Quadratic functions and the discriminantEdexcel A-Level Maths: Subtopic test

1. 1
   Consider the equation $x^2+kx+16=0$, where $k$ is a constant.

   (a)

   For which values of $k$ does the equation have a repeated root?

   [1 mark]

   • A$k=8$ only
   • B$k=8$ or $k=-8$
   • C$k=4$ or $k=-4$
   • D$k=64$ or $k=-64$

   (b)

   For which values of $k$ does the equation have no real roots?

   [1 mark]

   • A$k<-8$ or $k>8$
   • B$k<8$
   • C$-64<k<64$
   • D$-8<k<8$

   (c)

   Find the set of values of $k$ for which the equation has two distinct real roots.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $f$ is defined by $f(x)=2x^2-12x+7$ for $x\in\mathbb{R}$.

   (a)

   Which is the completed-square form of $f(x)$?

   [1 mark]

   • A$2(x-3)^2-2$
   • B$2(x+3)^2-11$
   • C$2(x-3)^2-11$
   • D$2(x-3)^2+7$

   (b)

   What are the coordinates of the turning point of the graph of $y=f(x)$?

   [1 mark]

   • A$(3,-11)$
   • B$(-3,-11)$
   • C$(3,11)$
   • D$(6,-11)$

   (c)

   Hence solve $f(x)=0$, giving your answers in exact form.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the equation $x^2+px+(p+3)=0$, where $p$ is a constant.

   (a)

   Find the set of values of $p$ for which the equation has real roots.

   [3 marks]

   (b)

   Find the value of $p$ for which the equation has equal roots and the repeated root is positive.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Each equation in this question can be reduced to a quadratic equation by a substitution. No calculator may be used.

   (a)

   Solve $2\cos^2x-\cos x-1=0$ for $0\leqslant x<360^\circ$.

   [6 marks]

   (b)

   Solve $2^{2x+1}-9\times2^x+4=0$.

   [6 marks]

   Total for question 4: 12 marks

End of questions