1. **Problem:** Two fair dice are thrown twice. Find the probability that the product of the numbers on the first throw and the sum of the numbers on the second throw is 8. 2. **Step 1:** Understand the problem. - First throw: product of the two dice. - Second throw: sum of the two dice. - We want the probability that the product from the first throw equals the sum from the second throw and that sum is 8. 3. **Step 2:** Find all pairs of dice rolls whose product equals 8. - Possible pairs: (1,8), (2,4), (4,2), (8,1) but dice only go from 1 to 6. - Valid pairs: (2,4) and (4,2). - Number of favorable outcomes for product = 8 is 2. 4. **Step 3:** Find all pairs of dice rolls whose sum equals 8. - Possible pairs: (2,6), (3,5), (4,4), (5,3), (6,2). - Number of favorable outcomes for sum = 8 is 5. 5. **Step 4:** Total possible outcomes for each throw = $6 \times 6 = 36$. 6. **Step 5:** Probability of product = 8 on first throw = $\frac{2}{36} = \frac{1}{18}$. 7. **Step 6:** Probability of sum = 8 on second throw = $\frac{5}{36}$. 8. **Step 7:** Since throws are independent, total probability = $\frac{1}{18} \times \frac{5}{36} = \frac{5}{648}$. --- 9. **Problem:** Find the third term of the geometric progression (G.P): $\sqrt{2} - 1$, $3 - 2\sqrt{2}$, ... 10. **Step 1:** Recall the formula for the $n^{th}$ term of a G.P: $a_n = a_1 r^{n-1}$. 11. **Step 2:** Identify $a_1 = \sqrt{2} - 1$. 12. **Step 3:** Find common ratio $r = \frac{a_2}{a_1} = \frac{3 - 2\sqrt{2}}{\sqrt{2} - 1}$. 13. **Step 4:** Simplify $r$: $$r = \frac{3 - 2\sqrt{2}}{\sqrt{2} - 1} \times \frac{\sqrt{2} + 1}{\sqrt{2} + 1} = \frac{(3 - 2\sqrt{2})(\sqrt{2} + 1)}{(\sqrt{2})^2 - 1^2} = \frac{(3 - 2\sqrt{2})(\sqrt{2} + 1)}{2 - 1} = (3 - 2\sqrt{2})(\sqrt{2} + 1)$$ 14. **Step 5:** Multiply numerator: $$3 \times \sqrt{2} + 3 \times 1 - 2\sqrt{2} \times \sqrt{2} - 2\sqrt{2} \times 1 = 3\sqrt{2} + 3 - 4 - 2\sqrt{2} = (3\sqrt{2} - 2\sqrt{2}) + (3 - 4) = \sqrt{2} - 1$$ 15. **Step 6:** So, $r = \sqrt{2} - 1$. 16. **Step 7:** Find third term: $$a_3 = a_1 r^{2} = (\sqrt{2} - 1)(\sqrt{2} - 1)^2 = (\sqrt{2} - 1)^3$$ 17. **Step 8:** Expand $(\sqrt{2} - 1)^3$ using binomial expansion: $$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$$ 18. **Step 9:** Substitute $a=\sqrt{2}$, $b=1$: $$ (\sqrt{2})^3 - 3(\sqrt{2})^2(1) + 3(\sqrt{2})(1)^2 - 1^3 = 2\sqrt{2} - 3 \times 2 + 3\sqrt{2} - 1 = 2\sqrt{2} - 6 + 3\sqrt{2} - 1 = 5\sqrt{2} - 7$$ 19. **Answer:** The third term is $5\sqrt{2} - 7$. --- 20. **Problem:** Points $P(-1, n-1)$, $Q(n, n-3)$, and $R(n-6, 3)$ are collinear. Find $n$. 21. **Step 1:** For points to be collinear, the slope between $P$ and $Q$ equals the slope between $Q$ and $R$. 22. **Step 2:** Calculate slope $m_{PQ}$: $$m_{PQ} = \frac{(n-3) - (n-1)}{n - (-1)} = \frac{n-3 - n + 1}{n + 1} = \frac{-2}{n+1}$$ 23. **Step 3:** Calculate slope $m_{QR}$: $$m_{QR} = \frac{3 - (n-3)}{(n-6) - n} = \frac{3 - n + 3}{n - 6 - n} = \frac{6 - n}{-6} = \frac{n - 6}{6}$$ 24. **Step 4:** Set slopes equal: $$\frac{-2}{n+1} = \frac{n - 6}{6}$$ 25. **Step 5:** Cross multiply: $$-2 \times 6 = (n - 6)(n + 1)$$ $$-12 = n^2 + n - 6n - 6 = n^2 - 5n - 6$$ 26. **Step 6:** Rearrange: $$n^2 - 5n - 6 + 12 = 0$$ $$n^2 - 5n + 6 = 0$$ 27. **Step 7:** Factor quadratic: $$(n - 2)(n - 3) = 0$$ 28. **Step 8:** Solutions: $$n = 2 \text{ or } n = 3$$ 29. **Answer:** $n = 2$ or $n = 3$. --- 30. **Problem:** Simplify $$\frac{729^{(1 - \frac{5}{6})m} + 243^{(m - \frac{1}{5})}}{27^{(\frac{5}{3}m - 1)}}$$ 31. **Step 1:** Express bases as powers of 3: - $729 = 3^6$ - $243 = 3^5$ - $27 = 3^3$ 32. **Step 2:** Rewrite expression: $$\frac{(3^6)^{(1 - \frac{5}{6})m} + (3^5)^{(m - \frac{1}{5})}}{(3^3)^{(\frac{5}{3}m - 1)}} = \frac{3^{6(1 - \frac{5}{6})m} + 3^{5(m - \frac{1}{5})}}{3^{3(\frac{5}{3}m - 1)}}$$ 33. **Step 3:** Simplify exponents: - $6(1 - \frac{5}{6})m = 6(\frac{1}{6})m = m$ - $5(m - \frac{1}{5}) = 5m - 1$ - $3(\frac{5}{3}m - 1) = 5m - 3$ 34. **Step 4:** Expression becomes: $$\frac{3^m + 3^{5m - 1}}{3^{5m - 3}}$$ 35. **Step 5:** Divide numerator terms by denominator: $$3^{m - (5m - 3)} + 3^{(5m - 1) - (5m - 3)} = 3^{m - 5m + 3} + 3^{5m - 1 - 5m + 3} = 3^{-4m + 3} + 3^{2}$$ 36. **Step 6:** Simplify: $$3^{3 - 4m} + 9$$ 37. **Answer:** The simplified expression is $3^{3 - 4m} + 9$. --- 38. **Problem:** The sum of the fifth and eighth terms of an arithmetic progression (A.P) is 28. Twice the seventh term minus the sum of the fifth and eighth terms is 25. Find: (a) first term (b) common difference (c) sum of the first eight terms 39. **Step 1:** Let first term be $a$ and common difference be $d$. 40. **Step 2:** Write terms: - $T_5 = a + 4d$ - $T_7 = a + 6d$ - $T_8 = a + 7d$ 41. **Step 3:** Given: $$T_5 + T_8 = 28 \Rightarrow (a + 4d) + (a + 7d) = 28 \Rightarrow 2a + 11d = 28$$ 42. **Step 4:** Also given: $$2T_7 - (T_5 + T_8) = 25$$ $$2(a + 6d) - [ (a + 4d) + (a + 7d) ] = 25$$ $$2a + 12d - 2a - 11d = 25$$ $$d = 25$$ 43. **Step 5:** Substitute $d=25$ into $2a + 11d = 28$: $$2a + 11 \times 25 = 28$$ $$2a + 275 = 28$$ $$2a = 28 - 275 = -247$$ $$a = -\frac{247}{2} = -123.5$$ 44. **Step 6:** Find sum of first 8 terms: $$S_8 = \frac{8}{2} [2a + (8 - 1)d] = 4 [2(-123.5) + 7 \times 25] = 4 [-247 + 175] = 4 \times (-72) = -288$$ 45. **Answer:** (a) $a = -123.5$ (b) $d = 25$ (c) $S_8 = -288$