OCR C1 (Core Mathematics 1)

Evaluate \(49^{\frac{1}{2}} + 8^{\frac{2}{3}}\). [3]

Find the set of values of \(x\) for which

1. \(6x - 11 > x + 4\), [2]
2. \(x^2 - 6x - 16 < 0\). [3]

1. Sketch on the same diagram the graphs of \(y = (x - 1)^2(x - 5)\) and \(y = 8 - 2x\). Label on your diagram the coordinates of any points where each graph meets the coordinate axes. [5]
2. Explain how your diagram shows that there is only one solution, \(\alpha\), to the equation $$(x - 1)^2(x - 5) = 8 - 2x.$$ [1]
3. State the integer, \(n\), such that $$n < \alpha < n + 1.$$ [1]

$$f(x) = x^2 - 10x + 17.$$

1. Express \(f(x)\) in the form \(a(x + b)^2 + c\). [3]
2. State the coordinates of the minimum point of the curve \(y = f(x)\). [1]
3. Deduce the coordinates of the minimum point of each of the following curves:
   1. \(y = f(x) + 4\), [2]
   2. \(y = f(2x)\). [2]

The points \(P\), \(Q\) and \(R\) have coordinates \((-5, 2)\), \((-3, 8)\) and \((9, 4)\) respectively.

1. Show that \(\angle PQR = 90°\). [4]

Given that \(P\), \(Q\) and \(R\) all lie on a circle,

1. find the coordinates of the centre of the circle, [3]
2. show that the equation of the circle can be written in the form $$x^2 + y^2 - 4x - 6y = k,$$ where \(k\) is an integer to be found. [3]

The straight line \(l_1\) has gradient \(\frac{3}{4}\) and passes through the point \(A (5, 3)\).

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

The straight line \(l_2\) has the equation \(3x - 4y + 3 = 0\) and intersects \(l_1\) at the point \(B\).

1. Find the coordinates of \(B\). [3]
2. Find the coordinates of the mid-point of \(AB\). [2]
3. Show that the straight line parallel to \(l_2\) which passes through the mid-point of \(AB\) also passes through the origin. [4]

\includegraphics{figure_8} The diagram shows the curve with equation \(y = 2 + 3x - x^2\) and the straight lines \(l\) and \(m\). The line \(l\) is the tangent to the curve at the point \(A\) where the curve crosses the \(y\)-axis.

1. Find an equation for \(l\). [5]

The line \(m\) is the normal to the curve at the point \(B\). Given that \(l\) and \(m\) are parallel,

1. find the coordinates of \(B\). [6]

The curve \(C\) has the equation $$y = 3 - x^{\frac{1}{2}} - 2x^{-\frac{1}{2}}, \quad x > 0.$$

1. Find the coordinates of the points where \(C\) crosses the \(x\)-axis. [4]
2. Find the exact coordinates of the stationary point of \(C\). [5]
3. Determine the nature of the stationary point. [2]
4. Sketch the curve \(C\). [2]