The discriminantEdexcel International A Level Maths: Subtopic test

1. 1
   Consider the equation $x^2-4x+7=0$.

   (a)

   What is the value of the discriminant of $x^2-4x+7$?

   [1 mark]

   • A$12$
   • B$44$
   • C$-20$
   • D$-12$

   (b)

   Which statement about the roots of $x^2-4x+7=0$ is correct?

   [1 mark]

   • AThe equation has two distinct real roots.
   • BThe equation has no real roots.
   • CThe equation has one repeated real root.
   • DThe equation has two equal roots, both negative.

   (c)

   Explain what the value of the discriminant tells you about the graph of $y=x^2-4x+7$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The equation $x^2+kx+9=0$, where $k$ is a positive constant, has equal roots.

   (a)

   Which equation must $k$ satisfy?

   [1 mark]

   • A$k^2-9=0$
   • B$k^2+36=0$
   • C$k^2-36=0$
   • D$k-36=0$

   (b)

   What is the value of $k$?

   [1 mark]

   • A$3$
   • B$9$
   • C$36$
   • D$6$

   (c)

   Solve $x^2+kx+9=0$ for this value of $k$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The line $y=2x+c$ and the curve $y=x^2-3x+4$, where $c$ is a constant.

   (a)

   Show that the $x$-coordinates of any points where the line meets the curve satisfy $x^2-5x+(4-c)=0$, and show that the discriminant of this equation is $4c+9$.

   [3 marks]

   (b)

   Given that the line is a tangent to the curve, find the value of $c$ and the coordinates of the point where the line touches the curve.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   $f(x)=kx^2+4x+(k-3)$, where $k$ is a non-zero constant.

   (a)

   Find the range of values of $k$ for which the equation $f(x)=0$ has two distinct real roots.

   [6 marks]

   (b)

   A student claims that $f(x)=0$ has real roots for every positive value of $k$.
   (i) Use $k=5$ to show that the student is wrong.
   (ii) Show that when $k=4$ the equation has a repeated root, and find it.
   (iii) When $k=-2$, state, with a reason, whether the graph of $y=f(x)$ meets the $x$-axis.

   [6 marks]

   Total for question 4: 12 marks

End of questions