Quadratic and non-linear simultaneous equationsIB MYP Maths Extended: Subtopic test

1. 1
   Consider the line $y=x+2$ and the curve $y=x^2$.

   (a)

   Eliminating $y$ gives a quadratic equation in $x$ for the points where the line meets the curve. Which equation is it?

   [1 mark]

   • A$x^2-x+2=0$
   • B$x^2-x-2=0$
   • C$x^2+x-2=0$
   • D$x^2+x+2=0$

   (b)

   Which pair gives both points where the line meets the curve?

   [1 mark]

   • A$(2,4)$ and $(1,3)$
   • B$(-2,4)$ and $(1,1)$
   • C$(2,4)$ and $(-1,-1)$
   • D$(2,4)$ and $(-1,1)$

   (c)

   The line is replaced by $y=2x-1$. Show that this new line meets the curve $y=x^2$ at exactly one point, and state the coordinates of that point.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the line $y=3x-4$ and the curve $y=x^2-4x+8$.

   (a)

   Which equation results from eliminating $y$ and collecting all terms on the left-hand side?

   [1 mark]

   • A$x^2-7x+12=0$
   • B$x^2-x+4=0$
   • C$x^2-7x+4=0$
   • D$x^2-x+12=0$

   (b)

   Which pair gives the points where the line meets the curve?

   [1 mark]

   • A$(3,0)$ and $(4,0)$
   • B$(-3,-13)$ and $(-4,-16)$
   • C$(3,5)$ and $(4,8)$
   • D$(3,5)$ and $(4,12)$

   (c)

   The line is changed to $y=2x-2$. Show that it does not meet the curve $y=x^2-4x+8$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A rectangular patio has length $x$ m and width $y$ m. Its perimeter is $20$ m and its area is $21$ m$^2$.

   (a)

   Show that $x^2-10x+21=0$.

   [3 marks]

   (b)

   Solve the equation to find the length and width of the patio, given that the length is greater than the width.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A footballer kicks a ball up a grassy slope. Using horizontal distance $x$ metres from the kicker, the height of the ball above the level ground is modelled by $y=-0.1x^2+1.2x$. The surface of the slope is modelled by the straight line $y=0.2x$.

   (a)

   (i) Solve the equations simultaneously to find where the ball lands on the slope.
   (ii) Write down the coordinates of the landing point.
   (iii) Find the greatest vertical distance between the ball and the slope.

   [6 marks]

   (b)

   A straight cable is to be fixed along the line $y=0.2x+2.5$. Use the discriminant to decide whether the ball hits the cable, and find the point of contact if it does.
   The cable is then raised to $y=0.2x+3$. Show that the ball now passes below it, and explain why a safety margin is still sensible when using this model.

   [6 marks]

   Total for question 4: 12 marks

End of questions