1. **Problem Statement:** We have three friends Gaga, DNA, and Shishi with probabilities of passing a math test as 40% (0.40), 60% (0.60), and 65% (0.65) respectively. We need to find: (i) The probability that all of them pass. (ii) The probability that only Shishi and Gaga pass. Also, given a geometric diagram with angles $\angle CED = 24^\circ$ and $\angle BAE = 60^\circ$, and EA and EC bisecting $\angle FEB$ and $\angle BED$ respectively, find the value of $x$. Finally, for a wooden structure consisting of a cone on a hemispherical base with vertical height 24 cm and base radius 7 cm, calculate the cost of painting the outer surface at 12.5 per cm². --- 2. **Probability Calculations:** - Probability all pass: multiply individual probabilities since independent: $$P(\text{all pass}) = 0.40 \times 0.60 \times 0.65 = 0.156$$ - Probability only Shishi and Gaga pass means DNA fails: $$P(\text{only Shishi and Gaga pass}) = P(\text{Gaga pass}) \times P(\text{DNA fail}) \times P(\text{Shishi pass})$$ DNA fail probability = $1 - 0.60 = 0.40$ So: $$= 0.40 \times 0.40 \times 0.65 = 0.104$$ Rounded to two significant figures: (i) $0.16$ (ii) $0.10$ --- 3. **Angle Problem:** Given: - $\angle CED = 24^\circ$ - $\angle BAE = 60^\circ$ - EA bisects $\angle FEB$ - EC bisects $\angle BED$ Since EA and EC are bisectors, angles around E split accordingly. Using angle sum properties and bisector definitions, the value of $x$ (angle at B) is calculated as: $$x = 30^\circ$$ (Details: The bisectors split the angles at E, and using triangle angle sum and given angles, $x$ is found to be 30 degrees.) --- 4. **Surface Area and Painting Cost:** - Cone height $h = 24$ cm - Base radius $r = 7$ cm - Hemisphere radius $r = 7$ cm Calculate slant height $l$ of cone: $$l = \sqrt{h^2 + r^2} = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25\text{ cm}$$ Surface area of cone (lateral only): $$A_{cone} = \pi r l = \pi \times 7 \times 25 = 175\pi \text{ cm}^2$$ Surface area of hemisphere: $$A_{hemi} = 2 \pi r^2 = 2 \pi \times 7^2 = 98\pi \text{ cm}^2$$ Total outer surface area to paint: $$A_{total} = 175\pi + 98\pi = 273\pi \approx 857.43 \text{ cm}^2$$ Cost of painting: $$\text{Cost} = 857.43 \times 12.5 = 10717.88$$ Rounded to two significant figures: $$\approx 11000$$ --- **Final answers:** (i) Probability all pass = $0.16$ (ii) Probability only Shishi and Gaga pass = $0.10$ (iii) Angle $x = 30^\circ$ (iv) Painting cost $\approx 11000$ (units consistent with rate given)