OCR C1 (Core Mathematics 1)

Find the value of \(y\) such that $$4^{y+1} = 8^{2y-1}.$$ [4]

Express \(\sqrt{22.5}\) in the form \(k\sqrt{10}\). [4]

A circle has the equation $$x^2 + y^2 + 8x - 4y + k = 0,$$ where \(k\) is a constant.

1. Find the coordinates of the centre of the circle. [2]

Given that the \(x\)-axis is a tangent to the circle,

1. Find the value of \(k\). [3]

$$\text{f}(x) = 4x - 3x^2 - x^3.$$

1. Fully factorise \(4x - 3x^2 - x^3\). [3]
2. Sketch the curve \(y = \text{f}(x)\), showing the coordinates of any points of intersection with the coordinate axes. [3]

1. Find in exact form the coordinates of the points where the curve \(y = x^2 - 4x + 2\) crosses the \(x\)-axis. [4]
2. Find the value of the constant \(k\) for which the straight line \(y = 2x + k\) is a tangent to the curve \(y = x^2 - 4x + 2\). [4]

Some ink is poured onto a piece of cloth forming a stain that then spreads. The area of the stain, \(A\) cm\(^2\), after \(t\) seconds is given by $$A = (p + qt)^2,$$ where \(p\) and \(q\) are positive constants. Given that when \(t = 0\), \(A = 4\) and that when \(t = 5\), \(A = 9\),

1. find the value of \(p\) and show that \(q = \frac{1}{5}\), [5]
2. find \(\frac{\mathrm{d}A}{\mathrm{d}t}\) in terms of \(t\), [3]
3. find the rate at which the area of the stain is increasing when \(t = 15\). [2]

The curve \(C\) has the equation \(y = x^2 + 2x + 4\).

1. Express \(x^2 + 2x + 4\) in the form \((x + p)^2 + q\) and hence state the coordinates of the minimum point of \(C\). [4]

The straight line \(l\) has the equation \(x + y = 8\).

1. Sketch \(l\) and \(C\) on the same set of axes. [3]
2. Find the coordinates of the points where \(l\) and \(C\) intersect. [4]

$$\text{f}(x) \equiv \frac{(x-4)^2}{2x^{\frac{1}{2}}}, \quad x > 0.$$

1. Find the values of the constants \(A\), \(B\) and \(C\) such that $$\text{f}(x) = Ax^{\frac{3}{2}} + Bx^{\frac{1}{2}} + Cx^{-\frac{1}{2}}.$$ [3]
2. Show that $$\text{f}'(x) = \frac{3x^2 - 8x - 16}{4x^{\frac{3}{2}}}.$$ [5]
3. Find the coordinates of the stationary point of the curve \(y = \text{f}(x)\). [3]

\includegraphics{figure_9} The diagram shows the parallelogram \(ABCD\). The points \(A\) and \(B\) have coordinates \((-1, 3)\) and \((3, 4)\) respectively and lie on the straight line \(l_1\).

1. Find an equation for \(l_1\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]

The points \(C\) and \(D\) lie on the straight line \(l_2\) which has the equation \(x - 4y - 21 = 0\).

1. Show that the distance between \(l_1\) and \(l_2\) is \(k\sqrt{17}\), where \(k\) is an integer to be found. [7]
2. Find the area of parallelogram \(ABCD\). [2]