OCR C1 (Core Mathematics 1)

1. (i) Calculate the discriminant of \(2 x ^ { 2 } + 8 x + 8\).
   (ii) State the number of real roots of the equation \(2 x ^ { 2 } + 8 x + 8 = 0\).
2. Find the set of values of \(x\) for which

$$( x - 1 ) ( x - 2 ) < 20 .$$

1. (i) Solve the equation

$$x ^ { \frac { 3 } { 2 } } = 27 .$$ (ii) Express \(\left( 2 \frac { 1 } { 4 } \right) ^ { - \frac { 1 } { 2 } }\) as an exact fraction in its simplest form.

4. Differentiate with respect to \(x\) $$\frac { 6 x ^ { 2 } - 1 } { 2 \sqrt { x } } .$$

5. The diagram shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve has a maximum at \(( - 3,4 )\) and a minimum at \(( 1 , - 2 )\). Showing the coordinates of any turning points, sketch on separate diagrams the curves with equations

1. \(\quad y = 2 \mathrm { f } ( x )\),
2. \(y = - \mathrm { f } ( x )\).

6. \(f ( x ) = 2 x ^ { 2 } - 4 x + 1\).

1. Find the values of the constants \(a\), \(b\) and \(c\) such that $$\mathrm { f } ( x ) = a ( x + b ) ^ { 2 } + c$$
2. State the equation of the line of symmetry of the curve \(y = \mathrm { f } ( x )\).
3. Solve the equation \(\mathrm { f } ( x ) = 3\), giving your answers in exact form.

7. A curve has the equation $$y = x ^ { 3 } + a x ^ { 2 } - 15 x + b$$ where \(a\) and \(b\) are constants. Given that the curve is stationary at the point \(( - 1,12 )\),

1. find the values of \(a\) and \(b\),
2. find the coordinates of the other stationary point of the curve.

8. The circle \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 8 y + k = 0$$ where \(k\) is a constant. Given that the point with coordinates \(( - 6,5 )\) lies on \(C\),

1. find the value of \(k\),
2. find the coordinates of the centre and the radius of \(C\). A straight line which passes through the point \(A ( 2,3 )\) is a tangent to \(C\) at the point \(B\).
3. Find the length \(A B\) in the form \(k \sqrt { 3 }\).

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

1. Show that the gradient of the curve at \(P\) is - 2 .
2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(y = m x + c\).
3. Find the coordinates of the point where the normal to the curve at \(P\) intersects the curve again.

10. The straight line \(l _ { 1 }\) has equation \(2 x + y - 14 = 0\) and crosses the \(x\)-axis at the point \(A\).

1. Find the coordinates of \(A\). The straight line \(l _ { 2 }\) is parallel to \(l _ { 1 }\) and passes through the point \(B ( - 6,6 )\).
2. Find an equation for \(l _ { 2 }\) in the form \(y = m x + c\). The line \(l _ { 2 }\) crosses the \(x\)-axis at the point \(C\).
3. Find the coordinates of \(C\). The point \(D\) lies on \(l _ { 1 }\) and is such that \(C D\) is perpendicular to \(l _ { 1 }\).
4. Show that \(D\) has coordinates \(( 5,4 )\).
5. Find the area of triangle \(A C D\).