Questions
Q1.Find the value of(a) (1)(b) (2)(Total 3 marks)
Q2.(i) Express(5 8 )(1 + 2 )in the form a + b 2 , where a and b
are integers.(3)(ii) Express 80 + in the form c 5, where c is an
integer.(3)(Total 6 marks)
Q3.
where p and q are integers.
(a) Find the value of p and the value of q.(3)(b) Calculate the
discriminant of 4x 5 x2(2)(c) On the axes on page 17, sketch the
curve with equation y = 4x 5 x2 showing clearly the coordinates of
any points where the curve crosses the coordinate axes.(3)
(Total 8 marks)
Q4.
where k is a real constant.(a) Find the discriminant of f(x) in
terms of k.(2)(b) Show that the discriminant of f(x) can be
expressed in the form (k + a)2 + b, where a and b are integers to
be found.(2)(c) Show that, for all values of k, the equation f(x) =
0 has real roots.(2)(Total 6 marks)
Q5.Solve the simultaneous equations
(7) (Total for question = 7 marks)
Q6.Find the set of values of x for which
(a) 3(x 2) < 8 2x(2)(b) (2x 7)(1 + x) < 0(3)(c) both 3(x
2) < 8 2xand (2x 7)(1 + x) < 0(1)(Total 6 marks)
Q7.The straight line L1 passes through the points (1, 3) and
(11, 12).
(a) Find an equation for L1 in the form ax + by + c = 0,where a,
b and c are integers.(4)
The line L2 has equation 3y + 4x 30 = 0.
(b) Find the coordinates of the point of intersection of L1 and
L2.(3)(Total 7 marks)
Q9.
A cuboid has a rectangular cross-section where the length of the
rectangle is equal to twice its width, x cm, as shown in Figure 2.
The volume of the cuboid is 81 cubic centimetres.(a) Show that the
total length, L cm, of the twelve edges of the cuboid is given
by
(3)(b) Use calculus to find the minimum value of L.(6)(c)
Justify that the value of L that you have found is a
minimum.(2)(Total 11 marks)
Q10.The volume V cm3 of a box, of height x cm, is given byV =
4x(5 x)2, 0 < x < 5(a) Find .(4)(b) Hence find the maximum
volume of the box.(4)(c) Justify that the volume that you found in
part (b) is a maximum.(2)(Total 10 marks)
Q11.The curve with equationy = x2 32(x) + 20, x > 0has a
stationary point P.Use calculus
(a) to find the coordinates of P,(6)(b) to determine the nature
of the stationary point P.(3)(Total 9 marks)
Q12.A curve with equation y = f(x) passes through the point (4,
25).Given thatf'(x) = 38x2 10x + 1, x > 0(a) find f(x),
simplifying each term.(5)(b) Find an equation of the normal to the
curve at the point (4, 25).Give your answer in the form ax + by + c
= 0, where a, b and c are integers to be found.(5)
(Total 10 marks)
Mark Scheme
Q1.
Q2.
Q3.
Q4.
Q5.
Q6.
Q7.
Q8.
Q9.
Q10.
Q11.
Q12.
