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Maths Test 008

Maths Test 008

Note
Test 008 covers chapter 7 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 3 August 2020.
The test total is 140 marks.

1. Simplify the fractions.                                                                          (20)
   
   
   1. \( \frac{3x^4 - 4x^3 - 6x}{2x} \)
      
      
   2. \( \frac{2x^2 + 2x - 40}{x^2 + 2x - 15} \)
      
      
   3. \( \frac{2x^2 + 7x + 6}{x^2 - 3x - 10} \)
      
      
   4. \( \frac{2x^2 - 5x - 3}{2x^2 - 9x + 9} \)
      
      
   5. \( \frac{6x^3 + 3x^2 - 84x}{6x^2 - 33x + 42} \)
      
      


2. Divide each polynomial                                                               (20)
   1. \( x^3 + x^2 - 7x - 15 \) by \( (x - 3) \)
   2. \( -5x^3 - 27x^2 + 23x + 30 \) by \( (x + 6) \)
   3. \( 4x^4 - 6x^3 + 10x^2 - 11x - 6 \) by \( (2x - 3) \)
   4. \( 2x^3 - 17x + 3 \) by \( (x + 3) \)
   5. \( 4x^4 - 3x^3 + 11x^2 - x - 1 \) by \( (4x + 1) \)


3. Given \( f(x) = 3x^3 - 14x^2 - 47x - 14 \),                                                     (10)
   1. find the remainder when \( f(x) \) is divided by \( 2x - 1 \)
   2. find the remainder when \( f(x) \) is divided by \( x + 2 \)
   3. factorise completely


4. Divide then factorise completely                                                               (10)
   1. \( x^3 - 1 \) by \( (x - 1) \)
   2. \( x^4 - 16 \) by \( (x + 2) \)


5. Given \( f(x) = 2x^3 + 3x^2 - 8x + 3 \),                                                     (20)
   1. show that \( (2x - 1) \) is a factor
   2. factorise completely
   3. write down the roots of \( f(x) = 0 \)


6. \( f(x) = 4x^3 + 4x^2 - 11x - 6 \)                                                     (20)
   1. use long division to show that \( (x + 2) \) is a factor of \( f(x) \)
   2. use the factor theorem to show that \( (x + 2) \) is a factor of \( f(x) \)
   3. factorise \( f(x) \) completely
   4. write down all the solutions to \( f(x) = 0 \)
   5. sketch \( f(x) \)


7. Use deduction for each question.                                                     (20)
   1. Prove that \( -3n^2 - 4n + 10 \) is negative for all values of n.
   2. The equation \( kx^2 + 3kx + 1 = 0 \), where k is a constant, has no real roots. Use deduction to prove that \( k \) satisfies the inequality \( 0 \lt k \lt \frac{4}{9} \).
   3. Prove that \( (x - \frac{2}{x})^3 \equiv x^3 - 6x + \frac{12}{x} - \frac{8}{x^3} \)


8. Find a counter-example to disprove each of the following statements.                                                     (20)
   1. if \( n \) is a positive integer, then \( n^4 - n \) is divisible by 4.
   2. \( 2n^2 - 6n + 1 \) is positive for all values of \( n \)
   3. \( a + \frac{1}{a} \ge 2 \)
   4. \( p + q \ge \sqrt{4pq} \)
   5. integers always have an odd number of factors

End of Maths Test 008