OCR C1 (Core Mathematics 1)

1. Find the value of \(y\) such that

$$4 ^ { y + 3 } = 8$$

1. Express

$$\frac { 2 } { 3 \sqrt { 5 } + 7 }$$ in the form \(a + b \sqrt { 5 }\) where \(a\) and \(b\) are rational.

3. A circle has the equation $$x ^ { 2 } + y ^ { 2 } - 6 y - 7 = 0$$

1. Find the coordinates of the centre of the circle.
2. Find the radius of the circle.

4. (i) Express \(x ^ { 2 } + 6 x + 7\) in the form \(( x + a ) ^ { 2 } + b\).
(ii) State the coordinates of the vertex of the curve \(y = x ^ { 2 } + 6 x + 7\).

5. Solve the simultaneous equations $$\begin{aligned} & x + y = 2 \\ & 3 x ^ { 2 } - 2 x + y ^ { 2 } = 2 \end{aligned}$$

6. The diagram shows the curve with equation \(y = 3 x - x ^ { \frac { 3 } { 2 } } , x \geq 0\). The curve meets the \(x\)-axis at the origin and at the point \(A\) and has a maximum at the point \(B\).

1. Find the \(x\)-coordinate of \(A\).
2. Find the coordinates of \(B\).

7. (i) Calculate the discriminant of \(x ^ { 2 } - 6 x + 12\).
(ii) State the number of real roots of the equation \(x ^ { 2 } - 6 x + 12 = 0\) and hence, explain why \(x ^ { 2 } - 6 x + 12\) is always positive.
(iii) Show that the line \(y = 8 - 2 x\) is a tangent to the curve \(y = x ^ { 2 } - 6 x + 12\).

8. $$f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$$

1. Show that $$( x + 1 ) ( x - 3 ) ( x - 4 ) \equiv x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12 .$$
2. Sketch the curve \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.
3. Showing the coordinates of any points of intersection with the coordinate axes, sketch on separate diagrams the curves
   1. \(\quad y = \mathrm { f } ( x + 3 )\),
   2. \(y = \mathrm { f } ( - x )\).

9. A curve has the equation \(y = \frac { x } { 2 } + 3 - \frac { 1 } { x } , x \neq 0\). The point \(A\) on the curve has \(x\)-coordinate 2 .

1. Find the gradient of the curve at \(A\).
2. Show that the tangent to the curve at \(A\) has equation $$3 x - 4 y + 8 = 0$$ The tangent to the curve at the point \(B\) is parallel to the tangent at \(A\).
3. Find the coordinates of \(B\).

10. The straight line \(l\) has gradient 3 and passes through the point \(A ( - 6,4 )\).

1. Find an equation for \(l\) in the form \(y = m x + c\). The straight line \(m\) has the equation \(x - 7 y + 14 = 0\).
   Given that \(m\) crosses the \(y\)-axis at the point \(B\) and intersects \(l\) at the point \(C\),
2. find the coordinates of \(B\) and \(C\),
3. show that \(\angle B A C = 90 ^ { \circ }\),
4. find the area of triangle \(A B C\).