OCR C1 (Core Mathematics 1)

1. Express \(\frac{21}{\sqrt{7}}\) in the form \(k\sqrt{7}\). [2]
2. Express \(8^{-1}\) as an exact fraction in its simplest form. [2]

Find \(\frac{dy}{dx}\) when

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

1. Express \(x^2 - 10x + 27\) in the form \((x + p)^2 + q\). [3]
2. Sketch the curve with equation \(y = x^2 - 10x + 27\), showing on your sketch
   1. the coordinates of the vertex of the curve,
   2. the coordinates of any points where the curve meets the coordinate axes. [3]

The straight line \(l_1\) has gradient 2 and passes through the point with coordinates \((4, -5)\).

1. Find an equation for \(l_1\) in the form \(y = mx + c\). [2]

The straight line \(l_2\) is perpendicular to the line with equation \(3x - y = 4\) and passes through the point with coordinates \((3, 0)\).

1. Find an equation for \(l_2\). [3]
2. Find the coordinates of the point where \(l_1\) and \(l_2\) intersect. [3]

Given that the equation $$4x^2 - kx + k - 3 = 0,$$ where \(k\) is a constant, has real roots,

1. show that $$k^2 - 16k + 48 \geq 0, \quad [2]$$
2. find the set of possible values of \(k\), [3]
3. state the smallest value of \(k\) for which the roots are equal and solve the equation when \(k\) takes this value. [3]

The points \(P\) and \(Q\) have coordinates \((-2, 6)\) and \((4, -1)\) respectively. Given that \(PQ\) is a diameter of circle \(C\),

1. find the coordinates of the centre of \(C\), [2]
2. show that \(C\) has the equation $$x^2 + y^2 - 2x - 5y - 14 = 0. \quad [5]$$

The point \(R\) has coordinates \((2, 7)\).

1. Show that \(R\) lies on \(C\) and hence, state the size of \(\angle PRQ\) in degrees. [2]

1. Describe fully the single transformation that maps the graph of \(y = f(x)\) onto the graph of \(y = f(x - 1)\). [2]
2. Showing the coordinates of any points of intersection with the coordinate axes and the equations of any asymptotes, sketch the graph of \(y = \frac{1}{x-1}\). [3]
3. Find the \(x\)-coordinates of any points where the graph of \(y = \frac{1}{x-1}\) intersects the graph of \(y = 2 + \frac{1}{x}\). Give your answers in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are rational. [5]

\includegraphics{figure_8} The diagram shows the curve \(C\) with the equation \(y = x^3 + 3x^2 - 4x\) and the straight line \(l\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\). [3]

The line \(l\) is the tangent to \(C\) at \(O\).

1. Find an equation for \(l\). [4]
2. Find the coordinates of the point where \(l\) intersects \(C\) again. [4]

The curve with equation \(y = 2x^3 - 8x^{\frac{1}{3}}\) has a minimum at the point \(A\).

1. Find \(\frac{dy}{dx}\). [3]
2. Find the \(x\)-coordinate of \(A\). [3]

The point \(B\) on the curve has \(x\)-coordinate 2.

1. Find an equation for the tangent to the curve at \(B\) in the form \(y = mx + c\). [6]