1. **Question 8:** Given the Venn diagram with sets A and B inside universal set U, values are: - A only: 6 and 4 (total 10) - B only: 3 and 5 (total 8) - Intersection A ∩ B: 1 and 2 (total 3) - Outside A and B but inside U: 7 and 8 (total 15) (i) Find $m(A)$: Sum all elements in set A including intersection. $$m(A) = 6 + 4 + 1 + 2 = 13$$ (ii) Calculate $P = \frac{m(A)}{m(U)}$: $m(U)$ is sum of all numbers inside U: $$m(U) = 6 + 4 + 1 + 2 + 3 + 5 + 7 + 8 = 36$$ Hence, $$P = \frac{13}{36}$$ (iii) Find $m(A \cup B)$: Union includes all elements in A, B and their intersection: $$m(A \cup B) = 6 + 4 + 1 + 2 + 3 + 5 = 21$$ (iv) Find $(A \cap B) \cup (B \cap A)$: Since intersection is commutative, this is just $A \cap B$: $$m(A \cap B) = 1 + 2 = 3$$ --- 2. **Question 9:** (a) Construct a tangent line at the point $T$ on the circle. - The radius from the center meets the tangent line at $T$ at a right angle ($90^\circ$). - The angle between another line segment inside the circle and the radius is $a$, so the angle between tangent and this segment at $T$ is $90^\circ - a$ by the complementary angle property. (b) Given circle with center $O$, and angles: (i) To find angle $a$: Use the triangle angle sum property inside the circle. Given angles include $110^\circ$ and angle $a$, so $$a + 110^\circ + other\ angle = 180^\circ$$ Assuming other angle is $h$, then: (ii) To find $h$ using the triangle angle sum, $$h = 180^\circ - 110^\circ - a = 70^\circ - a$$ --- 3. **Question 10:** (a) Evaluate the integral $$\int (4x - \tan x) \, dx$$ Step 1: Use linearity to split the integral: $$\int 4x \, dx - \int \tan x \, dx$$ Step 2: Integrate each separately: $$\int 4x \, dx = 2x^2 + C_1$$ $$\int \tan x \, dx = - \ln|\cos x| + C_2$$ Therefore, $$\int (4x - \tan x) \, dx = 2x^2 + \ln|\cos x| + C$$ where $C = C_1 + C_2$ is a constant. (b) (i) Sketch the curves $y=4x$ (a straight line through origin with slope 4) and $y=\tan x$ (periodic function with vertical asymptotes at $x=\pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \ldots$). (ii) Shade the area between these curves represented by the integral. The area is between the two curves where $4x > \tan x$ over the domain considered.