DoingMaths - Free maths worksheets
  • Home
  • Algebra
    • Algebraic expressions
    • Algebraic equations
    • Expanding brackets
    • Index notation
    • Inequalities
    • Quadratic equations
    • Sequences
    • Simultaneous equations
    • Straight line graphs
    • Substitution
  • Shapes, space and measures
    • Angles
    • Circles
    • Circle theorems
    • Compound measures
    • Construction
    • Distance/speed-time graphs
    • Length, area and volume
    • Metric and Imperial conversions
    • Metric units of measurement
    • Proof
    • Pythagoras' Theorem
    • Scale factors, similarity and congruence
    • Symmetry and reflection
    • Time
    • Trigonometry
  • Number
    • Primary Addition and Subtraction
    • Addition and subtraction
    • Basic number work
    • Compound percentage change
    • Decimals
    • Fractions
    • Fractions, decimals and percentages
    • Money
    • Multiplication and division
    • Percentages
    • Ratio and Proportion
    • Rounding
    • Standard form
  • Statistics and Probability
    • Averages and the Range
    • Collecting data
    • Box plots
    • Pie charts
    • Probability
  • More
    • Starters >
      • Puzzles and riddles
      • Maths Wordsearches
      • More starter ideas
    • Internet articles
    • A-Level Maths Paper Walkthroughs >
      • A-Level Maths, June 2018, Pure Paper 1 Question Walkthroughs
      • A-Level Maths, June 2018, Pure Paper 2, Question Walkthroughs
      • A-Level Maths, June 2018, Statistics and Mechanics, Question Walkthroughs
      • A-Level Maths, June 2019, Pure Paper 1, Question Walkthroughs
    • Mathematician of the Month
    • DoingMaths video channel
    • DoingMaths Shop
    • Contact us
    • Privacy policy

Edexcel A-Level Maths, Pure Paper 1, June 2019 Exam Question Walkthroughs

Pure Paper 1, 2019, Q1

f(x) = 3x^3 + 2ax^2 – 4x + 5a
​Given that (x + 3) is a factor of f(x), find the value of the constant a.

Pure Paper 1, 2019, Q2

Figure 1 shows a plot of part of the curve with equation y = cos x where x is measured in radians. Diagram 1, on the opposite page, is a copy of Figure 1.
(a) Use Diagram 1 to show why the equation cos x − 2x − 1/2 = 0 has only one real root, giving a reason for your answer.
Given that the root of the equation is alpha, and that alpha is small,
​b) use the small angle approximation for cos x to estimate the value of alpha to 3 decimal places.

Pure Paper 1, 2019, Q3

y = 5x^2+10x / (x+1)^2
(a) Show that dy/dx = A / (x+q)^n where A and n are constants to be found.
​(b) Hence deduce the range of values for x for which dy/dx is less than 0.

Pure Paper 1, 2019, Q4

Find the first three terms, in ascending powers of x, of the binomial expansion of 1/ _/(4 − x) giving each coefficient in its simplest form.
The expansion can be used to find an approximation to 2.
Possible values of x that could be substituted into this expansion are: x = –14, x = 2, x = − 1
b) Without evaluating your expansion, (i) state, giving a reason, which of the three values of x should not be used (ii) state, giving a reason, which of the three values of x would lead to the most accurate approximation to 2.

Pure Paper 1, 2019, Q5

f(x) = 2x^2 + 4x + 9
(a) Write f(x) in the form a(x + b)^2 + c, where a, b and c are integers to be found.
(b) Sketch the curve with equation y = f(x) showing any points of intersection with the coordinate axes and the coordinates of any turning point.
(c) (i) Describe fully the transformation that maps the curve with equation y = f(x) onto the curve with equation y = g(x) where g(x) = 2(x – 2)^2 + 4x – 3
​(ii) Find the range of the function h(x)= 21/(2x^2 + 4x + 9)

Pure Paper 1, 2019, Q6

Solve, for –180° < theta < 180°, the equation
5 sin 2theta= 9 tan theta
giving your answers, where necessary, to one decimal place.
(b) Deduce the smallest positive solution to the equation
5 sin (2x – 50°) = 9 tan (x – 25°)

Pure Paper 1, 2019, Q7

In a simple model, the value, £V, of a car depends on its age, t, in years. The following information is available for car A its value when new is £20 000 its value after one year is £16 000.
(a) Use an exponential model to form, for car A, a possible equation linking V with t. The value of car A is monitored over a 10-year period. Its value after 10 years is £2 000.

​(b) Evaluate the reliability of your model in light of this information.
The following information is available for car B: it has the same value, when new, as car A; its value depreciates more slowly than that of car A.
​(c) Explain how you would adapt the equation found in (a) so that it could be used to model the value of car B.

Pure Paper 1, 2019, Q8

Figure 2 shows a sketch of part of the curve with equation y = x(x + 2)(x – 4). The region R1 shown shaded in Figure 2 is bounded by the curve and the negative x-axis.
(a) Show that the exact area of R1 is 20/3
The region R2 also shown shaded in Figure 2 is bounded by the curve, the positive x-axis and the line with equation x = b, where b is a positive constant between 0 and 4. Given that the area of R1 is equal to the area of R2;
b) verify that b satisfies the equation (b + 2)^2 (3b^2 – 20b + 20) = 0
The roots of the equation 3b^2 – 20b + 20 = 0 are 1.225 and 5.442 to 3 decimal places. The value of b is therefore 1.225 to 3 decimal places.
​(c) Explain, with the aid of a diagram, the significance of the root 5.442

Pure Paper 1, 2019, Q9

Given that a > b > 0 and that a and b satisfy the equation log a – log b = log(a – b)
(a) show that a = b^2 / b - 1
(b) Write down the full restriction on the value of b, explaining the reason for this restriction.

Pure Paper 1, 2019, Q10

(i) Prove that for all n in the natural numbers, n^2 + 2 is not divisible by 4
(ii) “Given x in the real numbers, the value of |3x – 28| is greater than or equal to the value of (x – 9).”
​State, giving a reason, if the above statement is always true, sometimes true or never true.

Pure Paper 1, 2019, Q11

A competitor is running a 20 kilometre race. She runs each of the first 4 kilometres at a steady pace of 6 minutes per kilometre. After the first 4 kilometres, she begins to slow down. In order to estimate her finishing time, the time that she will take to complete each subsequent kilometre is modelled to be 5% greater than the time that she took to complete the previous kilometre.
Using the model, (a) show that her time to run the first 6 kilometres is estimated to be 36 minutes 55 seconds,
(b) show that her estimated time, in minutes, to run the rth kilometre, for r between 5 and 20 is 6 x 1.05^(r–4)
​(c) estimate the total time, in minutes and seconds, that she will take to complete the race.
Powered by Create your own unique website with customizable templates.
  • Home
  • Algebra
    • Algebraic expressions
    • Algebraic equations
    • Expanding brackets
    • Index notation
    • Inequalities
    • Quadratic equations
    • Sequences
    • Simultaneous equations
    • Straight line graphs
    • Substitution
  • Shapes, space and measures
    • Angles
    • Circles
    • Circle theorems
    • Compound measures
    • Construction
    • Distance/speed-time graphs
    • Length, area and volume
    • Metric and Imperial conversions
    • Metric units of measurement
    • Proof
    • Pythagoras' Theorem
    • Scale factors, similarity and congruence
    • Symmetry and reflection
    • Time
    • Trigonometry
  • Number
    • Primary Addition and Subtraction
    • Addition and subtraction
    • Basic number work
    • Compound percentage change
    • Decimals
    • Fractions
    • Fractions, decimals and percentages
    • Money
    • Multiplication and division
    • Percentages
    • Ratio and Proportion
    • Rounding
    • Standard form
  • Statistics and Probability
    • Averages and the Range
    • Collecting data
    • Box plots
    • Pie charts
    • Probability
  • More
    • Starters >
      • Puzzles and riddles
      • Maths Wordsearches
      • More starter ideas
    • Internet articles
    • A-Level Maths Paper Walkthroughs >
      • A-Level Maths, June 2018, Pure Paper 1 Question Walkthroughs
      • A-Level Maths, June 2018, Pure Paper 2, Question Walkthroughs
      • A-Level Maths, June 2018, Statistics and Mechanics, Question Walkthroughs
      • A-Level Maths, June 2019, Pure Paper 1, Question Walkthroughs
    • Mathematician of the Month
    • DoingMaths video channel
    • DoingMaths Shop
    • Contact us
    • Privacy policy