Q&A: practice questions with answers

Concept: Compare distance d with r₁ + r₂ and |r₁ − r₂|.

1. r₁ + r₂ = 8, |r₁ − r₂| = 0, d = 5
2. 0 < 5 < 8 → circles intersect at two points.
3. Intersecting circles have only 2 (direct) common tangents.

Yaad rakhein: d > sum → 4; d = sum → 3; between → 2; d = difference → 1; d < difference → 0.

Answer: (A) 2

Concept: $\text{Diagonal}=\sqrt{l^2+b^2}$.

1. $d=\sqrt{{15}^2+{20}^2}=\sqrt{225+400}$
2. $=\sqrt{625}=25$ cm

Trick: 15 : 20 : 25 = 5 × (3 : 4 : 5) — a Pythagorean triplet, no calculation needed.

Answer: (A) 25 cm

Concept: $\text{Volume}=l\times b\times h$.

1. $V=100\times 70\times 60$
2. $=100\times 4200=4,20,000$ cm³

Trick: $7\times 6=42$, then add the four zeros → 420000.

Answer: (B) 420000

Concept: Volume of pyramid $V=\frac{1}{3}\times \text{base area}\times \text{height}$.

1. Base area = $4\times 4=16$ m².
2. $V=\frac{1}{3}\times 16\times 6=$ 32 m³.

Yaad rakhein: Pyramid = one-third of the prism with the same base and height.

Answer: (B) $32m^3$

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

1. Area ABC : area DEF = 25 : 49
2. $\frac{150}{\text{area DEF}}=\frac{25}{49}$ → $\text{area DEF}=\frac{150\times 49}{25}$
3. $=6\times 49=294$ cm²

Trick: $150÷25=6$ → one unit = 6 → 49 units = 294.

Answer: (D) $294{\mathrm{cm}}^2$

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

1. $d=15$, $r_1-r_2=6-2=4$
2. $L=\sqrt{225-16}=\sqrt{209}$
3. $\sqrt{196}=14$, $\sqrt{225}=15$ → $\sqrt{209}\approx 14.45$ units

Trick: Direct → minus $(r_1-r_2)$; transverse → plus $(r_1+r_2)$.

Answer: (C) 14.45 units

Concept: $\cot A\times \tan A=1$, so use $x^3+\frac{1}{x^3}={(x+\frac{1}{x})}^3-3(x+\frac{1}{x})$.

1. Let $x=\tan A$, then $\cot A=\frac{1}{x}$ and $x+\frac{1}{x}=8$.
2. $x^3+\frac{1}{x^3}=8^3-3\times 8$
3. $=512-24=488$

Trick: If $x+\frac{1}{x}=k$, then $x^3+\frac{1}{x^3}=k^3-3k$.

Answer: (C) 488

Concept: For 2 years: $\text{CI}-\text{SI}=P\times {\left(\frac{R}{100}\right)}^2$.

1. $=35000\times {\left(\frac{15}{100}\right)}^2$
2. $=35000\times \frac{225}{10000}$
3. $=3.5\times 225=$ ₹787.5

Trick: Difference = interest on one year's interest: SI per year = 5250; 15% of 5250 = 787.5.

Answer: (A) Rs. 787.5

Concept: Centroid divides a median 2 : 1; the three medians split the triangle into 6 equal areas.

1. $RO=\frac{2}{3}\times 24=16$ cm; $OE=\frac{1}{3}\times 30=10$ cm
2. Area ROE = $\frac{1}{2}\times 16\times 10\times \sin {60}^{\circ }=40\sqrt{3}$ cm²
3. Area PQR = $6\times 40\sqrt{3}=240\sqrt{3}$ cm²

Trick: Shortcut: $\text{Area}=\left(\frac{4}{3}\right)\times \frac{1}{2}\times m_1\times m_2\times \sin \theta =\left(\frac{2}{3}\right)\times 24\times 30\times \left(\frac{\sqrt{3}}{2}\right)=240\sqrt{3}$.

Answer: (B) $240\sqrt{3}{\mathrm{cm}}^2$

Concept: $\text{Catch-up time}=\frac{\text{gap}}{\text{difference of speeds}}$.

1. Gap closes at 1.25 km/h (policeman 5.75 minus thief 4.5).
2. Gap 2400 m = 2.4 km → time = $\frac{2.4}{1.25}\text{h}=1.92\text{h}$
3. In 1.92 h the thief runs $1.92\times 4.5\text{km}=8.64\text{km}=8640\text{m}$

Trick: $\text{Thief's distance}=\text{gap}\times \frac{\text{thief speed}}{\text{speed difference}}=2400\times \frac{4.5}{1.25}=2400\times 3.6=8640$.

Answer: (B) 8,640 metres

Concept: $V=\pi r^{2}h,\text{CSA}=2\pi rh\Rightarrow \text{CSA}=\frac{2V}{r}$.

1. $h=1540÷(\frac{22}{7}\times 49)=1540÷154=10\text{cm}$
2. $\text{CSA}=2\times \frac{22}{7}\times 7\times 10=440{\text{cm}}^2$

Trick: $\text{CSA}=\frac{2V}{r}=\frac{2\times 1540}{7}=440$ — no need to find h.

Answer: (A) $440{\mathrm{cm}}^2$

Concept: $(\sqrt{x}+\sqrt{y}{)}^2=x+y+2\sqrt{xy}$.

1. $a=\sqrt{5}+\sqrt{2}$
2. $a^2=5+2+2\sqrt{10}=7+2\sqrt{10}$

Trick: Rational part = sum of numbers under roots (5 + 2); surd part = $2\sqrt{\text{product}}$.

Answer: (B) $7+2\sqrt{10}$

Concept: $1-{\sin }^{2}x={\cos }^{2}x$.

1. $\frac{1-{\sin }^{2}x}{\cos x}=\frac{{\cos }^{2}x}{\cos x}$
2. $=\cos x$

Trick: Put $x=0$: $\text{LHS}=\frac{1}{1}=1$; among options only $\cos 0=1$.

Answer: (A) $\cos x$

Concept: $\frac{B}{D}=\frac{B}{C}\times \frac{C}{D}$; A : B is not needed.

1. $\frac{B}{D}=\frac{6}{7}\times \frac{8}{9}=\frac{48}{63}$
2. $=\frac{16}{21}$

Trick: Cancel before multiplying: $\frac{6}{9}=\frac{2}{3}$ → $2\times 8:7\times 3=16:21$.

Answer: (D) 16 : 21

Concept: Build the right triangle from $\cos \theta$, then find ${\sec }^2$ and ${\csc }^2$.

1. $\cos \theta =\frac{3}{5}$ → 3-4-5 triangle → $\sin \theta =\frac{4}{5}$
2. ${\sec }^2\theta =\frac{25}{9}$; ${\csc }^2\theta =\frac{25}{16}$
3. $\frac{25}{9}-\frac{25}{16}=25\times \frac{7}{144}=\frac{175}{144}$

Trick: Take 25 common: $25(\frac{1}{9}-\frac{1}{16})=25\times \frac{7}{144}$.

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

Concept: Mughals were Turco-Mongol: Timurid on father's side, Mongol on mother's side.

1. Babur descended from Timur through his father and from Genghis Khan through his mother.
2. The word 'Mughal' itself comes from 'Mongol'.
3. Babur is the founder, not an ancestor; Mahmud of Ghazni and Alauddin Khalji belong to other dynasties.

Yaad rakhein: Father → Timur (5th generation); Mother → Genghis Khan (14th generation) — Babur, Baburnama.

Answer: (B) Genghis Khan

Concept: Bhakti movement of Karnataka in the 12th century.

1. Led by Basavanna (minister of Kalachuri king Bijjala), with Allama Prabhu and Akkamahadevi.
2. Opposed caste and ritualism; teachings in Kannada vachanas.
3. Assam – Sankaradeva; Bengal – Chaitanya; Maharashtra – Varkari saints.

Yaad rakhein: Lingayats wear an ishtalinga; Basavanna founded the 'Anubhava Mantapa' at Kalyana.

Answer: (B) Karnataka

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

1. Art. 51A(i): to safeguard public property and abjure violence ✓
2. Obeying elders, local self-government, honouring the national animal are not in the list.

Yaad rakhein: Added by the 42nd Amendment (1976) on Swaran Singh Committee advice; 11th duty (education of children 6–14) by the 86th Amendment (2002).

Answer: (D) Protecting public property

Concept: DPSP (Part IV, Art. 36–51) are non-justiciable but fundamental in governance.

1. Art. 37: not enforceable by any court → I is correct.
2. Art. 37 also makes it the State's duty to apply them in making laws → II is correct.

Yaad rakhein: DPSP idea borrowed from the Irish Constitution; Fundamental Rights are justiciable, DPSP are not.

Answer: (D) Both I and II

Concept: SAARC has 8 members.

1. Founded 8 Dec 1985 at Dhaka by 7 nations including Pakistan and Nepal.
2. Afghanistan joined in 2007 as the 8th member.
3. Both Pakistan and Nepal remain members.

Yaad rakhein: Mnemonic for members — 'BBIMNPSA': Bangladesh, Bhutan, India, Maldives, Nepal, Pakistan, Sri Lanka, Afghanistan. Secretariat: Kathmandu.

Answer: (D) Both I and II