Year 1 Pure 1 Chapter 2: Quadratics - Exam Questions (Total Marks 59)
1.	Given that the equation kx2 + 12x + k = 0, where k is a positive constant, has equal roots, find the value of k.

(Total 4 marks)
2.	The equation x2 + 3px + p = 0, where p is a non-zero constant, has equal roots.
	Find the value of p.

(Total 4 marks)
 3.	The equation x2 + 2px + (3p + 4) = 0, where p is a positive constant, has equal roots.
(a)	Find the value of p.
(4)

(b)	For this value of p, solve the equation x2 + 2px + (3p + 4) = 0.

(2)
(Total 6 marks)

 4.	(a)	Show that x2 + 6x + 11 can be written as
		(x + p)2 + q
	where p and q are integers to be found.
(2)

(b)	In the space below, sketch the curve with equation y = x2 + 6x + 11, showing clearly any intersections with the coordinate axes.
(2)

(c)	Find the value of the discriminant of x2 + 6x + 11

(2)
(Total 6 marks)
5.	(a)	Solve the equation 4x2 + 12x = 0.
(3)

 f(x) = 4x2 + 12x + c,
	where c is a constant.
(b)	Given that f(x) = 0 has equal roots, find the value of c and hence solve f(x) = 0.

(4)
(Total 7 marks)
6.	f(x) = x2 – kx + 9, where k is a constant.
(a)	Find the set of values of k for which the equation f(x) = 0 has equal real roots.
(4)

	Given that k = 4,
(b)	express f(x) in the form (x – p)2 + q, where p and q are constants to be found,
(3)

(c)	write down the minimum value of f(x) and the value of x for which this occurs.

(2)
(Total 9 marks)
7.	Given that
f(x) = x2 – 6x + 18,   x ³ 0,
(a)	express f(x) in the form (x – a)2 + b, where a and b are integers.
(3)

	The curve C with equation y = f(x), x ³ 0, meets the y-axis at P and has a minimum point at Q.
(b)	Sketch the graph of C, showing the coordinates of P and Q.
(4)

	The line y = 41 meets C at the point R.
(c)	Find the x-coordinate of R, giving your answer in the form p + qÖ2, where p and q are integers.
(5)

(Total 12 marks)
8.	
	x2 + 2x + 3 º (x + a)2 + b.
(a)	Find the values of the constants a and b.
(2)

(b)	In the space provided below, sketch the graph of y = x2 + 2x + 3, indicating clearly the coordinates of any intersections with the coordinate axes.
(3)

(c)	Find the value of the discriminant of x2 + 2x + 3.  Explain how the sign of the discriminant relates to your sketch in part (b).
(2)

	The equation x2 + kx + 3 = 0, where k is a constant, has equal roots.
 (d)	Find the values of k, giving your answer in surd form.

(4)
(Total 11 marks)