1. **Problem (a):** Find two numbers given their arithmetic mean (AM) is 5 and geometric mean (GM) is 4. 2. **Formula and rules:** - Arithmetic mean of two numbers $a$ and $b$ is $\frac{a+b}{2}$. - Geometric mean of two numbers $a$ and $b$ is $\sqrt{ab}$. 3. **Step-by-step solution:** - Given $\frac{a+b}{2} = 5 \Rightarrow a+b=10$. - Given $\sqrt{ab} = 4 \Rightarrow ab=16$. - We want to find $a$ and $b$ such that $a+b=10$ and $ab=16$. - Consider the quadratic equation $x^2 - (a+b)x + ab = 0$ which becomes $x^2 - 10x + 16 = 0$. - Solve using quadratic formula: $x = \frac{10 \pm \sqrt{10^2 - 4 \times 16}}{2} = \frac{10 \pm \sqrt{100 - 64}}{2} = \frac{10 \pm \sqrt{36}}{2}$. - So, $x = \frac{10 \pm 6}{2}$ gives $x=8$ or $x=2$. - Therefore, the two numbers are 8 and 2. 4. **Problem (b):** Find the sum of an infinite geometric series with first term $a=1$ and common ratio $r=\frac{1}{2}$. 5. **Formula:** Sum of infinite geometric series $S = \frac{a}{1-r}$ for $|r|<1$. 6. **Calculation:** - $S = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2$. 7. **Problem (c):** A group has 4 girls and 7 boys. Find the number of ways to select a team of 5 members under different conditions. 8. **Total number of ways to select 5 members from 11:** $\binom{11}{5} = 462$. 9. (i) **No girls:** Select all 5 from 7 boys. - Number of ways: $\binom{7}{5} = 21$. 10. (ii) **At least one boy and one girl:** - Total ways: 462 - Subtract teams with all girls: $\binom{4}{5} = 0$ (impossible) - Subtract teams with all boys: $\binom{7}{5} = 21$ - So, ways = $462 - 0 - 21 = 441$. 11. (iii) **At least three girls:** - Possible girl counts: 3, 4 - For 3 girls: $\binom{4}{3} \times \binom{7}{2} = 4 \times 21 = 84$ - For 4 girls: $\binom{4}{4} \times \binom{7}{1} = 1 \times 7 = 7$ - Total ways = $84 + 7 = 91$. 12. **Problem (d):** A coin is tossed three times. Event $P$: No head appears. 13. **Total outcomes:** $2^3 = 8$. 14. **Event $P$ means all tails:** Only 1 outcome (TTT). 15. **Probability of $P$:** $\frac{1}{8}$. **Final answers:** - (a) The two numbers are 8 and 2. - (b) Sum of infinite series is 2. - (c)(i) 21 ways, (ii) 441 ways, (iii) 91 ways. - (d) Probability of no head is $\frac{1}{8}$.