Edexcel January 2019 Pure Maths Paper 2

Edexcel January 2019 Pure Maths Mock Paper 2 (pdf)

Edexcel January 2019 Pure Maths Mock Paper 2 Mark Scheme (pdf)

1. (a) Given that θ is small and in radians, show that the equation
   cos θ - sin (1/2 θ) + 2 tan θ = 11/10
   can be written as
   5θ² – 15θ + 1 ≈ 0
   The solutions of the equation
   5θ² – 15θ + 1 = 0
   are 0.068 and 2.932, correct to 3 decimal places.
   (b) Comment on the validity of each of these values as approximate solutions to equation (I).
   
   
2. A curve has parametric equations
   x = 6t + 1
   y = 5 - 4/3 t, t ≠ 0
   Show that the Cartesian equation of the curve can be expressed in the form
   where a, b and k are constants to be found.
   
   
3. A curve has equation
   y = x² + kx + 14 - 8/(x - 5)
   where k is a constant.
   Given that the curve has a stationary point P, where x = 3
   (a) show that k = – 8
   (b) Determine the nature of the stationary point P, giving a reason for your answer.
   (c) Show that the curve has a point of inflection where x = 7
   The curve passes through the points (4.5, 14.25) and (5.5, –15.75)
   Jane uses this information to write down the following
   As there is a change of sign, the curve cuts the x-axis in the interval (4.5, 5.5)
   (d) Explain the error in Jane’s reasoning.
   
   
4. Figure 1 shows a sketch of part of the curve with equation y = f(x), where
   f(x) = x³ – 6x² + 7x + 2
   The curve cuts the x-axis at the points P, Q and R, as shown in Figure 1.
   The coordinates of Q are (2, 0)
   (a) Write f(x) as a product of two algebraic factors.
   (b) Find, giving your answer in simplest form,
   (i) the exact x coordinate of P,
   (ii) the exact x coordinate of R.
   (c) Deduce the number of real solutions, for –π ≤ θ ≤ 12π, to the equation
   sin³ θ – 6sin² θ + 7 sin θ + 2 = 0
   justifying your answer.
   
   

5. Figure 2 shows part of a graph with equation y = f(x), where
   f(x) = 7 – |3x – 5|
   The finite region R, shown shaded in Figure 2, is bounded by the graph with equation
   y = f(x) and the x-axis.
   (a) Find the area of R, giving your answer in simplest form.
   The equation
   7 – |3x – 5| = k
   where k is a constant, has two distinct real solutions.
   (b) Write down the range of possible values for k.
   
   
6. f(x) = (2 + kx)^–4
   where k is a positive constant
   The binomial expansion of f(x), in ascending powers of x, up to and including the term
   in x², is
   where A is a constant.
   (a) Find the value of A, giving your answer in simplest form.
   (b) Determine, giving a reason for your answer, whether the binomial expansion for f(x)
   is valid when x = 1/10
   
   
7. Figure 3 shows a plot of part of the curve with equation y = f(x), where
   f(x) = 2/x - e^x + 2x²
   The curve cuts the x-axis at the point A, where x = α, and at the point B, where x = β, as shown in Figure 3.
   (a) Show that α lies between –1.5 and –1
   (b) The iterative formula
   with x₁ = –1 can be used to estimate the value of α.
   (i) Find the value of x₃ to 4 decimal places.
   (ii) Find the value of α correct to 2 decimal places.
   The value of β lies in the interval [1.5, 3]
   A student takes 3 as her first approximation to β.
   Given f(3) = –1.4189 and fʹ(3) = –8.3078 to 4 decimal places,
   (c) apply the Newton-Raphson method once to f(x) to obtain a second approximation to β.
   Give your answer to 2 decimal places.
   A different student takes a starting value of 1.5 as his first approximation to β.
   (d) Use Figure 3 to explain whether or not the Newton-Raphson method with this starting
   value gives a good second approximation to β.
   
   
8. A circle with centre (9, –6) touches the x-axis as shown in Figure 4.
   (a) Write down an equation for the circle.
   A line l is parallel to the x-axis.
   The line l cuts the circle at points P and Q.
   Given that the distance PQ is 8
   (b) find the two possible equations for l.
   
   
9. The amount of antibiotic, y milligrams, in a patient’s bloodstream, t hours after the antibiotic was first given, is modelled by the equation
   y = ab^t
   where a and b are constants.
   (a) Show that this equation can be written in the form
   log₁₀ y = t log₁₀ b + c
   expressing the constant c in terms of a.
   A doctor measures the amount of antibiotic in the patient’s bloodstream at regular intervals for the first 5 hours after the antibiotic was first given.
   She plots a graph of log10 y against t and finds that the points on the graph lie close to a straight line passing through the point (0, 2.23) with gradient – 0.076
   (b) Estimate, to 2 significant figures, the value of a and the value of b.
   With reference to this model,
   (c) (i) give a practical interpretation of the value of the constant a,
   (ii) give a practical interpretation of the value of the constant b.
   (d) Use the model to estimate the time taken, after the antibiotic was first given, for the amount of antibiotic in the patient’s bloodstream to fall to 30milligrams. Give your answer, in hours, correct to one decimal place.
   (e) Comment on the reliability of your estimate in part (d).
   
   
10. A biologist conducted an experiment to investigate the growth of mould on a slice of bread.
    The biologist measured the surface area of bread, A cm², covered by mould at times, t days, after the start of the experiment.
    Initially 9.00cm² of the bread was covered by mould and 6 days later, 56.25cm² of the bread was covered by mould.
    In the biologist’s model, the rate of increase of the surface area of bread covered by mould, at any time t days, is proportional to the square root of that area.
    By forming and solving a differential equation,
    (a) show that the biologist’s model leads to the equation
    A = (3/4 t + 3)²
    The biologist’s full set of results are shown in the table below.
    Use the last four measurements from Table 1 to
    (b) (i) evaluate the biologist’s model,
    (ii) suggest a possible explanation of the results.
    
    
11. Figure 5 shows a sketch of the curve with equation y = f(x), where
    The curve has a minimum turning point at P and a maximum turning point at Q, as shown in Figure 5.
    (a) Show that the x coordinate of P and the x coordinate of Q are solutions of the equation cos 2x = 1/3
    (b) Hence find, to 2 decimal places, the x coordinate of the maximum turning point on the curve with equation
    (i) f = f(3x) + 5
    (ii) -f(1/4 x)
    

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