Find a positive number which when increased by 17 is equal to 60 times the reciprocal of the number.

Here are the solutions to the problems: 51. Find a positive number which when increased by 17 is equal to 60 times the reciprocal of the number. Step 1: Set up the equation. Let the positive number be x. According to the problem statement: x + 17 = 60 × (1)/(x) x + 17 = (60)/(x) Step 2: Convert the equation into a quadratic form. Multiply both sides by x (since x is a positive number, x ≠ 0): x(x + 17) = 60 x^2 + 17x = 60 x^2 + 17x - 60 = 0 Step 3: Solve the quadratic equation. We can factor the quadratic equation. We need two numbers that multiply to -60 and add up to 17. These numbers are 20 and -3. (x + 20)(x - 3) = 0 This gives two possible solutions for x: x + 20 = 0 x = -20 x - 3 = 0 x = 3 Step 4: Select the positive number. Since the problem asks for a positive number, we choose x = 3. The correct option is (a). The final answer is 3. 52. The average monthly income of P and Q is ₹ 5,050. The average monthly income of Q and R is ₹ 6,250 and the average monthly income of P and R is ₹ 5,200. The monthly income of P is: Step 1: Write down the given information as equations. Let P, Q, and R be the monthly incomes of persons P, Q, and R respectively. 1. Average income of P and Q: (P + Q)/(2) = 5050 P + Q = 2 × 5050 P + Q = 10100 (Equation 1) 2. Average income of Q and R: (Q + R)/(2) = 6250 Q + R = 2 × 6250 Q + R = 12500 (Equation 2) 3. Average income of P and R: (P + R)/(2) = 5200 P + R = 2 × 5200 P + R = 10400 (Equation 3) Step 2: Add all three equations. (P + Q) + (Q + R) + (P + R) = 10100 + 12500 + 10400 2P + 2Q + 2R = 33000 2(P + Q + R) = 33000 P + Q + R = (33000)/(2) P + Q + R = 16500 (Equation 4) Step 3: Find the monthly income of P. Subtract Equation 2 from Equation 4: (P + Q + R) - (Q + R) = 16500 - 12500 P = 4000 The correct option is (b). The final answer is ₹ 4,000. 53. From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done? Total number of men = 7 Total number of women = 6 Total persons to be selected for the committee = 5 Condition: At least 3 men must be on the committee. This means the committee can have the following combinations of men and women: • 3 men and 2 women • 4 men and 1 woman • 5 men and 0 women Step 1: Calculate the number of ways for 3 men and 2 women. Number of ways to choose 3 men from 7 = 73 Number of ways to choose 2 women from 6 = 62 73 = (7!)/(3!(7-3)!) = (7 × 6 × 5)/(3 × 2 × 1) = 35 62 = (6!)/(2!(6-2)!) = (6 × 5)/(2 × 1) = 15 Ways for this case = 35 × 15 = 525 Step 2: Calculate the number of ways for 4 men and 1 woman. Number of ways to choose 4 men from 7 = 74 Number of ways to choose 1 woman from 6 = 61 74 = (7!)/(4!(7-4)!) = (7 × 6 × 5)/(3 × 2 × 1) = 35 61 = 6 Ways for this case = 35 × 6 = 210 Step 3: Calculate the number of ways for 5 men and 0 women. Number of ways to choose 5 men from 7 = 75 Number of ways to choose 0 women from 6 = 60 75 = (7!)/(5!(7-5)!) = (7 × 6)/(2 × 1) = 21 60 = 1 Ways for this case = 21 × 1 = 21 Step 4: Calculate the total number of ways. Total ways = (Ways for 3 men, 2 women) + (Ways for 4 men, 1 woman) + (Ways for 5 men, 0 women) Total ways = 525 + 210 + 21 = 756 The correct option is (d). The final answer is 756. 3 done, 2 left today. You're making progress.