A curve has equation y = 5e^(2x−1)+e. The tangent to the curve at the point where x = 1 cuts the x-axis at the point P. Find the equation of the tangent in the form y = mx+c, where m and c are exact values, and hence find the x-coordinate of P.

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

1. [M2] dy/dx = 10e^(2x-1) [+0] oe
2. [M2] M1 for dy/dx = ke^(2x-1), k≠ 10
3. [M2] or SCM1 for dy/dx = 10e^(2x-1) + c where c is algebraic or numerical
4. [A1] dy/dx |_(x=1) = 10e and y = 6e
5. [A1] = 10e and y = 6e
6. [M1] FT their dy/dx |_(x=1) and their y provided M1 or SCM1 awarded and a value is found
7. [M1] y = 10ex + c and 6e = 10e + c
8. [M1] FT their dy/dx |_(x=1) and their y
9. [A1] y = 10ex - 4e isw
10. [A1] [x-coordinate of P = ] 0.4 oe, isw
11. [A1] dep on correct equation of tangent with exact values