Q&A: practice questions with answers

Concept: Apply the first discount, then find what percentage the second cut is.

1. After 20% off: $30000\times 0.8=$ ₹24,000.
2. Second cut = $24000-16800=$ ₹7,200.
3. Rate = $\frac{7200}{24000}\times 100=30\%$.

Trick: $\frac{16800}{30000}=0.56=0.8\times 0.7$ → second factor 0.7 → 30% off.

Answer: (A) 30%

Concept: Discount % is always on the marked price.

1. Take CP = 100 → MP = 120, SP = 108.
2. Discount = $120-108=12$.
3. Rate = $\frac{12}{120}\times 100=10\%$.

Trick: Discount % = $\frac{M-P}{100+M}\times 100=\frac{20-8}{120}\times 100=10\%$.

Answer: (D) 10%

Concept: Find alcohol and water separately from each part, then add.

1. From A (2 : 1), 9 L → alcohol 6 L, water 3 L.
2. From B (4 : 3), 14 L → alcohol 8 L, water 6 L.
3. Total: alcohol 14 L, water 9 L → 14 : 9.

Trick: Total 23 L = 14 + 9 – only options with parts adding to 23 are 14 : 9 and 9 : 14; alcohol is the larger part.

Answer: (B) 14 : 9

Concept: Area $\propto r^2$, volume $\propto r^3$ for similar solids.

1. Surface area ($3\pi r^2$ or $2\pi r^2$) $\propto r^2$ → $2^2=4$ → 4 : 1.
2. Volume $\left(\frac{2}{3}\pi r^3\right)$ $\propto r^3$ → $2^3=8$ → 8 : 1.

Trick: Scale factor k: lengths $\times k$, areas $\times k^2$, volumes $\times k^3$ – for any shape.

Answer: (A) 4 : 1 and 8 : 1

Concept: Equal volumes on the same base radius R.

1. Hemisphere: $(\frac{2}{3})\pi R^3$; Cone: $(\frac{1}{3})\pi R^{2}h$.
2. $(\frac{1}{3})\pi R^{2}h=(\frac{2}{3})\pi R^3\to h=2R$.

Trick: Cone : hemisphere : cylinder of same r and h = r have volumes 1 : 2 : 3 → cone needs $h=2R$ to match the hemisphere.

Answer: (B) 2R

Concept: Volume stays the same when a solid is melted and recast.

1. Sphere: $(\frac{4}{3})\pi \times 8^3=(\frac{4}{3})\pi \times 512$.
2. 4 cones: $4\times (\frac{1}{3})\pi \times 4^2\times h=(\frac{4}{3})\pi \times 16h$.
3. $16h=512\to h=32$ cm.

Trick: Cancel $(\frac{4}{3})\pi$ on both sides at once: $r^3=(\text{cone }{r}^2)\times h\to 512=16h$.

Answer: (B) 32 cm

Concept: LSA of a regular pyramid = $\frac{1}{2}\times$ base perimeter $\times$ slant height.

1. Perimeter = 30 cm, slant height = 10 cm.
2. LSA = $\frac{1}{2}\times 30\times 10=150$ cm².

Trick: Same as 4 triangles: each base 7.5, height 10 → $4\times \frac{1}{2}\times 7.5\times 10=150$.

Answer: (C) 150 cm²

Concept: Area ratio = $(\text{radius ratio}{)}^2$.

1. Radius 2 : 5 → area 4 : 25.
2. Larger = $100\pi \times \frac{25}{4}=625\pi$ cm².

Trick: Smaller 100π → $r^2=100\to r=10$ → larger $r=25$ → $625\pi$.

Answer: (C) 625π cm²

Concept: The centroid cuts each median in 2 : 1, the longer part towards the vertex.

1. O is the centroid (meeting point of medians).
2. $PO=\frac{2}{3}\times PS=\frac{2}{3}\times 15=10$ cm.

Trick: Median ÷ 3 = 5 → vertex side gets 2 parts (10), side part 1 (5).

Answer: (A) 10 cm

Concept: RHS: right angle, equal hypotenuse and one equal side.

1. ∠Q = ∠Y = 90° (Right angle).
2. PR = XZ (Hypotenuse), PQ = XY (Side).
3. So ΔPQR ≅ ΔXYZ by RHS.

Trick: SAS would need the included angle between the two equal sides – here the 90° angle is not between PQ and PR.

Answer: (A) Yes, by RHS

Concept: In a right triangle, leg² = (its projection on hypotenuse) × hypotenuse.

1. Right angle at B lies on the circle → AC is the diameter = 26 cm.
2. $A{B}^2=AD\times AC\to 100=AD\times 26$.
3. $AD=\frac{100}{26}\approx 3.85\approx 3.8$ cm.

Trick: Pattern: $A{B}^2=AD·AC,B{C}^2=CD·AC,B{D}^2=AD·CD$ – the 'projection rule'.

Answer: (A) 3.8 cm

Concept: Number of common tangents depends on how the circles are placed.

1. Externally touching: 2 direct tangents + 1 tangent at the contact point = 3.

Trick: Separate 4 → touch externally 3 → intersect 2 → touch internally 1 → one inside other 0.

Answer: (A) There are three common tangents to the circles

Concept: Direct common tangent length = $\sqrt{d^2-(R-r{)}^2}$, d = distance between centres.

1. Touching externally → centre distance $d=R+r$.
2. Using $a^2-b^2=(a+b)(a-b)$: $(R+r{)}^2$ minus $(R-r{)}^2$ equals $(2R)(2r)=4Rr$.
3. So $D=\sqrt{4Rr}=2\sqrt{Rr}$.

Trick: D is twice the geometric mean of the radii – e.g. $R=9,r=4\to D=2\times 6=12$.

Answer: (C) D = 2$\sqrt{\mathrm{Rr}}$

Concept: Angle at centre = 2 × angle at circumference; the perpendicular from the centre bisects the chord.

1. Central angle = $2\times {30}^{\circ }={60}^{\circ }$; half of it = 30°.
2. Half chord = 5 cm, opposite the 30° angle at O.
3. $\tan {30}^{\circ }=\frac{5}{d}\to d=\frac{5}{1/\sqrt{3}}=5\sqrt{3}$ cm.

Trick: 60° central angle makes an equilateral triangle: radius = chord = 10 → distance = $\left(\frac{\sqrt{3}}{2}\right)\times 10=5\sqrt{3}$.

Answer: (A) 5$\sqrt{3}$ cm

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

1. Sector = $\frac{60}{360}\times \pi \times 4^2=\frac{16\pi }{6}=\frac{8\pi }{3}$.
2. 60° with equal radii → equilateral triangle of side 4: area = $\left(\frac{\sqrt{3}}{4}\right)\times 16=4\sqrt{3}$.
3. Segment = $\frac{8\pi }{3}-4\sqrt{3}{\text{cm}}^2$.

Trick: For 60°, segment = $r^2(\frac{\pi }{6}-\frac{\sqrt{3}}{4})\to 16(\frac{\pi }{6}-\frac{\sqrt{3}}{4})=\frac{8\pi }{3}-4\sqrt{3}$.

Answer: (A) ($\frac{8\pi }{3}$) − 4$\sqrt{3}$

Concept: A word for writing in a secret code.

1. Cipher = a secret system of writing that hides the message.
2. Epistle = a formal letter; Diatribe = a bitter verbal attack; Manuscript = a handwritten/unpublished text.

Yaad rakhein: Decipher = to decode a cipher; cryptography = the science of codes.

Answer: (A) Cipher

Concept: A general official pardon, usually for political offences.

1. Amnesty = official pardon for people convicted of political offences.
2. Reprieve = delay/cancellation of a punishment (often a death sentence); Parole = conditional early release; Mercy = kindness in general.

Yaad rakhein: Amnesty International – a human-rights body; root 'amnesia' = forgetting (the offence is 'forgotten').

Answer: (A) Amnesty

Concept: Choose the phrasal verb that fits the logic of the sentence.

1. Investors usually avoid risky stocks.
2. Shy away from = avoid out of fear or caution → fits.
3. Lean towards = prefer (opposite sense); brush over = treat lightly; walk into = fall into a trap.

Trick: 'Tend to' + 'high-risk' signals caution → look for an 'avoid' meaning.

Answer: (C) shy away from

Concept: Find the word closest in meaning to 'explain' (make clear).

1. Illuminate = throw light on, make clear → synonym.
2. Confuse and darken are opposites; destroy is unrelated.

Yaad rakhein: Elucidate, clarify, expound, illuminate – all mean 'explain'.

Answer: (B) Illuminate

Concept: Find the word opposite to 'agree'.

1. Demur = raise an objection, show reluctance → opposite of agree.
2. Concur and accept mean agree; allow = permit.

Yaad rakhein: Concur/assent = agree; demur/dissent = disagree.

Answer: (B) Demur