Edexcel Jan 2021 IAL Pure Maths WMA11/01

Edexcel Jan 2021 IAL Pure Maths WMA11/01 question paper

Edexcel Jan 2021 IAL Pure Maths WMA11/01 mark scheme

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1. A curve has equation
   (a) Find, in simplest form, dy/dx
   The point P lies on the curve and has x coordinate 1/2
   (b) Find an equation of the normal to the curve at P, writing your answer in the form ax + by + c = 0, where a, b and c are integers to be found
2. A tree was planted.
   Exactly 3 years after it was planted, the height of the tree was 2m.
   Exactly 5 years after it was planted, the height of the tree was 2.4m.
   Given that the height, H metres, of the tree, t years after it was planted, can be modelled by the equation
   H³ = pt² + q
   where p and q are constants,
   (a) find, to 3 significant figures where necessary, the value of p and the value of q
   Exactly T years after the tree was planted, its height was 5m.
   (b) Find the value of T according to the model, giving your answer to one decimal place
3. Figure 1 shows a sketch of part of the curve C₁ with equation y = 4cos x°
   The point P and the point Q lie on C₁ and are shown in Figure 1.
   (a) State
   (i) the coordinates of P,
   (ii) the coordinates of Q.
   The curve C₂ has equation y = 4 cos x° + k, where k is a constant.
   Curve CC₂
   has a minimum y value of –1
   The point R is the maximum point on CC₂ with the smallest positive x coordinate.
   (b) State the coordinates of R.
4. The points P and Q, as shown in Figure 2, have coordinates (–2, 13) and (4, –5) respectively.
   The straight line l passes through P and Q.
   (a) Find an equation for l, writing your answer in the form y = mx + c, where m and c are integers to be found.
   The quadratic curve C passes through P and has a minimum point at Q.
   (b) Find an equation for C.
   The region R, shown shaded in Figure 2, lies in the second quadrant and is bounded by C and l only.
   (c) Use inequalities to define region R.
5. Figure 3 shows the plan view of a viewing platform at a tourist site.
   The shape of the viewing platform consists of a sector ABCOA of a circle, centre O, joined to a triangle AOD.
   Given that

• OA = OC = 6m
• AD = 14m
• angle ADC = 0.43 radians
• angle AOD is an obtuse angle
• OCD is a straight line
  find
  (a) the size of angle AOD, in radians, to 3 decimal places,
  (b) the length of arc ABC, in metres, to one decimal place,
  (c) the total area of the viewing platform, in m², to one decimal place

6. (a) Sketch the curve with equation
   (b) On a separate diagram, sketch the curve with equation
   stating the coordinates of the point of intersection with the x-axis and, in terms of k, the equation of the horizontal asymptote.
   (c) Find the range of possible values of k for which the curve with equation
7. In this question you must show all stages of your working.
   Solutions relying on calculator technology are not acceptable.
   f(x) = 2x - 3√x - 5, x > 0
   (a) Solve the equation
   f(x) = 9
   (b) Solve the equation
   f"(x) = 6
   
8. Figure 4 shows a sketch of part of the curve C with equation y = f(x), where f(x) = (3x – 2)²(x – 4)
   (a) Deduce the values of x for which f(x) > 0
   (b) Expand f(x) to the form
   ax³ + bx² + cx + d
   where a, b, c and d are integers to be found.
   The line l, also shown in Figure 4, passes through the y intercept of C and is parallel to the x-axis.
   The line l cuts C again at points P and Q, also shown in Figure 4.
   (c) Using algebra and showing your working, find the length of line PQ. Write your answer in the form k√3, where k is a constant to be found.
9. Find
   giving your answer in simplest form.
   (ii) A curve C has equation y = f(x).
   Given

• fʹ(x) = x² + ax + b where a and b are constants
• the y intercept of C is –8
• the point P(3,–2) lies on C
• the gradient of C at P is 2
  find, in simplest form, f(x).

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