1. **Problem statement:** We have several algebraic expressions and geometric questions to solve. --- 2. **Part a:** Simplify polynomial $A$. Given $A = x^2 + y^2 + xy$, it is already simplified. --- 3. **Part b:** Determine the degree of polynomial $A$. The degree of a polynomial is the highest sum of exponents in any term. For $A = x^2 + y^2 + xy$, the degrees of terms are: - $x^2$: degree 2 - $y^2$: degree 2 - $xy$: degree $1+1=2$ So the degree of $A$ is 2, not 3 as stated. --- 4. **Part c:** Define $C = A + B$ and simplify. Given $C = x^2 + y^2 - 2xy$ after simplification. --- 5. **Part d:** Evaluate $C$ at $x=6$, $y=7$. Calculate: $$C = 6^2 + 7^2 - 2 \times 6 \times 7 = 36 + 49 - 84 = 1$$ --- 6. **Multiple choice question 12:** Rectangle with width $x+3$ and length $x-3$, $x>3$. Area $= (x+3)(x-3) = x^2 - 9$ (m²). --- 7. **Essay question 1a:** Group monomials by like terms. Monomials: - $3.5y$, $y$, $-5y$ (like terms with $y$) - $4x^3 y^{2/3}$, $9x^3 y^{2/3}$ (like terms) - $-0.5x^2 y$ (unique) - $\frac{3}{4} x y^{3/4}$ (unique) --- 8. **Essay question 1b:** Perform division: $$\frac{10x^3 y^2 - 5x^2 y + 25xy}{5xy} = \frac{10x^3 y^2}{5xy} - \frac{5x^2 y}{5xy} + \frac{25xy}{5xy} = 2x^2 y - x + 5$$ --- 9. **Essay question 2a:** Given $M = 2x^2 + 4xy - 2y^2 - 4$, $N = 3x^2 - 2xy + 2y^2$. Find $P = M + N$: $$P = (2x^2 + 4xy - 2y^2 - 4) + (3x^2 - 2xy + 2y^2) = (2x^2 + 3x^2) + (4xy - 2xy) + (-2y^2 + 2y^2) - 4 = 5x^2 + 2xy - 4$$ --- 10. **Essay question 2b:** Evaluate $P$ at $x=1$, $y=-2$: $$P = 5(1)^2 + 2(1)(-2) - 4 = 5 - 4 - 4 = -3$$ --- 11. **Essay question 3a:** Given triangle $ABC$, $M$ between $B$ and $C$, $N$ on $AB$, $P$ on $AC$ such that $MN \parallel AC$ and $MP \parallel AB$. Quadrilateral $ANMP$ has both pairs of opposite sides parallel, so it is a parallelogram. --- 12. **Essay question 3b:** For $ANMP$ to be a rhombus, $M$ must be positioned so that all sides are equal in length. --- 13. **Essay question 3c:** For $ANMP$ to be a square, triangle $ABC$ must be right-angled and isosceles with $M$ at the midpoint of $BC$. --- **Final answers:** - a) $A = x^2 + y^2 + xy$ - b) Degree of $A$ is 2 - c) $C = x^2 + y^2 - 2xy$ - d) $C(6,7) = 1$ - Question 12: Area $= x^2 - 9$ - 1a) Groups: $(3.5y, y, -5y)$; $(4x^3 y^{2/3}, 9x^3 y^{2/3})$; $-0.5x^2 y$; $(3/4) x y^{3/4}$ - 1b) Result: $2x^2 y - x + 5$ - 2a) $P = 5x^2 + 2xy - 4$ - 2b) $P(1,-2) = -3$ - 3a) $ANMP$ is a parallelogram - 3b) $M$ positioned for $ANMP$ to be a rhombus - 3c) $ABC$ right isosceles, $M$ midpoint of $BC$ for $ANMP$ square