OCR C1 (Core Mathematics 1) 2014 June

Express \(5x^2 + 10x + 2\) in the form \(p(x + q)^2 + r\), where \(p\), \(q\) and \(r\) are integers. [4]

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

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

Find the real roots of the equation \(4x^4 + 3x^2 - 1 = 0\). [5]

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

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

Solve the following inequalities.

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

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

1. \(\frac{\text{d}y}{\text{d}x}\), [4]
2. \(\frac{\text{d}^2y}{\text{d}x^2}\). [2]

\(A\) is the point \((5, 7)\) and \(B\) is the point \((-1, -5)\).

1. Find the coordinates of the mid-point of the line segment \(AB\). [2]
2. Find an equation of the line through \(A\) that is perpendicular to the line segment \(AB\), giving your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers. [5]

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

1. Verify that the curve has a stationary point when \(x = 1\). [5]
2. Determine the nature of this stationary point. [2]
3. The tangent to the curve at this stationary point meets the \(y\)-axis at the point \(Q\). Find the coordinates of \(Q\). [2]

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. [3]
2. The point \(P\) has coordinates \((6, k)\) and lies inside the circle. Find the set of possible values of \(k\). [5]
3. Prove that the line \(2y = x\) does not meet the circle. [4]

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

1. Sketch the curve, giving the coordinates of all points of intersection with the axes. [3]
2. Find an equation of the tangent to the curve at the point where \(x = -1\). Give your answer in the form \(ax + by + c = 0\). [9]