FunctionsIB Maths: Applications and Interpretation HL: Topic test

1. 1
   A straight ramp is modelled by the line $l$ with equation $2x-3y+12=0$, where $x$ and $y$ are measured in metres.

   (a)

   Find the gradient of $l$.

   [1 mark]

   • A$-\frac23$
   • B$\frac23$
   • C$2$
   • D$\frac32$

   (b)

   A support beam is perpendicular to the ramp. What is the gradient of the beam?

   [1 mark]

   • A$\frac23$
   • B$\frac32$
   • C$-\frac23$
   • D$-\frac32$

   (c)

   Find the equation of the line parallel to $l$ that passes through the point $(6,1)$. Give your answer in the form $y=mx+c$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $f$ is defined by $f(x)=4-\sqrt{x+2}$.

   (a)

   Which is the largest possible domain of $f$?

   [1 mark]

   • A$x\ge-2$
   • B$x\le4$
   • C$x\ge4$
   • D$x\le-2$

   (b)

   Which is the range of $f$?

   [1 mark]

   • A$f(x)\ge-2$
   • B$f(x)\ge4$
   • C$f(x)\le4$
   • D$f(x)\le-2$

   (c)

   Find $f^{-1}(x)$ and state its domain.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $g$ is defined by $g(x)=\dfrac{6}{x-2}+1$ for $x\ne2$. A student uses a GDC to graph $y=g(x)$ and will transfer a sketch of it to paper.

   (a)

   Write down the equation of each asymptote and the coordinates of the $y$-intercept, so that they can be labelled on the sketch.

   [3 marks]

   (b)

   Use your GDC to find the values of $x$ for which $g(x)<x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A footbridge arch spans a river. The height $y$ metres of the arch above the water is modelled by $y=ax^2+bx+c$ for $0\le x\le16$, where $x$ metres is the horizontal distance from the left bank. The arch is $1$ m above the water at the left bank ($x=0$), $5.8$ m above the water at $x=4$, and $1$ m above the water at $x=16$.

   (a)

   Set up three equations and use your GDC to find the values of $a$, $b$ and $c$.

   [6 marks]

   (b)

   Using the model, (i) find the greatest height of the arch above the water and the value of $x$ at which it occurs; (ii) find the greatest width, in metres, of the part of the arch that is at least $6.5$ m above the water; (iii) a student uses the model to predict the height at $x=20$. Comment on this prediction.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The functions $f$ and $g$ are defined by $f(x)=3x-2$ and $g(x)=x^2+1$, for $x\in\mathbb{R}$.

   (a)

   Find $(f\circ g)(x)$.

   [1 mark]

   • A$3x^2+3$
   • B$9x^2-12x+5$
   • C$3x^2+1$
   • D$x^2+3x-1$

   (b)

   Find $f^{-1}(g(2))$.

   [1 mark]

   • A$\frac73$
   • B$13$
   • C$\frac{25}{9}$
   • D$\frac{1}{13}$

   (c)

   Solve $(g\circ f)(x)=10$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The graph of $y=f(x)$ has a maximum point at $(4,5)$.

   (a)

   Which are the coordinates of the maximum point of the graph of $y=f(x+1)-4$?

   [1 mark]

   • A$(5,9)$
   • B$(3,1)$
   • C$(5,1)$
   • D$(3,9)$

   (b)

   The graph of $y=f(x)$ is transformed to give the graph of $y=-f(2x)$. Which are the coordinates of the image of the point $(4,5)$?

   [1 mark]

   • A$(8,5)$
   • B$(2,5)$
   • C$(8,-5)$
   • D$(2,-5)$

   (c)

   The graph of $y=f(x)$ is reflected in the $y$-axis and then translated $3$ units to the right. Write down the equation of the resulting graph and the coordinates of its maximum point.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The metabolic rate $R$ watts (W) of a mammal of mass $m$ kg is modelled by $R=km^n$, where $k$ and $n$ are constants. A biologist plots $\ln R$ against $\ln m$ for several mammals and finds that the line of best fit is $\ln R=0.75\ln m+4.1$.

   (a)

   Find the value of $n$ and the value of $k$, giving $k$ to $3$ significant figures.

   [3 marks]

   (b)

   Use the model to (i) predict the metabolic rate of a mammal of mass $400$ kg; (ii) find the mass of a mammal with metabolic rate $1000$ W. Use your GDC.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The power output $P$ megawatts (MW) of a wind turbine at wind speed $v$ m s$^{-1}$ is modelled by $P(v)=av^3$ for $0\le v<12$, and by $P(v)=3.6$ for $v\ge12$, where $a$ is a constant. Electricity sells for $60$ EUR per MW per hour, so the hourly income $I$ EUR is $I(P)=60P$.

   (a)

   (i) Find the value of $a$ that makes the model continuous at $v=12$. (ii) Find the wind speed at which the output is $2$ MW. (iii) By what factor does the output multiply when the wind speed doubles from $5$ m s$^{-1}$ to $10$ m s$^{-1}$?

   [6 marks]

   (b)

   Use $a=\frac{1}{480}$ and consider $0\le v<12$. (i) Find $(I\circ P)(v)$. (ii) Find the inverse of $I\circ P$, and state its domain. (iii) State what the inverse represents in this context.

   [6 marks]

   Total for question 8: 12 marks

End of questions