OCR C1 (Core Mathematics 1) 2013 January

1. Solve the equation \(x^2 - 6x - 2 = 0\), giving your answers in simplified surd form. [3]
2. Find the gradient of the curve \(y = x^2 - 6x - 2\) at the point where \(x = -5\). [2]

Solve the equations

1. \(3^n = 1\), [1]
2. \(t^{-3} = 64\), [2]
3. \((8p^6)^{\frac{1}{3}} = 8\). [3]

1. Sketch the curve \(y = (1 + x)(2 - x)(3 + x)\), giving the coordinates of all points of intersection with the axes. [3]
2. Describe the transformation that transforms the curve \(y = (1 + x)(2 - x)(3 + x)\) to the curve \(y = (1 - x)(2 + x)(3 - x)\). [2]

1. Solve the simultaneous equations $$y = 2x^2 - 3x - 5, \quad 10x + 2y + 11 = 0.$$ [5]
2. What can you deduce from the answer to part (i) about the curve \(y = 2x^2 - 3x - 5\) and the line \(10x + 2y + 11 = 0\)? [1]

1. Simplify \((x + 4)(5x - 3) - 3(x - 2)^2\). [3]
2. The coefficient of \(x^2\) in the expansion of $$(x + 3)(x + k)(2x - 5)$$ is \(-3\). Find the value of the constant \(k\). [3]

1. The line joining the points \((-2, 7)\) and \((-4, p)\) has gradient 4. Find the value of \(p\). [3]
2. The line segment joining the points \((-2, 7)\) and \((6, q)\) has mid-point \((m, 5)\). Find \(m\) and \(q\). [3]
3. The line segment joining the points \((-2, 7)\) and \((d, 3)\) has length \(2\sqrt{13}\). Find the two possible values of \(d\). [4]

Find \(\frac{dy}{dx}\) in each of the following cases:

1. \(y = \frac{(3x)^2 \times x^4}{x}\), [3]
2. \(y = ^3\sqrt{x}\), [3]
3. \(y = \frac{1}{2x^3}\). [2]

The quadratic equation \(kx^2 + (3k - 1)x - 4 = 0\) has no real roots. Find the set of possible values of \(k\). [7]

A circle with centre \(C\) has equation \(x^2 + y^2 - 2x + 10y - 19 = 0\).

1. Find the coordinates of \(C\) and the radius of the circle. [3]
2. Verify that the point \((7, -2)\) lies on the circumference of the circle. [1]
3. Find the equation of the tangent to the circle at the point \((7, -2)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]

Find the coordinates of the points on the curve \(y = \frac{1}{3}x^3 + \frac{9}{x}\) at which the tangent is parallel to the line \(y = 8x + 3\). [10]