OCR C1 (Core Mathematics 1)

\begin{enumerate} \item (i) Express \(\frac { 21 } { \sqrt { 7 } }\) in the form \(k \sqrt { 7 }\).
(ii) Express \(8 ^ { - \frac { 1 } { 3 } }\) as an exact fraction in its simplest form. \item Find \(\frac { d y } { d x }\) when

1. \(y = x - 2 x ^ { 2 }\),
2. \(y = \frac { 3 } { x ^ { 2 } }\). \item (a) Express \(x ^ { 2 } - 10 x + 27\) in the form \(( x + p ) ^ { 2 } + q\).
   (b) Sketch the curve with equation \(y = x ^ { 2 } - 10 x + 27\), showing on your sketch
3. the coordinates of the vertex of the curve,
4. the coordinates of any points where the curve meets the coordinate axes. \item The straight line \(l _ { 1 }\) has gradient 2 and passes through the point with coordinates \(( 4 , - 5 )\).
5. Find an equation for \(l _ { 1 }\) in the form \(y = m x + c\). \end{enumerate} The straight line \(l _ { 2 }\) is perpendicular to the line with equation \(3 x - y = 4\) and passes through the point with coordinates \(( 3,0 )\).
6. Find an equation for \(l _ { 2 }\).
7. Find the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect.

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

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

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

1. find the coordinates of the centre of \(C\),
2. show that \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 2 x - 5 y - 14 = 0$$ The point \(R\) has coordinates (2, 7).
3. Show that \(R\) lies on \(C\) and hence, state the size of \(\angle P R Q\) in degrees.

7. (i) Describe fully the single transformation that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { f } ( x - 1 )\).
(ii) Showing the coordinates of any points of intersection with the coordinate axes and the equations of any asymptotes, sketch the graph of \(y = \frac { 1 } { x - 1 }\).
(iii) 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.

8.
\includegraphics[max width=\textwidth, alt={}, center]{98667bd4-a612-4b16-a75b-8d8637e5976d-3_611_828_251_392} The diagram shows the curve \(C\) with the equation \(y = x ^ { 3 } + 3 x ^ { 2 } - 4 x\) 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 \(B\). The line \(l\) is the tangent to \(C\) at \(O\).
2. Find an equation for \(l\).
3. Find the coordinates of the point where \(l\) intersects \(C\) again.

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

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the \(x\)-coordinate of \(A\). The point \(B\) on the curve has \(x\)-coordinate 2 .
3. Find an equation for the tangent to the curve at \(B\) in the form \(y = m x + c\).