Q&A: practice questions with answers

Concept: For similar triangles, the ratio of areas = square of the ratio of corresponding sides.

1. $\frac{\text{ar}(XYZ)}{\text{ar}(LMN)}={\left(\frac{YZ}{MN}\right)}^2=\frac{16}{25}$
2. So $\frac{YZ}{MN}=\frac{4}{5}$
3. $MN=\frac{5}{4}\times 2.4=3\text{cm}$

Trick: Take square roots of the area ratio: $16:25\to 4:5$, then scale the side.

Answer: (A) $3$ cm

Concept: Speed stays the same, so time is proportional to distance.

1. Speed $=\frac{56000\text{m}}{35\times 60\text{s}}=\frac{56000}{2100}=\frac{80}{3}\text{m/s}$
2. Time for 720 m $=720÷\frac{80}{3}=\frac{720\times 3}{80}=27\text{s}$

Trick: $56\text{km}$ in $35$ min $=1.6\text{km/min}=96\text{km/h}$; $96\times \frac{5}{18}=\frac{80}{3}\text{m/s}$.

Answer: (B) $27$

Concept: Use BODMAS: bracket first, then 'of', then division, then addition.

1. $274+830=1104$
2. $1104\text{ of }\frac{7}{4}=276\times 7=1932$
3. $\frac{92736}{1932}=48$
4. $48+513=561$

Trick: Check unit digits: $1932\times 48$ ends in $6$, matching $92736$.

Answer: (C) $561$

Concept: $(1+\sin \theta )(1-\sin \theta )={\cos }^2\theta$ and $(1+\cos \theta )(1-\cos \theta )={\sin }^2\theta$.

1. Expression $=\frac{{\cos }^2\theta }{{\sin }^2\theta }={\cot }^2\theta$
2. $\cot \theta =\frac{10}{3}$
3. ${\cot }^2\theta =\frac{100}{9}$

Trick: Such products always reduce to ${\cot }^2\theta$ — just invert and square $\tan \theta$.

Answer: (C) $\frac{100}{9}$

Concept: Average $=\frac{\text{sum}}{\text{number of cities}}$.

1. Science total $=2560+2650+2640+1250+2100=11200$
2. Average $=\frac{11200}{5}=2240$

Trick: Use the 'Total' row directly: $11200÷5$.

Answer: (D) $2240$

Concept: Same distance $d$ each way; total time = sum of the two times.

1. $\frac{d}{12}+\frac{d}{8}=\frac{15}{4}$ h
2. $\frac{2d+3d}{24}=\frac{15}{4}\Rightarrow 5d=90\Rightarrow d=18$ km
3. Total distance $=2d=36$ km

Trick: Average speed $=\frac{2\times 12\times 8}{20}=9.6\text{km/h}$; $9.6\times 3.75=36$ km.

Answer: (D) $36$

Concept: With $\tan \theta =\frac{7}{24}$, take a right triangle with sides 7, 24, 25.

1. $\cos \theta =\frac{24}{25},\sin \theta =\frac{7}{25}$
2. ${\cos }^2\theta -{\sin }^2\theta =\frac{576-49}{625}=\frac{527}{625}$

Trick: $\cos 2\theta =\frac{1-{\tan }^2\theta }{1+{\tan }^2\theta }=\frac{576-49}{576+49}$.

Answer: (D) $\frac{527}{625}$

Concept: Cutting squares of side $x$ from each corner: box $=(l-2x)\times (b-2x)\times x$.

1. Length $=60-10=50$ cm, breadth $=25-10=15$ cm, height $=5$ cm
2. Volume $=50\times 15\times 5=3750{\text{cm}}^3$

Trick: Subtract twice the cut from each side; the cut becomes the height.

Answer: (A) $3750{\text{cm}}^3$

Concept: Put all three equal to $k$ and express A, B, C in terms of $k$.

1. $0.4A=k\Rightarrow A=\frac{5k}{2}$; $0.6B=k\Rightarrow B=\frac{5k}{3}$; $\frac{C}{6}=k\Rightarrow C=6k$
2. Multiply by 6: $A:B:C=15:10:36$

Trick: Take $k=6$ (LCM-friendly): $A=15,B=10,C=36$.

Answer: (A) 15 : 10 : 36

Concept: $\%\text{ increase}=\frac{\text{new}-\text{old}}{\text{old}}\times 100$.

1. 2017: 500; 2018: 1000
2. Increase $=\frac{1000-500}{500}\times 100=100\%$

Trick: The value doubled, so the increase is $100\%$.

Answer: (C) $100\%$

Concept: Open brackets from the innermost: ( ), then { }, then [ ].

1. $(12-10-4)=-2$
2. $\{10+(-2)\}=8$
3. $[30+8]=38$
4. $35-38=-3$

Trick: Always remove the innermost bracket first and watch the sign.

Answer: (A) $-3$

Concept: Effective CP = purchase price + transport; $\text{SP}=\text{CP}\times \frac{130}{100}$.

1. $\text{CP}=\frac{1400\times 100}{130}=1076.92$
2. Transport $=1076.92-700=376.92\approx 377$

Trick: Find total CP from SP and profit %, then subtract the purchase price.

Answer: (C) ₹377

Concept: Direct common tangent $=\sqrt{d^2-(r_1-r_2{)}^2}$; for circles touching externally $d=r_1+r_2$, so it becomes $2\sqrt{r_1{r}_2}$.

1. $d=35+25=60$ cm
2. $L=\sqrt{{60}^2-{10}^2}=\sqrt{3500}$
3. $L=2\sqrt{35\times 25}=2\sqrt{875}\approx 59.16$ cm

Trick: Touching circles: direct tangent $=2\sqrt{r_1{r}_2}$.

Answer: (B) $59.16$

Concept: ${(a+\frac{1}{a})}^2=a^2+\frac{1}{a^2}+2$; apply it twice going down.

1. Let $a=4x^2$: ${(4x^2+\frac{1}{4x^2})}^2=14159+2=14161\Rightarrow 4x^2+\frac{1}{4x^2}=119$
2. ${(2x+\frac{1}{2x})}^2=119+2=121$
3. $2x+\frac{1}{2x}=11$

Trick: Add 2 and take the square root at each step: $14159\to 119\to 11$.

Answer: (B) $11$

Concept: Downstream speed $=u+c$, upstream speed $=u-c$.

1. $u=c+4\Rightarrow$ upstream speed $=4\text{km/h}$; upstream time $=\frac{35}{4}=8.75$ h
2. Downstream time $=10-8.75=1.25$ h $\Rightarrow u+c=\frac{45}{1.25}=36$
3. $u+c=36,u-c=4\Rightarrow u=20\text{km/h}$

Trick: The 'more than current' value is directly the upstream speed.

Answer: (D) $20\text{km/h}$

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

1. Discount $=82500-79200=3300$
2. $\frac{3300}{82500}\times 100=4\%$

Trick: $1\%$ of 82500 is 825; $3300=4\times 825$.

Answer: (B) $4\%$

Concept: $\frac{M_1{D}_1}{W_1}=\frac{M_2{D}_2}{W_2}$.

1. $\frac{24\times 25}{90}=\frac{27\times D}{162}$
2. $D=\frac{24\times 25\times 162}{90\times 27}=40$ days
3. 'How many more days' than 25: $40-25=15$

Trick: Note the wording 'more days' — compare with the original 25 days.

Answer: (C) $15$

Concept: Surface area of sphere $=4\pi r^2$.

1. $r=14$ cm
2. $4\times \frac{22}{7}\times 14\times 14=2464{\text{cm}}^2$

Trick: Also $=\pi d^2=\frac{22}{7}\times 784=2464$.

Answer: (D) $2464$

Concept: $(a+b{)}^2-(a-b{)}^2=4ab$.

1. Numerator $=4\times 525\times 175$
2. $\frac{4\times 525\times 175}{525\times 175}=4$

Trick: Spot the pattern — no need to calculate ${700}^2$ or ${350}^2$.

Answer: (B) $4$

Concept: Present $=\text{Past}\times (1+\frac{R}{100})$.

1. Past $=\frac{142560}{1.08}$
2. $=132000$

Trick: Check: $132000\times 1.08=142560$.

Answer: (D) 1,32,000