AQA Further Paper 2 2020 June — Question 12 6 marks

Exam BoardAQA
ModuleFurther Paper 2 (Further Paper 2)
Year2020
SessionJune
Marks6
TopicIntegration by Parts
12
  1. Given that \(I = \int _ { a } ^ { b } \mathrm { e } ^ { 2 t } \sin t \mathrm {~d} t\), show that $$I = \left[ q \mathrm { e } ^ { 2 t } \sin t + r \mathrm { e } ^ { 2 t } \cos t \right] _ { a } ^ { b }$$ where \(q\) and \(r\) are rational numbers to be found.
    [0pt] [6 marks]
    12
  2. A small object is initially at rest. The subsequent motion of the object is modelled by the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} t } + v = 5 \mathrm { e } ^ { t } \sin t$$ where \(v\) is the velocity at time \(t\).
    Find the speed of the object when \(t = 2 \pi\), giving your answer in exact form.
    13Charlotte is trying to solve this mathematical problem:
    Find the general solution of the differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \frac { \mathrm { d } y } { \mathrm {~d} x } - 2 y = 10 \mathrm { e } ^ { - 2 x }\)
    Charlotte's solution starts as follows:
    Particular integral: \(y = \lambda \mathrm { e } ^ { - 2 x }\)
    so \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2 \lambda \mathrm { e } ^ { - 2 x }\)
    and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 4 \lambda \mathrm { e } ^ { - 2 x }\)
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