Mathematics

Let the smallest even number be x. The four consecutive even numbers are x, x+2, x+4, and x+6. Their sum is given as 292, so we set up the equation: x + (x+2) + (x+4) + (x+6) = 292. Combining like terms, we get 4x + 12 = 292. Subtracting 12 from both sides gives 4x = 280. Dividing both sides by 4, we find x = 70. Therefore, the smallest number is 70, which corresponds to option C.

Let the amounts received by P, Q, and R be p, q, and r respectively. Given p + q + r = 1200. P gets half of the total amount received by Q and R: p = 1/2 (q + r). Q gets one-third of the total amount received by P and R: q = 1/3 (p + r). Solving these equations gives r = 400.

The word "TRIANGLE" has 8 letters with 5 consonants (T, R, N, G, L) and 3 vowels (I, A, E). The number of ways to select 3 consonants out of 5 is C(5,3) = 10, and the number of ways to select 2 vowels out of 3 is C(3,2) = 3. Multiplying these gives 10 × 3 = 30 ways.

The number of ways to form a group of 5 men and 2 women from 7 men and 3 women is calculated using combinations. The number of ways to choose 5 men out of 7 is C(7,5) = 21, and the number of ways to choose 2 women out of 3 is C(3,2) = 3. Since these choices are independent, multiply them: 21 × 3 = 63. Therefore, the correct answer is 63, which corresponds to option A.

Let the actual number of children in the school be x. Each child was supposed to get 2 bananas, so total bananas = 2x. On the day, 360 children were absent, so number of children present = x - 360. Each present child got 2 extra bananas, so each got 2 + 2 = 4 bananas. Total bananas distributed = 4(x - 360). Since total bananas remain the same, 2x = 4(x - 360). Solving: 2x = 4x - 1440 => 1440 = 2x => x = 720. Although 720 is not among the options, the correct mathematical solution is 720. The closest option given is 600 (option A), which is likely a typographical error in the options. The verified correct answer is 720 children in the school.

Let the original two-digit number be 10y + x, where y is the tens digit and x is the units digit. The sum of the digits of the number obtained by interchanging the digits is x + y = 13. The difference between the original number and the interchanged number is (10y + x) - (10x + y) = 9(y - x) = 45, so y - x = 5. From the two equations, x + y = 13 and y - x = 5, adding them gives 2y = 18, so y = 9. Substituting y = 9 into x + y = 13 gives x = 4. Therefore, the original number is 10y + x = 10*4 + 9 = 49. The interchanged number is 94, and the difference 94 - 49 = 45. The sum of digits of the interchanged number (94) is 9 + 4 = 13. Hence, the original number is 49 (option A).

The total number of ways to select 10 representatives from 22 people (12 seniors + 10 juniors) is 22C10. The number of groups with no seniors (only juniors) is 10C10 = 1. Therefore, the number of groups with at least one senior is 22C10 - 1.

Using the rule of allegation, the ratio of the quantities of the cheaper rice (Rs. 6 per kg) to the dearer rice (Rs. 8.75 per kg) to obtain a mixture costing Rs. 7.50 per kg is calculated as (8.75 - 7.50) : (7.50 - 6) = 1.25 : 1.5, which simplifies to 5 : 6. Therefore, the correct ratio is 5 : 6, corresponding to option A.

The word "THURSDAY" has 8 distinct letters. The number of different words (permutations) that can be formed using all 8 letters is 8! (8 factorial), which equals 40,320.

Let the five consecutive even numbers in set x be x, x+2, x+4, x+6, x+8. Their sum is 5x + 20 = 440, so 5x = 420 and x = 84. The least number of set x is 84. Double this is 168. The second least number of the new set is 121 less than 168, so 168 - 121 = 47. Let the new set be five consecutive integers: y, y+1, y+2, y+3, y+4, where y+1 = 47, so y = 46. The sum of the new set is y + (y+1) + (y+2) + (y+3) + (y+4) = 5y + 10 = 5*46 + 10 = 230 + 10 = 240. Therefore, the correct answer is B (240).