P2: DifferentiationEdexcel International A Level Maths: Topic test

1. 1
   The curve $C$ has equation $y=x^3-12x+5$.

   (a)

   Which gives the $x$-coordinates of the stationary points of $C$?

   [1 mark]

   • A$x=-2$ and $x=2$
   • B$x=-4$ and $x=4$
   • C$x=2$ only
   • D$x=-\sqrt{12}$ and $x=\sqrt{12}$

   (b)

   Which statement describes the stationary point of $C$ at $x=2$?

   [1 mark]

   • AIt is a maximum, with $y=-11$.
   • BIt is a minimum, with $y=-11$.
   • CIt is a minimum, with $y=21$.
   • DIt is a maximum, with $y=21$.

   (c)

   Find the set of values of $x$ for which $y$ is decreasing.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A rectangular sheet of card measures $24$ cm by $15$ cm. A square of side $x$ cm is cut from each corner and the sides are folded up to make an open tray of volume $V$ cm$^3$.

   (a)

   Which expression gives $V$ in terms of $x$?

   [1 mark]

   • A$x^3-39x^2+360x$
   • B$4x^2-78x+360$
   • C$4x^3-78x^2+360x$
   • D$4x^3-78x^2+360$

   (b)

   For which value of $x$ in the interval $0<x<7.5$ is $V$ stationary?

   [1 mark]

   • A$10$
   • B$13$
   • C$5$
   • D$3$

   (c)

   Show that the stationary value of $V$ found in part (b) is a maximum.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=x^4-8x^2+3$.

   (a)

   Find the coordinates of the stationary points of $C$.

   [3 marks]

   (b)

   Determine the nature of each stationary point, and hence find the values of $k$ for which $x^4-8x^2+3=k$ has four distinct real roots.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=ax^3+bx^2-12x$, where $a$ and $b$ are constants. $C$ has stationary points at $x=1$ and $x=-2$.

   (a)

   Find the values of $a$ and $b$.

   [6 marks]

   (b)

   Using your values of $a$ and $b$, find the coordinates and nature of each stationary point, and the values of $x$ for which $y$ is increasing.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   $\mathrm{f}(x)=x^2+\dfrac{16}{x}$, for $x>0$.

   (a)

   Which expression is $\mathrm{f}'(x)$?

   [1 mark]

   • A$2x+\dfrac{16}{x^2}$
   • B$2x-\dfrac{16}{x^2}$
   • C$2x-\dfrac{16}{x}$
   • D$2x-16x^{2}$

   (b)

   For which values of $x$ is $\mathrm{f}$ a decreasing function?

   [1 mark]

   • A$0<x<2$
   • B$x>2$
   • C$0<x<2.52$
   • D$x>0$

   (c)

   Find the minimum value of $\mathrm{f}(x)$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Two positive numbers $x$ and $y$ satisfy $x+y=12$. The product $P=x^2y$ is to be as large as possible.

   (a)

   Which expression gives $P$ in terms of $x$ only?

   [1 mark]

   • A$12x-x^2$
   • B$12x^2+x^3$
   • C$\dfrac{x^2}{12-x}$
   • D$12x^2-x^3$

   (b)

   For which value of $x$ in the interval $0<x<12$ does $P$ take its greatest value?

   [1 mark]

   • A$0$
   • B$6$
   • C$8$
   • D$12$

   (c)

   Find the greatest value of $P$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A theatre sells $(400-20p)$ tickets for each performance when the ticket price is £$p$, where $0<p<20$. Each ticket costs the theatre £5 to provide, and £$P$ is the profit per performance.

   (a)

   Show that $P=-20p^2+500p-2000$ and find the ticket price which gives a stationary value of $P$.

   [3 marks]

   (b)

   Show that this price gives the maximum profit, and find the maximum profit per performance and the number of tickets sold at that price.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A closed box has a square base of side $x$ cm and height $h$ cm. Its total surface area is $600$ cm$^2$ and its volume is $V$ cm$^3$.

   (a)

   Show that $V=150x-\frac12x^3$, and find the value of $x$ for which $V$ is stationary and the corresponding value of $V$.

   [6 marks]

   (b)

   Show that this value of $V$ is a maximum. By considering where $V$ is increasing and decreasing, explain why a box of volume $800$ cm$^3$ can be made in two different shapes, but a box of volume $1200$ cm$^3$ cannot.

   [6 marks]

   Total for question 8: 12 marks

End of questions