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Q31. A chocolate bar has m x n tiles. How many cuts are needed to break it into individual tiles without stacking?
Each cut increases the number of pieces by exactly one. To go from 1 piece to mn pieces, you need mn - 1 cuts.
Q32. A circle drawn in the x-y coordinate plane passes through the origin and has chords of lengths 8 units and 7 units on the x and y axes, respectively. The coordinates of its centre are:
If a circle passes through the origin (0,0) and intercepts the axes at (8,0) and (0,7), the line connecting these two points is a diameter. The center is the midpoint of the diameter: ( (8+0)/2 , (0+7)/2 ) = (4, 3.5).
Q33. The probability that a ticketless traveler is caught during a trip is 0.1. If the traveler makes 4 trips, the probability that he/she will be caught during at least one of the trips is:
The probability of not being caught in a single trip is 1 - 0.1 = 0.9. The probability of not being caught in all 4 trips is (0.9)^4. The probability of being caught at least once is the complement: 1 - (0.9)^4.
Q34. Choose the four-digit number where the product of the 1st and 4th digits is 40 and the middle digits is 28. The 1000s digit is as much less than the unit digit as the 100s is less than the 10s.
Digits a, b, c, d: a*d=40 and b*c=28. Possible for a,d: (5,8) or (8,5). For b,c: (4,7) or (7,4). Condition: d-a = c-b. For 5478: 8-5=3 and 7-4=3. This matches.
Q35. 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.
Q36. A new tyre can be used for at most 90 km. What is the maximum distance (in km) that can be covered by a three wheeled vehicle carrying one spare wheel, all four tyres being new?
Each of the 4 tyres has a lifespan of 90 km, giving a total of 360 tyre-kilometers available. The vehicle requires 3 tyres on the ground at any given time. By rotating the spare evenly, the maximum distance the vehicle can travel is 360 / 3 = 120 km.
Q37. 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.
Q38. What is the maximum number of cylindrical pencils of 0.5 cm diameter that can be stood in a square shaped stand of 5 cm x 5 cm inner cross section?
If stacked in a simple grid, 10x10 = 100 pencils fit. However, using hexagonal closest packing allows rows to nestle into the gaps of the adjacent row. The first row holds 10. The next row holds 9. The vertical distance between nested rows is 0.5 * sin(60) = 0.433 cm. Within 5cm, you can fit 11 rows (6 rows of 10, 5 rows of 9), totaling 60 + 45 = 105 pencils.
Q39. 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.
Q40. The number of diagonals of a convex deodecagon (12-gon) is:
The formula for the number of diagonals in an n-sided polygon is n(n-3)/2. For n=12, it is 12(9)/2 = 54.