AQA C1 (Core Mathematics 1) 2009 June

1 The line \(A B\) has equation \(3 x + 5 y = 11\).

1. 1. Find the gradient of \(A B\).
   2. The point \(A\) has coordinates (2,1). Find an equation of the line which passes through the point \(A\) and which is perpendicular to \(A B\).
2. The line \(A B\) intersects the line with equation \(2 x + 3 y = 8\) at the point \(C\). Find the coordinates of \(C\).

2

1. Express \(\frac { 5 + \sqrt { 7 } } { 3 - \sqrt { 7 } }\) in the form \(m + n \sqrt { 7 }\), where \(m\) and \(n\) are integers.
2. The diagram shows a right-angled triangle. The hypotenuse has length \(2 \sqrt { 5 } \mathrm {~cm}\). The other two sides have lengths \(3 \sqrt { 2 } \mathrm {~cm}\) and \(x \mathrm {~cm}\). Find the value of \(x\).

3 The curve with equation \(y = x ^ { 5 } + 20 x ^ { 2 } - 8\) passes through the point \(P\), where \(x = - 2\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Verify that the point \(P\) is a stationary point of the curve.
3. 1. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(P\).
   2. Hence, or otherwise, determine whether \(P\) is a maximum point or a minimum point.
4. Find an equation of the tangent to the curve at the point where \(x = 1\).

4

1. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - x + 6\).
   1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
   2. Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
   3. Express \(\mathrm { p } ( x ) = x ^ { 3 } - x + 6\) in the form \(( x + 2 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
   4. The equation \(\mathrm { p } ( x ) = 0\) has one root equal to - 2 . Show that the equation has no other real roots.
2. The curve with equation \(y = x ^ { 3 } - x + 6\) is sketched below.
   \includegraphics[max width=\textwidth, alt={}, center]{5f1ff5fa-b6e8-4c4f-aef7-63eb947b299f-3_529_702_945_667} The curve cuts the \(x\)-axis at the point \(A ( - 2,0 )\) and the \(y\)-axis at the point \(B\).
   1. State the \(y\)-coordinate of the point \(B\).
   2. Find \(\int _ { - 2 } ^ { 0 } \left( x ^ { 3 } - x + 6 \right) \mathrm { d } x\).
   3. Hence find the area of the shaded region bounded by the curve \(y = x ^ { 3 } - x + 6\) and the line \(A B\).

5 A circle with centre \(C\) has equation $$( x - 5 ) ^ { 2 } + ( y + 12 ) ^ { 2 } = 169$$

1. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle.
2. 1. Verify that the circle passes through the origin \(O\).
   2. Given that the circle also passes through the points \(( 10,0 )\) and \(( 0 , p )\), sketch the circle and find the value of \(p\).
3. The point \(A ( - 7 , - 7 )\) lies on the circle.
   1. Find the gradient of \(A C\).
   2. Hence find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

6

1. 1. Express \(x ^ { 2 } - 8 x + 17\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Hence write down the minimum value of \(x ^ { 2 } - 8 x + 17\).
   3. State the value of \(x\) for which the minimum value of \(x ^ { 2 } - 8 x + 17\) occurs.
      (1 mark)
2. The point \(A\) has coordinates (5,4) and the point \(B\) has coordinates ( \(x , 7 - x\) ).
   1. Expand \(( x - 5 ) ^ { 2 }\).
   2. Show that \(A B ^ { 2 } = 2 \left( x ^ { 2 } - 8 x + 17 \right)\).
   3. Use your results from part (a) to find the minimum value of the distance \(A B\) as \(x\) varies.

7 The curve \(C\) has equation \(y = k \left( x ^ { 2 } + 3 \right)\), where \(k\) is a constant.
The line \(L\) has equation \(y = 2 x + 2\).

1. Show that the \(x\)-coordinates of any points of intersection of the curve \(C\) with the line \(L\) satisfy the equation $$k x ^ { 2 } - 2 x + 3 k - 2 = 0$$
2. The curve \(C\) and the line \(L\) intersect in two distinct points.
   1. Show that $$3 k ^ { 2 } - 2 k - 1 < 0$$
   2. Hence find the possible values of \(k\).