UPSC CAPF

A point (x, y) is strictly inside this triangle if it satisfies the following conditions:
1. It must be to the right of the line x=0: x > 0.
2. It must be below the line y=100: y < 100. 3. It must be above the line y=x: y > x.

Combining these, we are looking for integer coordinates (x, y) such that 0 < x < y < 100. Let's iterate through possible integer values for x. Since x > 0 and x < y < 100, the smallest possible integer value for x is 1. If x = 1, y must be an integer such that 1 < y < 100. Possible y values are 2, 3, ..., 99. The number of such y values is 99 - 2 + 1 = 98. If x = 2, y must be an integer such that 2 < y < 100. Possible y values are 3, 4, ..., 99. The number of such y values is 99 - 3 + 1 = 97. If x = 3, y must be an integer such that 3 < y < 100. Possible y values are 4, 5, ..., 99. The number of such y values is 99 - 4 + 1 = 96. ... What is the largest possible integer value for x? Since x < y < 100, the largest possible integer value for y is 99. This requires x to be at least 1 less than 99, i.e., x < 99. So, the largest possible integer value for x is 98. If x = 98, y must be an integer such that 98 < y < 100. The only possible y value is 99. The number of such y values is 1. (99 - 99 + 1 = 1) The total number of integer points inside the triangle is the sum of the number of possible y values for each x from 1 to 98. Total points = (99 - 1) + (99 - 2) + (99 - 3) + ... + (99 - 98) Total points = 98 + 97 + 96 + ... + 1 This is the sum of the first 98 positive integers. The formula for the sum of the first n positive integers is n(n+1)/2. Here, n = 98. Sum = 98 * (98 + 1) / 2 = 98 * 99 / 2 = 49 * 99. 49 * 99 = 49 * (100 - 1) = 4900 - 49 = 4851. The number of points inside the triangle with integer coordinates is 4851.

The second wheel rolls on the larger circle (R2). Let its radius be r2. The distance covered by its point of contact is the circumference of the larger circle, which is 2πR2. This distance is also equal to N times the circumference of the second wheel, which is N * 2πr2.
So, 2πR2 = N * 2πr2
R2 = N * r2

We know R2 = 11000 cm and N = 1000/3.
11000 = (1000/3) * r2
r2 = 11000 * (3 / 1000)
r2 = 11 * 3 = 33 cm.

The radius of the other wheel is 33 cm.

Let’s test if this interpretation makes sense by assuming statement C is indeed the “not correct” one. This means C&F&H = 0. The five regions could then be, for example: C only, F only, H only, C&F only, C&H only, F&H only (this is 6, so one more must be 0). Or perhaps C only, F only, H only, C&F only, F&H only (omitting C&H only and C&F&H). Or C only, F only, H only, C&F only, C&H only (omitting F&H only and C&F&H).

If C&F&H = 0, then statement C is false. Let’s see if the other statements *could* be true under various configurations of 5 non-empty regions (where C&F&H=0):
A) “Every hockey player plays football”: Requires H only = 0 and C&H only = 0. Possible if the 5 regions are, for example, F only, C only, C&F only, F&H only, some other.
B) “Every cricket player plays either football or hockey”: Requires C only = 0. Possible.
D) “There are some football players who play neither cricket nor hockey”: Requires F only > 0. Possible.

Since statement C (C&F&H > 0) directly contradicts the implication that the region C&F&H is one of the *missing* regions among the five non-empty ones, statement C is the most likely to be the “not correct” one under a plausible interpretation of the five given areas. The wording “five disjoint bounded areas… labelled with number of players belonging to that area” strongly suggests these are the *only* regions within the shapes that contain players. If the C&F&H intersection is not one of these five, its count is 0.