1.Let f (x) = , x 4 and g (x) = x2, x .(a)Find (g f ) (3).(b)Find f 1(x).(c)Write down the domain of f 1.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................(Total 6 marks)

2.Consider two different quadratic functions of the form f (x) = 4x2 qx + 25. The graph of each function has its vertex on the x-axis.(a)Find both values of q.(b)For the greater value of q, solve f (x) = 0.(c)Find the coordinates of the point of intersection of the two graphs.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................(Total 6 marks)

3.Let f (x) = ln (x + 2), x 2 and g (x) = e(x4), x 0.(a)Write down the x-intercept of the graph of f.(b)(i)Write down f (1.999).(ii)Find the range of f.(c)Find the coordinates of the point of intersection of the graphs of f and g.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................(Total 6 marks)

4.The graph of a function f is shown in the diagram below. The point A (1, 1) is on the graph, and y = 1 is a horizontal asymptote.

(a)Let g (x) = f (x 1) + 2. On the diagram, sketch the graph of g.(b)Write down the equation of the horizontal asymptote of g.(c)Let A be the point on the graph of g corresponding to point A. Write down the coordinates of A...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................(Total 6 marks)5.The functions f and g are defined by .(a)Find an expression for (f g) (x).(b)Show that f l (18) + gl (18) = 22.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................(Total 6 marks)

6.The function f is defined by for 3 < x < 3.(a)On the grid below, sketch the graph of f.

(b)Write down the equation of each vertical asymptote.(c)Write down the range of the function f.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................(Total 6 marks)

7.The quadratic function f is defined by f (x) = 3x2 12x + 11.(a)Write f in the form f (x) = 3(x h)2 k.(b)The graph of f is translated 3 units in the positive x-direction and 5 units in the positive y-direction. Find the function g for the translated graph, giving your answer in the form g(x)= 3(x p)2 + q.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................(Total 6 marks)

8.In the diagram below, the points O(0, 0) and A(8, 6) are xed. The angle varies as the point P(x, 10) moves along the horizontal line y = 10.

Diagram to scale (a)(i)Show that (ii)Write down a similar expression for OP in terms of x.(2)

(b)Hence, show that

(3) (c)Find, in degrees, the angle when x = 8.(2)

(d)Find the positive value of x such that .(4)Let the function f be dened by

(e)Consider the equation f (x) = 1.(i)Explain, in terms of the position of the points O, A, and P, why thisequation has a solution.(ii)Find the exact solution to the equation.(5)(Total 16 marks)

9.Consider the line L with equation y + 2x = 3. The line L1 is parallel to L and passes through the point (6, 4).(a)Find the gradient of L1.(b)Find the equation of L1 in the form y = mx + b.(c)Find the x-coordinate of the point where line L1 crosses the x-axis.Working:

Answers:(a)..................................................................(b)..................................................................(c)..................................................................

(Total 6 marks)

10.The function f is given by f (x) = e(x11) 8.(a)Find f 1(x).(b)Write down the domain of f l(x).Working:

Answers:(a)..................................................................(b)..................................................................

(Total 6 marks)

11.The graph of y = f (x) is shown in the diagram.

(a)On each of the following diagrams draw the required graph,(i)y = 2 f (x);

(ii)y = f (x 3).

(b)The point A (3, 1) is on the graph of f. The point A is the corresponding point on the graph of y = f (x) + 1. Find the coordinates of A.Working:

Answer:(b)..................................................................

(Total 6 marks)

12.The equation of a curve may be written in the form y = a(x p)(x q). The curve intersects the x-axis at A(2, 0) and B(4, 0). The curve of y = f (x) is shown in the diagram below.

(i)Write down the value of p and of q.(ii)Given that the point (6, 8) is on the curve, find the value of a.(iii)Write the equation of the curve in the form y = ax2 + bx + c.(5)13.Let f (x) = loga x, x 0.(a)Write down the value of(i)f (a);(ii)f (1);(iii)f (a4 ).................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................(3)

(b)The diagram below shows part of the graph of f.

On the same diagram, sketch the graph of f1.(3)(Total 6 marks)

14.The following diagram shows part of the graph of f (x) = 5 x2 with vertex V (0, 5).Its image y = g (x) after a translation with vector has vertex T (3, 6).

(a)Write down the value of(i)h;(ii)k.(2) (b)Write down an expression for g (x).(2)

(c)On the same diagram, sketch the graph of y = g (x).(2)............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................(Total 6 marks)

15.Let f (x) = ln (x + 5) + ln 2, for x 5.(a)Find f 1(x).(4)Let g (x) = ex.(b)Find (g f) (x), giving your answer in the form ax + b, where a, b, .........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................(3)(Total 7 marks)