1. **Simplify:** $\frac{(3x^2y)(4xy^3)}{6xy^2}$ Multiply the numerators: $3x^2y \times 4xy^3 = 12x^{2+1}y^{1+3} = 12x^3y^4$ Divide by denominator: $\frac{12x^3y^4}{6xy^2} = 2x^{3-1}y^{4-2} = 2x^2y^2$ 2. **Factorise completely:** $x^2 - 25$ Recognize difference of squares: $x^2 - 5^2 = (x-5)(x+5)$ 3. **Simplify:** $(2x - 3)(x + 4) - (x - 5)(x - 2)$ Expand each: $(2x - 3)(x + 4) = 2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12$ $(x - 5)(x - 2) = x^2 - 2x - 5x + 10 = x^2 - 7x + 10$ Subtract: $(2x^2 + 5x - 12) - (x^2 - 7x + 10) = 2x^2 + 5x - 12 - x^2 + 7x - 10 = x^2 + 12x - 22$ 4. **Solve for x:** $2x + 5 = 17$ Subtract 5: $2x = 12$ Divide by 2: $x = 6$ --- 5. **Number sequence:** 6; 10; 14; 18; ... a) Next two terms: Common difference $d = 10 - 6 = 4$ Next terms: $18 + 4 = 22$, then $22 + 4 = 26$ b) Common difference: $4$ c) $n$th term $T_n = a + (n-1)d = 6 + (n-1)4 = 4n + 2$ d) Find $n$ when $T_n = 150$ $4n + 2 = 150 \Rightarrow 4n = 148 \Rightarrow n = 37$ --- 6. **Rectangle area and perimeter:** length $l=12$ cm, breadth $b=8$ cm Area $A = l \times b = 12 \times 8 = 96$ cm$^2$ Perimeter $P = 2(l + b) = 2(12 + 8) = 40$ cm 7. **Cylinder volume:** radius $r=2.5$ m, height $h=4$ m Volume $V = \pi r^2 h = \pi \times (2.5)^2 \times 4 = \pi \times 6.25 \times 4 = 25\pi$ m$^3$ Approximate: $25 \times 3.1416 = 78.54$ m$^3$ 8. **Convert area:** $3.6$ 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$^2$ 9. **Polygon sides from interior angles sum:** sum $S = 900^\circ$ Formula: $S = (n-2) \times 180^\circ$ $900 = (n-2) \times 180 \Rightarrow n-2 = \frac{900}{180} = 5 \Rightarrow n = 7$ **Final answers:** a) $2x^2y^2$ b) $(x-5)(x+5)$ c) $x^2 + 12x - 22$ d) $x=6$ a) Next terms: $22, 26$ b) $4$ c) $T_n = 4n + 2$ d) $n=37$ a) Area $= 96$ cm$^2$, Perimeter $= 40$ cm b) Volume $= 25\pi \approx 78.54$ m$^3$ c) $36,000$ cm$^2$ d) $7$ sides