Mathematics

The first 10 odd numbers are 1, 3, 5, 7, 9, 11, 13, 15, 17, and 19. Their sum is 100. The average is calculated as the sum divided by the number of terms, which is 100 / 10 = 10. Therefore, the correct answer is option B (10).

The average of the first 20 natural numbers is calculated by summing them (1+2+...+20 = 210) and dividing by 20, which equals 10.5.

The average of 35 numbers is 25, so their total sum is 35 × 25 = 875. If each number is multiplied by 5, the new sum becomes 875 × 5 = 4375. The new average is then 4375 ÷ 35 = 125.

Three years ago, the total age of the six members was 6 × 19 = 114 years. Today, each member is 3 years older, so total age of original six members is 114 + 6 × 3 = 132 years. Let the boy's age be x. The average age today is (132 + x) / 7. Since the average remains 19, (132 + x) / 7 = 19, so 132 + x = 133, giving x = 1 year. However, this contradicts the calculation. Rechecking: 3 years ago average = 19, so total age = 114. Today, original members aged 3 years each, total 114 + 18 = 132. Average remains 19 with 7 members: total age = 7 × 19 = 133. So, boy's age = 133 - 132 = 1 year. Therefore, the correct answer is A (1 year). The initial answer was A, which is correct. The explanation clarifies the calculation.

The original average is 23. Increasing each of the 10 numbers by 4 increases the total sum by 10 × 4 = 40. The new average is (original total + 40) ÷ 10 = (10 × 23 + 40) ÷ 10 = (230 + 40) ÷ 10 = 270 ÷ 10 = 27.

Doubling each student's marks doubles the total sum of marks, so the average also doubles. Since the original average is 36, the new average is 36 × 2 = 72.

1 inch is exactly equal to 2.54 centimeters, which is the standard conversion factor used internationally.

Isaac Newton, along with Gottfried Wilhelm Leibniz, is credited as a founder of calculus. Newton developed the fundamental principles of calculus independently, earning him the title 'Father of Calculus.'

Let the original average expenditure per student be x. Initially, total expenditure = 100x. After accommodating 20 more students, average decreases by Rs.5, so new average = x - 5. New total expenditure = 120(x - 5) = 120x - 600. Given total expenditure increased by Rs.400, so 120x - 600 = 100x + 400. Solving gives x = Rs.50. Therefore, new total expenditure = 120 * (50 - 5) = 120 * 45 = Rs.5400.

Let the average before the 17th innings be x. Total runs before 17th innings = 16x. After scoring 85 in the 17th innings, the new average is x + 3. So, (16x + 85)/17 = x + 3. Solving gives x = 33, so the new average is 36.