1. **Question 8: Venn Diagram** (i) Find $m(A)$: The elements inside circle A are 6, 4 and also the intersection elements 1, 2. So, $$m(A) = 6 + 4 + 1 + 2 = 13.$$ (ii) Find $P = \frac{m(A)}{m(U)}$: The universal set U contains all elements: 6,4,1,2,3,5,7,8. So, $$m(U) = 6 + 4 + 1 + 2 + 3 + 5 + 7 + 8 = 36.$$ Therefore, $$P = \frac{13}{36}.$$ (iii) Find $(A \cup B)^c$: $A \cup B$ contains all elements in A or B or their intersection: 6,4,1,2,3,5 Elements in $U$ not in $A \cup B$ are 7,8. So, $$(A \cup B)^c = \{7,8\}.$$ (iv) Find $(A^c \cap B) \cup (B^c \cap A)$: $A^c \cap B$ are elements in B but not in A: 3,5 $B^c \cap A$ are elements in A but not in B: 6,4 Their union is $$\{3,5\} \cup \{6,4\} = \{3,4,5,6\}.$$ 2. **Question 9: Circle and Angles** (a) Construct a tangent line at point $T$ on the circle with center $O$: The tangent at $T$ is perpendicular to the radius $OT$. Hence, the angle between the radius and tangent line is $90^\circ - \alpha$ as given. (b) Find angles given circle with center $O$ and triangle inscribed: (i) To find $\alpha$: In triangle with angles $\alpha$, $110^\circ$, and $h$, sum of angles is $180^\circ$: $$\alpha + 110 + h = 180.$$ (ii) Find $h$: Using circle properties, angle at center $O$ opposite to arc equals twice the inscribed angle. We find $\alpha = 35^\circ$ and thus, $$h = 35^\circ.$$ 3. **Question 10: Integral and Curves** (a) Evaluate $$\int (4x - \tan x) \, dx$$: $$\int 4x \, dx = 2x^2 + C_1,$$ $$\int - \tan x \, dx = - ( - \ln|\cos x| ) + C_2 = \ln|\cos x| + C_2.$$ Therefore, $$\int (4x - \tan x) \, dx = 2x^2 + \ln|\cos x| + C.$$ (b) (i) Sketch curves $y=4x$ and $y=\tan x$ on same axes. (ii) The shaded area between the two curves corresponds to the integral computed in (a), between the relevant limits. Final answers: (i) $m(A) = 13$ (ii) $P = \frac{13}{36}$ (iii) $(A \cup B)^c = \{7,8\}$ (iv) $(A^c \cap B) \cup (B^c \cap A) = \{3,4,5,6\}$ (ix)(a) Tangent line at $T$ makes angle $90^\circ - \alpha$ with radius. (ix)(b)(i) $\alpha = 35^\circ$ (ix)(b)(ii) $h = 35^\circ$ (x)(a) $$\int (4x - \tan x) \, dx = 2x^2 + \ln|\cos x| + C.$$