Mathematics

The vector b is incorrectly written as '3jk + 3k', which is not a valid vector expression. Assuming it means b = 3j + 3k, the projection of a on b is (a·b)/|b|. Calculating with b = 3j + 3k, a·b = 0 + 3 + 3 = 6, |b| = √(0^2 + 3^2 + 3^2) = √18 = 3√2, so projection = 6/(3√2) = 2/√2. This matches option A, not B. Since the original b is invalid, and the correct projection is 2/√2, the correct answer should be A if b is corrected. Due to the original error in b, the best choice is 'None of these' (D).

The population increases by 700 each year for 5 years, so total increase = 700 × 5 = 3500. The population after 5 years is 25000, so the initial population = 25000 - 3500 = 21500.

The multiplicative inverse of a complex number z = a + bi is 1/z = (a - bi) / (a^2 + b^2). For 6 - 5i, the inverse is (6 + 5i) / (6^2 + (-5)^2) = (6 + 5i) / 61. The imaginary part is therefore 5/61.

The work done by a force vector F in moving an object from point P to Q is the dot product of F and the displacement vector PQ. Here, F = i + j + k and displacement PQ = Q - P = (1-1, 2-1, 2-1) = (0,1,1). The dot product is (1)(0) + (1)(1) + (1)(1) = 0 + 1 + 1 = 2.

The total number of balls is 9 + 8 + 3 = 20. The number of balls that are not orange is 9 + 8 = 17. Therefore, the probability of selecting a ball that is not orange is 17/20.