Find the value of x if 58 = x mod 8. Find the gradient of a line joining the points (2, 5) and (3, 1). If p varies inversely as the square root of (m + 2) and p = 12 when m = 7, find m when p = 6.

Here are the solutions to the problems: 22. Find the value of x if 58 x 8. Step 1: To find x, we need to find the remainder when 58 is divided by 8. 58 ÷ 8 Step 2: Perform the division. 58 = 8 × 7 + 2 The remainder is 2. Therefore, x = 2. The correct option is A. The value of x is 2. 23. Find the gradient of a line joining the points (2, 5) and (3, 1). Step 1: Identify the coordinates of the two points. Let (x_1, y_1) = (2, 5) and (x_2, y_2) = (3, 1). Step 2: Use the formula for the gradient (slope) of a line: m = (y_2 - y_1)/(x_2 - x_1). m = (1 - 5)/(3 - 2) Step 3: Calculate the gradient. m = (-4)/(1) m = -4 The calculated gradient is -4. However, -4 is not among the given options (A. 5, B. 6, C. 6, D. 5, E. 7). It is highly probable that there is a typo in the question. If we assume the second point was (3, 10) instead of (3, 1), then the gradient would be: m = (10 - 5)/(3 - 2) = (5)/(1) = 5 Based on this assumption, the correct option would be A or D. We will choose A. The correct option is A, assuming the second point was intended to be (3, 10). The gradient is 5. 24. If p varies inversely as the square root of (m+2) and p=12 when m=7, find m when p=6. Step 1: Write the variation equation. p = (k)/(sqrt(m+2)) where k is the constant of proportionality. Step 2: Use the initial conditions (p=12 when m=7) to find k. 12 = (k)/(sqrt(7+2)) 12 = (k)/(sqrt(9)) 12 = (k)/(3) k = 12 × 3 k = 36 Step 3: Write the specific variation equation. p = (36)/(sqrt(m+2)) Step 4: Use the new value (p=6) to find m. 6 = (36)/(sqrt(m+2)) Step 5: Solve for m. sqrt(m+2) = (36)/(6) sqrt(m+2) = 6 Square both sides: (sqrt(m+2))^2 = 6^2 m+2 = 36 m = 36 - 2 m = 34 The correct option is E. The value of m is 34. 3 done, 2 left today. You're making progress.