AQA C1 (Core Mathematics 1) 2007 January

1 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 7 x + k$$ where \(k\) is a constant.

1. 1. Given that \(x + 2\) is a factor of \(\mathrm { p } ( x )\), show that \(k = 10\).
   2. Express \(\mathrm { p } ( x )\) as the product of three linear factors.
2. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
3. Sketch the curve with equation \(y = x ^ { 3 } - 4 x ^ { 2 } - 7 x + 10\), indicating the values where the curve crosses the \(x\)-axis and the \(y\)-axis. (You are not required to find the coordinates of the stationary points.)

2 The line \(A B\) has equation \(3 x + 5 y = 8\) and the point \(A\) has coordinates (6, -2).

1. 1. Find the gradient of \(A B\).
   2. Hence find an equation of the straight line which is perpendicular to \(A B\) and which passes through \(A\).
2. The line \(A B\) intersects the line with equation \(2 x + 3 y = 3\) at the point \(B\). Find the coordinates of \(B\).
3. The point \(C\) has coordinates \(( 2 , k )\) and the distance from \(A\) to \(C\) is 5 . Find the two possible values of the constant \(k\).

3

1. Express \(\frac { \sqrt { 5 } + 3 } { \sqrt { 5 } - 2 }\) in the form \(p \sqrt { 5 } + q\), where \(p\) and \(q\) are integers.
2. 1. Express \(\sqrt { 45 }\) in the form \(n \sqrt { 5 }\), where \(n\) is an integer.
   2. Solve the equation $$x \sqrt { 20 } = 7 \sqrt { 5 } - \sqrt { 45 }$$ giving your answer in its simplest form.

4 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 12 y + 12 = 0\).

1. By completing the square, express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle.
3. Show that the circle does not intersect the \(x\)-axis.
4. The line with equation \(x + y = 4\) intersects the circle at the points \(P\) and \(Q\).
   1. Show that the \(x\)-coordinates of \(P\) and \(Q\) satisfy the equation $$x ^ { 2 } + 3 x - 10 = 0$$
   2. Given that \(P\) has coordinates (2,2), find the coordinates of \(Q\).
   3. Hence find the coordinates of the midpoint of \(P Q\).

5 The diagram shows an open-topped water tank with a horizontal rectangular base and four vertical faces. The base has width \(x\) metres and length \(2 x\) metres, and the height of the tank is \(h\) metres.
\includegraphics[max width=\textwidth, alt={}, center]{33da89e2-f74f-4d5a-8bbd-ceaa728b6c34-4_403_410_477_792} The combined internal surface area of the base and four vertical faces is \(54 \mathrm {~m} ^ { 2 }\).

1. 1. Show that \(x ^ { 2 } + 3 x h = 27\).
   2. Hence express \(h\) in terms of \(x\).
   3. Hence show that the volume of water, \(V \mathrm {~m} ^ { 3 }\), that the tank can hold when full is given by $$V = 18 x - \frac { 2 x ^ { 3 } } { 3 }$$
2. 1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
   2. Verify that \(V\) has a stationary value when \(x = 3\).
3. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) and hence determine whether \(V\) has a maximum value or a minimum value when \(x = 3\).
   (2 marks)

6 The curve with equation \(y = 3 x ^ { 5 } + 2 x + 5\) is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{33da89e2-f74f-4d5a-8bbd-ceaa728b6c34-5_428_563_372_740} The curve cuts the \(x\)-axis at the point \(A ( - 1,0 )\) and cuts the \(y\)-axis at the point \(B\).

1. 1. State the coordinates of the point \(B\) and hence find the area of the triangle \(A O B\), where \(O\) is the origin.
   2. Find \(\int \left( 3 x ^ { 5 } + 2 x + 5 \right) \mathrm { d } x\).
   3. Hence find the area of the shaded region bounded by the curve and the line \(A B\).
2. 1. Find the gradient of the curve with equation \(y = 3 x ^ { 5 } + 2 x + 5\) at the point \(A ( - 1,0 )\).
   2. Hence find an equation of the tangent to the curve at the point \(A\).

7 The quadratic equation \(( k + 1 ) x ^ { 2 } + 12 x + ( k - 4 ) = 0\) has real roots.

1. Show that \(k ^ { 2 } - 3 k - 40 \leqslant 0\).
2. Hence find the possible values of \(k\).