Questions — AQA C1 (156 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
AQA C1 2011 June Q2
2
    1. Express \(\sqrt { 48 }\) in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
    2. Simplify \(\frac { \sqrt { 48 } + 2 \sqrt { 27 } } { \sqrt { 12 } }\), giving your answer as an integer.
  2. Express \(\frac { 1 - 5 \sqrt { 5 } } { 3 + \sqrt { 5 } }\) in the form \(m + n \sqrt { 5 }\), where \(m\) and \(n\) are integers.
AQA C1 2011 June Q3
3 The volume, \(V \mathrm {~m} ^ { 3 }\), of water in a tank after time \(t\) seconds is given by $$V = \frac { t ^ { 3 } } { 4 } - 3 t + 5$$
  1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} t }\).
    1. Find the rate of change of volume, in \(\mathrm { m } ^ { 3 } \mathrm {~s} ^ { - 1 }\), when \(t = 1\).
    2. Hence determine, with a reason, whether the volume is increasing or decreasing when \(t = 1\).
    1. Find the positive value of \(t\) for which \(V\) has a stationary value.
    2. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} t ^ { 2 } }\), and hence determine whether this stationary value is a maximum value or a minimum value.
      (3 marks)
AQA C1 2011 June Q4
4
  1. Express \(x ^ { 2 } + 5 x + 7\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
    (3 marks)
  2. A curve has equation \(y = x ^ { 2 } + 5 x + 7\).
    1. Find the coordinates of the vertex of the curve.
    2. State the equation of the line of symmetry of the curve.
    3. Sketch the curve, stating the value of the intercept on the \(y\)-axis.
  3. Describe the geometrical transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 5 x + 7\).
AQA C1 2011 June Q5
5 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + 3\).
  1. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
  2. Use the Factor Theorem to show that \(x + 1\) is a factor of \(\mathrm { p } ( x )\).
    1. Express \(\mathrm { p } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + 3\) in the form \(( x + 1 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
    2. Hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.
AQA C1 2011 June Q6
6 The curve with equation \(y = x ^ { 3 } - 2 x ^ { 2 } + 3\) is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{c44c229e-44b2-4799-9c9c-bfccdd09d450-4_590_787_365_625} The curve cuts the \(x\)-axis at the point \(A ( - 1,0 )\) and passes through the point \(B ( 1,2 )\).
  1. Find \(\int _ { - 1 } ^ { 1 } \left( x ^ { 3 } - 2 x ^ { 2 } + 3 \right) \mathrm { d } x\).
    (5 marks)
  2. Hence find the area of the shaded region bounded by the curve \(y = x ^ { 3 } - 2 x ^ { 2 } + 3\) and the line \(A B\).
    (3 marks)
AQA C1 2011 June Q7
7 Solve each of the following inequalities:
  1. \(\quad 2 ( 4 - 3 x ) > 5 - 4 ( x + 2 )\);
  2. \(\quad 2 x ^ { 2 } + 5 x \geqslant 12\).
AQA C1 2011 June Q8
8 A circle has centre \(C ( 3 , - 8 )\) and radius 10 .
  1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
  2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis.
  3. The tangent to the circle at the point \(A\) has gradient \(\frac { 5 } { 2 }\). Find an equation of the line \(C A\), giving your answer in the form \(r x + s y + t = 0\), where \(r , s\) and \(t\) are integers.
  4. The line with equation \(y = 2 x + 1\) intersects the circle.
    1. Show that the \(x\)-coordinates of the points of intersection satisfy the equation $$x ^ { 2 } + 6 x - 2 = 0$$
    2. Hence show that the \(x\)-coordinates of the points of intersection are of the form \(m \pm \sqrt { n }\), where \(m\) and \(n\) are integers.
AQA C1 2012 June Q1
1 Express \(\frac { 5 \sqrt { 3 } - 6 } { 2 \sqrt { 3 } + 3 }\) in the form \(m + n \sqrt { 3 }\), where \(m\) and \(n\) are integers.
(4 marks)
AQA C1 2012 June Q2
2 The line \(A B\) has equation \(4 x - 3 y = 7\).
    1. Find the gradient of \(A B\).
    2. Find an equation of the straight line that is parallel to \(A B\) and which passes through the point \(C ( 3 , - 5 )\), giving your answer in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers.
  2. The line \(A B\) intersects the line with equation \(3 x - 2 y = 4\) at the point \(D\). Find the coordinates of \(D\).
  3. The point \(E\) with coordinates \(( k - 2,2 k - 3 )\) lies on the line \(A B\). Find the value of the constant \(k\).
AQA C1 2012 June Q3
3 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 5 x - 6$$
    1. Use the Factor Theorem to show that \(x + 1\) is a factor of \(\mathrm { p } ( x )\).
    2. Express \(\mathrm { p } ( x )\) as the product of three linear factors.
  2. Verify that \(\mathrm { p } ( 0 ) > \mathrm { p } ( 1 )\).
  3. Sketch the curve with equation \(y = x ^ { 3 } + 2 x ^ { 2 } - 5 x - 6\), indicating the values where the curve crosses the \(x\)-axis.
AQA C1 2012 June Q4
4 The diagram shows a solid cuboid with sides of lengths \(x \mathrm {~cm} , 3 x \mathrm {~cm}\) and \(y \mathrm {~cm}\).
\includegraphics[max width=\textwidth, alt={}, center]{dbc25177-4a28-480f-93d5-41acb2a2d28c-3_349_472_376_769} The total surface area of the cuboid is \(32 \mathrm {~cm} ^ { 2 }\).
    1. Show that \(3 x ^ { 2 } + 4 x y = 16\).
    2. Hence show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cuboid is given by $$V = 12 x - \frac { 9 x ^ { 3 } } { 4 }$$
  2. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
    1. Verify that a stationary value of \(V\) occurs when \(x = \frac { 4 } { 3 }\).
    2. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) and hence determine whether \(V\) has a maximum value or a minimum value when \(x = \frac { 4 } { 3 }\).
AQA C1 2012 June Q5
5
    1. Express \(x ^ { 2 } - 3 x + 5\) in the form \(( x - p ) ^ { 2 } + q\).
    2. Hence write down the equation of the line of symmetry of the curve with equation \(y = x ^ { 2 } - 3 x + 5\).
  2. The curve \(C\) with equation \(y = x ^ { 2 } - 3 x + 5\) and the straight line \(y = x + 5\) intersect at the point \(A ( 0,5 )\) and at the point \(B\), as shown in the diagram below.
    \includegraphics[max width=\textwidth, alt={}, center]{dbc25177-4a28-480f-93d5-41acb2a2d28c-4_471_707_653_676}
    1. Find the coordinates of the point \(B\).
    2. Find \(\int \left( x ^ { 2 } - 3 x + 5 \right) \mathrm { d } x\).
    3. Find the area of the shaded region \(R\) bounded by the curve \(C\) and the line segment \(A B\).
AQA C1 2012 June Q6
6 The circle with centre \(C ( 5,8 )\) touches the \(y\)-axis, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{dbc25177-4a28-480f-93d5-41acb2a2d28c-5_485_631_370_715}
  1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
    1. Verify that the point \(A ( 2,12 )\) lies on the circle.
    2. Find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(s x + t y + u = 0\), where \(s , t\) and \(u\) are integers.
  3. The points \(P\) and \(Q\) lie on the circle, and the mid-point of \(P Q\) is \(M ( 7,12 )\).
    1. Show that the length of \(C M\) is \(n \sqrt { 5 }\), where \(n\) is an integer.
    2. Hence find the area of triangle \(P C Q\).
AQA C1 2012 June Q7
7 The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), of a curve \(C\) at the point \(( x , y )\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 20 x - 6 x ^ { 2 } - 16$$
    1. Show that \(y\) is increasing when \(3 x ^ { 2 } - 10 x + 8 < 0\).
    2. Solve the inequality \(3 x ^ { 2 } - 10 x + 8 < 0\).
  2. The curve \(C\) passes through the point \(P ( 2,3 )\).
    1. Verify that the tangent to the curve at \(P\) is parallel to the \(x\)-axis.
    2. The point \(Q ( 3 , - 1 )\) also lies on the curve. The normal to the curve at \(Q\) and the tangent to the curve at \(P\) intersect at the point \(R\). Find the coordinates of \(R\).
      (7 marks)
AQA C1 2013 June Q1
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 \(D\).
AQA C1 2013 June Q2
2
    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.
AQA C1 2013 June Q3
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.
    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 \(C\).
AQA C1 2013 June Q4
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 \(x = - 3\).
AQA C1 2013 June Q5
5
    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.
AQA C1 2013 June Q6
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)
AQA C1 2013 June Q7
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 \(k\).
AQA C1 2014 June Q1
7 marks
1 The point \(A\) has coordinates \(( - 1,2 )\) and the point \(B\) has coordinates \(( 3 , - 5 )\).
    1. Find the gradient of \(A B\).
    2. Hence find an equation of the line \(A B\), giving your answer in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers.
  2. The midpoint of \(A B\) is \(M\).
    1. Find the coordinates of \(M\).
    2. Find an equation of the line which passes through \(M\) and which is perpendicular to \(A B\). [3 marks]
  3. The point \(C\) has coordinates \(( k , 2 k + 3 )\). Given that the distance from \(A\) to \(C\) is \(\sqrt { 13 }\), find the two possible values of the constant \(k\).
    [0pt] [4 marks]
AQA C1 2014 June Q2
4 marks
2 A rectangle has length \(( 9 + 5 \sqrt { 3 } ) \mathrm { cm }\) and area \(( 15 + 7 \sqrt { 3 } ) \mathrm { cm } ^ { 2 }\).
Find the width of the rectangle, giving your answer in the form \(( m + n \sqrt { 3 } ) \mathrm { cm }\), where \(m\) and \(n\) are integers.
[0pt] [4 marks]
AQA C1 2014 June Q3
4 marks
3 A curve has equation \(y = 2 x ^ { 5 } + 5 x ^ { 4 } - 1\).
  1. Find:
    1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
    2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
  2. The point on the curve where \(x = - 1\) is \(P\).
    1. Determine whether \(y\) is increasing or decreasing at \(P\), giving a reason for your answer.
    2. Find an equation of the tangent to the curve at \(P\).
  3. The point \(Q ( - 2,15 )\) also lies on the curve. Verify that \(Q\) is a maximum point of the curve.
    [0pt] [4 marks]
AQA C1 2014 June Q4
3 marks
4
    1. Express \(16 - 6 x - x ^ { 2 }\) in the form \(p - ( x + q ) ^ { 2 }\) where \(p\) and \(q\) are integers.
    2. Hence write down the maximum value of \(16 - 6 x - x ^ { 2 }\).
    1. Factorise \(16 - 6 x - x ^ { 2 }\).
    2. Sketch the curve with equation \(y = 16 - 6 x - x ^ { 2 }\), stating the values of \(x\) where the curve crosses the \(x\)-axis and the value of the \(y\)-intercept.
      [0pt] [3 marks]