Questions C1 (1442 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 2012 January Q3
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
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
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
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 \(x ^ { 2 } + y ^ { 2 } + 14 x - 10 y + 49 = 0\).
  5. Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
  6. Write down:
    1. the coordinates of \(C\);
    2. the radius of the circle.
  7. Sketch the circle.
  8. 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
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
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
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
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
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
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
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 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
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
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
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
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 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
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 Solve each of the following inequalities:
  1. \(3 ( x - 1 ) > 3 - 5 ( x + 6 )\);
  2. \(\quad x ^ { 2 } - x - 6 < 0\).
AQA C1 2005 June Q8
8 A line has equation \(y = m x - 1\), where \(m\) is a constant.
A curve has equation \(y = x ^ { 2 } - 5 x + 3\).
  1. Show that the \(x\)-coordinate of any point of intersection of the line and the curve satisfies the equation $$x ^ { 2 } - ( 5 + m ) x + 4 = 0$$
  2. Find the values of \(m\) for which the equation \(x ^ { 2 } - ( 5 + m ) x + 4 = 0\) has equal roots.
    (4 marks)
  3. Describe geometrically the situation when \(m\) takes either of the values found in part (b).
    (1 mark)
AQA C1 2006 June Q1
1 The point \(A\) has coordinates \(( 1,7 )\) and the point \(B\) has coordinates \(( 5,1 )\).
    1. Find the gradient of the line \(A B\).
    2. Hence, or otherwise, show that the line \(A B\) has equation \(3 x + 2 y = 17\).
  2. The line \(A B\) intersects the line with equation \(x - 4 y = 8\) at the point \(C\). Find the coordinates of \(C\).
  3. Find an equation of the line through \(A\) which is perpendicular to \(A B\).
AQA C1 2006 June Q2
2
  1. Express \(x ^ { 2 } + 8 x + 19\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
  2. Hence, or otherwise, show that the equation \(x ^ { 2 } + 8 x + 19 = 0\) has no real solutions.
  3. Sketch the graph of \(y = x ^ { 2 } + 8 x + 19\), stating the coordinates of the minimum point and the point where the graph crosses the \(y\)-axis.
  4. Describe geometrically the transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 8 x + 19\).
AQA C1 2006 June Q3
3 A curve has equation \(y = 7 - 2 x ^ { 5 }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find an equation for the tangent to the curve at the point where \(x = 1\).
  3. Determine whether \(y\) is increasing or decreasing when \(x = - 2\).
AQA C1 2006 June Q4
4
  1. Express \(( 4 \sqrt { 5 } - 1 ) ( \sqrt { 5 } + 3 )\) in the form \(p + q \sqrt { 5 }\), where \(p\) and \(q\) are integers.
  2. Show that \(\frac { \sqrt { 75 } - \sqrt { 27 } } { \sqrt { 3 } }\) is an integer and find its value.
AQA C1 2006 June Q5
5 The curve with equation \(y = x ^ { 3 } - 10 x ^ { 2 } + 28 x\) is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{f2c95d73-d3fe-48f7-af07-84f12bb06727-3_483_899_402_568} The curve crosses the \(x\)-axis at the origin \(O\) and the point \(A ( 3,21 )\) lies on the curve.
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Hence show that the curve has a stationary point when \(x = 2\) and find the \(x\)-coordinate of the other stationary point.
    1. Find \(\int \left( x ^ { 3 } - 10 x ^ { 2 } + 28 x \right) \mathrm { d } x\).
    2. Hence show that \(\int _ { 0 } ^ { 3 } \left( x ^ { 3 } - 10 x ^ { 2 } + 28 x \right) \mathrm { d } x = 56 \frac { 1 } { 4 }\).
    3. Hence determine the area of the shaded region bounded by the curve and the line \(O A\).