Q&A: practice questions with answers

Concept: $\text{Area}=\frac{1}{2}\times (\text{sum of parallel sides})\times \text{height}$.

1. $98=\frac{1}{2}\times (12+16)\times h$
2. $98=14h$
3. $h=$ 7 cm

Trick: Half the sum of bases (14) is the 'average width'; $98÷14=7$.

Answer: (B) 7

Concept: $\text{TSA}=2(lb+bh+hl)$; cost = area × rate.

1. $lb=10,bh=3.75,hl=6$ → sum 19.75
2. $\text{TSA}=2\times 19.75=39.5{\text{m}}^2$
3. $\text{Cost}=39.5\times 12=$ ₹474

Trick: Cost must be 12 × (something ending in .5) – only 474 is $12\times 39.5$.

Answer: (C) Rs. 474

Concept: $V=\frac{1}{3}\times \text{base area}\times \text{height}$; equilateral triangle area = $\frac{\sqrt{3}}{4}{a}^2$.

1. $\text{Base}=\frac{\sqrt{3}}{4}\times 100=25\sqrt{3}{\text{cm}}^2$
2. $V=\frac{1}{3}\times 25\sqrt{3}\times 9=75\sqrt{3}$
3. $=75\times 1.732\approx$ 129.9 cm³

Yaad rakhein: Volume of pyramid/cone = $\frac{1}{3}$ of the prism/cylinder with same base and height.

Answer: (A) $129.9{\mathrm{cm}}^3$

Concept: Equal base angles ⇒ $PQ=QR=10$; altitude to hypotenuse = half the hypotenuse.

1. $PR=10\sqrt{2}\to P{R}^2=200$
2. $QM=\frac{PR}{2}=5\sqrt{2}\to Q{M}^2=50$
3. $P{R}^2-Q{M}^2=200-50=$ 150

Trick: $PR=2QM\Rightarrow P{R}^2-Q{M}^2=4Q{M}^2-Q{M}^2=3Q{M}^2=3\times 50=150$.

Answer: (A) 150

Concept: Medians of a triangle meet at the centroid.

1. Median: vertex to midpoint of opposite side.
2. Meeting point = centroid, dividing each median 2 : 1 from the vertex.
3. Circumcentre – perpendicular bisectors; incentre – angle bisectors; orthocentre – altitudes.

Trick: M-C: Medians → Centroid; A-O: Altitudes → Orthocentre.

Answer: (B) Centroid

Concept: Set $\frac{1}{3}\pi r^{2}h=\frac{1}{2}\times \pi R^2(2h)$.

1. $\frac{1}{3}\times 144\times h=R^2\times h$
2. $R^2=48$
3. $R=\sqrt{48}=$ $4\sqrt{3}$ cm

Yaad rakhein: Cone volume = $\frac{1}{3}\pi r^{2}h$; cylinder = $\pi r^{2}h$.

Answer: (B) $4\sqrt{3}\mathrm{cm}$

Concept: (r₁ − r₂) < d < (r₁ + r₂) means the circles intersect at two points.

1. Intersecting circles: no transverse tangent possible.
2. Only 2 direct common tangents exist → 2.

Yaad rakhein: Apart: 4, touching outside: 3, intersecting: 2, touching inside: 1, one inside other: 0.

Answer: (B) 2

Concept: Write in sin and cos.

1. $\cot x=\frac{\cos x}{\sin x},\csc x=\frac{1}{\sin x}$
2. $\cot x÷\csc x=\frac{\cos x}{\sin x}\times \sin x$
3. $=$ $\cos x$

Trick: Put $x={45}^{\circ }$: $1÷\sqrt{2}=\frac{1}{\sqrt{2}}=\cos {45}^{\circ }$ ✓.

Answer: (A) $\cos x$

Concept: Distributed profit = 90% of total; P gets $\frac{5}{8}$ of it.

1. $\text{P's share}=\frac{5}{8}\times 90\%\text{ of }X=900$
2. $\text{Distributed}=900\times \frac{8}{5}=1440$
3. $X=1440÷0.9=$ ₹1600

Trick: Work backwards: 900 → $\times \frac{8}{5}$ → 1440 → $÷0.9$ → 1600.

Answer: (D) Rs. 1600

Concept: Apply each discount on the reduced price.

1. $1500\times 0.9=1350$
2. $1350\times 0.8=$ ₹1080

Trick: Net discount = $10+20-\frac{10\times 20}{100}=28\%$ → 72% of 1500 = 1080.

Answer: (B) ₹1080

Concept: Find one part's value from the known term.

1. 7 parts = 280 → 1 part = 40
2. Child = $4\times 40=$ 160

Trick: Child = $280\times \frac{4}{7}=160$.

Answer: (C) 160

Concept: $\text{Total}=\text{average}\times \text{number of items}$.

1. $\text{Required total}=52\times 5=260$
2. $\text{Current total}=40+55+60+45=200$
3. 5th test = $260-200=$ 60

Trick: Deviation method: current avg 50 is 2 short on 4 tests (−8); need $52+8=60$.

Answer: (C) 60

Concept: Internal tangent $L=\sqrt{d^2-(r_1+r_2{)}^2}$.

1. $r_1+r_2=10,L=24$
2. $d^2={24}^2+{10}^2=576+100=676$
3. $d=$ 26 units

Trick: 10, 24, 26 is 2 × (5, 12, 13) triplet.

Answer: (D) 26 unit

Concept: $\csc =\frac{\text{hypotenuse}}{\text{perpendicular}}$; $\sec =\frac{\text{hypotenuse}}{\text{base}}$.

1. $\csc x=\frac{5}{3}$ → P = 3, H = 5
2. $\text{Base}=\sqrt{25-9}=4$
3. $\sec x=\frac{5}{4}$

Trick: 3-4-5 triplet: missing side is 4 instantly.

Answer: (B) $\frac{5}{4}$

Concept: ${\sec }^2=1+{\tan }^2,{\csc }^2=1+{\cot }^2$.

1. $\cot \theta =\frac{3}{4}\to \tan \theta =\frac{4}{3}$
2. ${\sec }^2\theta =1+\frac{16}{9}=\frac{25}{9}$; ${\csc }^2\theta =1+\frac{9}{16}=\frac{25}{16}$
3. $\frac{25}{9}-\frac{25}{16}=25\times \frac{7}{144}=$ $\frac{175}{144}$

Trick: Answer must be positive with denominator 144 ($=9\times 16$) – only A and C; $25\times 7=175$ → A.

Answer: (A) $\frac{175}{144}$

Concept: Mongol threat forced Sultans to keep huge permanent armies at Delhi.

1. Alauddin Khalji: paid soldiers in cash, fixed market prices, introduced dagh (horse branding) and chehra (soldier's descriptive roll).
2. Muhammad bin Tughluq: raised a very large army (incl. the planned Khurasan campaign); paying it strained his treasury and led to token-currency experiments.
3. Both statements are correct.

Yaad rakhein: Alauddin's market control was run by the Diwan-i-Riyasat and Shahna-i-Mandi.

Answer: (D) Both I and II

Concept: Alvars were Tamil Vaishnava saints.

1. Their 4,000 hymns form the Nalayira Divya Prabandham, compiled by Nathamuni.
2. Silappadikaram and Manimekalai are Sangam-era epics; Tirukkural is Thiruvalluvar's ethical work.

Trick: Alvar → Vishnu → Divya Prabandham; Nayanar → Shiva → Tevaram/Tirumurai.

Answer: (B) Divya Prabandham

Concept: Fundamental Duties are listed in Article 51A (Part IV-A).

1. 51A(g): protect and improve the natural environment (forests, lakes, rivers, wildlife).
2. Paying taxes and voting are not in the list.
3. 51A(k) covers education for children aged 6–14, not up to 18.

Yaad rakhein: Duties added by the 42nd Amendment, 1976 (Swaran Singh Committee); 11th duty added by the 86th Amendment, 2002.

Answer: (D) Protecting and improving the natural environment

Concept: The 36th Amendment Act, 1975 made Sikkim a full State.

1. Inserted Article 371F with special provisions for Sikkim.
2. 35th Amendment (1974) had given only 'associate State' status.
3. 42nd (1976) is the 'mini-Constitution'; 38th dealt with emergency-related matters.

Trick: 35 = associate, 36 = full State – one step up.

Answer: (D) 36th Constitutional Amendment Act, 1975

Concept: SAARC has 8 members.

1. Members: Afghanistan, Bangladesh, Bhutan, India, Maldives, Nepal, Pakistan, Sri Lanka.
2. Iran is only an observer.

Trick: Mnemonic 'MBBS PAIN': Maldives, Bangladesh, Bhutan, Sri Lanka, Pakistan, Afghanistan, India, Nepal.

Answer: (B) Iran