A curve has parametric equations x = t^2 + 1, y = 2ln(t + 1), t > -1 (a) Find dy/dx in terms of t. (b) Find the equation of the normal to the curve at the point where t = 1. Give your answer in the form y = mx + c.

Mark scheme

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

1. B1: dx/dt = 2t.
2. B1: dy/dt = 2/(t + 1).
3. M1: Uses dy/dx = (dy/dt) / (dx/dt).
4. A1: dy/dx = 1/(t(t + 1)). (Simplification is required for A1)
5. M1: Substitutes t = 1 into dx/dt and dy/dt (or into their dy/dx expression) to find the gradient of the tangent.
6. A1ft: Correct gradient of the tangent (ft from their dy/dx). (e.g., if dy/dx = 1/(t(t+1)), then m_tangent = 1/2)
7. M1: Calculates the gradient of the normal using m_normal = -1/m_tangent.
8. M1: Finds the coordinates (x, y) of the point on the curve where t = 1. (x = 1^2 + 1 = 2, y = 2ln(1 + 1) = 2ln2).
9. A1: Correct equation of the normal in the form y = mx + c. (e.g., y = -2x + 4 + 2ln2)