Q&A: practice questions with answers

Concept: After n replacements, remaining acid = initial acid $\times {(1-\frac{x}{V})}^n$.

1. Initial acid = $\frac{3}{5}$ of 100 L = 60 L
2. Each operation keeps 90% of the acid: $60\times {\left(\frac{9}{10}\right)}^2=60\times 0.81=48.6$ L
3. Volume stays 100 L → acid = 48.6%

Trick: Acid % directly: $60\%\times 0.81=48.6\%$ – no need to track litres.

Answer: (D) 48.6%

Concept: Total work = efficiency × days; use efficiencies as units per day.

1. Ashish: 2 units/day for 40 days → total = 80 units
2. All three: $2+3+5=10$ units/day → 5 days = 50 units; left = 30
3. Ashish + Navin: 5 units/day → $30÷5=6$ days

Trick: Take the efficiency ratio itself as daily units – the whole problem becomes simple division.

Answer: (D) 6 days

Concept: Lateral surface area of a prism = base perimeter × height.

1. Base perimeter = $3+4+5=12$ cm
2. LSA = $12\times 8=96$ cm²

Trick: Lateral faces are three rectangles: $3\times 8+4\times 8+5\times 8=24+32+40=96$. Base area (6 cm² each) is not included.

Answer: (C) 96 cm²

Concept: Frustum volume = big cone − small (similar) cone cut from the top.

1. Similar cones: r/h same → small cone radius = $6\times \frac{6}{9}=4$ cm, height 6 cm
2. Big cone = $\frac{1}{3}\pi \times 36\times 9=108\pi$; small cone = $\frac{1}{3}\pi \times 16\times 6=32\pi$
3. Frustum = $108\pi -32\pi =76\pi$ cm³

Trick: Volumes of similar cones scale as $(\text{height ratio}{)}^3$: ${\left(\frac{2}{3}\right)}^3=\frac{8}{27}$ → frustum = $108\pi \times \frac{19}{27}=76\pi$.

Answer: (B) 76π cm³

Concept: $\tan A$ $=\frac{\text{perpendicular}}{\text{base}}$; complete the triangle with Pythagoras.

1. $\tan A=\frac{3}{4}$ → sides 3, 4, hypotenuse 5
2. $\sin A=\frac{3}{5}$, $\cos A=\frac{4}{5}$
3. $3\times \frac{3}{5}-4\times \frac{4}{5}=\frac{9-16}{5}=-\frac{7}{5}$

Trick: Divide by $\cos A$: $3\sin A-4\cos A=\cos A(3\tan A-4)=\frac{4}{5}\times (\frac{9}{4}-4)=-\frac{7}{5}$.

Answer: (A) −$\frac{7}{5}$

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

1. $3.5\times 180=630$
2. $630÷3.14\approx {200.63}^{\circ }$

Trick: 1 radian $\approx {57.3}^{\circ }$, so 3.5 rad $\approx {200}^{\circ }$ – only option (C) is near 200.

Answer: (C) 200.63°

Concept: If sin P = cos Q (acute angles), then P + Q = 90°.

1. 2x + 3x − 20° = 90°
2. 5x = 110° → x = 22°

Trick: Check: 2x = 44°, 3x − 20° = 46°, and 44° + 46° = 90° ✓.

Answer: (C) 22°

Concept: Exterior angle at a vertex = 180° − interior angle at that vertex.

1. Interior angle at the right-angled vertex = 90°
2. Exterior = 180° − 90° = 90°

Trick: The 60° is a distractor – the exterior angle at the right angle is always 90°, whatever the other angles.

Answer: (D) 90°

Concept: Orthocentre position depends on the type of triangle.

1. 105° > 90° → obtuse triangle.
2. Acute: inside; right: at the right-angle vertex; obtuse: outside.

Trick: Orthocentre and circumcentre both go outside for an obtuse triangle; centroid and incentre always stay inside.

Answer: (B) Outside the triangle

Concept: Multiply by the conjugate of the denominator: $(\sqrt{a}-\sqrt{b})(\sqrt{a}+\sqrt{b})=a-b$.

1. $\frac{\sqrt{6}+\sqrt{7}}{\sqrt{6}-\sqrt{7}}\times \frac{\sqrt{6}+\sqrt{7}}{\sqrt{6}+\sqrt{7}}$
2. Top: $(\sqrt{6}+\sqrt{7}{)}^2=6+7+2\sqrt{42}=13+2\sqrt{42}$; bottom: $6-7=-1$
3. Result = $-(13+2\sqrt{42})=-13-2\sqrt{42}$

Trick: Estimate: $\sqrt{6}\approx 2.45$, $\sqrt{7}\approx 2.65$ → $5.1÷(-0.2)\approx -25.5$; only $-13-2\sqrt{42}\approx -26$ is close.

Answer: (B) −13 − 2$\sqrt{42}$

Concept: A line parallel to one side divides the other two sides in the same ratio.

1. PS : SQ = PT : TR
2. 5 : 15 = 4 : TR → 1 : 3 = 4 : TR
3. TR = 12 cm

Trick: SQ is 3 times PS, so TR is 3 times PT: $3\times 4=12$.

Answer: (D) 12 cm

Concept: Reflex angle = 360° − given angle (it is the larger angle, between 180° and 360°).

1. Angle at centre = 105°
2. Reflex angle = 360° − 105° = 255°

Trick: A reflex angle is always more than 180° – of the options only 255°, 210° and 245° qualify; then just subtract.

Answer: (A) 255°

Concept: ${\sec }^{2}A-{\tan }^{2}A=1\Rightarrow (\sec A-\tan A)=\frac{1}{\sec A+\tan A}$.

1. $\sec A+\tan A=5\Rightarrow \sec A-\tan A=\frac{1}{5}$
2. Subtract: $2\tan A=5-\frac{1}{5}=\frac{24}{5}$
3. $\tan A=\frac{12}{5}$

Trick: Shortcut: if $\sec A+\tan A=k$, then $\tan A=\frac{k^2-1}{2k}=\frac{25-1}{10}=\frac{12}{5}$.

Answer: (C) $\frac{12}{5}$

Concept: Arc length = radius × angle (angle in radians).

1. Radius = $\frac{d}{2}$
2. Arc = $\frac{d}{2}\times \theta =\frac{d\theta }{2}$

Trick: Check with $\theta =2\pi$: $\frac{d\theta }{2}=\pi d$, which is the full circumference ✓.

Answer: (A) $\frac{\mathrm{d\theta }}{2}$

Concept: Take total work = LCM of days and subtract known efficiencies.

1. Total work = $\text{LCM}(6,12,18)=36$ units
2. $P+Q+R=\frac{36}{6}=6$ units/day; P = 3, Q = 2
3. $R=6-3-2=1$ unit/day → 36 days

Trick: LCM method avoids fractions: efficiencies 6, 3, 2 leave just 1 for R.

Answer: (D) 36 days

Concept: An -ing phrase after a main clause that shows the manner/result of the action is an adverbial participial phrase.

1. 'providing…' starts with the present participle of provide.
2. It explains in what way the medium allows independence – it modifies the whole action 'allows', not a noun.
3. It is not an absolute phrase (no separate subject of its own).

Trick: Ask 'How / with what result?' of the main verb – if the -ing phrase answers it, it is adverbial.

Answer: (B) Present participial phrase functioning as an adverb

Concept: A prepositional phrase that sets the circumstance for the verb works as an adverb.

1. 'Beyond the casual image…' begins with the preposition beyond.
2. It tells under what condition/context podcasters 'must negotiate' – so it modifies that verb phrase.
3. It does not rename a noun (not appositive) and does not act as a subject/object (not a noun).

Trick: Remove the phrase: the sentence stays complete – a sign it is a modifier (adverbial), not a core noun part.

Answer: (B) Adverbial phrase modifying "must negotiate"

Concept: Intricate = having many connected parts; complicated.

1. The passage follows the word with a long list: production quality, analytics, licensing, sponsorship, social media.
2. Such a many-part 'ecosystem' is complex and detailed.

Yaad rakhein: Intricate ≈ complicated, elaborate, convoluted; antonym – simple, plain.

Answer: (B) Complex and detailed

Concept: Egalitarian = based on equality; open to everyone.

1. The passage explains it: 'anyone with a microphone can… reach a global audience'.
2. Opposite idea = limited to a few → exclusive.
3. Democratic, inclusive, accessible are near-synonyms, not opposites.

Trick: Three options mean the same thing – the odd one in meaning (exclusive) is the antonym.

Answer: (A) Exclusive

Concept: The main tension is the gap between how podcasting looks and what it really demands.

1. Opening: podcasting was 'once dismissed as a hobby' and has a 'casual image'.
2. Body: success needs editing, marketing, analytics, contracts, finance – professional demands.
3. Sponsorship, local restrictions or audio vs engagement are only details, not the main conflict.

Trick: Look for contrast words – 'Beyond the casual image', 'Yet the reality' – they point to the core tension.

Answer: (A) The tension between the casual perception of podcasting and the professional demands of the medium