Q&A: practice questions with answers

Concept: A radius is perpendicular to the tangent at the point of contact; with vertically opposite angles at E, the two triangles are similar.

1. $DA⟂AC$ and $BC⟂AC$, so $\angle DAE=\angle BCE={90}^{\circ }$.
2. $\angle AED=\angle CEB$ (vertically opposite), hence $△DAE~△BCE$.
3. $\frac{DA}{AE}=\frac{BC}{EC}\Rightarrow \frac{8}{12}=\frac{x}{18}$
4. $x=\frac{8\times 18}{12}=12$

Trick: Radii are in the same ratio as the tangent pieces: $8:x=12:18$.

Answer: (A) $12$

Concept: If $a:b=c:d$, then $d$ is the fourth proportional: $d=\frac{b\times c}{a}$.

1. $10:12=15:x$
2. $x=\frac{12\times 15}{10}=\frac{180}{10}=18$

Trick: Fourth proportional = (2nd × 3rd) ÷ 1st.

Answer: (D) $18$

Concept: Total surface area of a solid hemisphere = curved surface $2\pi r^2$ + flat circular base $\pi r^2$ = $3\pi r^2$.

1. $r=11\text{cm}$
2. $\text{TSA}=3\pi r^2=3\pi \times 121=363\pi {\text{cm}}^2$

Yaad rakhein: Hemisphere: CSA = $2\pi r^2$, TSA = $3\pi r^2$.

Answer: (C) $363\pi {\text{cm}}^2$

Concept: Solve the innermost bracket first, then curly, then square brackets; ÷ and × left to right.

1. $(14\times 14)=196$
2. $\{13\times 14÷196\}=\frac{182}{196}=\frac{13}{14}$
3. $[14÷13\times \frac{13}{14}]=\frac{14}{13}\times \frac{13}{14}=1$
4. $5\times 1=5$

Trick: Everything inside the square bracket cancels to $1$, so the answer is just the $5$ outside.

Answer: (C) $5$

Concept: Valid votes = votes cast − blank votes; winner's + rival's votes = valid votes.

1. Let voters $=N$. Votes cast $=0.9N$; valid $=0.9N-60$
2. Winner $=0.46N$; rival $=0.46N-410$
3. $0.92N-410=0.9N-60\Rightarrow 0.02N=350\Rightarrow N=17500$

Trick: Check: $0.46\times 17500=8050$, rival $7640$, sum $15690=15750-60$ ✓.

Answer: (A) $17500$

Concept: Look down each month's column and check whether all four types (X, Y, Z, T) have the same value.

1. Jan: X = Y = Z = T = $25$ → all equal.
2. May: $20,30,30,30$ → X differs; March and June are clearly unequal.
3. Only January has equal production of all types ($4\times 25=100$).

Trick: If the total is divisible by 4, check that month first: Jan total $100=4\times 25$.

Answer: (A) January

Concept: Efficiency is inversely proportional to time; 4 times as fast means one-fourth the time.

1. Suresh: $20$ days, Arvind: $\frac{20}{4}=5$ days.
2. One day work together $=\frac{1}{5}+\frac{1}{20}=\frac{4+1}{20}=\frac{1}{4}$
3. Time together $=4$ days.

Trick: Efficiencies $4:1$, total work $=20\times 1=20$ units, together $\frac{20}{4+1}=4$ days.

Answer: (D) $4$

Concept: Arc length $l=\frac{\theta }{{360}^{\circ }}\times 2\pi r$.

1. $\frac{{72}^{\circ }}{{360}^{\circ }}=\frac{1}{5}$
2. $l=\frac{1}{5}\times 2\times \frac{22}{7}\times 35=\frac{1}{5}\times 220=44\text{cm}$

Trick: ${72}^{\circ }$ is one-fifth of the circle, so arc = circumference ÷ 5 $=220÷5$.

Answer: (C) $44\text{cm}$

Concept: Use $\beta =-(\alpha +\gamma )$ and expand.

1. ${\beta }^2={\alpha }^2+{\gamma }^2+2\alpha \gamma$
2. Numerator $=3({\alpha }^2+{\gamma }^2+2\alpha \gamma )+{\alpha }^2+{\gamma }^2=4{\alpha }^2+4{\gamma }^2+6\alpha \gamma$
3. Denominator $=2({\alpha }^2+{\gamma }^2+2\alpha \gamma )-\alpha \gamma =2{\alpha }^2+2{\gamma }^2+3\alpha \gamma$
4. Ratio $=\frac{2(2{\alpha }^2+2{\gamma }^2+3\alpha \gamma )}{2{\alpha }^2+2{\gamma }^2+3\alpha \gamma }=2$

Trick: Put $\alpha =1,\gamma =1,\beta =-2$: $\frac{12+1+1}{8-1}=\frac{14}{7}=2$.

Answer: (D) $2$

Concept: In a degree pie chart, amounts are proportional to angles, so the percentage can be found directly from degrees.

1. Tennis + Basketball $={35}^{\circ }+{40}^{\circ }={75}^{\circ }$
2. Hockey + Cricket $={94}^{\circ }+{135}^{\circ }={229}^{\circ }$
3. $\frac{75}{229}\times 100=32.75\%$ (the ₹21,000 for Football is not needed).

Trick: Compare angles, not rupees – the given amount is a distractor.

Answer: (D) $32.75\%$

Concept: Write $\csc \theta +\cot \theta =\frac{1+\cos \theta }{\sin \theta }$ and use $1-{\cos }^2\theta ={\sin }^2\theta$.

1. Expression $=\csc \theta (1-\cos \theta )·\frac{1+\cos \theta }{\sin \theta }$
2. $=\csc \theta ·\frac{1-{\cos }^2\theta }{\sin \theta }=\csc \theta ·\frac{{\sin }^2\theta }{\sin \theta }$
3. $=\csc \theta ·\sin \theta =1$

Trick: Put $\theta ={90}^{\circ }$: $1\times (1-0)\times (1+0)=1$.

Answer: (C) $1$

Concept: Convert everything to base 3. In BODMAS, 'of' is done before ÷ and ×.

1. $9^{18}=3^{36}$, ${27}^3=3^9$, $\sqrt{6561}=81=3^4$
2. 'of' first: $3^{14}\text{ of }{3}^9=3^{23}$
3. $3^{36}÷3^{23}=3^{13}$
4. $3^{13}\times 3^4=3^{17}$

Trick: Don't skip 'of' – treating it as ordinary × after ÷ gives the wrong $3^{35}$.

Answer: (D) $3^{17}$

Concept: Successive change: $a\%$ up and $b\%$ down gives net $a-b-\frac{ab}{100}$ %.

1. Let CP $=100$, MP $=135$.
2. SP $=135\times 0.85=114.75$
3. Profit $=14.75\%=14\frac{3}{4}\%$

Trick: $35-15-\frac{35\times 15}{100}=20-5.25=14.75\%$

Answer: (C) $14\frac{3}{4}\%$

Concept: Standard result: $(1+\cot A-\csc A)(1+\tan A+\sec A)=2$; and ${\sin }^{2}A+{\cos }^{2}A=1$.

1. Write in sin, cos: $\frac{\sin A+\cos A-1}{\sin A}·\frac{\cos A+\sin A+1}{\cos A}$
2. $=\frac{(\sin A+\cos A{)}^2-1}{\sin A\cos A}=\frac{2\sin A\cos A}{\sin A\cos A}=2$
3. Expression $=2-3(1)=-1$

Trick: Put $A={45}^{\circ }$: $(2-\sqrt{2})(2+\sqrt{2})=4-2=2$, then $2-3=-1$.

Answer: (A) $-1$

Concept: Longest rod in a cube = body diagonal $a\sqrt{3}$; the largest sphere inside a cube has diameter $=a$.

1. $a\sqrt{3}=42\sqrt{3}\Rightarrow a=42\text{m}$
2. Sphere radius $r=\frac{42}{2}=21\text{m}$
3. Surface area $=4\pi r^2=4\times \frac{22}{7}\times 21\times 21=5544{\text{m}}^2$

Trick: Diagonal $a\sqrt{3}$ → side $a$ → sphere radius $\frac{a}{2}$.

Answer: (A) $5544$

Concept: Add inside the bracket first, then multiply.

1. $13.49+12.51=26$
2. $26\times 13=338$

Trick: Decimals $\mathrm{.49}+\mathrm{.51}=1$ make a whole number – look for such pairs.

Answer: (D) $338$

Concept: Overall profit is on total cost price; required SP = total target SP − SP already received.

1. Total CP $=2\times 9000=₹18000$; target SP $=18000\times 1.18=₹21240$
2. First watch SP $=9000\times 0.9=₹8100$
3. Second watch SP $=21240-8100=₹13140$

Trick: Needed profit $=3240$ + loss on first $900$ = $4140$ profit on second: $9000+4140=13140$.

Answer: (D) $13,140$

Concept: Volume $\propto r^3$ and surface area $\propto r^2$.

1. $r_1^3:r_2^3=512:3375\Rightarrow r_1:r_2=8:15$ (cube roots).
2. Surface areas $=8^2:{15}^2=64:225$

Trick: Volume ratio → cube root → square: $\sqrt[3]{512}=8$, $\sqrt[3]{3375}=15$.

Answer: (D) $64:225$

Concept: $a^3-b^3=(a-b{)}^3+3ab(a-b)$.

1. $x-3y=4$; take $a=x,b=3y$.
2. $x^3-27y^3=(x-3y{)}^3+3·x·3y(x-3y)=64+9xy\times 4=64+36xy$
3. $x^3-27y^3-36xy=64$

Trick: Put $y=0$: $x=4$, so value $=4^3=64$.

Answer: (C) $64$

Concept: Percentage of a part $=\frac{\text{part}}{\text{total}}\times 100$.

1. Boys : Girls $=3:5$, total $=8$ parts.
2. Boys $=\frac{3}{8}\times 100=37.5\%$

Trick: $\frac{1}{8}=12.5\%$, so $\frac{3}{8}=3\times 12.5=37.5\%$.

Answer: (D) $37.5\%$