Q&A: practice questions with answers

Concept: Amount = 184% ⇒ SI = 84% of P in 6 years.

1. Rate = $84÷6=14\%$ p.a.
2. SI on 3500 for 3 years = $\frac{3500\times 14\times 3}{100}$
3. = ₹1470

Trick: 3 years at 14% = 42% of 3500 = 1470.

Answer: (D) Rs. 1470

Concept: Let shares be 4x and 5x; equate the interest difference.

1. Kavya: $4x\times 8\times \frac{5}{100}=1.6x$; Aryan: $5x\times 7\times \frac{5}{100}=1.75x$
2. $0.15x=2691\Rightarrow x=17940$
3. Distributed = $9x=161460$ = 90% of total
4. Total = $161460\times \frac{10}{9}=$ ₹1,79,400

Trick: Total = $2691÷0.15\times 9÷0.9=2691\times 60$ = 1,61,460 ÷ 0.9 – or check: 90% of 1,79,400 = 1,61,460 ✔

Answer: (B) Rs. 1,79,400

Concept: $\tan \theta =\frac{\text{height}}{\text{base}}$; use the Pythagorean triplet.

1. $\tan \theta =\frac{8}{15}$ ⇒ $\text{height}=8k,\text{base}=15k$
2. $\text{Ladder}=\sqrt{64+225}k=17k$
3. $17k=17\Rightarrow k=1\Rightarrow$ height = 8 m

Trick: 8-15-17 is a Pythagorean triplet – ladder already 17, so height is simply 8.

Answer: (C) 8 meters

Concept: Under CI, equal time blocks multiply money by the same factor.

1. 6-year factor = $\frac{9000}{4000}=\frac{9}{4}$
2. 3-year factor = $\sqrt{\frac{9}{4}}=\frac{3}{2}$
3. Amount = $3200\times \frac{3}{2}=$ ₹4,800

Trick: Half the time ⇒ square root of the growth factor.

Answer: (B) Rs. 4,800

Concept: Present ages 3x and 7x; after 6 years ratio is 3 : 5.

1. $\frac{3x+6}{7x+6}=\frac{3}{5}$
2. $15x+30=21x+18\Rightarrow x=2$
3. Son = 6, mother = 14

Trick: Test options: 6 : 14 = 3 : 7 and 12 : 20 = 3 : 5 ✔ – done.

Answer: (A) 6,14

Concept: Tangent–chord angle = inscribed angle in the alternate segment; centre angle = 2 × inscribed angle.

1. Angle between tangent at A and chord AC = 50° ⇒ inscribed angle on arc AC = 50°.
2. Angle at centre by minor arc AC = $2\times {50}^{\circ }=$ 100°.
3. The 85° (on chord AB) is extra data.

Trick: Tangent–chord angle × 2 = central angle of that chord, directly.

Answer: (C) 100°

Concept: For the same distance, time is inversely proportional to speed.

1. Train time = 6 h
2. Speed ratio 8 : 3 ⇒ time ratio 3 : 8
3. Car time = $6\times \frac{8}{3}=$ 16 hours

Trick: Same distance: ${\text{time}}_2={\text{time}}_1\times \frac{{\text{speed}}_1}{{\text{speed}}_2}$.

Answer: (B) 16 hours

Concept: Passed = $(100-\text{fail})\%$ of each group.

1. Boys passed = 70% of 5 = 3.5
2. Girls passed = 50% of 7 = 3.5
3. Ratio = 3.5 : 3.5 = 1 : 1

Trick: Directly: $5\times 70:7\times 50=350:350$.

Answer: (C) 1 : 1

Concept: sec A = cosec B ⇒ A + B = 90°.

1. sec θ = cosec 52°
2. θ + 52° = 90°
3. θ = 38°

Trick: Co-function pairs (sin–cos, tan–cot, sec–cosec) with equal values ⇒ angles add to 90°.

Answer: (B) 38°

Concept: Write everything as a power of 5 and compare exponents.

1. $(\sqrt{5}{)}^6=5^3;{25}^2=5^4\Rightarrow \text{LHS}=5^7$
2. $\text{RHS}=5^n\times 5\sqrt{5}=5^{n+\frac{3}{2}}$
3. $n+\frac{3}{2}=7\Rightarrow n=$ $\frac{11}{2}$

Trick: $\sqrt{5}=5^{\frac{1}{2}}$, so $5\sqrt{5}=5^{\frac{3}{2}}$ – convert roots to half-powers first.

Answer: (C) $\frac{11}{2}$

Concept: For $y=a-\sqrt{b}$ with $a^2-b=1$, $\frac{1}{y}=a+\sqrt{b}$.

1. $y=5-\sqrt{24};(5{)}^2-24=1$
2. $\frac{1}{y}=5+\sqrt{24}$
3. $y+\frac{1}{y}=$ 10

Trick: When $a^2-b=1$, $y+\frac{1}{y}=2a$ straight away ($2\times 5=10$).

Answer: (C) 10

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

1. Area ratio = $48:6=8$
2. Side ratio = $\sqrt{8}=2\sqrt{2}$
3. Larger side = $3\times 2\sqrt{2}=$ $6\sqrt{2}$ cm

Trick: $\text{Side scale}=\sqrt{\text{area scale}}$ – never multiply the side by 8.

Answer: (B) $6\sqrt{2}\mathrm{cm}$

Concept: Area of similar figures $\propto (\text{height}{)}^2$.

1. Height ratio 1 : 4 ⇒ area ratio 1 : 16
2. Trapezium = $16-1=15$ parts
3. Trapezium : large triangle = 15 : 16

Trick: Ratio k ⇒ remaining part : whole = $(1-k^2):1=\frac{15}{16}$.

Answer: (B) $15:16$

Concept: $a^2+b^2=(a+b{)}^2-2ab$.

1. $={12}^2-2\times 14$
2. $=144-28=$ 116

Yaad rakhein: Similarly $(a-b{)}^2=(a+b{)}^2-4ab=144-56=88$.

Answer: (D) 116

Concept: $\frac{x^3+y^3}{x+y}=x^2-xy+y^2$.

1. $x=\frac{16+8}{2}=12,y=4$
2. $x^2-xy+y^2=144-48+16$
3. = 112

Trick: $x^2-xy+y^2=\frac{(x+y{)}^2+3(x-y{)}^2}{4}=\frac{256+192}{4}=112$.

Answer: (C) 112

Concept: Many rights-and-judiciary features of our Constitution come from the US Constitution.

1. From USA: Fundamental Rights, Judicial Review, impeachment of the President, removal of SC/HC judges, Vice-President post.
2. UK gave parliamentary system and single citizenship; Canada – federation with strong Centre; Australia – Concurrent List.

Trick: Mnemonic "USA = Rights + Review + Removal" (FR, judicial review, impeachment).

Answer: (B) United States of America

Concept: Duties are in Article 51A (Part IV-A); DPSPs are Articles 36–51 (Part IV).

1. 1 ✔ – Fundamental Duties are listed in Article 51A.
2. 2 ✔ – respecting the Constitution, National Flag and Anthem is the first duty.
3. 3 ✘ – separating judiciary from executive is a DPSP (Article 50).

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

Answer: (A) 1 and 2 only

Concept: Host country of the next OIC Islamic Summit (2026).

1. The 16th Islamic Summit of the OIC is to be held in Azerbaijan in 2026.
2. The 15th summit was held in Banjul, The Gambia (2024).

Yaad rakhein: OIC: founded 1969, HQ Jeddah (Saudi Arabia), 57 members; India is not a member.

Answer: (A) Azerbaijan

Concept: WWF-India named a sports star as Goodwill Ambassador for Earth Hour 2024.

1. Badminton star P. V. Sindhu was the Goodwill Ambassador for Earth Hour India 2024.
2. Earth Hour is a WWF campaign – lights off for one hour on a Saturday in March.

Yaad rakhein: Earth Hour started in Sydney, Australia in 2007.

Answer: (A) P. V. Sindhu

Concept: Khorana's work on the genetic code.

1. 1968 Nobel in Physiology or Medicine – for decoding the genetic code and its role in protein synthesis.
2. Shared with Robert Holley and Marshall Nirenberg.

Yaad rakhein: Khorana also made the first artificial (synthetic) gene.

Answer: (A) Physiology or Medicine