1. Prove the commutative law of addition for matrices A and B where: $$A=\begin{bmatrix}3 & 4 \\ 2 & 5\end{bmatrix},\quad B=\begin{bmatrix}3 & 2 \\ 1 & 1\end{bmatrix}$$ Calculate $$A+B$$: $$A+B=\begin{bmatrix}3+3 & 4+2 \\ 2+1 & 5+1\end{bmatrix}=\begin{bmatrix}6 & 6 \\ 3 & 6\end{bmatrix}$$ Calculate $$B+A$$: $$B+A=\begin{bmatrix}3+3 & 2+4 \\ 1+2 & 1+5\end{bmatrix}=\begin{bmatrix}6 & 6 \\ 3 & 6\end{bmatrix}$$ Since $$A+B = B+A$$, commutative law holds. 2. Subtract $$5+2i$$ from $$7-6i$$: $$ (7-6i) - (5+2i) = 7-6i-5-2i = (7-5) + (-6i - 2i) = 2 - 8i $$ 3. Write $$\log_9 81 = 2$$ in exponential form: By definition, $$\log_b a = c \iff b^c = a$$, so: $$9^2 = 81$$ 4. Reduce $$\frac{4a + 12}{a^2 - 9}$$ to lowest terms: Factor numerator and denominator: $$4a + 12 = 4(a + 3)$$ $$a^2 - 9 = (a - 3)(a + 3)$$ Cancel common factor $$(a+3)$$: $$\frac{4(a+3)}{(a-3)(a+3)} = \frac{4}{a-3}$$ 5. Find $$a^2 + b^2 + c^2$$ if $$a + b + c = 4$$ and $$ab + bc + ca = -8$$: Recall: $$ (a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) $$ Substitute values: $$4^2 = a^2 + b^2 + c^2 + 2(-8)$$ $$16 = a^2 + b^2 + c^2 - 16$$ Add 16 to both sides: $$a^2 + b^2 + c^2 = 32$$ 6. Factorize $$x^4 + 64$$: Rewrite as sum of squares: $$x^4 + 8^2$$ Use sum of squares factorization (complex): $$x^4 + 64 = (x^2 + 4x + 8)(x^2 - 4x + 8)$$ 7. Find HCF by division of $$x^3 - 3x + 2$$ and $$x^3 - 5x^2 + 7x - 3$$: Using polynomial division and Euclidean algorithm, Divide $$x^3 - 5x^2 + 7x - 3$$ by $$x^3 - 3x + 2$$, remainder found; continue dividing until remainder zero; final divisor is HCF: Result HCF = $$x^2 - 2x + 1 = (x-1)^2$$ 8. Solve equation $$\frac{1}{5x - 9} = \frac{2}{7}$$: Cross multiply: $$7 = 2(5x - 9)$$ $$7 = 10x - 18$$ $$10x = 25$$ $$x = \frac{25}{10} = 2.5$$ 9. Show points x(1,2), y(3,4), z(0,1) form scalene triangle: Calculate distances: $$xy = \sqrt{(3-1)^2 + (4-2)^2} = \sqrt{4 + 4} = \sqrt{8}$$ $$yz = \sqrt{(3-0)^2 + (4-1)^2} = \sqrt{9 + 9} = \sqrt{18}$$ $$xz = \sqrt{(1-0)^2 + (2-1)^2} = \sqrt{1 + 1} = \sqrt{2}$$ Since all sides $$\sqrt{8}, \sqrt{18}, \sqrt{2}$$ are unequal, triangle is scalene. 10. Solve radical equation $$6 + 2\sqrt{a} = 12$$: Subtract 6: $$2\sqrt{a} = 6$$ Divide by 2: $$\sqrt{a} = 3$$ Square both sides: $$a = 9$$ 11. Prove Hypotenuse and one side congruent triangles are congruent: By RHS criterion for right triangles, if hypotenuse and one leg are equal, triangles are congruent. 12. Prove in parallelogram: (i) Opposite sides congruent: By definition and properties of parallelogram. (ii) Opposite angles congruent: By parallel lines and alternate interior angles properties. Q_count for Section-B: 12 Section-C 2. Prove any point equidistant from endpoints lies on right bisector: Use coordinate geometry or congruent triangles to show equal distance implies lying on perpendicular bisector. 3. Prove side opposite greater angle longer: Using triangle inequality and angle-side relationships. 4. Prove internal bisector divides opposite side proportionally: By similarity of triangles formed by bisector. 5. Ladder problem: Known ladder length $$3.9$$m and wall height $$3.1$$m. Find distance $$x$$ from foot of wall: By Pythagoras: $$x = \sqrt{3.9^2 - 3.1^2} = \sqrt{15.21 - 9.61} = \sqrt{5.6} \approx 2.366$$ meters. 6. Prove triangles on same base and altitude have equal area: Area formula $$= \frac{1}{2} \times base \times height$$ implies equal area. 7. Construct $$\triangle XYZ$$ with $$XY=5.6$$cm, $$\angle X=75^\circ$$ and $$\angle Y=45^\circ$$: Steps include drawing base, constructing angles at X and Y, locating point Z at intersection. Q_count for Section-C: 7