Integrating exponentials and trigonometric functionsAQA A-Level Maths: Subtopic test

1. 1
   A curve has gradient function $\frac{dy}{dx}=6e^{3x}$ and passes through the point $(0,5)$.

   (a)

   Find $\int 6e^{3x}\,dx$.

   [1 mark]

   • A$18e^{3x}+c$
   • B$2e^{3x}+c$
   • C$6e^{3x}+c$
   • D$\frac{6e^{4x}}{4}+c$

   (b)

   Find the equation of the curve.

   [1 mark]

   • A$y=2e^{3x}+5$
   • B$y=2e^{3x}-3$
   • C$y=2e^{3x}+3$
   • D$y=6e^{3x}-1$

   (c)

   Find the exact value of $\int_0^{\ln 2}6e^{3x}\,dx$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle moves on a straight line. At time $t$ seconds its velocity is $v=4\cos 2t$ m s$^{-1}$, where $2t$ is in radians. The particle starts at the origin, so its displacement is $s=0$ when $t=0$.

   (a)

   Find the displacement $s$ of the particle at time $t$.

   [1 mark]

   • A$s=-2\sin 2t$
   • B$s=8\sin 2t$
   • C$s=2\cos 2t$
   • D$s=2\sin 2t$

   (b)

   Find the displacement when $t=\frac{\pi}{4}$.

   [1 mark]

   • A$2$ m
   • B$0$ m
   • C$\sqrt2$ m
   • D$-2$ m

   (c)

   Find the total distance travelled by the particle between $t=0$ and $t=\frac{\pi}{2}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(x)=\frac{3}{x}-4e^{-2x}+5\cos\frac{x}{2}$ and $h(x)=\frac{3}{x}+5\cos\frac{x}{2}$, for $x>0$, where angles are in radians.

   (a)

   Find $\int f(x)\,dx$.

   [3 marks]

   (b)

   Find the exact value of $\int_{\pi}^{2\pi}h(x)\,dx$, giving your answer in the form $a\ln2+b$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Water flows into a tank at a rate $R$ litres per minute, where $R=20e^{-0.1t}+6\sin\frac{\pi t}{30}$ and $t$ is the time in minutes after the flow starts (the angle is in radians). A calculator may be used.

   (a)

   Find the volume of water that flows into the tank in the first 15 minutes.

   [6 marks]

   (b)

   The tank initially holds 50 litres and water is also removed at a constant 8 litres per minute.
   (i) Find the volume in the tank after 15 minutes.
   (ii) A student uses the model to predict the flow at $t=45$. Evaluate $R$ at $t=45$ and comment on the validity of the model.

   [6 marks]

   Total for question 4: 12 marks

End of questions