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Q31. 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.
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. If a plant with green leaves is kept in a dark room with only green light ON, which one of the following would we observe?
Green leaves appear green because they reflect green light while absorbing other wavelengths (red/blue) for photosynthesis. In a dark room illuminated only by green light, the plant will reflect almost all the light back, making it appear brighter than the non-reflective surroundings.
Q34. For a certain regular solid, Faces + Vertices = Edges + 2. For three such distinct objects, what is the total value of Faces + Vertices - Edges?
For one object, F + V - E = 2. For three distinct, non-touching objects, the total sum is 2 + 2 + 2 = 6.
Q35. There are small and large bacteria of the same species. If S is surface area and V is volume, then which of the following is correct?
For any geometric shape, Surface Area (S) scales with the square of the radius (r^2) while Volume (V) scales with the cube (r^3). Therefore, the surface-to-volume ratio (S/V) scales inversely with size (1/r). As an organism gets smaller, its surface-area-to-volume ratio increases.
Q36. The statement: 'The father of my son is the only child of your parents' is true in how many relations?
'Father of my son' is either the speaker (if male) or the speaker's husband. 'Only child of your parents' is the listener. This identifies a unique husband-wife relationship dynamic.
Q37. The difference between the squares of the ages of a father and son is 899. The age of the father when the son was born was:
F^2 - S^2 = 899 => (F-S)(F+S) = 31 * 29 or 899 * 1. F-S is the age of the father at the son's birth. If F-S = 29 and F+S = 31, then F=30, S=1. (F-S)=29 is a valid age.
Q38. For which values of A and B is sin A = cot B?
Evaluate the options: If A=0 and B=pi/2, then sin(0) = 0, and cot(pi/2) = cos(pi/2)/sin(pi/2) = 0/1 = 0. Both sides equal 0, making the equation true.
Q39. 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.
Q40. 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.