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Q1. Tiwari and Deo take 2 hours to do a job. Tiwari and Hari take 3 hours. Deo and Hari take 6 hours. Which statement is incorrect?
Work rates: T+D=1/2, T+H=1/3, D+H=1/6. Adding them: 2(T+D+H)=1. So T+D+H=1/2. Since T+D=1/2, H must be 0 (no work). Rates are: T=1/3 (3 hrs), D=1/6 (6 hrs), H=0. Thus, Hari being the fastest is incorrect.
Q2. A boy holds one end of a rope of length l and the other end is fixed to a thin pole of radius r (r<<l). Keeping the rope taut, the boy goes around the pole causing the rope to get wound around the pole. Each round takes 10 s. What is the speed with which the boy approaches the pole?
With each full revolution around the pole, the rope wraps around the circumference of the pole, shortening the distance between the boy and the pole by 2*pi*r. Since one round takes 10 seconds, the speed of approach is Distance / Time = (2*pi*r) / 10 = pi*r / 5.
Q3. A wheel barrow with unit spacing between its wheels is pushed along a semi-circular path of mean radius 10. The difference between distances covered by the inner and outer wheels is:
With a unit spacing (1) and mean radius 10, the outer radius is 10.5 and the inner is 9.5. Distance for a semi-circle is πr. Difference = π(10.5) - π(9.5) = π(10.5 - 9.5) = π.
Q4. In ponds where there is an overabundance of aquatic plants like water hyacinth, the fish population is found to be low, the reason being:
When invasive plants like water hyacinth overgrow and eventually die, their decomposition by aerobic bacteria consumes massive amounts of dissolved oxygen from the water (increasing the Biological Oxygen Demand). This lack of oxygen suffocates the fish population.
Q5. How many digits are there in 3^16 when it is expressed in the decimal form?
We can estimate this using logarithms: log10(3^16) = 16 * log10(3) ≈ 16 * 0.4771 = 7.6336. Since the characteristic is 7, the number of digits is 7 + 1 = 8.
Q6. A bicycle tube has a mean circumference of 200 cm and a circular cross-section diameter of 6 cm. What is the approximate volume of water required to fill it?
Volume of a torus = (Area of cross-section) * (Mean circumference). Radius of cross-section = 3 cm. Area = pi * 3^2 = 9pi. Volume = 9pi * 200 = 1800 pi.
Q7. The sum of two numbers is equal to sum of square of 11 and cube of 9. The larger number is 5^2 less than square of 25. What is the value of the sum of twice of 24 percent of the smaller number and half of the larger number?
Total Sum = 11^2 + 9^3 = 121 + 729 = 850. Larger number = 25^2 - 5^2 = 625 - 25 = 600. Therefore, the smaller number is 850 - 600 = 250. Twice of 24% of the smaller number = 2 * (0.24 * 250) = 120. Half of the larger number = 600 / 2 = 300. The sum is 120 + 300 = 420.
Q8. A vendor sells articles (cost price Rs.100). He sells at a premium for 8 months and at half that premium price for 4 months. He makes a net profit of 20% at the end of the year. What is the premium price?
Total cost for 12 items = 1200. A 20% profit means total revenue = 1440. Let P be the premium price. Equation: 8P + 4(P/2) = 1440 -> 8P + 2P = 1440 -> 10P = 1440. P = 144.
Q9. The smallest square floor which can be completely paved with tiles of size 8x6 without breaking any tile, needs n tiles. Find n.
To form a square, the side length must be a common multiple of the tile's dimensions (8 and 6). The Least Common Multiple (LCM) of 8 and 6 is 24. So, the smallest square floor is 24x24. The area of the floor is 576. The area of one tile is 48. The number of tiles (n) = 576 / 48 = 12.
Q10. To determine the number of parrots in a sparse population, an ecologist captures 30 parrots and puts rings around their necks and releases them. After a week he captures 40 parrots and finds that 8 of them have rings on their necks. What approximately is the parrot population?
This uses the Lincoln-Petersen mark-recapture index. Total Population (N) = (Number marked initially * Total caught in second sample) / Number of marked recaptures. N = (30 * 40) / 8 = 1200 / 8 = 150.