonyms of the word; the odd one out is the antonym.

Answer: (B) Authentic

Concept: 'Break the bank' = to cost too much / use up all one's money.

1. Example: Buying a phone shouldn't break the bank (shouldn't cost all your money).
2. It is about overspending — not winning, saving or borrowing.

Yaad rakhein: Mostly used in the negative: 'It won't break the bank' = it's affordable.

Answer: (A) Spend all one's money

Concept: 'Lay it on the line' = to say something clearly and honestly, without hiding anything.

1. Example: The manager laid it on the line — targets must be met.
2. 'Hide the truth' is the opposite; relaxing and playing carefully are unrelated.

Yaad rakhein: Similar idioms: 'call a spade a spade', 'speak one's mind', 'not mince words'.

Answer: (C) To speak frankly and honestly