Tangents and normalsAQA A-Level Maths: Subtopic test

1. 1
   The curve $C$ has equation $y=2x^3-5x+1$ and passes through the point $P$ where $x=2$.

   (a)

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

   [1 mark]

   • A$19$
   • B$7$
   • C$43$
   • D$24$

   (b)

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

   [1 mark]

   • A$y=19x-38$
   • B$y=19x-31$
   • C$y=19x+7$
   • D$y=-\frac{x}{19}+\frac{135}{19}$

   (c)

   Find the 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.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=x+\frac{12}{x}$ for $x\neq0$ and passes through the point $P(3,7)$.

   (a)

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

   [1 mark]

   • A$\frac73$
   • B$-3$
   • C$-\frac13$
   • D$3$

   (b)

   Find the point where the normal to $C$ at $P$ meets the $x$-axis.

   [1 mark]

   • A$(24,0)$
   • B$\left(\frac{16}{3},0\right)$
   • C$\left(-\frac73,0\right)$
   • D$\left(\frac23,0\right)$

   (c)

   Find the coordinates of the other point on $C$ at which the tangent is parallel to the tangent at $P$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=x^2+px+q$, where $p$ and $q$ are constants. The tangent to $C$ at the point $(1,2)$ has gradient $5$.

   (a)

   Find the values of $p$ and $q$.

   [3 marks]

   (b)

   The normal to $C$ at $(1,2)$ meets $C$ again at the point $Q$. Find the coordinates of $Q$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=x^3-3x^2-9x+4$ and meets the $y$-axis at the point $A$.

   (a)

   (i) Find the coordinates of the two points on $C$ at which the tangent is parallel to the line $y=15x-2$.
   (ii) Find the equation of the normal to $C$ at the point with the positive $x$-coordinate, giving your answer in the form $ax+by+c=0$ where $a$, $b$ and $c$ are integers.

   [6 marks]

   (b)

   The tangent to $C$ at $A$ meets $C$ again at the point $B$. Find the coordinates of $B$ and show that the tangent to $C$ at $B$ is parallel to the $x$-axis.

   [6 marks]

   Total for question 4: 12 marks

End of questions