FunctionsIB Maths: Applications and Interpretation SL: Topic test

1. 1
   A straight path is modelled by the line $L_1$ with equation $y=-\frac{2}{3}x+4$.

   (a)

   Write down the gradient of a line perpendicular to $L_1$.

   [1 mark]

   • A$\frac{3}{2}$
   • B$-\frac{3}{2}$
   • C$\frac{2}{3}$
   • D$-\frac{2}{3}$

   (b)

   A second path is parallel to $L_1$ and passes through the point $(6,1)$. Find the equation of the second path.

   [1 mark]

   • A$y=\frac{3}{2}x-8$
   • B$y=-\frac{2}{3}x+1$
   • C$y=-\frac{2}{3}x-3$
   • D$y=-\frac{2}{3}x+5$

   (c)

   Find the $x$-coordinate of the point where $L_1$ crosses the $x$-axis.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $h$ is defined by $h(x)=\frac{8}{x-3}+2$, for $x\neq3$.

   (a)

   Write down the equation of the vertical asymptote of the graph of $y=h(x)$.

   [1 mark]

   • A$x=-3$
   • B$x=3$
   • C$x=2$
   • D$y=3$

   (b)

   Write down the range of $h$.

   [1 mark]

   • A$h(x)\in\mathbb{R}$
   • B$h(x)\neq3$
   • C$h(x)\neq2$
   • D$h(x)>2$

   (c)

   Find the value of $h^{-1}(4)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The height $h$ metres of a stone arch above the water, at a horizontal distance $x$ metres from its left foot, is modelled by $h(x)=-0.25x^{2}+3x$, for $0\le x\le12$.

   (a)

   Use your GDC to find (i) the width of the arch where it meets the water, (ii) the maximum height of the arch above the water.

   [3 marks]

   (b)

   A boat is $4$ m wide and its highest point is $7.2$ m above the water. The boat travels under the arch with its centre line directly below the highest point of the arch. Determine whether the boat can pass under the arch.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A cinema offers two monthly plans. Plan P costs $C_P(n)=12+4.5n$ EUR for watching $n$ films in a month. Plan Q costs a flat $30$ EUR for up to $6$ films, and $6$ EUR for each further film after the sixth.

   (a)

   (i) State what the value $4.5$ represents in this context.
   (ii) State what the value $12$ represents in this context.
   (iii) Find the cost of Plan P for $8$ films.
   (iv) Find the number of films for which Plan P costs $75$ EUR, that is, find $C_P^{-1}(75)$.

   [6 marks]

   (b)

   (i) For $n>6$, write down and simplify an expression for the cost $C_Q(n)$ of Plan Q.
   (ii) Find the number of films, $n>6$, for which the two plans cost the same.
   (iii) Karim watches $10$ films in a month. Determine which plan is cheaper for him and by how much.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The concentration of a drug in a patient's blood is modelled by $C(t)=40e^{-0.2t}$ mg/L, where $t\ge0$ is the time in hours after the injection.

   (a)

   Write down the equation of the horizontal asymptote of the graph of $C$ against $t$.

   [1 mark]

   • A$t=0$
   • B$C=0$
   • C$C=40$
   • D$C=0.2$

   (b)

   Find the concentration $5$ hours after the injection.

   [1 mark]

   • A$14.7$ mg/L
   • B$39$ mg/L
   • C$0.00920$ mg/L
   • D$5.41$ mg/L

   (c)

   Find the time at which the concentration first falls to $10$ mg/L. Give your answer in hours, correct to 3 significant figures.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The height of a piston head above the base of an engine is modelled by $h(t)=4\cos(90t^\circ)+10$ cm, where $t$ is the time in milliseconds after the start of observation.

   (a)

   Find the period of the motion.

   [1 mark]

   • A$90$ ms
   • B$0.25$ ms
   • C$4$ ms
   • D$360$ ms

   (b)

   Find the minimum height of the piston head above the base.

   [1 mark]

   • A$4$ cm
   • B$10$ cm
   • C$14$ cm
   • D$6$ cm

   (c)

   Use your GDC, in degree mode, to find the height of the piston head at $t=1.5$ ms. Give your answer correct to 3 significant figures.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A section of a model roller coaster track has height $f(x)$ cm at a horizontal distance $x$ cm from the start, where $f(x)=x^{3}-6x^{2}+9x$ for $0\le x\le5$.

   (a)

   Use your GDC to find (i) the $x$-intercepts of the graph of $y=f(x)$, (ii) the coordinates of the local maximum point.

   [3 marks]

   (b)

   Sketch the graph of $y=f(x)$ for $0\le x\le5$. Label the axes and show clearly the intercepts, the local maximum and the end points of the graph.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A start-up company models its monthly profit, $P$ thousand USD, $n$ months after launch, by $P(n)=an^{2}+bn+c$. The profit was $-4.5$ in month $1$, and $6$ in months $4$ and $8$.

   (a)

   (i) Write down three equations in $a$, $b$ and $c$.
   (ii) Use your GDC to find the values of $a$, $b$ and $c$.
   (iii) Hence write down the value of $n$ at which the profit is a maximum.

   [6 marks]

   (b)

   In this part use the model $P(n)=-0.5n^{2}+6n-10$.
   (i) Find the first month $n$ in which the profit is positive.
   (ii) Find the maximum monthly profit, in USD.
   (iii) The company uses the model to predict the profit in month $24$. Find this prediction and comment on whether it is reasonable.

   [6 marks]

   Total for question 8: 12 marks

End of questions