The curve C has equation cos(y) = x² sin(x).

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. (a) M1: for differentiating cos(y) implicitly with respect to x.
2. -sin(y) dy/dx
3. M1: for attempting the product rule on x² sin(x).
4. 2x sin(x) + x²cos(x)
5. A1: for correct differentiation of both sides: -sin(y) dy/dx = 2x sin(x) + x²cos(x).
6. A1: for dy/dx = -(2x sin(x) + x²cos(x))/(sin(y)).
7. (b) M1: for substituting x = √(π/2) and y = π/2 into their dy/dx.
8. At (√(π/2), π/2), sin(π/2)=1 and cos(π/2)=0.
9. dy/dx = -(2√(π/2) sin(√(π/2)) + (π/2) cos(√(π/2)))/(sin(π/2)) = -(2√(π/2)(1) + (π/2)(0))/1 = -2√(π/2) = -√(2π).
10. A1: for the correct equation of the tangent: y - π/2 = -√(2π) (x - √(π/2)).