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1. Consider f(x) = 2kx2 – 4kx + 1, for k ≠ 0. The equation f(x) = 0 has two equal roots.
(a) Find the value of k.
(b) The line y = p intersects the graph of f. Find all possible values of p.
2. The following diagram shows part of the graph of a quadratic function f.
The x-intercepts are at (–4, 0) and (6, 0) and the y-intercept is at (0, 240).
(a) Write down f(x) in the form f(x) = –10(x – p)(x – q).
(b) Find another expression for f(x) in the form f(x) = –10(x – h)2 + k.
(c) Show that f(x) can also be written in the form f(x) = 240 + 20x – 10x2.
A particle moves along a straight line so that its velocity, v m s–1, at time t seconds is given by
v = 240 + 20t – 10t2, for 0 ≤ t ≤ 6.
(d) (i) Find the value of t when the speed of the particle is greatest.
(ii) Find the acceleration of the particle when its speed is zero.
3. Let f(x) = 8x – 2x2. Part of the graph of f is shown below.
(a) Find the x-intercepts of the graph.
(b) (i) Write down the equation of the axis of symmetry.
(ii) Find the y-coordinate of the vertex.
4. Let f(x) = p(x – q)(x – r). Part of the graph of f is shown below.
The graph passes through the points (–2, 0), (0, –4) and (4, 0).
(a) Write down the value of q and of r.
(b) Write down the equation of the axis of symmetry.
(c) Find the value of p.
5. The quadratic equation kx2 + (k – 3)x + 1 = 0 has two equal real roots.
(a) Find the possible values of k.
(b) Write down the values of k for which x2 + (k – 3)x + k = 0 has two equal real roots.
6. Let f (x) = 3(x + 1)2 – 12.
(a) Show that f (x) = 3x2 + 6x – 9.
(b) For the graph of f
(i) write down the coordinates of the vertex;
(ii) write down the equation of the axis of symmetry;
(iii) write down the y-intercept;
(iv) find both x-intercepts.
(c) Hence sketch the graph of f.
7. Let f (x) = 2x2 – 12x + 5.
(a) Express f(x) in the form f(x) = 2(x – h)2 – k.
(b) Write down the vertex of the graph of f.
(c) Write down the equation of the axis of symmetry of the graph of f.
(d) Find the y-intercept of the graph of f.
(e) The x-intercepts of f can be written as , where p, q, r .
Find the value of p, of q, and of r.
8. Part of the graph of the function y = d (x −m)2 + p is given in the diagram below.
The x-intercepts are (1, 0) and (5, 0). The vertex is V(m, 2).
(a) Write down the value of
(b) Find d.
9. Consider the function f (x) = 2x2 – 8x + 5.
(a) Express f (x) in the form a (x – p)2 + q, where a, p, q Î .
(b) Find the minimum value of f (x).
10. The diagram shows the graph of the function y = ax2 + bx + c.
Complete the table below to show whether each expression is positive, negative or zero.
b2 – 4ac
11. The diagram shows the parabola y = (7 – x)(l + x). The points A and C are the x-intercepts and the point B is the maximum point.
1. The area of the enclosed region shown in the diagram is defined by
y ³ x2 + 2, y £ ax + 2, where a > 0.
This region is rotated 360° about the x-axis to form a solid of revolution. Find, in terms of a, the volume of this solid of revolution.
(Total 4 marks)
2. Using the substitution u = x + 1, or otherwise, find the integral
3. When air is released from an inflated balloon it is found that the rate of decrease of the volume of the balloon is proportional to the volume of the balloon. This can be represented by the differential equation = – kv, where v is the volume, t is the time and k is the constant of proportionality.
(a) If the initial volume of the balloon is v0, find an expression, in terms of k, for the volume of the balloon at time t.
(b) Find an expression, in terms of k, for the time when the volume is
4. Consider the function f : x x – x2 for –1 £ x £ k, where 1 < k £ 3.
(a) Sketch the graph of the function f.
(b) Find the total finite area enclosed by the graph of f, the x-axis and the line x = k.
(Total 7 marks)
5. The area between the graph of y = ex and the x-axis from x = 0 to x = k (k > 0) is rotated through 360° about the x-axis. Find, in terms of k and e, the volume of the solid generated.
6. Find the real number k > 1 for which dx = .
7. In the diagram, PTQ is an arc of the parabola y = a2 – x2, where a is a positive constant, and PQRS is a rectangle. The area of the rectangle PQRS is equal to the area between the arc PTQ of the parabola and the x-axis.
Find, in terms of a, the dimensions of the rectangle.
8. Consider the function fk (x) = , where k Î
(a) Find the derivative of fk (x), x > 0.
(b) Find the interval over which f0 (x) is increasing.
The graph of the function fk (x) is shown below.
(c) (i) Show that the stationary point of fk (x) is at x = ek–1.
(ii) One x-intercept is at (0, 0). Find the coordinates of the other x-intercept.
(d) Find the area enclosed by the curve and the x-axis.
(e) Find the equation of the tangent to the curve at A.
(f) Show that the area of the triangular region created by the tangent and the
coordinate axes is twice the area enclosed by the curve and the x-axis.
(g) Show that the x-intercepts of fk (x) for consecutive values of k form a geometric sequence.
(Total 20 marks)
9. Find the values of a > 0, such that dx = 0.22.
(Total 3 marks)
10. Let f (x) = ln |x5 – 3x2|, –0.5 < x < 2, x ¹ a, x ¹ b; (a, b are values of x for which f (x) is not defined).
(a) (i) Sketch the graph of f (x), indicating on your sketch the number of zeros of f (x). Show also the position of any asymptotes.
(ii) Find all the zeros of f (x), (that is, solve f (x) = 0).
(b) Find the exact values of a and b.
(c) Find f (x), and indicate clearly where f¢ (x) is not defined.
(d) Find the exact value of the x-coordinate of the local maximum of f (x), for 0 < x < 1.5. (You may assume that there is no point of inflexion.)
(e) Write down the definite integral that represents the area of the region enclosed by f (x) and the x-axis. (Do not evaluate the integral.)
(Total 16 marks)
11. Calculate the area bounded by the graph of y = x sin (x2) and the x-axis, between x = 0 and the smallest positive x-intercept.