Q&A: practice questions with answers

Concept: HCF of two numbers also divides their difference, so test the difference first.

1. $1020-960=60$
2. $960÷60=16$ and $1020÷60=17$ — both exact.
3. 16 and 17 are co-prime, so nothing larger works: $\text{HCF}=60$.

Trick: HCF always divides the difference of the numbers — check the difference and its factors first.

Answer: (C) $60$

Concept: Identity: ${\text{cosec}}^{2}A-{\cot }^{2}A=1$.

1. Take 23 common: $23({\text{cosec}}^{2}A-{\cot }^{2}A)$
2. $=23\times 1=23$

Trick: Whenever ${\text{cosec}}^2$ and ${\cot }^2$ (or ${\sec }^2$ and ${\tan }^2$) appear with the same coefficient $k$, the value is just $k$.

Answer: (A) $23$

Concept: Net growth rate $=$ birth rate $-$ death rate; population grows like compound interest.

1. Net rate $=12.7\%-2.7\%=10\%$ per year.
2. After 3 years: $125000\times (1.1{)}^3=125000\times 1.331=166375$
3. Increase $=166375-125000=41375$

Trick: For 10% for 3 years, the total increase is $33.1\%$: $125000\times 0.331=41375$.

Answer: (B) $41,375$

Concept: Perimeter of rectangle $=2(l+b)$, perimeter of square $=4a$, area of square $=a^2$.

1. Rectangle perimeter $=2(10+8)=36\text{cm}$
2. Square: $4a=36\Rightarrow a=9\text{cm}$
3. Area $=9^2=81{\text{cm}}^2$

Trick: Side of the square $=\frac{l+b}{2}=\frac{10+8}{2}=9$.

Answer: (C) $81{\text{cm}}^2$

Concept: Sum of the angles of a triangle is ${180}^{\circ }$.

1. Let each equal angle be $x$.
2. $x+x+{70}^{\circ }={180}^{\circ }\Rightarrow 2x={110}^{\circ }$
3. $x={55}^{\circ }$

Trick: Each base angle $=\frac{{180}^{\circ }-\text{third angle}}{2}=\frac{{110}^{\circ }}{2}$.

Answer: (D) ${55}^{\circ }$

Concept: Multiply each ratio term by its own growth factor $(1+\frac{r}{100})$.

1. A: $2\times 1.3=2.6$
2. B: $3\times 1.2=3.6$
3. C: $4\times 1.1=4.4$
4. $2.6:3.6:4.4=26:36:44=13:18:22$

Trick: Use $2\times 130:3\times 120:4\times 110=260:360:440$, then divide by 20.

Answer: (C) $13:18:22$

Concept: $\text{Discount}\%=\frac{\text{MP}-\text{SP}}{\text{MP}}\times 100$

1. Discount $=18000-15840=2160$
2. $\text{Discount}\%=\frac{2160}{18000}\times 100=12\%$

Trick: $\frac{15840}{18000}=0.88$, so the buyer paid 88% — discount is 12%.

Answer: (C) $12\%$

Concept: $\text{Required}\%=\frac{\text{first value}}{\text{second value}}\times 100$

1. B in 2019 $=18$ lakh; C in 2021 $=17$ lakh.
2. $\frac{18}{17}\times 100=105.88\%$

Trick: Numerator bigger than denominator, so the answer must exceed 100% — only (C) or (D); $\frac{18}{17}\approx 1.0588$.

Answer: (C) $105.88\%$

Concept: Numbers leaving the same remainder $r$ on division by $a,b,c$ are of the form $\text{LCM}\times k+r$.

1. $\text{LCM}(12,16,24)=48$, so numbers are $48k+5$.
2. Between 550 and 700: $48\times 12+5=581$, $48\times 13+5=629$, $48\times 14+5=677$
3. Sum $=581+629+677=1887$

Trick: Sum of the three terms $=3\times \text{middle term}=3\times 629=1887$.

Answer: (B) $1887$

Concept: $\text{Total}=\text{Average}\times \text{Number}$; removed item $=$ old total $-$ new total.

1. Total of 6 baskets $=38\times 6=228$
2. Total of remaining 5 $=34\times 5=170$
3. Blue basket $=228-170=58$

Trick: Removed value $=$ new avg $+$ (drop in avg $\times$ old count) $=34+4\times 6=58$.

Answer: (D) $58$

Concept: Read the required sectors and simplify the ratio.

1. English $+$ Bhojpuri $=15+10=25$
2. Tamil $=5$
3. Ratio $=25:5=5:1$

Trick: Ratio questions on one chart never need the total (120) — the common multiplier cancels.

Answer: (D) $5:1$

Concept: $(a+b)(a-b)=a^2-b^2$

1. $(4y+\frac{3}{2})(4y-\frac{3}{2})=16y^2-\frac{9}{4}$
2. Compare with $16y^2-k$: $k=\frac{9}{4}$

Trick: $k$ is simply the square of the constant term: $(\frac{3}{2}{)}^2=\frac{9}{4}$.

Answer: (A) $\frac{9}{4}$

Concept: Find the total from the known category, then apply the percentage changes.

1. B $=15\%=1500\Rightarrow$ total $=10000$; D $=25\%=2500$
2. 1996: B $=1500\times 1.1=1650$
3. 1996: D $=2500\times 0.95=2375$
4. Total $=1650+2375=4025$

Trick: D $=\frac{25}{15}\times 1500=2500$ directly, without finding the total.

Answer: (A) $4025$

Concept: Curved surface area of a hemisphere $=2\pi r^2$.

1. $\text{CSA}=2\times \frac{22}{7}\times 21\times 21=2772{\text{cm}}^2$
2. 75% of it $=\frac{3}{4}\times 2772=2079$
3. Rounded to the nearest ten $=2080{\text{cm}}^2$

Trick: $\frac{3}{4}\times 2\pi r^2=1.5\pi r^2=1.5\times 1386=2079$.

Answer: (C) $2080$

Concept: Chain the percentages: total $\to$ votes cast $\to$ valid votes $\to$ candidate.

1. Let total votes $=T$. Cast $=0.65T$; valid $=0.97\times 0.65T$
2. Candidate $=0.65\times 0.97\times 0.65T=0.409825T$
3. $0.409825T=81965\Rightarrow T=200000$

Trick: Check options: $200000\times 0.65=130000\to \times 0.97=126100\to \times 0.65=81965$ ✓.

Answer: (A) $200000$

Concept: Moving in opposite directions, the separation grows at the sum of speeds.

1. Relative speed $=6+4=10\text{km/h}$
2. Distance after 4 h $=10\times 4=40\text{km}$

Trick: Opposite directions $\Rightarrow$ add speeds; same direction $\Rightarrow$ subtract.

Answer: (C) $40\text{km}$

Concept: For equal distances at speeds $x$ and $y$: $\text{Average speed}=\frac{2xy}{x+y}$.

1. $\frac{2\times 12\times 18}{12+18}=\frac{432}{30}$
2. $=14.4=14\frac{2}{5}\text{km/h}$

Trick: Average speed for a to-and-fro trip is always less than the simple average (here $15$), so (B) is a trap.

Answer: (A) $14\frac{2}{5}\text{km/h}$

Concept: Subjects have different maximum marks, so convert percentages to marks: overall $\%=\frac{\text{total marks}}{\text{total max}}\times 100$.

1. Mohan's marks: $0.70\times 300=210$, $0.75\times 300=225$, $0.80\times 150=120$, $0.65\times 300=195$, $0.85\times 200=170$
2. Total $=920$ out of $1250$
3. $\frac{920}{1250}\times 100=73.6\%$

Trick: Simple average of the percentages gives $75\%$ (not an option) — a weighted average is needed when maximum marks differ.

Answer: (D) 73.6 Percent

Concept: Difference of averages $=$ average of the differences.

1. Appeared total $=2500+3000+4500+1200=11200$; average $=2800$
2. Qualified total $=1350+2500+3900+850=8600$; average $=2150$
3. Difference $=2800-2150=650$

Trick: Differences city-wise: $1150+500+600+350=2600$; $\frac{2600}{4}=650$.

Answer: (C) $650$

Concept: $\text{SP}=\text{CP}+\text{CP}\times \frac{\text{Profit}\%}{100}$; here profit% $=$ CP $=x$.

1. $x+\frac{x^2}{100}=1200$
2. $x^2+100x-120000=0\Rightarrow (x+400)(x-300)=0$
3. $x=300$ (CP is positive)

Trick: Option check: CP $=300$, profit $=300\%$ of 300 $=900$; $300+900=1200$ ✓.

Answer: (D) ₹300