Simplify to standard form

3

1. Find the first three terms, in ascending powers of \(x\), in the expansion of
   (a) \(\quad ( 1 - x ) ^ { 6 }\),
   (b) \(( 1 + 2 x ) ^ { 6 }\).
2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \([ ( 1 - x ) ( 1 + 2 x ) ] ^ { 6 }\).

4. (a) Find the first four terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 2 - \frac { 1 } { 4 } x \right) ^ { 6 }$$ (b) Given that \(x\) is small, so terms in \(x ^ { 4 }\) and higher powers of \(x\) may be ignored, show $$\left( 2 - \frac { 1 } { 4 } x \right) ^ { 6 } + \left( 2 + \frac { 1 } { 4 } x \right) ^ { 6 } = a + b x ^ { 2 }$$ where \(a\) and \(b\) are constants to be found.

12

1. Expand \(( 1 + 2 x ) ^ { 6 }\), simplifying all the terms.
2. Hence find an expression for \(\mathrm { f } ( x ) = ( 1 + 2 x ) ^ { 6 } + ( 1 - 2 x ) ^ { 6 }\) in its simplest form.
3. Substituting \(x = 0.01\) into the first two terms of \(\mathrm { f } ( x )\) gives an approximate value, z for \(1.02 ^ { 6 } + 0.98 ^ { 6 }\). Find \(z\). By considering the value of the third term, comment on the accuracy of \(z\) as an approximation for \(1.02 ^ { 6 } + 0.98 ^ { 6 }\).

1 Simplify \(( 3 + 2 x ) ^ { 3 } - ( 3 - 2 x ) ^ { 3 }\).

10

1. Find the binomial expansion of \(( 2 x + 5 ) ^ { 4 }\), simplifying the terms.
2. Hence show that \(( 2 x + 5 ) ^ { 4 } - ( 2 x - 5 ) ^ { 4 }\) can be written as $$320 x ^ { 3 } + k x$$ where the value of the constant \(k\) is to be stated.
3. Verify that \(x = 2\) is a root of the equation $$( 2 x + 5 ) ^ { 4 } - ( 2 x - 5 ) ^ { 4 } = 3680 x - 800$$ and find the other possible values of \(x\).

7. (i) Expand \(( 2 + x ) ^ { 4 }\) in ascending powers of \(x\), simplifying each coefficient.
(ii) Find the integers \(A , B\) and \(C\) such that $$( 2 + x ) ^ { 4 } + ( 2 - x ) ^ { 4 } \equiv A + B x ^ { 2 } + C x ^ { 4 }$$ (iii) Find the real values of \(x\) for which $$( 2 + x ) ^ { 4 } + ( 2 - x ) ^ { 4 } = 136$$

1

1. Find the binomial expansion of \(( 3 + 2 x ) ^ { 5 }\), simplifying the terms.
2. Hence find the binomial expansion of \(( 3 + 2 x ) ^ { 5 } + ( 3 - 2 x ) ^ { 5 }\).

6

1. 1. Using the binomial expansion, or otherwise, express \(( 2 + x ) ^ { 3 }\) in the form \(8 + a x + b x ^ { 2 } + x ^ { 3 }\), where \(a\) and \(b\) are integers. (3 marks)
   2. Write down the expansion of \(( 2 - x ) ^ { 3 }\).
2. Hence show that \(( 2 + x ) ^ { 3 } - ( 2 - x ) ^ { 3 } = 24 x + 2 x ^ { 3 }\).
3. Hence show that the curve with equation $$y = ( 2 + x ) ^ { 3 } - ( 2 - x ) ^ { 3 }$$ has no stationary points.

6

1. Using the binomial expansion, or otherwise:
   1. express \(( 1 + x ) ^ { 3 }\) in ascending powers of \(x\);
   2. express \(( 1 + x ) ^ { 4 }\) in ascending powers of \(x\).
2. Hence, or otherwise:
   1. express \(( 1 + 4 x ) ^ { 3 }\) in ascending powers of \(x\);
   2. express \(( 1 + 3 x ) ^ { 4 }\) in ascending powers of \(x\).
3. Show that the expansion of $$( 1 + 3 x ) ^ { 4 } - ( 1 + 4 x ) ^ { 3 }$$ can be written in the form $$p x ^ { 2 } + q x ^ { 3 } + r x ^ { 4 }$$ where \(p , q\) and \(r\) are integers.

5

1. By using the binomial expansion, or otherwise, express \(( 1 + 2 x ) ^ { 4 }\) in the form $$1 + a x + b x ^ { 2 } + c x ^ { 3 } + 16 x ^ { 4 }$$ where \(a\), \(b\) and \(c\) are integers.
2. Hence show that \(( 1 + 2 x ) ^ { 4 } + ( 1 - 2 x ) ^ { 4 } = 2 + 48 x ^ { 2 } + 32 x ^ { 4 }\).
3. Hence show that the curve with equation $$y = ( 1 + 2 x ) ^ { 4 } + ( 1 - 2 x ) ^ { 4 }$$ has just one stationary point and state its coordinates.

5. (a) Given that \(( 2 + x ) ^ { 5 } + ( 2 - x ) ^ { 5 } = A + B x ^ { 2 } + C x ^ { 4 }\), find the values of the constants \(A , B\) and \(C\).
(b) Using the substitution \(y = x ^ { 2 }\) and your answers to part (a), solve, $$( 2 + x ) ^ { 5 } + ( 2 - x ) ^ { 5 } = 349$$

6. (a) Expand \(( 2 + x ) ^ { 4 }\) in ascending powers of \(x\), simplifying each coefficient.
(b) Find the integers \(A , B\) and \(C\) such that $$( 2 + x ) ^ { 4 } + ( 2 - x ) ^ { 4 } \equiv A + B x ^ { 2 } + C x ^ { 4 }$$ (c) Find the real values of \(x\) for which $$( 2 + x ) ^ { 4 } + ( 2 - x ) ^ { 4 } = 136$$

10. (a) Given that $$( 2 + x ) ^ { 5 } + ( 2 - x ) ^ { 5 } = A + B x ^ { 2 } + C x ^ { 4 }$$ find the values of the constants \(A , B\) and \(C\).
(b) Using the substitution \(y = x ^ { 2 }\) and your answers to part (a), solve, $$( 2 + x ) ^ { 5 } + ( 2 - x ) ^ { 5 } = 349$$ [P2 June 2001 Question 8]