1. **Simplify:** $\frac{3x^2y \times 4xy^3}{6xy^2}$ Step 1: Multiply numerator terms: $3x^2y \times 4xy^3 = 12x^{2+1}y^{1+3} = 12x^3y^4$ Step 2: Divide by denominator: $\frac{12x^3y^4}{6xy^2} = 2x^{3-1}y^{4-2} = 2x^2y^2$ 2. **Factorise completely:** $x^2 - 25$ Step 1: Recognize difference of squares: $a^2 - b^2 = (a-b)(a+b)$ Step 2: $x^2 - 25 = (x - 5)(x + 5)$ 3. **Simplify:** $(2x - 3)(x + 4) - (x - 5)(x - 2)$ Step 1: Expand each product: $(2x - 3)(x + 4) = 2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12$ $(x - 5)(x - 2) = x^2 - 2x - 5x + 10 = x^2 - 7x + 10$ Step 2: Subtract second from first: $(2x^2 + 5x - 12) - (x^2 - 7x + 10) = 2x^2 + 5x - 12 - x^2 + 7x - 10 = (2x^2 - x^2) + (5x + 7x) + (-12 - 10) = x^2 + 12x - 22$ 4. **Solve for $x$:** $2x + 5 = 17$ Step 1: Subtract 5 from both sides: $2x = 12$ Step 2: Divide both sides by 2: $x = 6$ 5. **Number Pattern:** Sequence: 6, 10, 14, 18, ... a) Next two terms: Common difference $d = 10 - 6 = 4$ Next terms: $18 + 4 = 22$, $22 + 4 = 26$ b) Common difference: $4$ c) $n$th term $T_n = a + (n-1)d = 6 + (n-1)4 = 4n + 2$ d) Find $n$ when $T_n = 150$ $150 = 4n + 2$ $4n = 148$ $n = 37$ 6. **Geometry and Measurement:** a) Rectangle length = 12 cm, breadth = 8 cm Area $= l \times b = 12 \times 8 = 96$ cm$^2$ Perimeter $= 2(l + b) = 2(12 + 8) = 40$ cm b) Cylinder radius $r = 2.5$ m, height $h = 4$ m Volume $= \pi r^2 h = \pi \times (2.5)^2 \times 4 = \pi \times 6.25 \times 4 = 25\pi \approx 78.54$ m$^3$ c) Convert $3.6$ m$^2$ to cm$^2$ $1$ m $= 100$ cm, so $1$ m$^2 = 100^2 = 10,000$ cm$^2$ $3.6$ m$^2 = 3.6 \times 10,000 = 36,000$ cm$^2$ d) Sum of interior angles $= 900^\circ$ Formula: Sum $= (n-2) \times 180^\circ$ $900 = (n-2) \times 180$ $n-2 = \frac{900}{180} = 5$ $n = 7$ **Final answers:** a) $2x^2y^2$ b) $(x-5)(x+5)$ c) $x^2 + 12x - 22$ d) $x=6$ Number pattern: a) $22, 26$ b) $4$ c) $T_n = 4n + 2$ d) $37$ Geometry: a) Area $= 96$ cm$^2$, Perimeter $= 40$ cm b) Volume $\approx 78.54$ m$^3$ c) $36,000$ cm$^2$ d) Number of sides $= 7$