P4: DifferentiationEdexcel International A Level Maths: Topic test

1. 1
   A curve $C$ has equation $2x^2+xy-y^2=8$.

   (a)

   Which of the following is $\frac{dy}{dx}$?

   [1 mark]

   • A$\frac{4x+y}{2y-x}$
   • B$\frac{4x+y}{x-2y}$
   • C$\frac{2x}{y}$
   • D$\frac{4x+y}{x+2y}$

   (b)

   Find the gradient of $C$ at the point $(3,5)$.

   [1 mark]

   • A$-\frac{17}{7}$
   • B$\frac{17}{13}$
   • C$\frac{17}{7}$
   • D$\frac{7}{17}$

   (c)

   Find an equation of the normal to $C$ at the point $(2,0)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve $C$ has parametric equations $x=t^3-3$, $y=3t^2+2$, where $t$ is a real parameter.

   (a)

   Which of the following is $\frac{dy}{dx}$, for $t\neq0$?

   [1 mark]

   • A$\frac t2$
   • B$\frac2t$
   • C$18t^3$
   • D$\frac{6t}{3t^2-3}$

   (b)

   Find the gradient of the normal to $C$ at the point where $t=4$.

   [1 mark]

   • A$\frac12$
   • B$2$
   • C$-\frac12$
   • D$-2$

   (c)

   Find an equation of the tangent to $C$ at the point where $t=-2$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The temperature $\theta$ °C of a cup of tea, $t$ minutes after it is poured, decreases at a rate proportional to the amount by which $\theta$ exceeds the room temperature of $20$ °C. When $\theta=80$ the temperature is decreasing at $3$ °C per minute.

   (a)

   Form a differential equation for $\theta$ in terms of $t$ and a positive constant $k$, and find the value of $k$.

   [3 marks]

   (b)

   Find the rate at which $\frac{d\theta}{dt}$ is changing with respect to $t$ when $\theta=60$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A curve $C$ has equation $x^2+4y^2=20$. The point $P(2,2)$ lies on $C$.

   (a)

   (i) Show that $\frac{dy}{dx}=-\frac{x}{4y}$.
   (ii) Find an equation of the tangent to $C$ at $P$.
   (iii) The tangent at $P$ meets the $x$-axis at $A$ and the $y$-axis at $B$. Find the area of triangle $OAB$, where $O$ is the origin.

   [6 marks]

   (b)

   The point $Q$ moves along $C$ so that its $x$-coordinate increases at a constant rate of $0.3$ units per second. When $Q$ is at $P$, find
   (i) the rate of change of the $y$-coordinate of $Q$,
   (ii) the rate of change of the distance $OQ$, giving your answer to $3$ significant figures.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The edge length $x$ mm of a metal cube increases at a constant rate of $0.02$ mm s$^{-1}$ as the cube is heated.

   (a)

   Find the rate of increase of the volume of the cube when $x=40$.

   [1 mark]

   • A$4800$ mm$^3$ s$^{-1}$
   • B$1280$ mm$^3$ s$^{-1}$
   • C$2.4$ mm$^3$ s$^{-1}$
   • D$96$ mm$^3$ s$^{-1}$

   (b)

   Find the value of $x$ at which the volume is increasing at $24$ mm$^3$ s$^{-1}$.

   [1 mark]

   • A$400$
   • B$20$
   • C$40$
   • D$8$

   (c)

   The surface area of the cube is $A=6x^2$ mm$^2$. Find the rate of increase of $A$ when $x=40$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A drug is infused into a patient's bloodstream at a constant rate of $8$ mg per hour. The body removes the drug at a rate proportional to the amount $x$ mg present in the bloodstream at time $t$ hours.

   (a)

   Which differential equation models $x$, where $k$ is a positive constant?

   [1 mark]

   • A$\frac{dx}{dt}=8+kx$
   • B$\frac{dx}{dt}=k(8-x)$
   • C$\frac{dx}{dt}=8-kx$
   • D$\frac{dx}{dt}=-kx$

   (b)

   The amount of drug stops changing when $x=20$. Find the value of $k$.

   [1 mark]

   • A$0.4$
   • B$2.5$
   • C$160$
   • D$12$

   (c)

   Given that $k=0.4$, find the rate at which the amount of drug is changing when $x=10$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A curve $C$ has parametric equations $x=4\cos t$, $y=\sin2t$, for $0\leq t<2\pi$.

   (a)

   Find $\frac{dy}{dx}$ in terms of $t$.

   [3 marks]

   (b)

   Find an equation of the tangent to $C$ at the point where $t=\frac\pi6$, giving your answer in the form $x+py=q$, where $p$ and $q$ are exact constants.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A curve $C$ has parametric equations $x=2t^2$, $y=t^3$, where $t$ is a real parameter.

   (a)

   (i) Find $\frac{dy}{dx}$ in terms of $t$, for $t\neq0$.
   (ii) Find an equation of the tangent to $C$ at the point where $t=2$, in the form $ax+by+c=0$ where $a$, $b$ and $c$ are integers.
   (iii) The normal to $C$ at the point where $t=2$ meets the $x$-axis at $A$. Find the coordinates of $A$.

   [6 marks]

   (b)

   A point $P$ moves along $C$ so that the parameter $t$ increases at a constant rate of $0.1$ units per second. Let $T$ be the time in seconds. When $t=2$, find the rate of change of
   (i) the $x$-coordinate of $P$,
   (ii) the $y$-coordinate of $P$,
   (iii) the gradient of $C$ at $P$.

   [6 marks]

   Total for question 8: 12 marks

End of questions