1. Given the equation $a - \frac{1}{a} = 3$, find the value of $\frac{5a}{a^{2} + 2a - 1}$. Step 1: Multiply both sides of the equation by $a$ to clear the fraction: $$a^{2} - 1 = 3a$$ Step 2: Rearrange terms: $$a^{2} - 3a - 1 = 0$$ Step 3: We want to find: $$\frac{5a}{a^{2} + 2a - 1}$$ Step 4: Notice $a^{2}$ from Step 2 is $3a + 1$: substitute in denominator: $$a^{2} + 2a - 1 = (3a + 1) + 2a - 1 = 5a$$ Step 5: So the expression is: $$\frac{5a}{5a} = 1$$ Answer: গ) 1 2. Sets $A = \{2, 3, 5\}$ and $B = \{4, 6\}$. The product set is $A \times B = \{(2,4), (2,6), (3,4), (3,6), (5,4), (5,6)\}$. The relation is $y = 2x$. Check pairs where $y = 2x$: For $x=2$: $y=4$ (in B) ✔ For $x=3$: $y=6$ (in B) ✔ For $x=5$: $y=10$ (not in B) ✘ Hence pairs satisfying are $\{(2,4), (3,6)\}$ Answer: গ) $\{(2,4), (3,6)\}$ 3. Statements: i. Function's graph is a mapping. ii. Each function is a relation. iii. Not all relations are functions. Analysis: i is true (graph represents mapping). ii is true (every function is a relation). iii is true (some relations are not functions). Answer: ঘ) i, ii ও iii 4. Factorize the expression: $$\frac{1}{2}x^{2} + \frac{7}{6}x + \frac{1}{3}$$ Step 1: Multiply entire expression by 6 to clear denominators: $$3x^{2} + 7x + 2$$ Step 2: Factor $3x^{2} + 7x + 2$: Find two numbers that multiply to $3*2=6$ and add to 7: 6 and 1. Step 3: Rewrite as: $$3x^{2} + 6x + x + 2 = 3x(x+2) + 1(x+2) = (3x + 1)(x + 2)$$ Step 4: Divide back by 6: $$\frac{1}{6}(3x + 1)(x + 2)$$ Answer: খ) $\frac{1}{6}(x + 2)(3x + 1)$ 5. Cost price = 200, profit = 25%, VAT = 10%. Find selling price including VAT. Step 1: Calculate selling price before VAT: $$200 + 0.25 \times 200 = 200 + 50 = 250$$ Step 2: Calculate VAT on selling price: $$10\% \text{ of } 250 = 25$$ Step 3: Selling price including VAT: $$250 + 25 = 275$$ Answer: খ) 275 টাকা 6. Given $x = 3 + 2\sqrt{2}$, find $x + \frac{1}{x}$. Step 1: Find $\frac{1}{x}$ by rationalizing denominator: $$\frac{1}{3 + 2\sqrt{2}} = \frac{3 - 2\sqrt{2}}{(3)^2 - (2\sqrt{2})^2} = \frac{3 - 2\sqrt{2}}{9 - 8} = 3 - 2\sqrt{2}$$ Step 2: Add $x + \frac{1}{x}$: $$(3 + 2\sqrt{2}) + (3 - 2\sqrt{2}) = 6$$ Answer: ক) 6