Edexcel C1 (Core Mathematics 1)

1. Solve the inequality

$$x ( 2 x + 1 ) \leq 6 .$$

1. The curve \(C\) has the equation \(y = ( x - a ) ^ { 2 }\) where \(a\) is a constant.

Given that $$\frac { \mathrm { d } y } { \mathrm { dx } } = 2 x - 6 ,$$

1. find the value of \(a\),
2. describe fully a single transformation that would map \(C\) onto the graph of \(y = x ^ { 2 }\).

4. (a) Find in exact form the coordinates of the points where the curve \(y = x ^ { 2 } - 4 x + 2\) crosses the \(x\)-axis.
(b) Find the value of the constant \(k\) for which the straight line \(y = 2 x + k\) is a tangent to the curve \(y = x ^ { 2 } - 4 x + 2\).

5. The curve \(C\) with equation \(y = ( 2 - x ) ( 3 - x ) ^ { 2 }\) crosses the \(x\)-axis at the point \(A\) and touches the \(x\)-axis at the point \(B\).

1. Sketch the curve \(C\), showing the coordinates of \(A\) and \(B\).
2. Show that the tangent to \(C\) at \(A\) has the equation $$x + y = 2 .$$

6. $$f ( x ) = 9 + 6 x - x ^ { 2 } .$$

1. Find the values of \(A\) and \(B\) such that $$\mathrm { f } ( x ) = A - ( x + B ) ^ { 2 }$$
2. State the maximum value of \(\mathrm { f } ( x )\).
3. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your answers in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are integers.
4. Sketch the curve \(y = \mathrm { f } ( x )\).

7. (a) An arithmetic series has a common difference of 7 . Given that the sum of the first 20 terms of the series is 530 , find

1. the first term of the series,
2. the smallest positive term of the series.
   (b) The terms of a sequence are given by $$u _ { n } = ( n + k ) ^ { 2 } , \quad n \geq 1 ,$$ where \(k\) is a positive constant.
   Given that \(u _ { 2 } = 2 u _ { 1 }\),
3. find the value of \(k\),
4. show that \(u _ { 3 } = 11 + 6 \sqrt { 2 }\).

8. The straight line \(l _ { 1 }\) passes through the point \(A ( - 2,5 )\) and the point \(B ( 4,1 )\).

1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers. The straight line \(l _ { 2 }\) passes through \(B\) and is perpendicular to \(l _ { 1 }\).
2. Find an equation for \(l _ { 2 }\). Given that \(l _ { 2 }\) meets the \(y\)-axis at the point \(C\),
3. show that triangle \(A B C\) is isosceles.

9. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\) where $$f ^ { \prime } ( x ) = 1 + \frac { 2 } { \sqrt { x } } , \quad x > 0$$ The straight line \(l\) has the equation \(y = 2 x - 1\) and is a tangent to \(C\) at the point \(P\).

1. State the gradient of \(C\) at \(P\).
2. Find the \(x\)-coordinate of \(P\).
3. Find an equation for \(C\).
4. Show that \(C\) crosses the \(x\)-axis at the point \(( 1,0 )\) and at no other point.