1. Solve the inequality $3 + x > 5$ and represent the integral solutions on the number line from -4 to 8. Step 1: Subtract 3 from both sides: $$3 + x > 5 \\ x > 5 - 3 \\ x > 2$$ Step 2: The integral solutions are integers greater than 2, so they are: $$3, 4, 5, 6, 7, 8$$ --- 2. There are 11 pens: 5 blue, 2 black, and the rest are red. Step 1: Calculate the number of red pens: $$11 - (5 + 2) = 11 - 7 = 4$$ Step 2: Probability of taking a red pen = \frac{\text{number of red pens}}{\text{total pens}} = \frac{4}{11}.$ --- 3. Circle with diameter 28 cm. (i) Check circumference. Formula: $$C = \pi d = 3.14 \times 28 = 87.92 \text{ cm (approximately)}$$ Given 176 cm is incorrect, so response = "x". (ii) Check area. Formula: $$A = \pi r^2 = 3.14 \times 14^2 = 3.14 \times 196 = 615.44 \text{ cm}^2 \approx 616 \text{ cm}^2$$ Given 616 cm$^2$ is accurate, so response = "\u2713". --- 4. Given $\angle AP\hat{C} = \angle BP\hat{D}$, show that $\angle AP\hat{B} = \angle CP\hat{D}$. Step 1: The problem involves angles around point $P$ with rays to points $A,B,C,D$. Step 2: By the properties of vertical or corresponding angles or by angle sum around $P$, the equality holds. Hence, $\angle AP\hat{B} = \angle CP\hat{D}$. --- 5. In triangle $ABC$, $PQ$ is drawn parallel to $BC$ through point $A$. Step: Since $PQ \parallel BC$, corresponding angles are equal, so the measure of $\angle BAC$ remains the same. Hence, $B\hat{A}C$ is as given by the figure or equal to $B\hat{P}Q$ if angles are given. --- 6. Find the bearings: (i) Bearing of $B$ from $A$: Given direction at $A$ is North and the bearing to $B$ is $30^\circ$ east of north. (ii) Bearing of $A$ from $B$: Bearing is reciprocal plus or minus 180$^\circ$: $$30^\circ + 180^\circ = 210^\circ$$ --- 7. Refrigerator sold for 72,810 with 10% discount. (i) Marked price = Selling price divided by (1 - discount rate): $$= \frac{72810}{1-0.10} = \frac{72810}{0.90} = 80899 (approx)$$ (ii) Discount = Marked price - Selling price $$= 80899 - 72810 = 8089$$ --- 8. Mean of five numbers $17, 15, 18, x, 13$ is 16. (i) Calculate $x$: $$\frac{17 + 15 + 18 + x + 13}{5} = 16 \\ 63 + x = 80 \\ x = 17$$ (ii) Range = Maximum - Minimum: Numbers are $17, 15, 18, 17, 13$ Max = 18, Min = 13 $$\text{Range} = 18 - 13 = 5$$