A-LevelMathematicsDifferentiation9709/32 · Feb/Mar 2023 · Q56 Marks
The parametric equations of a curve are x = te²ᵗ, y = t² + t + 3.
- (a)Show that dy/dx = e⁻²ᵗ[3]
- (b)Hence show that the normal to the curve, where t = −1, passes through the point (0, 3 − 1/e⁴).[3]
Mark scheme
Marking points from the official mark scheme (Feb/Mar 2023). The mark or mark type is shown in brackets.
- [B1] Obtain dx/dt = e²ᵗ + 2te²ᵗ OE
- [M1] Use dy/dx = (dy/dt) × (dt/dx)
- [A1] Obtain the given answer dy/dx = e⁻²ᵗ AG Need to see e²ᵗ (1 + 2t) in denominator.
- [B1] Obtain x = -e⁻² or -1/e² and y = 3 at t = -1
- [B1] Obtain gradient of normal = -e⁻² or -1/e²
- [B1] x = 0 substituted into equation of normal or use of gradients to give y = 3 - 1/e⁴ with no errors AG SC Decimals B0 B1 B0 -0.135.
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