Two fair dice are thrown. Find the probability of obtaining a sum of 7 or 11.

Here are the solutions to the problems: 6. Probability of obtaining a sum of 7 or 11 Step 1: Determine the total number of possible outcomes when two fair dice are thrown. Each die has 6 faces, so the total number of outcomes is 6 × 6 = 36. Step 2: List the outcomes that result in a sum of 7. The pairs are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). There are 6 outcomes that sum to 7. The probability of getting a sum of 7 is P(sum of 7) = (6)/(36). Step 3: List the outcomes that result in a sum of 11. The pairs are (5,6), (6,5). There are 2 outcomes that sum to 11. The probability of getting a sum of 11 is P(sum of 11) = (2)/(36). Step 4: Since the events are mutually exclusive (a sum cannot be both 7 and 11 simultaneously), add their probabilities to find the probability of obtaining a sum of 7 or 11. P(sum of 7 or 11) = P(sum of 7) + P(sum of 11) P(sum of 7 or 11) = (6)/(36) + (2)/(36) P(sum of 7 or 11) = (8)/(36) Step 5: Simplify the probability. P(sum of 7 or 11) = (2)/(9) The probability of obtaining a sum of 7 or 11 is (2)/(9). 7. (a) Find the position vector of P Step 1: Identify the given position vectors and the ratio. The position vector of point F is f = 3i - 2j. The position vector of point G is g = i + j. Point P divides FG in the ratio 5:3. This means m=5 and n=3. Step 2: Use the section formula for internal division. If a point P divides the line segment joining points F and G with position vectors f and g respectively, in the ratio m:n, then the position vector of P, p, is given by: p = nf + mgm+n Step 3: Substitute the given values into the formula. p = 3(3i - 2j) + 5(i + j)5+3 Step 4: Simplify the expression. p = (9i - 6j) + (5i + 5j)8 p = (9+5)i + (-6+5)j8 p = 14i - 1j8 p = (14)/(8)i - (1)/(8)j p = (7)/(4)i - (1)/(8)j The position vector of P is (7)/(4)i - (1)/(8)j.