OCR C1 (Core Mathematics 1) 2013 June

Express each of the following in the form \(a\sqrt{5}\), where \(a\) is an integer.

1. \(4\sqrt{15} \times \sqrt{3}\) [2]
2. \(\frac{20}{\sqrt{5}}\) [1]
3. \(5^{\frac{3}{2}}\) [1]

Solve the equation \(8x^6 + 7x^3 - 1 = 0\). [5]

It is given that \(f(x) = \frac{6}{x^2} + 2x\).

1. Find \(f'(x)\). [3]
2. Find \(f''(x)\). [2]

1. Express \(3x^2 + 9x + 10\) in the form \(3(x + p)^2 + q\). [3]
2. State the coordinates of the minimum point of the curve \(y = 3x^2 + 9x + 10\). [2]
3. Calculate the discriminant of \(3x^2 + 9x + 10\). [2]

1. Sketch the curve \(y = \frac{2}{x^2}\). [2]
2. The curve \(y = \frac{2}{x^2}\) is translated by 5 units in the negative \(x\)-direction. Find the equation of the curve after it has been translated. [2]
3. Describe a transformation that transforms the curve \(y = \frac{2}{x^2}\) to the curve \(y = \frac{1}{x^2}\). [2]

A circle \(C\) has equation \(x^2 + y^2 + 8y - 24 = 0\).

1. Find the centre and radius of the circle. [3]
2. The point \(A(2, 2)\) lies on the circumference of \(C\). Given that \(AB\) is a diameter of the circle, find the coordinates of \(B\). [2]

Solve the inequalities

1. \(3 - 8x > 4\), [2]
2. \((2x - 4)(x - 3) \leq 12\). [5]

\(A\) is the point \((-2, 6)\) and \(B\) is the point \((3, -8)\). The line \(l\) is perpendicular to the line \(x - 3y + 15 = 0\) and passes through the mid-point of \(AB\). Find the equation of \(l\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [7]

1. Sketch the curve \(y = 2x^2 - x - 6\), giving the coordinates of all points of intersection with the axes. [5]
2. Find the set of values of \(x\) for which \(2x^2 - x - 6\) is a decreasing function. [3]
3. The line \(y = 4\) meets the curve \(y = 2x^2 - x - 6\) at the points \(P\) and \(Q\). Calculate the distance \(PQ\). [4]

The curve \(y = (1 - x)(x^2 + 4x + k)\) has a stationary point when \(x = -3\).

1. Find the value of the constant \(k\). [7]
2. Determine whether the stationary point is a maximum or minimum point. [2]
3. Given that \(y = 9x - 9\) is the equation of the tangent to the curve at the point \(A\), find the coordinates of \(A\). [5]