22 November 2025 · Shift 2

Intersection of two lines The point where the two paths meet must satisfy both equations at the same time. y=-x+14 ... (i) and y=x-2 ... (ii) Equating the two expressions for y: -x+14=x-2. 14+2=x+x, so 2x=16 and x=8. Putting x=8 in (ii): y=8-2=6. So the two mosquitoes cross at (8,6). Hence, option (a) is the correct answer.

Square root unit digit A perfect square that ends in 4 must have a square root ending in 2 or in 8. ${120}^2=14400$ and ${130}^2=16900$, and 16384 lies between these, so the root lies between 120 and 130. The only candidates in that range are 122 and 128, and ${128}^2=16384$. So $\sqrt{16384}=128$, whose unit digit is 8. Hence, option (d) is the correct answer.

Partnership profit share Both partners keep their money in the business for the same time, so the profit is divided in the ratio of the capitals alone. P:Q=4000:6000=2:3. Total number of parts =2+3=5. Q's share $=\frac{3}{5}\times 2000=1200$, that is ₹1,200. Hence, option (c) is the correct answer.

Profit-sharing ratio When the partners invest for different periods, each partner's claim is capital $\times$ time. K's claim $=7000\times 5=35000$. L's claim $=4000\times 7=28000$. Required ratio =35000:28000=35:28=5:4. Hence, option (d) is the correct answer.

Correcting an average Sum of the 30 results as first calculated $=30\times 25=750$. One score of 55 was entered as 25, so the total is short by 55-25=30. Correct total =750+30=780. Correct average $=\frac{780}{30}=26$. As a check, the total rises by 30 spread over 30 results, so the average rises by exactly 1, from 25 to 26. Hence, option (a) is the correct answer.

Successive percentage increase Take the original price as 100 so that every change can be read directly as a percentage. After the first rise of 15%: $100\times \frac{115}{100}=115$. After the second rise of 25% on this new price: $115\times \frac{125}{100}=143.75$. Total increase =143.75-100=43.75 on an original 100, that is 43.75%. The shortcut $a+b+\frac{ab}{100}$ gives the same figure: $15+25+\frac{15\times 25}{100}=43.75$. Hence, option (b) is the correct answer.

Population growth A yearly rise of 20% multiplies the population by $\frac{120}{100}=\frac{6}{5}$ every year. Over 3 years the multiplier is ${\left(\frac{6}{5}\right)}^3=\frac{216}{125}$. If the population 3 years ago was P, then $P\times \frac{216}{125}=21600$. $P=21600\times \frac{125}{216}=100\times 125=12500$. Hence the population 3 years ago was 12,500, option (c).

Ratio from percentages Let the two numbers be x and y. The condition given is $\frac{25}{100}x=\frac{30}{100}y$. Multiplying both sides by 100: 25x=30y. Dividing both sides by the common factor 5: 5x=6y. So the two numbers stand in the ratio 5:6. Hence, option (b) is the correct answer.

Successive discounts Two discounts act one after the other: the second is worked out on the price left after the first. Price after the 20% discount $=1500\times \frac{80}{100}=1200$. Price after the further 10% discount $=1200\times \frac{90}{100}=1080$. So the woman finally pays ₹1,080. Hence, option (d) is the correct answer.

Marked price from discount A discount of 5% means the selling price is 100-5=95% of the marked price. So $\frac{95}{100}\times MP=950$. $MP=950\times \frac{100}{95}=10\times 100=1000$. A check: 5% of 1000 is 50, and 1000-50=950, exactly the given selling price. Hence the marked price is ₹1,000, option (c).

Trains - relative speed To pass a man the train has to cover only its own length, and it does so relative to the man. Relative speed $=\frac{\mathrm{length}\mathrm{of}\mathrm{train}}{\mathrm{time}}=\frac{240}{12}=20$ m/s 20 m/s $=20\times \frac{18}{5}=72$ km/h Both move in the same direction, so relative speed = speed of train - speed of man. Speed of train =72+12=84 km/h Hence, option (a) is the correct answer.

Train crossing a tunnel Convert the speed: 54 km/h $=54\times \frac{5}{18}=15$ m/s. Distance covered in 40 seconds $=15\times 40=600$ m. While crossing a tunnel a train covers (length of tunnel + its own length). 400+L=600 L=600-400=200 m Hence, option (d) is the correct answer.

Hemisphere curved surface Only the dome is painted, so the area needed is the curved surface of the hemisphere. Formula: curved surface of a hemisphere $=2\pi r^2$ Diameter =14 m, so $r=\frac{14}{2}=7$ m. Area $=2\times \frac{22}{7}\times 7\times 7$ $=2\times 22\times 7=308$ square metres Hence, option (c) is the correct answer.

Area from coordinates A(1,1) and B(6,1) have the same y-coordinate, so AB is a horizontal side and can be taken as the base. Base AB=6-1=5 units. D(3,4) and C(8,4) lie on the line y=4, so the opposite side is parallel to AB and the height is the gap between the two lines. Height =4-1=3 units. Area of a parallelogram $=\mathrm{base}\times \mathrm{height}=5\times 3=15$ square units Hence, option (c) is the correct answer.

Tiling - area division Bring both measurements to the same unit: 8 m =800 cm and 6 m =600 cm. Area of the hall $=800\times 600=480000$ square cm. Area of one tile $=80\times 60=4800$ square cm. Number of tiles $=\frac{480000}{4800}=100$ Hence, option (b) is the correct answer.

Cuboid - number of pieces Volume of the large block $=72\times 36\times 24=62208$ cubic cm. Volume of one small block $=12\times 6\times 8=576$ cubic cm. Number of blocks $=\frac{62208}{576}=108$ The count is exact because each edge of the large block is a whole multiple of the matching edge of the small block: $72÷12=6$, $36÷6=6$, $24÷8=3$, and $6\times 6\times 3=108$. Hence, option (d) is the correct answer.

Cuboid volume - filling rate Volume of the tank $=\mathrm{length}\times \mathrm{breadth}\times \mathrm{depth}$ $=1.5\times 0.8\times 0.5=0.6$ cubic metres 1 cubic metre =1000 litres, so the tank holds $0.6\times 1000=600$ litres. Time taken =2 hours $=2\times 60=120$ minutes. Rate $=\frac{600}{120}=5$ litres per minute Hence, option (a) is the correct answer.

Pyramid lateral surface The four faces are congruent triangles, each with base 8 m and slant height 5 m. Formula: lateral surface of a pyramid $=\frac{1}{2}\times \mathrm{perimeter}\mathrm{of}\mathrm{base}\times \mathrm{slant}\mathrm{height}$ Perimeter of the square base $=4\times 8=32$ m Lateral surface $=\frac{1}{2}\times 32\times 5=80$ So $80m^2$ of canvas is needed. Hence, option (b) is the correct answer.

Volume of a pyramid Check the base: $6^2+8^2=36+64=100={10}^2$, so the right angle is between the sides 6 m and 8 m. Area of the base $=\frac{1}{2}\times 6\times 8=24$ square metres Formula: volume of a pyramid $=\frac{1}{3}\times \mathrm{base}\mathrm{area}\times \mathrm{height}$ $=\frac{1}{3}\times 24\times 9$ $=8\times 9=72m^3$ Hence, option (c) is the correct answer.

Cone - height from CSA Formula: curved surface area of a cone $=\pi rl$, where l is the slant height. $\frac{22}{7}\times 7\times l=550$ $22l=550\Rightarrow l=25$ cm In a cone the radius, height and slant height form a right triangle: $l^2=r^2+h^2$. $h=\sqrt{{25}^2-7^2}=\sqrt{625-49}=\sqrt{576}=24$ cm Hence, option (d) is the correct answer.

Volume of a prism Test the base triangle: $9^2+{12}^2=81+144=225={15}^2$, so it is right-angled with legs 9 cm and 12 cm. Area of the base $=\frac{1}{2}\times 9\times 12=54$ square cm Formula: volume of a prism $=\mathrm{area}\mathrm{of}\mathrm{base}\times \mathrm{length}$ $=54\times 20=1080{\mathrm{cm}}^3$ Hence, option (b) is the correct answer.

Sphere inside a hemisphere Let the radius of the hemisphere be r. The biggest sphere that fits sits on the flat face with its centre on the axis, so it touches the flat face and the curved face; its radius is $\frac{r}{2}$. Volume of the hemisphere $=\frac{2}{3}\pi r^3$ Volume of the sphere $=\frac{4}{3}\pi {\left(\frac{r}{2}\right)}^3=\frac{4}{3}\pi \times \frac{r^3}{8}=\frac{\pi r^3}{6}$ Volume of the remaining solid $=\frac{2}{3}\pi r^3-\frac{1}{6}\pi r^3=\frac{4-1}{6}\pi r^3=\frac{\pi r^3}{2}$ Required ratio $=\frac{\pi r^3}{6}:\frac{\pi r^3}{2}=\frac{1}{6}:\frac{1}{2}=1:3$ Hence, option (c) is the correct answer.

Fourth proportional If a, b, c are given, the fourth proportional d satisfies a:b::c:d, i.e. $\frac{a}{b}=\frac{c}{d}$. Here a=3, b=9, c=6, so $\frac{3}{9}=\frac{6}{d}$. Cross-multiplying, $d=\frac{b\times c}{a}=\frac{9\times 6}{3}=\frac{54}{3}=18$ Hence, option (a) is the correct answer.

Average of squares Formula: sum of the squares of the first n natural numbers $=\frac{n\left(n+1\right)\left(2n+1\right)}{6}$ For n=8: sum $=\frac{8\times 9\times 17}{6}=\frac{1224}{6}=204$ So $B=\frac{204}{8}=25.5$ Doubling a number makes its square four times as large, because ${\left(2x\right)}^2=4x^2$. Every squared value is multiplied by 4, so the average is also multiplied by 4. New average $=4B=4\times 25.5=102$ Hence, option (b) is the correct answer.

Average - replacement The group still has 5 members, so a rise of 3 in the average means a rise of $5\times 3=15$ years in the total age. The only change in the total is that a member aged 20 years is replaced by the new member. Age of new person - age of person who left =15 Age of new person =20+15=35 years Hence, option (a) is the correct answer.