P1: DifferentiationEdexcel International A Level Maths: Topic test

1. 1
   A curve has equation $y=5x^3-\dfrac{4}{x}+2x^{\frac12}$ for $x>0$.

   (a)

   Which of the following is $\dfrac{\mathrm{d}y}{\mathrm{d}x}$?

   [1 mark]

   • A$15x^2-\frac{4}{x^2}+\frac{1}{\sqrt x}$
   • B$15x^2+\frac{4}{x^2}+\frac{1}{2\sqrt x}$
   • C$15x^2+\frac{4}{x^2}+\frac{1}{\sqrt x}$
   • D$15x^2+\frac{4}{x}+\frac{1}{\sqrt x}$

   (b)

   Find the gradient of the curve at the point where $x=4$.

   [1 mark]

   • A$240.75$
   • B$240.25$
   • C$241.5$
   • D$240.5$

   (c)

   Find $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=(3x-1)(x+4)$.

   (a)

   Which of the following is $\dfrac{\mathrm{d}y}{\mathrm{d}x}$?

   [1 mark]

   • A$3$
   • B$6x+11$
   • C$6x+12$
   • D$6x^2+11x$

   (b)

   Find the gradient of $C$ at the point where $x=-1$.

   [1 mark]

   • A$-17$
   • B$17$
   • C$-5$
   • D$5$

   (c)

   Find the value of $x$ at which the gradient of $C$ is $17$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve $C$ has equation $y=x^3-5x^2+4x+6$. The point $P(2,2)$ lies on $C$.

   (a)

   Find an equation of the tangent to $C$ at $P$.

   [3 marks]

   (b)

   Find an equation of the normal to $C$ at $P$, giving your answer in the form $ax+by+c=0$ where $a$, $b$ and $c$ are integers, and find the coordinates of the point where the normal meets the $x$-axis.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The height $h$ metres of a drone above the ground, $t$ seconds after take-off, is modelled by $h=2t^3-15t^2+36t$ for $0\le t\le6$.

   (a)

   (i) Find $\dfrac{\mathrm{d}h}{\mathrm{d}t}$.
   (ii) Find the rate of change of the height of the drone when $t=1$.
   (iii) Find the values of $t$ at which the drone is momentarily not changing height.

   [6 marks]

   (b)

   (i) Find $\dfrac{\mathrm{d}^2h}{\mathrm{d}t^2}$.
   (ii) Find the value of $\dfrac{\mathrm{d}^2h}{\mathrm{d}t^2}$ when $t=2$.
   (iii) Find an equation of the tangent to the graph of $h$ against $t$ at the point where $t=1$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The function $f$ is defined by $f(x)=\dfrac{x^2-6x+8}{2x}$ for $x\neq0$.

   (a)

   Which of the following is $f'(x)$?

   [1 mark]

   • A$\frac12+\frac{4}{x^2}$
   • B$\frac{2x-6}{2x}$
   • C$\frac12-\frac{4}{x}$
   • D$\frac12-\frac{4}{x^2}$

   (b)

   Find $f'(2)$.

   [1 mark]

   • A$\frac32$
   • B$-\frac12$
   • C$-\frac32$
   • D$\frac12$

   (c)

   Find the values of $x$ for which $f'(x)=0$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The curve $C$ has equation $y=2x^3-5x+1$. The point $P(2,7)$ lies on $C$.

   (a)

   Find the gradient of $C$ at $P$.

   [1 mark]

   • A$19$
   • B$-\frac1{19}$
   • C$24$
   • D$7$

   (b)

   Which of the following is an equation of the normal to $C$ at $P$?

   [1 mark]

   • A$19x-y-31=0$
   • B$x-19y+131=0$
   • C$x+19y-135=0$
   • D$19x+y-45=0$

   (c)

   Find the coordinates of the point where the tangent to $C$ at $P$ meets the $y$-axis.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The curve $C$ has equation $y=x^{\frac32}-6x+5$ for $x>0$. The point $A$ on $C$ has $x$-coordinate $4$.

   (a)

   Show that the tangent to $C$ at $A$ has equation $3x+y-1=0$.

   [3 marks]

   (b)

   The normal to $C$ at $A$ meets the $x$-axis at the point $B$. Find the coordinates of $B$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The curve $C$ has equation $y=x^2+\dfrac{a}{x}$ for $x>0$, where $a$ is a constant. The gradient of $C$ at the point $P$ where $x=2$ is $\frac72$.

   (a)

   (i) Find the value of $a$.
   (ii) Find an equation of the tangent to $C$ at $P$, giving your answer in the form $px+qy+r=0$, where $p$, $q$ and $r$ are integers.

   [6 marks]

   (b)

   Given that $a=2$:
   (i) find $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}$;
   (ii) find the value of $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}$ at $P$;
   (iii) find an equation of the normal to $C$ at $P$, giving your answer in the form $px+qy+r=0$, where $p$, $q$ and $r$ are integers.

   [6 marks]

   Total for question 8: 12 marks

End of questions