OCR C1 (Core Mathematics 1)

1. Express \(\sqrt { 50 } + 3 \sqrt { 8 }\) in the form \(k \sqrt { 2 }\).
2. Find the coordinates of the stationary point of the curve with equation

$$y = x + \frac { 4 } { x ^ { 2 } } .$$

3. The diagram shows the curve with equation \(y = x ^ { 3 } + a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are constants. The curve crosses the \(x\)-axis at the point \(( - 1,0 )\) and touches the \(x\)-axis at the point \(( 3,0 )\). Show that \(a = - 5\) and find the values of \(b\) and \(c\).

4. 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 \(y = x ^ { 2 }\).

5. The straight line \(l _ { 1 }\) has the equation \(3 x - y = 0\). The straight line \(l _ { 2 }\) has the equation \(x + 2 y - 4 = 0\).

1. Sketch \(l _ { 1 }\) and \(l _ { 2 }\) on the same diagram, showing the coordinates of any points where each line meets the coordinate axes.
2. Find, as exact fractions, the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect.

6. (a) Given that \(y = 2 ^ { x }\), find expressions in terms of \(y\) for

1. \(2 ^ { x + 2 }\),
2. \(2 ^ { 3 - x }\).
   (b) Show that using the substitution \(y = 2 ^ { x }\), the equation $$2 ^ { x + 2 } + 2 ^ { 3 - x } = 33$$ can be rewritten as $$4 y ^ { 2 } - 33 y + 8 = 0$$ (c) Hence solve the equation $$2 ^ { x + 2 } + 2 ^ { 3 - x } = 33$$

1. The point \(A\) has coordinates ( 4,6 ).

Given that \(O A\), where \(O\) is the origin, is a diameter of circle \(C\),

1. find an equation for \(C\). Circle \(C\) crosses the \(x\)-axis at \(O\) and at the point \(B\).
2. Find the coordinates of \(B\).
3. Find an equation for the tangent to \(C\) at \(B\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.

8. (i) Express \(3 x ^ { 2 } - 12 x + 11\) in the form \(a ( x + b ) ^ { 2 } + c\).
(ii) Sketch the curve with equation \(y = 3 x ^ { 2 } - 12 x + 11\), showing the coordinates of the minimum point of the curve. Given that the curve \(y = 3 x ^ { 2 } - 12 x + 11\) crosses the \(x\)-axis at the points \(A\) and \(B\),
(iii) find the length \(A B\) in the form \(k \sqrt { 3 }\).

9. A curve has the equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 7 x\).

1. Show that the curve only crosses the \(x\)-axis at one point. The point \(P\) on the curve has coordinates \(( 3,3 )\).
2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers. The normal to the curve at \(P\) meets the coordinate axes at \(Q\) and \(R\).
3. Show that triangle \(O Q R\), where \(O\) is the origin, has area \(28 \frac { 1 } { 8 }\).