3. Let mappings F= (F1, F2) R² → R² and G = (G1, G2): R² → R² be defined by F₁(x1, x2) = x² + x2, G₁(y1, y2)=sin(y2), F2(x1, x2, x3) = x1 + x2, G2(y1, y2) = cos(y1). (i) Find the composition mapping GoF: R2 → R². (ii) By using the chain rule, find the derivative of the mapping Go F, that is [5 Marks] D(GF)(x1, x2). [15 Marks] (iii) Give reasons why the mapping GoF is differentiable at every x = R². [5 Marks]

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8. Consider the function y = c₁e² cos 2x + c₂e sin 2x + c3 + C₁x + c5€¯ + c6е 2x (a) r²(r - 1)(r-2) (r + 1) (r + 2)² = 0 (b) r²(r 1)²(r - 2)(r + 1)² (r + 2)² = 0 (c) r²(r− 1)² (r² + 4)(r + 1)(r + 2)² = 0 (d) r(r² − 2r+5)(r + 1)(r + 2)² = 0 (e) r²(r² - 2r+5)(r+1)(r+ 2)² = 0 ✓ + C7xe -2x with the c, are arbitrary constants. This y is the general solution to a certain homogeneous linear first-order ODE. Which of the following is its auxiliary equation?

do b) d) f) H)

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