Edexcel January 2019 Pure Maths Paper 1

Edexcel January 2019 Pure Maths Mock Paper 1 (pdf)

Edexcel January 2019 Pure Maths Mock Paper 1 Mark Scheme (pdf)

1. Figure 1 shows part of the curve with equation y = xe^1/5x2 for x ≥ 0
   The finite region R, shown shaded in Figure 1, is bounded by the curve, the y-axis, the x-axis, and the line with equation x = 2
   The table below shows corresponding values of x and y for y = xe^1/5x2
   (a) Use the trapezium rule, with all the values of y in the table, to find an estimate for the area of R, giving your answer to 2 decimal places.
   (b) Use your answer to part (a) to deduce an estimate for giving your answers to 2 decimal places.
   
   
2. The shape AOCBA, shown in Figure 2, consists of a sector AOB of a circle centre O joined to a triangle BOC.
   The points A, O and C lie on a straight line with AO = 7.5cm and OC = 8.5cm.
   The size of angle AOB is 1.2 radians.
   Find, in cm, the perimeter of the shape AOCBA, giving your answer to one decimal place.
   
   
3. The equation
   3x² + k = 5x + 2
   where k is a constant, has no real roots.
   Find the range of possible values for k.
   
   
4. The function f is defined by
   f(x) = 12x/(3x + 4)
   (a) Find the range of f.
   (b) Find f^–1.
   (c) (c) Show that
   ff(x) = 9x/(3x + 1)
   (d) Show that ff(x) = 7/2 has no solutions.
   

5. A curve has equation
   y = 4x² – 5x
   The curve passes through the point P(2, 6)
   Use differentiation from first principles to find the value of the gradient of the curve at P.
   
   
6. Figure 3 shows a sketch of part of the curve with equation
   The finite region R, shown shaded in Figure 3, is bounded by the curve, the y-axis, the x-axis, and the line with equation x = 2
   (a) Use the substitution u = x^{1/2 e} to show that the area of R can be given by
   where a and b are constants to be found.
   (b) Hence use algebraic integration to show that the exact area of R is 3ln(5e/(e+4))
   
   
7. (a) Express 3 sin θ – 4cos θ in the form Rsin(θ – α), where R > 0 and 0 < α < 90°
   State the value of R and give the value of α to 2 decimal places.
   The temperature in a greenhouse, G°C, is modelled by the equation
   G = 17 + 3 sin(15t)° – 4 cos(15t)° 0 ≤ t ≤ 17
   where t is the time in hours after 5a.m.
   (b) Find, according to this model,
   (i) the maximum temperature in the greenhouse,
   (ii) the time, after midday, when the temperature in the greenhouse is 20°C.
   Give your answer to the nearest minute.
   
   
8. (i) Show that y² – 4y + 7 is positive for all real values of y.
   (ii) Bobby claims that
   e^3x ≥ e^2x
   Determine whether Bobby’s claim is always true, sometimes true or never true,
   justifying your answer.
   (iii) Elsa claims that
   ‘if n² is even, then n must be even’
   Use proof by contradiction to show that Elsa’s claim is true.
   (iv)Ying claims that
   ‘the sum of two different irrational numbers is irrational’
   Determine whether Ying’s claim is always true, sometimes true or never true,
   justifying your answer.
   
   
9. (a) Show that
   where k is a constant to be found.
   (b) Hence explain why the equation
   has no real solutions.
   
   
10. Figure 4 shows a bowl with a circular cross-section.
    Initially the bowl is empty. Water begins to flow into the bowl.
    At time t seconds after the water begins to flow into the bowl, the height of the water in the bowl is h cm.
    The volume of water, Vcm³, in the bowl is modelled as
    V = 4πh(h + 6) 0 ≤ h ≤ 25
    The water flows into the bowl at a constant rate of 80π cm³ s^–1
    (a) Show that, according to the model, it takes 36 seconds to fill the bowl with water from empty to a height of 24 cm.
    (b) Find, according to the model, the rate of change of the height of the water, in cms^–1, when t = 8
    
    
11. (i) Given that
    y = a^x
    where a is a positive constant, show that
    dy/dx = a^xln a
    (ii) Given that
    x = 2 tan y,
    show that
    dy/dx = k(4 + x²)
    where k is a constant to be found.
    
    
12. A suspension bridge cable PQR hangs between the tops of two vertical towers,
    AP and BR, as shown in Figure 5.
    A walkway AOB runs between the bases of the towers, directly under the cable.
    The towers are 100m apart and each tower is 24m high.
    At the point O, midway between the towers, the cable is 4m above the walkway.
    The points P, Q, R, A, O and B are assumed to lie in the same vertical plane and AOB is assumed to be horizontal.
    Figure 6 shows a symmetric quadratic curve PQR used to model this cable.
    Given that O is the origin,
    (a) find an equation for the curve PQR.
    Lee can safely inspect the cable up to a height of 12m above the walkway.
    A defect is reported on the cable at a location 19m horizontally from one of the towers.
    (b) Determine whether, according to the model, Lee can safely inspect this defect.
    (c) Give a reason why this model may not be suitable to determine whether Lee can safely inspect this defect.
    
    
13. Given that p is a positive constant,
    (a) show that
    where k is a constant to be found,
    (b) show that
    (c) Hence find the set of values of p for which
    giving your answer in set notation.
    
    
14. Figure 7 shows a sketch of the curve with equation
    y = 4xe^–2x x ≥ 0
    The line l is the normal to the curve at the point P(1, 4e^–2)
    The finite region R, shown shaded in Figure 7, is bounded by the curve, the line l, and the x-axis.
    Find the exact value of the area of R.
    
    
15. Relative to a fixed origin O, the points A and B are such that
    and the points C and D are such that
    (a) Find the position vector of the point D.
    Given that ABCD is a trapezium,
    (b) find the value of p.
    

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