Edexcel IALMathematicsGeneral Applications9 Marks

The curve C has equation y = 2e^x + 3. The line L has equation y = 7.

(a)Find the x-coordinate of the point where the curve C intersects the line L.[3]
(b)Find the area of the finite region R bounded by the curve C, the line L, the y-axis and the line x = ln 2. Give your answer in the form A + B ln C, where A, B, C are integers.[6]

Mark scheme

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

(a) M1 for setting the curve equation equal to the line equation: 2e^x + 3 = 7.
(a) A1 for correctly isolating the exponential term: 2e^x = 4 or e^x = 2.
(a) A1 for finding the x-coordinate: x = ln 2.
(b) M1 for setting up the correct definite integral for the area between the functions, i.e., ∫(Upper function - Lower function) dx, with correct limits identified.
(b) A1 for correctly determining the integrand: ∫( 7 - (2e^x + 3) ) dx = ∫(4 - 2e^x) dx.
(b) M1 for correctly integrating their function: 4x - 2e^x.
(b) A1 for using the correct limits of integration: 0 and ln 2.
(b) M1 for correctly substituting the limits and subtracting: [4(ln2) - 2e^(ln2)] - [4(0) - 2e^0].
(b) A1 for the final area: (4ln2 - 4) - (0 - 2) = 4ln2 - 2 (or -2 + 4ln2), which is in the form A + B ln C with A=-2, B=4, C=2.

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.