Q&A: practice questions with answers

Concept: Cire perdue (lost-wax) is a metal-casting technique.

1. A wax model is coated with clay, heated so the wax melts out, and molten metal is poured into the mould.
2. The famous Dancing Girl of Mohenjo-daro is a bronze statue made by this method.

Yaad rakhein: Lost-wax → bronze → Dancing Girl (Mohenjo-daro).

Answer: (B) Bronze

Concept: The French East India Company was formed in 1664.

1. Francois Martin founded Pondicherry in 1674 on land granted by the ruler of Valikondapuram.
2. Pondicherry later became the headquarters of the French in India.

Yaad rakhein: French: Company 1664 → Surat factory 1668 → Pondicherry 1674.

Answer: (D) 1674

Concept: For an odd number $n$ of observations, median = ${\left(\frac{n+1}{2}\right)}^{\text{th}}$ value after arranging in order.

1. Ascending: 41, 44, 45, 55, 58, 67, 69, 71, 78, 88, 95.
2. $n=11$, so median = $\frac{11+1}{2}=6^{\text{th}}$ value.
3. $6^{\text{th}}$ value = 67.

Trick: Strike one number from each end alternately; the one left in the middle is the median.

Answer: (C) 67

Concept: Mean $=\frac{\sum fx}{\sum f}$.

1. $\sum fx=19\times 22+31\times 35+54\times 64+69\times 34+95\times 57$
2. $=418+1085+3456+2346+5415=12720$
3. $\sum f=22+35+64+34+57=212$
4. Mean $=\frac{12720}{212}=60$

Trick: Check unit digits: $12720$ ends in 0 and $212\times 60=12720$ – exact.

Answer: (A) 60

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

1. $407=P{\left(\frac{10}{100}\right)}^2=\frac{P}{100}$
2. $P=407\times 100=40700$

Trick: At 10% for 2 years, difference = 1% of the principal.

Answer: (A) ₹40,700

Concept: Production $\propto$ (manpower) $\times$ (hours). To keep production constant, hours must rise in inverse ratio.

1. Manpower falls 10%: factor $\frac{9}{10}$.
2. Hours must become $\frac{10}{9}$ times.
3. Increase $=\frac{10-9}{9}\times 100=\frac{100}{9}\%=11\frac{1}{9}\%$.

Trick: A fall of $\frac{1}{10}$ needs a rise of $\frac{1}{9}$ (denominator minus numerator).

Answer: (C) $11\frac{1}{9}\%$

Concept: Base area $=\pi r^2$; curved surface area $=\pi rl$.

1. $\pi r^2=3136\pi \Rightarrow r^2=3136\Rightarrow r=56\text{cm}$
2. $\text{CSA}=\pi \times 56\times 65$
3. $=3640\pi {\text{cm}}^2$

Trick: $56\times 65=56\times 60+56\times 5=3360+280=3640$.

Answer: (A) $3640\pi$

Concept: When a transversal cuts two parallel lines, angles in matching positions at the two intersections are called corresponding angles.

1. In the figure, ∠1 and ∠2 occupy the same relative position (upper-right) at each intersection.
2. Corresponding angles axiom: if the lines are parallel, each pair of corresponding angles is equal.
3. Co-interior angles (not corresponding) are supplementary.

Trick: Corresponding & alternate angles → equal; co-interior → sum ${180}^{\circ }$.

Answer: (C) equal

Concept: Successive increases multiply: $P\times (1+\frac{r_1}{100})(1+\frac{r_2}{100})$.

1. $100000\times \frac{105}{100}=105000$
2. $105000\times \frac{120}{100}=126000$

Trick: Net % $=5+20+\frac{5\times 20}{100}=26\%$ → 1,26,000.

Answer: (A) 1,26,000

Concept: Let brother = $B$, sister = $S$ and form two equations.

1. $3B=5S+25$ and $5S=B-5$.
2. Substitute: $3B=(B-5)+25\Rightarrow 2B=20\Rightarrow B=10$.
3. $5S=5\Rightarrow S=1$; difference $=10-1=9$.

Trick: Substitute the whole term $5S$ directly – no need to find $S$ first.

Answer: (D) 9

Concept: Sum = average $\times$ count.

1. Sum of 9 numbers $=9\times 17=153$.
2. Sum of 7 numbers $=7\times 17=119$.
3. Remaining two: $153-119=34$, average $=\frac{34}{2}=17$.

Trick: If the average of a part equals the overall average, the rest also has the same average.

Answer: (A) 17

Concept: Area of a regular hexagon of side $s$ $=\frac{3\sqrt{3}}{2}{s}^2$.

1. Square side $=\sqrt{2304}=48$, perimeter $=4\times 48=192\text{cm}$.
2. Hexagon side $s=\frac{192}{6}=32\text{cm}$.
3. Area $=\frac{3\sqrt{3}}{2}\times {32}^2=\frac{3\sqrt{3}}{2}\times 1024=1536\sqrt{3}{\text{cm}}^2$.

Trick: Hexagon = 6 equilateral triangles: $6\times \frac{\sqrt{3}}{4}{s}^2$.

Answer: (A) $1536\sqrt{3}$

Concept: Upstream speed $=B-S$, downstream speed $=B+S$.

1. Upstream speed $=\frac{33.2}{83/60}=\frac{33.2\times 60}{83}=24\text{km/h}$.
2. $B:S=8:5$ → $B-S=3$ parts $=24$ → 1 part $=8$: $B=64$, $S=40$, downstream $=104\text{km/h}$.
3. Time $=\frac{47.1}{24}+\frac{55.9}{104}=1.9625+0.5375=2.5$ hours.

Trick: $33.2÷83=0.4\text{km/min}=24\text{km/h}$.

Answer: (B) 2.5

Concept: Take CP $=100$; then MP and SP follow directly from the percentages.

1. CP $=100$ → MP $=126$ (26% above CP).
2. Profit 13% → SP $=113$.
3. Discount $=\frac{126-113}{126}\times 100=\frac{1300}{126}\approx 10.32\%$.

Trick: The ₹568 is not needed – percentages alone decide the discount.

Answer: (B) 10.32%

Concept: $(x+y+z{)}^2=x^2+y^2+z^2+2xy+2yz+2zx$.

1. Squares: $49a^2+36b^2+16c^2$.
2. $2(7a)(6b)=84ab$, $2(6b)(4c)=48bc$, $2(7a)(4c)=56ac$.
3. Sum: $49a^2+36b^2+16c^2+84ab+48bc+56ac$.

Trick: Only check the cross terms: $84ab$, $48bc$, $56ac$ – only option (C) has all three.

Answer: (C) $49a^2+36b^2+16c^2+84ab+48bc+56ac$

Concept: Use one-hour work (rates) and subtract.

1. $C=\frac{1}{27}-\frac{1}{63}=\frac{7-3}{189}=\frac{4}{189}$
2. $B=\frac{1}{42}-\frac{4}{189}=\frac{9-8}{378}=\frac{1}{378}$
3. B alone takes 378 hours.

Trick: LCM method: total work 378 units → A = 6, A+C = 14, B+C = 9 → C = 8, B = 1 → 378 h.

Answer: (D) 378

Concept: $a:b::c:d\Rightarrow ad=bc$; mean proportional: $y^2=$ product of extremes.

1. $(29+x)(8+x)=(31+x)(7+x)$ → $232+37x=217+38x$ → $x=15$.
2. $5x=75$, $2x-3=27$.
3. $y^2=75\times 27=2025\Rightarrow y=45$ ($y>0$).

Trick: $75\times 27=25\times 81$, so $y=5\times 9=45$.

Answer: (C) 45

Concept: Gain $=\text{SP}-\text{CP}$, Loss $=\text{CP}-\text{SP}$. Here gain $=1.25\times$ loss.

1. $8845-x=1.25(x-1987)$
2. $8845+2483.75=2.25x\Rightarrow x=\frac{11328.75}{2.25}=5035$
3. SP for 20% gain $=5035\times \frac{120}{100}=6042$

Trick: Gain : Loss $=5:4$, so CP divides the gap $8845-1987=6858$ in ratio $5:4$: $\text{CP}=1987+\frac{4}{9}\times 6858=5035$.

Answer: (A) ₹6,042

Concept: $a^m\times a^n=a^{m+n}$ and ${\left(\frac{2}{11}\right)}^n={\left(\frac{11}{2}\right)}^{-n}$.

1. LHS $={\left(\frac{11}{2}\right)}^{12}\times {\left(\frac{11}{2}\right)}^{-2}\times {\left(\frac{11}{2}\right)}^{16}={\left(\frac{11}{2}\right)}^{26}$
2. $={\left(\frac{2}{11}\right)}^{-26}$
3. So $9m+9=-26\Rightarrow 9m=-35\Rightarrow m=-\frac{35}{9}$

Trick: Flip the base → change the sign of the power.

Answer: (B) $-\frac{35}{9}$

Concept: Average $=\frac{\text{Sum}}{\text{Number}}$.

1. Sum $=40+49+56+63+37=245$
2. Average $=\frac{245}{5}=49$ kg

Trick: Take 49 as base: deviations $-9,0,+7,+14,-12$ sum to 0 → average 49.

Answer: (A) 49