Q&A: practice questions with answers

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

Solution:

1. $16x^4-72x^2{y}^2+81y^4=(4x^2-9y^2{)}^2$, so its $\sqrt{}=4x^2-9y^2$.
2. $4x^2-12xy+9y^2=(2x-3y{)}^2$, so its $\sqrt{}=2x-3y$ (positive since $2x>3y$).
3. Ratio = $\frac{(2x-3y)(2x+3y)}{2x-3y}=2x+3y$.
4. Expression = $(2x+3y)-(2x-3y)=6y$.

Trick: Put $x=3,y=1$ ($2x>3y$): $\frac{\sqrt{1296-648+81}}{\sqrt{36-36+9}}-3=\frac{27}{3}-3=6=6y$. Only 6y fits.

Answer: (B) $6y$

Concept: Profit ratio = (capital × time) ratio.

Solution:

1. Kirti: $27,000\times 12$; Vaishnavi joins after 7 months, so works 5 months: $V\times 5$.
2. $27,000\times 12:5V=8:9$.
3. $5V=27,000\times 12\times \frac{9}{8}=3,64,500$.
4. $V=$ ₹72,900.

Trick: Kirti's 3,24,000 is 8 parts → 1 part = 40,500; Vaishnavi = $9\times 40,500÷5=$ ₹72,900.

Answer: (C) ₹72,900

Concept: $SI=\frac{P\times R\times T}{100}$.

Solution:

1. $SI=\frac{75,000\times 5\times 5}{100}$
2. $=750\times 25=$ ₹18,750.

Trick: 5% for 5 years = 25% of principal; 25% of 75,000 = $\frac{75,000}{4}=$ ₹18,750.

Answer: (D) ₹18,750

Concept: $P=\frac{SI\times 100}{R\times T}$.

Solution:

1. Total rate = $6\times 2=12\%$.
2. $P=\frac{720\times 100}{12}=$ ₹6,000.

Trick: 12% of P = 720 → $1\%=60\to 100\%=6,000$.

Answer: (C) 6000

Concept: Form equations from present and future ages.

Solution:

1. $L+T=24$.
2. $L+2=3(T+2)\to L=3T+4$.
3. $3T+4+T=24\to T=5$.
4. $L=24-5=19$ years.

Trick: Option check: $L=19\to T=5$; after 2 years 21 and 7, and $21=3\times 7$ ✓.

Answer: (C) 19

Concept: Use LCM as total work; efficiency = work per day.

Solution:

1. Total work = LCM(12, 8) = 24 units; A = 2/day, B = 3/day.
2. 2 days together: $2\times 5=10$ units.
3. Remaining 14 units by A: $14÷2=7$ days.
4. Total = $2+7=9$ days.

Trick: LCM method avoids fractions: work = 24, A = 2, B = 3.

Answer: (B) 9

Concept: Each exterior angle of a regular n-gon = $\frac{{360}^{\circ }}{n}$.

Solution:

1. Each exterior angle = $\frac{{360}^{\circ }}{24}={15}^{\circ }$.
2. 8 angles = $8\times {15}^{\circ }={120}^{\circ }$.

Trick: Sum of 8 of 24 angles = $(\frac{8}{24})\times {360}^{\circ }=\frac{{360}^{\circ }}{3}={120}^{\circ }$.

Answer: (D) 120°

Concept: Solve brackets first; 'of' means multiply.

Solution:

1. $\frac{3}{4}÷0.25=\frac{3}{4}\times 4=3$.
2. $3+\frac{1}{2}=\frac{7}{2}$.
3. $\frac{2}{5}$ of $\frac{7}{2}=\frac{7}{5}$.
4. $\frac{7}{5}-\frac{1}{3}=\frac{21}{15}-\frac{5}{15}=\frac{16}{15}$.

Trick: $÷0.25=\times 4$; this one step clears the decimal.

Answer: (D) $\frac{16}{15}$

Concept: Output ∝ machines × hours ($\frac{M_1{H}_1}{W_1}=\frac{M_2{H}_2}{W_2}$).

Solution:

1. Rate = $\frac{120}{6\times 8}=2.5$ items per machine-hour.
2. Output = $9\times 4\times 2.5=90$ items.

Trick: $120\times (\frac{9}{6})\times (\frac{4}{8})=120\times \frac{3}{2}\times \frac{1}{2}=90$.

Answer: (C) 90

Concept: In $\Delta PQR$ with $\angle Q={90}^{\circ }$, $P+R={90}^{\circ }$, so $\sin P=\cos R,\cos P=\sin R,\sec P=cosecR$.

Solution:

1. $\tan R=\frac{1}{3}\to PQ=1,QR=3,PR=\sqrt{10};\sin R=\frac{1}{\sqrt{10}},\cos R=\frac{3}{\sqrt{10}}$.
2. $\sin P=\cos R=\frac{3}{\sqrt{10}},\cos P=\sin R=\frac{1}{\sqrt{10}},\sec P=cosecR=\sqrt{10}$.
3. Numerator = $\sqrt{10}(\frac{9}{\sqrt{10}}-\frac{3}{\sqrt{10}})=6$; Denominator = $\sqrt{10}(\frac{4}{\sqrt{10}}-\frac{1}{\sqrt{10}})=3$.
4. Value = $1+\frac{6}{3}=3$.

Trick: sec P = cosec R cancel; $\sin P=\cos R$, so fraction = $\frac{3\cos R-\cos R}{4\sin R-\sin R}=\frac{2\cos R}{3\sin R}=\frac{2}{3\tan R}=2$. Answer $1+2=3$.

Answer: (D) 3

Concept: $\text{Mode}=3\text{Median}-2\text{Mean}$.

Solution:

1. $60=3\times 65-2\text{Mean}$.
2. $2\text{Mean}=195-60=135$.
3. Mean = 67.5.

Trick: $\text{Mean}=\frac{3\times \text{Median}-\text{Mode}}{2}=\frac{195-60}{2}=67.5$.

Answer: (D) 67.5

Concept: Translate words into an equation.

Solution:

1. $\frac{n}{3}+55=83\to \frac{n}{3}=28$.
2. $n=84$.
3. Digit sum = $8+4=12$.

Yaad rakhein: Always answer what is asked — here the digit sum, not the number 84.

Answer: (C) 12

Concept: Apply BODMAS; 'of' is multiplication done before ÷.

Solution:

1. Numerator: 5 of 6 = 30; $\{36÷3\}=12$.
2. $30-12+2=20$.
3. Denominator: 3 of 3 = 9; $9÷9=1$.
4. Value = $\frac{20}{1}=20$.

Yaad rakhein: Order: Brackets → Of → Division → Multiplication → Addition → Subtraction.

Answer: (B) 20

Concept: Average = Sum ÷ Count.

Solution:

1. Average = $240÷6=40$.

Yaad rakhein: Sum = Average × Count — useful for reverse questions.

Answer: (B) 40

Concept: Joining solids does not change total volume.

Solution:

1. Volume of one cube = $4^3=64{\text{cm}}^3$.
2. Cuboid volume = $6\times 64=384{\text{cm}}^3$.

Yaad rakhein: Volume adds up when solids are joined, but surface area decreases (joined faces get hidden).

Answer: (B) 384 cm3

Concept: Add the pair rates: $2(A+B+C)$ = sum of pair rates.

Solution:

1. Work = $\text{LCM}(10,12,15)=60$ units.
2. $A+B=6,B+C=5,A+C=4$ units/day.
3. $2(A+B+C)=15\to A+B+C=7.5$/day.
4. Time = $60÷7.5=8$ days.

Trick: Time together = $\frac{2xyz}{xy+yz+zx}=\frac{2\times 1800}{120+180+150}=\frac{3600}{450}=8$.

Answer: (C) 8 days

Concept: $\text{SP}=\text{CP}\times (1+\text{markup})\times (1-\text{discount})$.

Solution:

1. $\text{SP}=\text{CP}\times 1.36\times 0.75=1.02\text{CP}$ → profit 2%.
2. $2\%$ of $\text{CP}=134\to \text{CP}=$ ₹6,700.
3. List price = $1.36\times 6,700=$ ₹9,112.

Trick: Net % = $36-25-\frac{36\times 25}{100}=2\%$ profit; then $\text{LP}=134\times 50\times 1.36$.

Answer: (A) 9,112

Concept: Income = Expenditure + Savings.

Solution:

1. Initial income = $12,000+8,000+15,000=$ ₹35,000.
2. New income = $35,000\times 1.15=$ ₹40,250.
3. New food = 13,200; new rent = 9,600; total = 22,800.
4. New savings = $40,250-22,800=$ ₹17,450.

Trick: Track only the changes: income +5,250, expenses $+1,200+1,600=+2,800$ → savings = $15,000+2,450=17,450$.

Answer: (B) ₹17,450

Concept: Distinct prime factors are added once each.

Solution:

1. $84=2^2\times 3\times 7$.
2. Prime factors: 2, 3, 7.
3. Sum = $2+3+7=12$.

Yaad rakhein: 'Prime factors' means distinct primes; counting 2 twice would give 14 (a trap option).

Answer: (D) 12

Concept: Overall % = sum of (share × profit%).

Solution:

1. Shares: 60%, then 40% of 40% = 16%, rest 24%.
2. $60\%\times 21=12.6$.
3. $16\%\times 46=7.36$; $24\%\times (-29)=-6.96$.
4. $x=12.6+7.36-6.96=13$.

Trick: Take 100 goods: 60, 16, 24 → $\frac{60\times 21+16\times 46-24\times 29}{100}=\frac{1260+736-696}{100}=13$.

Answer: (B) 13