28 November 2025 · Shift 2

Profit percentage Formula: Profit % $=\frac{\mathrm{Profit}}{\mathrm{Cost}\mathrm{price}}\times 100$. Here the profit is ₹30 and the cost price is ₹120. Profit % $=\frac{30}{120}\times 100=\frac{1}{4}\times 100$. $=25\%$. Hence option (d) is correct.

Successive discounts Take the marked price as 100 and apply the two discounts one after the other. After 10% off: $100\times 0.9=90$; after a further 20% off: $90\times 0.8=72$. The customer pays 72 instead of 100, so the total fall is 100-72=28, i.e. $28\%$. The same result comes from the formula $a+b-\frac{ab}{100}=10+20-\frac{10\times 20}{100}=28\%$. Hence option (b) is correct.

Simple interest Formula: SI $=\frac{P\times R\times T}{100}$, where P is the principal, R the rate and T the time. Here P=6000, R=8 and T=3. SI $=\frac{6000\times 8\times 3}{100}=\frac{144000}{100}$. =1440. So the simple interest is ₹1,440 and option (a) is correct.

Compound interest At 20% per annum the amount is multiplied by 1.2 at the end of every year. Formula: $A=P{\left(1+\frac{R}{100}\right)}^T=5000\times {\left(1.2\right)}^3$. $A=5000\times 1.728=8640$. CI =A-P=8640-5000=3640. So the compound interest is ₹3,640 and option (c) is correct.

Angles at a point Joining the centre to all 18 vertices cuts the pizza into 18 equal triangular slices. The angles at the centre lie side by side and together make one complete turn. Each slice has a central angle of $\frac{{360}^{\circ }}{18}={20}^{\circ }$. Sum $=18\times {20}^{\circ }={360}^{\circ }$, and this total stays ${360}^{\circ }$ whatever the number of sides. Hence option (c) is correct.

Terminating decimals A terminating decimal is one that stops after a finite number of digits, such as 0.75 or 2.4. Any such number can be written as a ratio of two integers: $0.75=\frac{75}{100}=\frac{3}{4}$. A number of the form $\frac{p}{q}$ with integers p,q and $q\ne 0$ is by definition rational, so every terminating decimal is rational. It need not be a whole number, since 0.75 is not one, and it can never be irrational because the decimal form of an irrational number never terminates. Hence option (c) is correct.

Volume of hollow cylinder The material of the pipe is the outer cylinder minus the hollow inner cylinder. Formula: Volume $=\pi \left(R^2-r^2\right)h$, with R=7 cm, r=5 cm and h=1.5 m =150 cm. Volume $=\pi \left(7^2-5^2\right)\times 150=\pi \left(49-25\right)\times 150$. $=\pi \times 24\times 150=3600\pi {\mathrm{cm}}^3$. Hence option (c) is correct.

Surface area of prism Each box is a prism: an equilateral triangle of side 10 cm at the top and at the bottom, and three rectangles round the sides. Area of one triangular face $=\frac{\sqrt{3}}{4}{a}^2=\frac{1.732}{4}\times {10}^2=43.3{\mathrm{cm}}^2$. Top and bottom together $=2\times 43.3=86.6{\mathrm{cm}}^2$. Lateral area = perimeter of base $\times$ height $=\left(3\times 10\right)\times 20=600{\mathrm{cm}}^2$. Total surface area of a box $=600+86.6=686.6{\mathrm{cm}}^2$. Hence option (c) is correct.

Ratio of volumes The height of a hemisphere of radius r is r itself, so the cylinder also has radius r and height r. Volume of hemisphere $=\frac{2}{3}\pi r^3$. Volume of cylinder $=\pi r^{2}h=\pi r^2\times r=\pi r^3$. Ratio $=\frac{2}{3}\pi r^3:\pi r^3=\frac{2}{3}:1=2:3$. Hence option (c) is correct.

Area of parallelogram Formula: area of a parallelogram $=ab\sin \theta$, where a and b are two adjacent sides and $\theta$ the angle between them. Area $=18\times 12\times \sin {30}^{\circ }=216\times \frac{1}{2}=108{\mathrm{cm}}^2$. The height asked for stands on the longer side, so use Area = base $\times$ height with base =18 cm. $108=18\times h$, which gives $h=\frac{108}{18}=6$ cm. So the area is $108{\mathrm{cm}}^2$ and the height is 6 cm, and option (c) is correct.

Sphere recast into hemispheres Melting keeps the volume unchanged, so first find how many hemispheres the ball gives. Volume of the sphere $=\frac{4}{3}\pi R^3=\frac{4}{3}\pi {\left(14\right)}^3$. Volume of one hemisphere $=\frac{2}{3}\pi r^3=\frac{2}{3}\pi {\left(7\right)}^3$. Number of hemispheres $=\frac{\frac{4}{3}\pi {\left(14\right)}^3}{\frac{2}{3}\pi {\left(7\right)}^3}=2\times {\left(\frac{14}{7}\right)}^3=2\times 8=16$. Curved surface area of one hemisphere $=2\pi r^2=2\times \frac{22}{7}\times {\left(7\right)}^2=308$ cm${}^2$. Combined curved surface area $=16\times 308=4928$ cm${}^2$. Hence option (d) is correct.

Square-root equation The condition given is $\sqrt{25}+\sqrt{9+y^2}=8$. $\sqrt{25}=5$, so $5+\sqrt{9+y^2}=8$. $\sqrt{9+y^2}=8-5=3$. Squaring both sides: $9+y^2=9$. $y^2=0$, therefore y=0. Hence option (a) is correct.

Partnership profit share When the partners invest for different periods, the profit is divided in the ratio of (investment × time). Chirag: $8000\times 9=72000$. Sahil: $9000\times 4=36000$. Ratio of profit =72000:36000=2:1. Chirag's share $=\frac{2}{2+1}\times 21000=\frac{2}{3}\times 21000=14000$. So Chirag receives ₹14,000; hence option (b) is correct.

Trigonometric ratios $\cot \theta =1$ means $\tan \theta =1$, which gives $\theta ={45}^{\circ }$. At ${45}^{\circ }$: $\sec {45}^{\circ }=\sqrt{2}$ and $\cos {45}^{\circ }=\frac{1}{\sqrt{2}}$. $\sec \theta +\cos \theta =\sqrt{2}+\frac{1}{\sqrt{2}}=\frac{2+1}{\sqrt{2}}=\frac{3}{\sqrt{2}}$. Hence option (b) is correct.

Sign of ratios by quadrant $\cos x=\frac{4}{5}$ gives base =4 and hypotenuse =5. By Pythagoras the perpendicular $=\sqrt{5^2-4^2}=\sqrt{25-16}=3$. In Quadrant IV only cosine (and secant) are positive, so $\sin x=-\frac{3}{5}$. $\tan x=\frac{\sin x}{\cos x}=\frac{-3/5}{4/5}=-\frac{3}{4}$. Hence option (c) is correct.

HCF word problem The greatest equal piece must divide both lengths exactly, so it is the HCF of the two lengths. Work in centimetres: 1.8 m =180 cm and 1.2 m =120 cm. $180=2^2\times 3^2\times 5$ and $120=2^3\times 3\times 5$. HCF $=2^2\times 3\times 5=60$ cm =0.6 m. Hence option (b) is correct.

Evaporation from a solution Salt in the solution $=20\%$ of $50=\frac{20}{100}\times 50=10$ L, so water =40 L. Only water evaporates, so the 10 L of salt stays the same. Let the new volume be V; then $25\%$ of V=10. $V=\frac{10\times 100}{25}=40$ L. Water evaporated =50-40=10 L. Hence option (a) is correct.

Ratio and proportion Orange : pineapple : soda =5:3:2, so the mixture has 5+3+2=10 equal parts. 10 parts =20 L, therefore 1 part $=\frac{20}{10}=2$ L. Soda water is 2 parts $=2\times 2=4$ L. Hence option (c) is correct.

Mixing three vessels Take the liquid in A, B and C as 2, 3 and 5 units — these already match the given ratio 2:3:5. A has milk : water =1:1, so of its 2 units, milk =1 and water =1. B has milk : water =2:1, so of its 3 units, milk =2 and water =1. C has milk : water =3:2, so of its 5 units, milk =3 and water =2. Total milk =1+2+3=6 and total water =1+1+2=4. Required ratio =6:4=3:2. Hence option (d) is correct.

Fraction of a journey Let the total distance be x km. By bike $=\frac{x}{5}$, so the remaining distance $=x-\frac{x}{5}=\frac{4x}{5}$. On foot $=\frac{2}{3}\times \frac{4x}{5}=\frac{8x}{15}$. By bus $=\frac{4x}{5}-\frac{8x}{15}=\frac{12x-8x}{15}=\frac{4x}{15}$. This bus part is 12 km, so $\frac{4x}{15}=12$. $x=\frac{12\times 15}{4}=45$. The total distance is 45 km; hence option (a) is correct.

Complementary angles For complementary angles, $\sin \theta =\cos \left({90}^{\circ }-\theta \right)$. Given $\sin A=\cos B$, rewrite the left side as $\cos \left({90}^{\circ }-A\right)$. So $\cos \left({90}^{\circ }-A\right)=\cos B$, which gives ${90}^{\circ }-A=B$. Therefore $A+B={90}^{\circ }$ — the two angles are complementary. Hence option (a) is correct.

Surds — difference of squares Use the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$ with $a=\sqrt{13}$ and b=3. $\left(\sqrt{13}+3\right)\left(\sqrt{13}-3\right)={\left(\sqrt{13}\right)}^2-{\left(3\right)}^2$. =13-9=4. Hence option (c) is correct.

Simultaneous surd equations $\sqrt{x}+\sqrt{y}=14$ … (i) $\sqrt{x}-\sqrt{y}=8$ … (ii) Adding (i) and (ii): $2\sqrt{x}=22\Rightarrow \sqrt{x}=11\Rightarrow x=121$. Subtracting (ii) from (i): $2\sqrt{y}=6\Rightarrow \sqrt{y}=3\Rightarrow y=9$. x+y=121+9=130. Shortcut: $x+y=\frac{{\left(14\right)}^2+{\left(8\right)}^2}{2}=\frac{196+64}{2}=130$. Hence option (d) is correct.

Simple interest Simple interest $=\frac{P\times R\times T}{100}$, so with the same sum and rate each extra year adds one year's interest. Difference $=\frac{P\times 12\times 4}{100}-\frac{P\times 12\times 3}{100}=\frac{P\times 12\times 1}{100}$. Therefore $\frac{12P}{100}=72$. $P=\frac{72\times 100}{12}=600$. The sum is ₹600; hence option (a) is correct.

Average of four numbers Sum = average $\times$ count, so $M_1+M_2+M_3=3\times 40=120$. Similarly $M_2+M_3+M_4=3\times 34=102$. Since $M_1=36$, we get $M_2+M_3=120-36=84$. Put this in the second sum: $84+M_4=102$. $M_4=102-84=18$. Hence option (d) is correct.