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AQA C1 2011 January Q4
12 marks Moderate -0.8
4 The curve sketched below passes through the point \(A ( - 2,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{889639d6-0a31-4569-8370-1e72291a0c47-3_538_734_365_662} The curve has equation \(y = 14 - x - x ^ { 4 }\) and the point \(P ( 1,12 )\) lies on the curve.
    1. Find the gradient of the curve at the point \(P\).
    2. Hence find the equation of the tangent to the curve at the point \(P\), giving your answer in the form \(y = m x + c\).
    1. Find \(\int _ { - 2 } ^ { 1 } \left( 14 - x - x ^ { 4 } \right) \mathrm { d } x\).
    2. Hence find the area of the shaded region bounded by the curve \(y = 14 - x - x ^ { 4 }\) and the line \(A P\).
      (2 marks)
AQA C1 2011 January Q5
13 marks Moderate -0.8
5
    1. Sketch the curve with equation \(y = x ( x - 2 ) ^ { 2 }\).
    2. Show that the equation \(x ( x - 2 ) ^ { 2 } = 3\) can be expressed as $$x ^ { 3 } - 4 x ^ { 2 } + 4 x - 3 = 0$$
  2. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 4 x - 3\).
    1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x + 1\).
    2. Use the Factor Theorem to show that \(x - 3\) is a factor of \(\mathrm { p } ( x )\).
    3. Express \(\mathrm { p } ( x )\) in the form \(( x - 3 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
  3. Hence show that the equation \(x ( x - 2 ) ^ { 2 } = 3\) has only one real root and state the value of this root.
AQA C1 2011 January Q6
13 marks Easy -1.2
6 A circle has centre \(C ( - 3,1 )\) and radius \(\sqrt { 13 }\).
    1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
    2. Hence find the equation of the circle in the form $$x ^ { 2 } + y ^ { 2 } + m x + n y + p = 0$$ where \(m , n\) and \(p\) are integers.
  2. The circle cuts the \(y\)-axis at the points \(A\) and \(B\). Find the distance \(A B\).
    1. Verify that the point \(D ( - 5 , - 2 )\) lies on the circle.
    2. Find the gradient of \(C D\).
    3. Hence find an equation of the tangent to the circle at the point \(D\).
AQA C1 2011 January Q7
11 marks Standard +0.3
7
    1. Express \(4 - 10 x - x ^ { 2 }\) in the form \(p - ( x + q ) ^ { 2 }\).
    2. Hence write down the equation of the line of symmetry of the curve with equation \(y = 4 - 10 x - x ^ { 2 }\).
  2. The curve \(C\) has equation \(y = 4 - 10 x - x ^ { 2 }\) and the line \(L\) has equation \(y = k ( 4 x - 13 )\), where \(k\) is a constant.
    1. Show that the \(x\)-coordinates of any points of intersection of the curve \(C\) with the line \(L\) satisfy the equation $$x ^ { 2 } + 2 ( 2 k + 5 ) x - ( 13 k + 4 ) = 0$$
    2. Given that the curve \(C\) and the line \(L\) intersect in two distinct points, show that $$4 k ^ { 2 } + 33 k + 29 > 0$$
    3. Solve the inequality \(4 k ^ { 2 } + 33 k + 29 > 0\).
AQA C1 2012 January Q1
11 marks Moderate -0.8
1 The point \(A\) has coordinates (6, -4) and the point \(B\) has coordinates (-2, 7).
  1. Given that the point \(O\) has coordinates \(( 0,0 )\), show that the length of \(O A\) is less than the length of \(O B\).
    1. Find the gradient of \(A B\).
    2. Find an equation of the line \(A B\) in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers.
  3. The point \(C\) has coordinates \(( k , 0 )\). The line \(A C\) is perpendicular to the line \(A B\). Find the value of the constant \(k\).
AQA C1 2012 January Q2
10 marks Moderate -0.8
2
  1. Factorise \(x ^ { 2 } - 4 x - 12\).
  2. Sketch the graph with equation \(y = x ^ { 2 } - 4 x - 12\), stating the values where the curve crosses the coordinate axes.
    1. Express \(x ^ { 2 } - 4 x - 12\) in the form \(( x - p ) ^ { 2 } - q\), where \(p\) and \(q\) are positive integers.
    2. Hence find the minimum value of \(x ^ { 2 } - 4 x - 12\).
  4. The curve with equation \(y = x ^ { 2 } - 4 x - 12\) is translated by the vector \(\left[ \begin{array} { r } - 3 \\ 2 \end{array} \right]\). Find an equation of the new curve. You need not simplify your answer.
AQA C1 2012 January Q3
9 marks Easy -1.2
3
    1. Simplify \(( 3 \sqrt { 2 } ) ^ { 2 }\).
    2. Show that \(( 3 \sqrt { 2 } - 1 ) ^ { 2 } + ( 3 + \sqrt { 2 } ) ^ { 2 }\) is an integer and find its value.
  2. Express \(\frac { 4 \sqrt { 5 } - 7 \sqrt { 2 } } { 2 \sqrt { 5 } + \sqrt { 2 } }\) in the form \(m - \sqrt { n }\), where \(m\) and \(n\) are integers.
AQA C1 2012 January Q4
16 marks Moderate -0.8
4 The curve with equation \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\) is sketched below. The point \(O\) is at the origin and the curve passes through the points \(A ( - 1,0 )\) and \(B ( 1,4 )\). \includegraphics[max width=\textwidth, alt={}, center]{91170a77-e266-4c81-89ee-1fc29a538485-3_447_752_438_653}
  1. Given that \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\), find:
    1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\);
    2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  2. Find an equation of the tangent to the curve at the point \(A ( - 1,0 )\).
  3. Verify that the point \(B\), where \(x = 1\), is a minimum point of the curve.
  4. The curve with equation \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\) is sketched below. The point \(O\) is at the origin and the curve passes through the points \(A ( - 1,0 )\) and \(B ( 1,4 )\). \includegraphics[max width=\textwidth, alt={}, center]{91170a77-e266-4c81-89ee-1fc29a538485-3_451_757_1736_648}
    1. Find \(\int _ { - 1 } ^ { 1 } \left( x ^ { 5 } - 3 x ^ { 2 } + x + 5 \right) \mathrm { d } x\).
    2. Hence find the area of the shaded region bounded by the curve between \(A\) and \(B\) and the line segments \(A O\) and \(O B\).
AQA C1 2012 January Q5
8 marks Moderate -0.8
5 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } + c x ^ { 2 } + d x - 12\), where \(c\) and \(d\) are constants.
  1. When \(\mathrm { p } ( x )\) is divided by \(x + 2\), the remainder is - 150 . Show that \(2 c - d + 65 = 0\).
  2. Given that \(x - 3\) is a factor of \(\mathrm { p } ( x )\), find another equation involving \(c\) and \(d\).
  3. By solving these two equations, find the value of \(c\) and the value of \(d\).
AQA C1 2012 January Q6
7 marks Moderate -0.8
6 A rectangular garden is to have width \(x\) metres and length \(( x + 4 )\) metres.
  1. The perimeter of the garden needs to be greater than 30 metres. Show that \(2 x > 11\).
  2. The area of the garden needs to be less than 96 square metres. Show that \(x ^ { 2 } + 4 x - 96 < 0\).
  3. Solve the inequality \(x ^ { 2 } + 4 x - 96 < 0\).
  4. Hence determine the possible values of the width of the garden. \(7 \quad\) A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 14 x - 10 y + 49 = 0\).
    1. 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. Sketch the circle.
      4. A line has equation \(y = k x + 6\), where \(k\) is a constant.
        1. Show that the \(x\)-coordinates of any points of intersection of the line and the circle satisfy the equation \(\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0\).
        2. The equation \(\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0\) has equal roots. Show that $$12 k ^ { 2 } - 7 k - 12 = 0$$
        3. Hence find the values of \(k\) for which the line is a tangent to the circle.
AQA C1 2013 January Q1
11 marks Moderate -0.8
1 The point \(A\) has coordinates \(( - 3,2 )\) and the point \(B\) has coordinates \(( 7 , k )\).
The line \(A B\) has equation \(3 x + 5 y = 1\).
    1. Show that \(k = - 4\).
    2. Hence find the coordinates of the midpoint of \(A B\).
  2. Find the gradient of \(A B\).
  3. A line which passes through the point \(A\) is perpendicular to the line \(A B\). Find an equation of this line, giving your answer in the form \(p x + q y + r = 0\), where \(p , q\) and \(r\) are integers.
  4. The line \(A B\), with equation \(3 x + 5 y = 1\), intersects the line \(5 x + 8 y = 4\) at the point \(C\). Find the coordinates of \(C\).
AQA C1 2013 January Q2
8 marks Moderate -0.8
2 A bird flies from a tree. At time \(t\) seconds, the bird's height, \(y\) metres, above the horizontal ground is given by $$y = \frac { 1 } { 8 } t ^ { 4 } - t ^ { 2 } + 5 , \quad 0 \leqslant t \leqslant 4$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} t }\).
    1. Find the rate of change of height of the bird in metres per second when \(t = 1\).
    2. Determine, with a reason, whether the bird's height above the horizontal ground is increasing or decreasing when \(t = 1\).
    1. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\) when \(t = 2\).
    2. Given that \(y\) has a stationary value when \(t = 2\), state whether this is a maximum value or a minimum value.
AQA C1 2013 January Q3
8 marks Easy -1.3
3
    1. Express \(\sqrt { 18 }\) in the form \(k \sqrt { 2 }\), where \(k\) is an integer.
    2. Simplify \(\frac { \sqrt { 8 } } { \sqrt { 18 } + \sqrt { 32 } }\).
  2. Express \(\frac { 7 \sqrt { 2 } - \sqrt { 3 } } { 2 \sqrt { 2 } - \sqrt { 3 } }\) in the form \(m + \sqrt { n }\), where \(m\) and \(n\) are integers.
AQA C1 2013 January Q4
12 marks Moderate -0.8
4
    1. Express \(x ^ { 2 } - 6 x + 11\) in the form \(( x - p ) ^ { 2 } + q\).
    2. Use the result from part (a)(i) to show that the equation \(x ^ { 2 } - 6 x + 11 = 0\) has no real solutions.
  2. A curve has equation \(y = x ^ { 2 } - 6 x + 11\).
    1. Find the coordinates of the vertex of the curve.
    2. Sketch the curve, indicating the value of \(y\) where the curve crosses the \(y\)-axis.
    3. Describe the geometrical transformation that maps the curve with equation \(y = x ^ { 2 } - 6 x + 11\) onto the curve with equation \(y = x ^ { 2 }\).
AQA C1 2013 January Q5
10 marks Moderate -0.8
5 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18$$
  1. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x + 1\).
    1. Use the Factor Theorem to show that \(x - 3\) is a factor of \(\mathrm { p } ( x )\).
    2. Express \(\mathrm { p } ( x )\) as a product of linear factors.
  3. Sketch the curve with equation \(y = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18\), stating the values of \(x\) where the curve meets the \(x\)-axis.
AQA C1 2013 January Q6
8 marks Moderate -0.5
6 The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), of a curve at the point \(( x , y )\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 10 x ^ { 4 } - 6 x ^ { 2 } + 5$$ The curve passes through the point \(P ( 1,4 )\).
  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. Find the equation of the curve.
AQA C1 2013 January Q7
10 marks Standard +0.3
7 A circle with centre \(C ( - 3,2 )\) has equation $$x ^ { 2 } + y ^ { 2 } + 6 x - 4 y = 12$$
  1. Find the \(y\)-coordinates of the points where the circle crosses the \(y\)-axis.
  2. Find the radius of the circle.
  3. The point \(P ( 2,5 )\) lies outside the circle.
    1. Find the length of \(C P\), giving your answer in the form \(\sqrt { n }\), where \(n\) is an integer.
    2. The point \(Q\) lies on the circle so that \(P Q\) is a tangent to the circle. Find the length of \(P Q\).
AQA C1 2013 January Q8
8 marks Moderate -0.3
8 A curve has equation \(y = 2 x ^ { 2 } - x - 1\) and a line has equation \(y = k ( 2 x - 3 )\), where \(k\) is a constant.
  1. Show that the \(x\)-coordinate of any point of intersection of the curve and the line satisfies the equation $$2 x ^ { 2 } - ( 2 k + 1 ) x + 3 k - 1 = 0$$
  2. The curve and the line intersect at two distinct points.
    1. Show that \(4 k ^ { 2 } - 20 k + 9 > 0\).
    2. Find the possible values of \(k\).
AQA C1 2005 June Q1
12 marks Easy -1.2
1 The point \(A\) has coordinates \(( 6,5 )\) and the point \(B\) has coordinates \(( 2 , - 1 )\).
  1. Find the coordinates of the midpoint of \(A B\).
  2. Show that \(A B\) has length \(k \sqrt { 13 }\), where \(k\) is an integer.
    1. Find the gradient of the line \(A B\).
    2. Hence, or otherwise, show that the line \(A B\) has equation \(3 x - 2 y = 8\).
  4. The line \(A B\) intersects the line with equation \(2 x + y = 10\) at the point \(C\). Find the coordinates of \(C\).
AQA C1 2005 June Q2
10 marks Easy -1.2
2
  1. Express \(x ^ { 2 } - 6 x + 16\) in the form \(( x - p ) ^ { 2 } + q\).
  2. A curve has equation \(y = x ^ { 2 } - 6 x + 16\). Using your answer from part (a), or otherwise:
    1. find the coordinates of the vertex (minimum point) of the curve;
    2. sketch the curve, indicating the value where the curve crosses the \(y\)-axis;
    3. state the equation of the line of symmetry of the curve.
  3. Describe geometrically the transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } - 6 x + 16\).
AQA C1 2005 June Q3
10 marks Moderate -0.8
3 A circle has centre \(C ( 2 , - 1 )\) and radius 5 . The point \(P\) has coordinates \(( 6,2 )\).
  1. Write down an equation of the circle.
  2. Verify that the point \(P\) lies on the circle.
  3. Find the gradient of the line \(C P\).
    1. Find the gradient of a line which is perpendicular to \(C P\).
    2. Hence find an equation for the tangent to the circle at the point \(P\).
AQA C1 2005 June Q4
15 marks Moderate -0.3
4 The curve with equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{3729de55-7139-4f41-8584-640f173c0e09-3_444_588_411_717} The curve touches the \(x\)-axis at the point \(A ( 1,0 )\) and cuts the \(x\)-axis at the point \(B\).
    1. Use the factor theorem to show that \(x - 3\) is a factor of $$\mathrm { p } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3$$
    2. Hence find the coordinates of \(B\).
  2. The point \(M\), shown on the diagram, is a minimum point of the curve with equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3\).
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Hence determine the \(x\)-coordinate of \(M\).
  3. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 1\).
    1. Find \(\int \left( x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3 \right) \mathrm { d } x\).
    2. Hence determine the area of the shaded region bounded by the curve and the coordinate axes.
AQA C1 2005 June Q5
5 marks Easy -1.2
5 Express each of the following in the form \(m + n \sqrt { 3 }\), where \(m\) and \(n\) are integers:
  1. \(( \sqrt { 3 } + 1 ) ^ { 2 }\);
  2. \(\frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 }\).
AQA C1 2005 June Q6
7 marks Moderate -0.8
6 The cubic polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = ( x - 2 ) \left( x ^ { 2 } + x + 3 \right)\).
  1. Show that \(\mathrm { p } ( x )\) can be written in the form \(x ^ { 3 } + a x ^ { 2 } + b x - 6\), where \(a\) and \(b\) are constants whose values are to be found.
  2. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x + 1\).
    (2 marks)
  3. Prove that the equation \(( x - 2 ) \left( x ^ { 2 } + x + 3 \right) = 0\) has only one real root and state its value.
    (3 marks)
AQA C1 2005 June Q7
7 marks Easy -1.2
7 Solve each of the following inequalities:
  1. \(3 ( x - 1 ) > 3 - 5 ( x + 6 )\);
  2. \(\quad x ^ { 2 } - x - 6 < 0\).