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Maths Test 005

Maths Test 005

Note
Test 005 covers chapter 4 of the Edexcel Maths AS course.
There is no time limit - the average person should complete the test in an hour and a half.
The test will remain available until midnight on 24 July 2020.
The test total is 120 marks.

1. Sketch the following equations and show the points where they cross the coordinate axes           (20)
   1. \( y = (x - 3)(x + 1)(2 - x) \)
   2. \( y = -x(2x + 1)(3 - x) \)
   3. \( y = (x - 2)^2(1 - x) \)
   4. \( y = (x + 5)^3 \)
   5. \( y = (x - 1)(x^2 + 1) \)


2. Sketch each equation below, showing points of intersection on the sketch           (20)
   1. \( y = 12x^3 - 3x \)
   2. \( y = x^3 + 2x^2 + x \)
   3. \( y = - (x - \frac{1}{2})^3 \)
   4. \( y = (x - 10)(x^2 -2x) + 12x \)
   5. \( y = 3x + 2x^2 - x^3 \)


3. Sketch the following equations and show the points where they cross the coordinate axes           (20)
   1. \( y = (x + 2)(x - 1)(2 - x)(x + 1) \)
   2. \( y = x^2(4x + 1)(3 - x) \)
   3. \( y = (3 - x)^4 \)
   4. \( y = (x + 1)(x - 2)(x^2 - 4x + 3) \)
   5. \( y = (x - 2)^2(3x^2 + 7x - 6) \)


4. Sketch the two equations on the same set of axes and estimate the points of intersection           (20)
   1. \( y = x^2(1 - x) \) and \( y = -\frac{2}{x} \)
   2. \( y = x(x - 4) \) and \( y = (x - 3)^3 \)
   3. \( y = 4 \) and \( y = x(x - 1)(x + 2)(x + 3)^2 \)
   4. \( y = x^2(x - 1)(x + 1) \) and \( y = \frac{1}{3}x^3 + 1 \)


5. Consider the equations \( f(x) = \frac{4}{x^2}\) and \( g(x) = 3x + 7 \).           (20)
   1. Sketch \( f(x) \) and \( g(x) \) on the same axes.
   2. Write down the number of real solutions for the equation \( \frac{4}{x^2} = 3x + 7 \).
   3. Show that the equation can be written as \( (x + 1)(x + 2)(3x - 2) = 0 \).
   4. Determine the exact coordinates of the points of intersection.


6. Given the equations \( f(x) = \frac{1}{2x}\), \( g(x) = (2x)^2(2x - 3) \), and \( 3h(x) = -(x)^2(2 - x)^2 \), sketch           (20)
   1. \( f(-x) \) and \( \frac{1}{2}g(3x) \)
   2. \( -g(x + 2) \) and \( h(2x) + 1 \)
   3. \( f(x + 2) + 1 \) and \( h( \frac{x}{4}) - 1 \)
   4. \( 3g(x + 1) - 2 \) and \( 3f(2 - x) \)

End of Maths Test 005