Q&A: practice questions with answers

Concept: Take total work $=$ LCM of the days; one-day work is additive.

1. Total work $=\text{LCM}(10,15,5)=30$ units
2. A $=3$, B $=2$, A+B+C $=6$ units/day
3. C $=6-3-2=1$ unit/day $\Rightarrow \frac{30}{1}=30$ days

Trick: $\frac{1}{C}=\frac{1}{5}-\frac{1}{10}-\frac{1}{15}=\frac{1}{30}$.

Answer: (C) $30$

Concept: Successive deductions multiply: remaining $=x\times (1-\frac{20}{100})(1-\frac{10}{100})$.

1. $x\times 0.8\times 0.9=0.72x$
2. $0.72x=7200\Rightarrow x=10000$

Trick: 72% of the sum $=7200\Rightarrow$ 100% $=10000$.

Answer: (B) Rs. 10,000

Concept: Greatest $n$-digit number divisible by given numbers $=$ largest multiple of their LCM within $n$ digits.

1. $\text{LCM}(15,24,40)=120$
2. $9999÷120$ gives remainder $39$
3. $9999-39=9960$

Trick: Last digit must be 0 (divisible by 10 via 40) and number divisible by 8 and 3: 9960 ✓, 9990 fails for 8.

Answer: (A) $9960$

Concept: Take the common factor $(m-n)$ from the numerator.

1. Numerator $=(m-n)[(m^2+n^2)-(m-n{)}^2]$
2. $=(m-n)[m^2+n^2-m^2+2mn-n^2]=(m-n)·2mn$
3. Denominator $=m^{2}n-m{n}^2=mn(m-n)$
4. Value $=\frac{2mn(m-n)}{mn(m-n)}=2$

Trick: Put $m=2,n=1$: $\frac{5\times 1-1}{4-2}=\frac{4}{2}=2$.

Answer: (C) $2$

Concept: $x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)$

1. $(x+y+z{)}^2=64=20+2(xy+yz+zx)\Rightarrow xy+yz+zx=22$
2. Value $=8\times (20-22)=8\times (-2)=-16$

Trick: Find $xy+yz+zx$ first from $\frac{(\text{sum}{)}^2-\text{sum of squares}}{2}=\frac{64-20}{2}=22$.

Answer: (D) $-16$

Concept: Solve brackets first, then powers, then $÷,\times$ (left to right), then $-$.

1. Bracket: $256-196-40=20$
2. ${12}^3=1728$; $1728\times 20÷2=17280$
3. $9^4=6561$; $17280-6561=10719$

Trick: ${16}^2-{14}^2=(16-14)(16+14)=60$ — quick bracket value.

Answer: (C) $10719$

Concept: $\%\text{more}=\frac{\text{difference}}{\text{base}}\times 100$, base $=$ the value after "than".

1. Mountains in Punjab $=40$; rivers in Odisha $=10$ (thousand km)
2. $\frac{40-10}{10}\times 100=300\%$

Trick: $40$ is 4 times $10$, i.e. 400% OF it, but 300% MORE than it — (A) is the trap.

Answer: (B) $300\%$

Concept: If PT is a tangent and PAB a secant from an external point P: $P{T}^2=PA\times PB$.

1. ${12}^2=PA\times 24\Rightarrow PA=\frac{144}{24}=6\text{cm}$
2. $AB=PB-PA=24-6=18\text{cm}$

Trick: Don't stop at PA $=6$ (not asked) — subtract from PB.

Answer: (B) $18\text{cm}$

Concept: Difference of averages $=\frac{\text{difference of totals}}{\text{number of years}}$.

1. Wheat total $=162+196+187+189=734$
2. Oats total $=131+116+103+101=451$
3. Difference $=\frac{734-451}{4}=\frac{283}{4}=70.75\text{kg}$

Trick: Year-wise differences $31+80+84+88=283$; divide by 4.

Answer: (C) $70.75$

Concept: $\cos (A-B)=\cos A\cos B+\sin A\sin B$; in the second quadrant cosine is negative.

1. $\cos A=-\sqrt{1-\frac{1}{9}}=-\frac{2\sqrt{2}}{3}$
2. $\cos B=-\sqrt{1-\frac{1}{25}}=-\frac{2\sqrt{6}}{5}$
3. $\cos A\cos B=\frac{4\sqrt{12}}{15}=\frac{8\sqrt{3}}{15}$; $\sin A\sin B=\frac{1}{15}$
4. $\cos (A-B)=\frac{8\sqrt{3}+1}{15}$

Trick: Both cosines negative, so their product is positive — answer has $+1$.

Answer: (C) $\frac{8\sqrt{3}+1}{15}$

Concept: Divisible by 36 $\Rightarrow$ divisible by 4 (last two digits) and by 9 (digit sum).

1. By 4: last two digits $y4$ divisible by 4 $\Rightarrow y$ even; smallest natural $y=2$
2. By 9: digit sum $=29+x+y=31+x\Rightarrow x=5$
3. $\frac{x}{y}-\frac{y}{x}=\frac{5}{2}-\frac{2}{5}=\frac{21}{10}=2\frac{1}{10}$

Trick: $y4$ is a multiple of 4 only for $y=0,2,4,6,8$; $y=0$ is not natural, so $y=2$.

Answer: (B) $2\frac{1}{10}$

Concept: Radius $⟂$ tangent at the point of contact; triangles ORT and PST are similar (AA).

1. $\angle ORT=\angle PST={90}^{\circ }$ and $\angle OTR=\angle PTS$ (vertically opposite)
2. $\frac{OR}{RT}=\frac{PS}{ST}\Rightarrow \frac{10}{16}=\frac{x}{24}$
3. $x=\frac{10\times 24}{16}=15\text{cm}$

Trick: Radii are in the same ratio as the tangent lengths: $10:x=16:24$.

Answer: (A) $15$

Concept: Moving towards each other, the gap closes at the sum of the speeds.

1. Relative speed $=80+82=162\text{km/h}$
2. Time $=1\text{ h }30\text{ min}=1.5\text{h}$
3. Distance $=162\times 1.5=243\text{km}$

Trick: $162\times 1.5=162+81=243$.

Answer: (C) $243\text{km}$

Concept: $\sin (A+B)=\sin A\cos B+\cos A\sin B$

1. $\frac{\sin A\cos B+\cos A\sin B}{\sin A\cos B}$
2. $=1+\frac{\cos A}{\sin A}·\frac{\sin B}{\cos B}$
3. $=1+\cot A\tan B$

Trick: Split the fraction term by term — the first term always becomes 1.

Answer: (A) $1+\cot A\tan B$

Concept: Students opting Biology $=$ enrolled $\times$ percentage.

1. School D: enrolled $=800$, Biology $=20\%$
2. $800\times \frac{20}{100}=160$

Trick: The male : female ratio is extra data here — only enrolled $\times$ percentage is needed.

Answer: (B) $160$

Concept: Total surface area of a solid cylinder $=2\pi r(r+h)$.

1. $2\times \frac{22}{7}\times 7\times (7+8)$
2. $=44\times 15=660{\text{cm}}^2$

Trick: With $r=7$ and $\pi =\frac{22}{7}$, $2\pi r=44$ — multiply by $(r+h)$.

Answer: (B) $660{\text{cm}}^2$

Concept: 'of' is solved before division and multiplication; then $÷,\times$ left to right.

1. 'of' first: $6\text{ of }7=42$, $4\text{ of }3=12$
2. $72-42÷42\times 12÷4\times 3-5$
3. $42÷42=1$; $1\times 12=12$; $12÷4=3$; $3\times 3=9$
4. $72-9-5=58$

Trick: Never treat 'of' as ordinary $\times$ in left-to-right order — it binds first.

Answer: (C) $58$

Concept: Wages are shared in the ratio of efficiencies (work done), i.e. inverse ratio of days.

1. Efficiency A : B $=\frac{1}{10}:\frac{1}{15}=3:2$
2. B's share $=\frac{2}{5}\times 1250=500$

Trick: Ratio of shares $=$ reverse of days: $15:10=3:2$.

Answer: (B) Rs. 500

Concept: $\text{HCF}\times \text{LCM}=x\times y$

1. $7\times 441=49\times y$
2. $y=\frac{3087}{49}=63$

Trick: $\frac{441}{49}=9$, so $y=9\times 7=63$; check: $\text{HCF}(49,63)=7$ ✓.

Answer: (D) $63$

Concept: Mean proportional of $a$ and $b$ $=\sqrt{ab}$.

1. $\sqrt{0.04\times 0.36}=\sqrt{0.04}\times \sqrt{0.36}$
2. $=0.2\times 0.6=0.12$

Trick: Take square roots separately: $\sqrt{0.04}=0.2$, $\sqrt{0.36}=0.6$.

Answer: (D) $0.12$