O-LevelAdditional MathematicsDifferentiation and integration4037/22 · Oct/Nov 2023 · Q79 Marks

A particle is travelling in a straight line. Its displacement, s metres, from the origin at time t seconds is given by s = 1.5e^(2t) + 2e^(-2t) - t.

(a)Find expressions for the velocity, v ms⁻¹, and acceleration, a ms⁻², of the particle.[3]
(b)Find the time, T seconds, when the particle is at rest.[4]
(c)Find the acceleration of the particle at time T seconds.[2]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

[B2] B1 for 3e^(2t) or -4e^(-2t) for Velocity: 3e^(2t) - 4e^(-2t) - 1 isw
[B1] FT me^(2t) + ne^(-2t) + k where m, n and k are constants for Acceleration: 6e^(2t) + 8e^(-2t) isw
[B1] 3e^(4t) - e^(2t) - 4 = 0 or 3(e^(2t))² - e^(2t) - 4 = 0
[M1] FT their 3-term quadratic in e^(2t) oe for (3e^(2t)-4)(e^(2t)+1) = 0
[A1] e^(2t) = 4/3 nfww
[A1] 1/2 ln(4/3) oe, isw or 0.144 or 0.1438[41...] rot to 4 or more dp and no other solutions
[M1] FT pe^(2t) + qe^(-2t) where p and q are non-zero constants and their positive 1/2 ln(4/3) from part (b) for 6e^(2t) + 8e^(-2t) or 6(4/3) + 8(3/4)
[A1] 14 nfww

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.

Topic: Differentiation and integration, O-Level Additional Mathematics (syllabus 4037). Structured question, 9 marks.
Source: O-Level Additional Mathematics, Oct/Nov 2023 Paper 2 Variant 2, question 7.

Questions and mark schemes are from past examination papers and remain the copyright of the exam board. How Oracle Prep builds its content