AQA C1 (Core Mathematics 1) 2013 June

1 The line \(A B\) has equation \(3 x - 4 y + 5 = 0\).

1. The point with coordinates \(( p , p + 2 )\) lies on the line \(A B\). Find the value of the constant \(p\).
2. Find the gradient of \(A B\).
3. The point \(A\) has coordinates ( 1,2 ). The point \(C ( - 5 , k )\) is such that \(A C\) is perpendicular to \(A B\). Find the value of \(k\).
4. The line \(A B\) intersects the line with equation \(2 x - 5 y = 6\) at the point \(D\). Find the coordinates of \(D\).

2

1. 1. Express \(\sqrt { 48 }\) in the form \(n \sqrt { 3 }\), where \(n\) is an integer.
   2. Solve the equation $$x \sqrt { 12 } = 7 \sqrt { 3 } - \sqrt { 48 }$$ giving your answer in its simplest form.
2. Express \(\frac { 11 \sqrt { 3 } + 2 \sqrt { 5 } } { 2 \sqrt { 3 } + \sqrt { 5 } }\) in the form \(m - \sqrt { 15 }\), where \(m\) is an integer.

3 A circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 10 x + 14 y + 25 = 0$$

1. Write the equation of \(C\) in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$ where \(a , b\) and \(k\) are integers.
2. Hence, for the circle \(C\), write down:
   1. the coordinates of its centre;
   2. its radius.
3. 1. Sketch the circle \(C\).
   2. Write down the coordinates of the point on \(C\) that is furthest away from the \(x\)-axis.
4. Given that \(k\) has the same value as in part (a), describe geometrically the transformation which maps the circle with equation \(( x + 1 ) ^ { 2 } + y ^ { 2 } = k\) onto the circle \(C\).

4

1. The polynomial \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 3 } - 4 x + 15\).
   1. Use the Factor Theorem to show that \(x + 3\) is a factor of \(\mathrm { f } ( x )\).
   2. Express \(\mathrm { f } ( x )\) in the form \(( x + 3 ) \left( x ^ { 2 } + p x + q \right)\), where \(p\) and \(q\) are integers.
2. A curve has equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 60 x + 7\).
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Show that the \(x\)-coordinates of any stationary points of the curve satisfy the equation $$x ^ { 3 } - 4 x + 15 = 0$$
   3. Use the results above to show that the only stationary point of the curve occurs when \(x = - 3\).
   4. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = - 3\).
   5. Hence determine, with a reason, whether the curve has a maximum point or a minimum point when \(x = - 3\).

5

1. 1. Express \(2 x ^ { 2 } + 6 x + 5\) in the form \(2 ( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
   2. Hence write down the minimum value of \(2 x ^ { 2 } + 6 x + 5\).
2. The point \(A\) has coordinates \(( - 3,5 )\) and the point \(B\) has coordinates \(( x , 3 x + 9 )\).
   1. Show that \(A B ^ { 2 } = 5 \left( 2 x ^ { 2 } + 6 x + 5 \right)\).
   2. Use your result from part (a)(ii) to find the minimum value of the length \(A B\) as \(x\) varies, giving your answer in the form \(\frac { 1 } { 2 } \sqrt { n }\), where \(n\) is an integer.

6 A curve has equation \(y = x ^ { 5 } - 2 x ^ { 2 } + 9\). The point \(P\) with coordinates \(( - 1,6 )\) lies on the curve.

1. Find the equation of the tangent to the curve at the point \(P\), giving your answer in the form \(y = m x + c\).
2. The point \(Q\) with coordinates \(( 2 , k )\) lies on the curve.
   1. Find the value of \(k\).
   2. Verify that \(Q\) also lies on the tangent to the curve at the point \(P\).
3. The curve and the tangent to the curve at \(P\) are sketched below.
   \includegraphics[max width=\textwidth, alt={}, center]{aa42b4fd-1e37-48b8-90ee-269916c4db2c-4_721_887_936_589}
   1. Find \(\int _ { - 1 } ^ { 2 } \left( x ^ { 5 } - 2 x ^ { 2 } + 9 \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve and the tangent to the curve at \(P\).
      (3 marks)

7 The quadratic equation $$( 2 k - 7 ) x ^ { 2 } - ( k - 2 ) x + ( k - 3 ) = 0$$ has real roots.

1. Show that \(7 k ^ { 2 } - 48 k + 80 \leqslant 0\).
2. Find the possible values of \(k\).