OCR C1 (Core Mathematics 1) 2006 June

The points \(A(1, 3)\) and \(B(4, 21)\) lie on the curve \(y = x^2 + x + 1\).

1. Find the gradient of the line \(AB\). [2]
2. Find the gradient of the curve \(y = x^2 + x + 1\) at the point where \(x = 3\). [2]

1. Evaluate \(27^{\frac{2}{3}}\). [2]
2. Express \(5\sqrt{5}\) in the form \(5^n\). [1]
3. Express \(\frac{1 - \sqrt{5}}{3 + \sqrt{5}}\) in the form \(a + b\sqrt{5}\). [3]

1. Express \(2x^2 + 12x + 13\) in the form \(a(x + b)^2 + c\). [4]
2. Solve \(2x^2 + 12x + 13 = 0\), giving your answers in simplified surd form. [3]

1. By expanding the brackets, show that $$(x - 4)(x - 3)(x + 1) = x^3 - 6x^2 + 5x + 12.$$ [3]
2. Sketch the curve $$y = x^3 - 6x^2 + 5x + 12,$$ giving the coordinates of the points where the curve meets the axes. Label the curve \(C_1\). [3]
3. On the same diagram as in part (ii), sketch the curve $$y = -x^3 + 6x^2 - 5x - 12.$$ Label this curve \(C_2\). [2]

Solve the inequalities

1. \(1 < 4x - 9 < 5\), [3]
2. \(y^2 \geq 4y + 5\). [5]

1. Solve the equation \(x^4 - 10x^2 + 25 = 0\). [4]
2. Given that \(y = \frac{2}{5}x^5 - \frac{20}{3}x^3 + 50x + 3\), find \(\frac{dy}{dx}\). [2]
3. Hence find the number of stationary points on the curve \(y = \frac{2}{5}x^5 - \frac{20}{3}x^3 + 50x + 3\). [2]

1. Solve the simultaneous equations $$y = x^2 - 5x + 4, \quad y = x - 1.$$ [4]
2. State the number of points of intersection of the curve \(y = x^2 - 5x + 4\) and the line \(y = x - 1\). [1]
3. Find the value of \(c\) for which the line \(y = x + c\) is a tangent to the curve \(y = x^2 - 5x + 4\). [4]

A cuboid has a volume of \(8 \text{m}^3\). The base of the cuboid is square with sides of length \(x\) metres. The surface area of the cuboid is \(A \text{m}^2\).

1. Show that \(A = 2x^2 + \frac{32}{x}\). [3]
2. Find \(\frac{dA}{dx}\). [3]
3. Find the value of \(x\) which gives the smallest surface area of the cuboid, justifying your answer. [4]

The points \(A\) and \(B\) have coordinates \((4, -2)\) and \((10, 6)\) respectively. \(C\) is the mid-point of \(AB\). Find

1. the coordinates of \(C\), [2]
2. the length of \(AC\), [2]
3. the equation of the circle that has \(AB\) as a diameter, [3]
4. the equation of the tangent to the circle in part (iii) at the point \(A\), giving your answer in the form \(ax + by = c\). [5]