OCR C1 (Core Mathematics 1) 2014 June

1 Express \(5 x ^ { 2 } + 10 x + 2\) in the form \(p ( x + q ) ^ { 2 } + r\), where \(p , q\) and \(r\) are integers.

2 Express each of the following in the form \(k \sqrt { 3 }\), where \(k\) is an integer.

1. \(\frac { 6 } { \sqrt { 3 } }\)
2. \(10 \sqrt { 3 } - 6 \sqrt { 27 }\)
3. \(3 ^ { \frac { 5 } { 2 } }\)

3 Find the real roots of the equation \(4 x ^ { 4 } + 3 x ^ { 2 } - 1 = 0\).

4 The curve \(y = \mathrm { f } ( x )\) passes through the point \(P\) with coordinates \(( 2,5 )\).

1. State the coordinates of the point corresponding to \(P\) on the curve \(y = \mathrm { f } ( x ) + 2\).
2. State the coordinates of the point corresponding to \(P\) on the curve \(y = \mathrm { f } ( 2 x )\).
3. Describe the transformation that transforms the curve \(y = \mathrm { f } ( x )\) to the curve \(y = \mathrm { f } ( x + 4 )\).

5 Solve the following inequalities.

1. \(5 < 6 x + 3 < 14\)
2. \(x ( 3 x - 13 ) \geqslant 10\)

6 Given that \(y = 6 x ^ { 3 } + \frac { 4 } { \sqrt { x } } + 5 x\), find

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   \(7 \quad A\) is the point \(( 5,7 )\) and \(B\) is the point \(( - 1 , - 5 )\).
3. Find the coordinates of the mid-point of the line segment \(A B\).
4. Find an equation of the line through \(A\) that is perpendicular to the line segment \(A B\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

8 A curve has equation \(y = 3 x ^ { 3 } - 7 x + \frac { 2 } { x }\).

1. Verify that the curve has a stationary point when \(x = 1\).
2. Determine the nature of this stationary point.
3. The tangent to the curve at this stationary point meets the \(y\)-axis at the point \(Q\). Find the coordinates of \(Q\).

9 A circle with centre \(C\) has equation \(( x - 2 ) ^ { 2 } + ( y + 5 ) ^ { 2 } = 25\).

1. Show that no part of the circle lies above the \(x\)-axis.
2. The point \(P\) has coordinates \(( 6 , k )\) and lies inside the circle. Find the set of possible values of \(k\).
3. Prove that the line \(2 y = x\) does not meet the circle.

10 A curve has equation \(y = ( x + 2 ) ^ { 2 } ( 2 x - 3 )\).

1. Sketch the curve, giving the coordinates of all points of intersection with the axes.
2. Find an equation of the tangent to the curve at the point where \(x = - 1\). Give your answer in the form \(a x + b y + c = 0\). \section*{OCR}