5.11 Further integrationIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   Water flows into a tank at a rate of $r(t)=6\sqrt{t}$ litres per minute, where $t$ is the time in minutes since the flow started. The tank is empty when $t=0$.

   (a)

   Find $\displaystyle\int6\sqrt{t}\,dt$.

   [1 mark]

   • A$3t^{-1/2}+c$
   • B$6t^{3/2}+c$
   • C$9t^{3/2}+c$
   • D$4t^{3/2}+c$

   (b)

   Find the volume of water that flows into the tank between $t=1$ and $t=4$.

   [1 mark]

   • A$18$ litres
   • B$28$ litres
   • C$32$ litres
   • D$4$ litres

   (c)

   Find the time taken for the volume of water in the tank to reach $500$ litres.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The population of a colony of bacteria changes at a rate $\dfrac{dP}{dt}=\dfrac{12}{3t+2}$ thousand per hour, where $t$ is the time in hours and $t\ge0$. At $t=0$ the population is $5$ thousand.

   (a)

   Find $\displaystyle\int\frac{12}{3t+2}\,dt$.

   [1 mark]

   • A$4\ln(3t+2)+c$
   • B$12\ln(3t+2)+c$
   • C$-\dfrac{4}{(3t+2)^2}+c$
   • D$36\ln(3t+2)+c$

   (b)

   Find the increase in the population between $t=0$ and $t=6$.

   [1 mark]

   • A$4\ln18\approx11.6$ thousand
   • B$12\ln10\approx27.6$ thousand
   • C$4\ln10\approx9.21$ thousand
   • D$4\ln20\approx12.0$ thousand

   (c)

   Find the time at which the population reaches $20$ thousand.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle moves in a straight line. At time $t$ seconds its velocity is $v=4t\sin(t^2)$ m s$^{-1}$, where the angle is in radians. The displacement $s$ from the starting point is $0$ when $t=0$.

   (a)

   Find an expression for $s$ in terms of $t$.

   [3 marks]

   (b)

   Find the first time after $t=0$ at which the particle is at rest, and its displacement from the starting point at that time.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A curve $y=f(x)$, defined for $0\le x<\dfrac{\pi}{2}$, passes through the point $(0,\,4)$. Its gradient function is $f'(x)=\dfrac{\sin x}{\cos x}+\dfrac{2}{\cos^2x}$.

   (a)

   (i) Use the substitution $u=\cos x$ to show that $\displaystyle\int\frac{\sin x}{\cos x}\,dx=-\ln(\cos x)+c$.
   (ii) Hence find $f(x)$.

   [6 marks]

   (b)

   (i) Find the exact value of $\displaystyle\int_0^{\pi/4}f'(x)\,dx$, giving your answer in the form $a+b\ln2$.
   (ii) Hence find the exact value of $f\left(\frac{\pi}{4}\right)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions