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Maths Test 009

Maths Test 009

Note
Test 009 covers chapter 8 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 6 August 2020.
The test total is 160 marks.

1. Use Pascal's Triangle to find the expansion of                                                     (20)
   1. \( ( p + q)^5 \)
   2. \( ( 4 - \frac{1}{2}x)^4 \)
   3. \( (5ab - 3y)^3 \)
   4. \( (3x^2 - 2y)^5 \)


2. Make use of Pascal's triangle to complete each problem.                     (20)
   1. Fully expand \( (1 + 4x)(2 - 3x)^4 \)
   2. Use the expansion of \( (2 + y)^3 \) to expand \( (2 + x - x^2)^3 \)
   3. Find the value of a \( (2 + ax)^3 \) if the coefficient of \( x^2 \) is 54
   4. Find the coefficient of \( x^2 \) for \( (p - 2x)^4 \).


3. Use factorials to work out the following for Pascal's triangle.           (20)
   1. The 5th number on the 12th row of Pascal's triangle.
   2. Find the next 4 values in the 14th row which starts as       1       13       78      
   3. Write out the 6th row.
   4. Given \( (1 + 3x)^{13} \) , find the coefficient of \(x^4 \).


4. Use the binomial theorem to find the expansion of           (20)
   1. \( (3x - y)^6 \)
   2. \( (1 + x)^8 \)
   3. \( (x + \frac{1}{x})^10 \)
   4. \( (2 - x)^5 \)


5. Given \( (2 + x)(3 - 2x)^7 \), find the coefficient of           (10)
   1. \( x^5\)
   2. \( x^2 \)
   3. \( x^8 \)


6. \( h(x) = (1 + kx)^{10} \), where \( k \) is a constant.           (10)
   1. Find the coefficient of \( x^5 \) in terms of \( k \).
   2. If the coefficient of \( x^3 \) in the binomial expansion of \( h(x) \) is 15, find \( k \).


7. When \( (1 - 2x)^p \) is expanded, the coefficient of \( x^2 \) is 40. Given that \( p \gt 0 \), find           (10)
   1. the value of \( p \).
   2. the coefficient of \( x \)
   3. the coefficient of \( x^3 \)


8. \( f(x) = ( 1 + px)^{15} \), where \( p \) is a non-zero constant. find           (10)
   1. Write the first five terms in ascending powers of \( x \).
   2. Given that the coefficient of \( x \) is \( -q \) and the coefficient of \( x^2 \) is \( 5q \), find the value of \( p \) and the value of \( q \).


9. In the binomial expansion of \( ( 1 + x)^{30} \), the coefficients of \( x^8 \), \( x^9 \), and \( x^{10} \), are \( p \), \( q \) and \( r \) respectively, find           (10)
   1. the value of \( \frac{p}{q} \)
   2. the value of \( \frac{r}{p} \)
   3. the value of \( \frac{r}{q} \)


10. Given \( f(x) = (1 - 5x) ^{30} \)           (20)
    1. find the first four terms, in ascending powers of \( x \), in the binomial expansion of \( f(x) \).
    2. use the first four terms of \( f(x) \) to estimate the value of \( (0.995)^{30} \), giving the answer to 6 decimal places.
    3. determine the actual value of \( (0.995)^{30} \) using a calculator.
    4. calculate the percentage error between the accurate and estimate values of \( (0.995)^{30} \).


11. If \( x \) is so small that terms of \( x^3 \) and higher can be ignored, and \( (2 - x)(3 + x)^4 \approx a + bx + cx^2 \) find the values of the constants \( a \), \( b \), and \( c \)           (10)

End of Maths Test 009