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 2007 January Q7
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 \(k\).
AQA C1 2008 January Q1
1 The triangle \(A B C\) has vertices \(A ( - 2,3 ) , B ( 4,1 )\) and \(C ( 2 , - 5 )\).
  1. Find the coordinates of the mid-point of \(B C\).
    1. Find the gradient of \(A B\), in its simplest form.
    2. Hence find an equation of the line \(A B\), giving your answer in the form \(x + q y = r\), where \(q\) and \(r\) are integers.
    3. Find an equation of the line passing through \(C\) which is parallel to \(A B\).
  3. Prove that angle \(A B C\) is a right angle.
AQA C1 2008 January Q2
2 The curve with equation \(y = x ^ { 4 } - 32 x + 5\) has a single stationary point, \(M\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find the \(x\)-coordinate of \(M\).
    1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
    2. Hence, or otherwise, determine whether \(M\) is a maximum or a minimum point.
  4. Determine whether the curve is increasing or decreasing at the point on the curve where \(x = 0\).
AQA C1 2008 January Q3
3
  1. Express \(5 \sqrt { 8 } + \frac { 6 } { \sqrt { 2 } }\) in the form \(n \sqrt { 2 }\), where \(n\) is an integer.
  2. Express \(\frac { \sqrt { 2 } + 2 } { 3 \sqrt { 2 } - 4 }\) in the form \(c \sqrt { 2 } + d\), where \(c\) and \(d\) are integers.
AQA C1 2008 January Q4
4 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 10 y + 20 = 0\).
  1. By completing the square, express this equation in the form $$x ^ { 2 } + ( y - b ) ^ { 2 } = k$$
  2. Write down:
    1. the coordinates of \(C\);
    2. the radius of the circle, leaving your answer in surd form.
  3. A line has equation \(y = 2 x\).
    1. Show that the \(x\)-coordinate of any point of intersection of the line and the circle satisfies the equation \(x ^ { 2 } - 4 x + 4 = 0\).
    2. Hence show that the line is a tangent to the circle and find the coordinates of the point of contact, \(P\).
  4. Prove that the point \(Q ( - 1,4 )\) lies inside the circle.
AQA C1 2008 January Q5
5
  1. Factorise \(9 - 8 x - x ^ { 2 }\).
  2. Show that \(25 - ( x + 4 ) ^ { 2 }\) can be written as \(9 - 8 x - x ^ { 2 }\).
  3. A curve has equation \(y = 9 - 8 x - x ^ { 2 }\).
    1. Write down the equation of its line of symmetry.
    2. Find the coordinates of its vertex.
    3. Sketch the curve, indicating the values of the intercepts on the \(x\)-axis and the \(y\)-axis.
AQA C1 2008 January Q6
6
  1. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 7 x - 6\).
    1. Use the Factor Theorem to show that \(x + 1\) is a factor of \(\mathrm { p } ( x )\).
    2. Express \(\mathrm { p } ( x ) = x ^ { 3 } - 7 x - 6\) as the product of three linear factors.
  2. The curve with equation \(y = x ^ { 3 } - 7 x - 6\) is sketched below.
    \includegraphics[max width=\textwidth, alt={}, center]{de4f827d-f237-488a-9177-3d85d0cb1771-4_403_762_651_641} The curve cuts the \(x\)-axis at the point \(A\) and the points \(B ( - 1,0 )\) and \(C ( 3,0 )\).
    1. State the coordinates of the point \(A\).
    2. Find \(\int _ { - 1 } ^ { 3 } \left( x ^ { 3 } - 7 x - 6 \right) \mathrm { d } x\).
    3. Hence find the area of the shaded region bounded by the curve \(y = x ^ { 3 } - 7 x - 6\) and the \(x\)-axis between \(B\) and \(C\).
    4. Find the gradient of the curve \(y = x ^ { 3 } - 7 x - 6\) at the point \(B\).
    5. Hence find an equation of the normal to the curve at the point \(B\).
AQA C1 2008 January Q7
7 The curve \(C\) has equation \(y = x ^ { 2 } + 7\). The line \(L\) has equation \(y = k ( 3 x + 1 )\), where \(k\) is a constant.
  1. Show that the \(x\)-coordinates of any points of intersection of the line \(L\) with the curve \(C\) satisfy the equation $$x ^ { 2 } - 3 k x + 7 - k = 0$$
  2. The curve \(C\) and the line \(L\) intersect in two distinct points. Show that $$9 k ^ { 2 } + 4 k - 28 > 0$$
  3. Solve the inequality \(9 k ^ { 2 } + 4 k - 28 > 0\).
AQA C1 2007 June Q1
1 The points \(A\) and \(B\) have coordinates \(( 6 , - 1 )\) and \(( 2,5 )\) respectively.
    1. Show that the gradient of \(A B\) is \(- \frac { 3 } { 2 }\).
    2. Hence find an equation of the line \(A B\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
    1. Find an equation of the line which passes through \(B\) and which is perpendicular to the line \(A B\).
    2. The point \(C\) has coordinates ( \(k , 7\) ) and angle \(A B C\) is a right angle. Find the value of the constant \(k\).
AQA C1 2007 June Q2
2
  1. Express \(\frac { \sqrt { 63 } } { 3 } + \frac { 14 } { \sqrt { 7 } }\) in the form \(n \sqrt { 7 }\), where \(n\) is an integer.
  2. Express \(\frac { \sqrt { 7 } + 1 } { \sqrt { 7 } - 2 }\) in the form \(p \sqrt { 7 } + q\), where \(p\) and \(q\) are integers.
AQA C1 2007 June Q3
3
    1. Express \(x ^ { 2 } + 10 x + 19\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
    2. Write down the coordinates of the vertex (minimum point) of the curve with equation \(y = x ^ { 2 } + 10 x + 19\).
    3. Write down the equation of the line of symmetry of the curve \(y = x ^ { 2 } + 10 x + 19\).
    4. Describe geometrically the transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 10 x + 19\).
  2. Determine the coordinates of the points of intersection of the line \(y = x + 11\) and the curve \(y = x ^ { 2 } + 10 x + 19\).
AQA C1 2007 June Q4
4 A model helicopter takes off from a point \(O\) at time \(t = 0\) and moves vertically so that its height, \(y \mathrm {~cm}\), above \(O\) after time \(t\) seconds is given by $$y = \frac { 1 } { 4 } t ^ { 4 } - 26 t ^ { 2 } + 96 t , \quad 0 \leqslant t \leqslant 4$$
  1. Find:
    1. \(\frac { \mathrm { d } y } { \mathrm {~d} t }\);
      (3 marks)
    2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\).
      (2 marks)
  2. Verify that \(y\) has a stationary value when \(t = 2\) and determine whether this stationary value is a maximum value or a minimum value.
    (4 marks)
  3. Find the rate of change of \(y\) with respect to \(t\) when \(t = 1\).
  4. Determine whether the height of the helicopter above \(O\) is increasing or decreasing at the instant when \(t = 3\).
AQA C1 2007 June Q5
5 A circle with centre \(C\) has equation \(( x + 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 25\).
  1. Write down:
    1. the coordinates of \(C\);
    2. the radius of the circle.
    1. Verify that the point \(N ( 0 , - 2 )\) lies on the circle.
    2. Sketch the circle.
    3. Find an equation of the normal to the circle at the point \(N\).
  3. The point \(P\) has coordinates (2, 6).
    1. Find the distance \(P C\), leaving your answer in surd form.
    2. Find the length of a tangent drawn from \(P\) to the circle.
AQA C1 2007 June Q6
6
  1. The polynomial \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 3 } + 4 x - 5\).
    1. Use the Factor Theorem to show that \(x - 1\) is a factor of \(\mathrm { f } ( x )\).
    2. Express \(\mathrm { f } ( x )\) in the form \(( x - 1 ) \left( x ^ { 2 } + p x + q \right)\), where \(p\) and \(q\) are integers.
    3. Hence show that the equation \(\mathrm { f } ( x ) = 0\) has exactly one real root and state its value.
  2. The curve with equation \(y = x ^ { 3 } + 4 x - 5\) is sketched below.
    \includegraphics[max width=\textwidth, alt={}, center]{23f34515-3373-4644-a8a1-82b45809d934-4_505_959_868_529} The curve cuts the \(x\)-axis at the point \(A ( 1,0 )\) and the point \(B ( 2,11 )\) lies on the curve.
    1. Find \(\int \left( x ^ { 3 } + 4 x - 5 \right) \mathrm { d } x\).
    2. Hence find the area of the shaded region bounded by the curve and the line \(A B\).
AQA C1 2007 June Q7
7 The quadratic equation $$( 2 k - 3 ) x ^ { 2 } + 2 x + ( k - 1 ) = 0$$ where \(k\) is a constant, has real roots.
  1. Show that \(2 k ^ { 2 } - 5 k + 2 \leqslant 0\).
    1. Factorise \(2 k ^ { 2 } - 5 k + 2\).
    2. Hence, or otherwise, solve the quadratic inequality $$2 k ^ { 2 } - 5 k + 2 \leqslant 0$$
AQA C1 2008 June Q1
1 The straight line \(L\) has equation \(y = 3 x - 1\) and the curve \(C\) has equation $$y = ( x + 3 ) ( x - 1 )$$
  1. Sketch on the same axes the line \(L\) and the curve \(C\), showing the values of the intercepts on the \(x\)-axis and the \(y\)-axis.
  2. Show that the \(x\)-coordinates of the points of intersection of \(L\) and \(C\) satisfy the equation \(x ^ { 2 } - x - 2 = 0\).
  3. Hence find the coordinates of the points of intersection of \(L\) and \(C\).
AQA C1 2008 June Q2
2 It is given that \(x = \sqrt { 3 }\) and \(y = \sqrt { 12 }\).
Find, in the simplest form, the value of:
  1. \(x y\);
  2. \(\frac { y } { x }\);
  3. \(( x + y ) ^ { 2 }\).
AQA C1 2008 June Q3
3 Two numbers, \(x\) and \(y\), are such that \(3 x + y = 9\), where \(x \geqslant 0\) and \(y \geqslant 0\). It is given that \(V = x y ^ { 2 }\).
  1. Show that \(V = 81 x - 54 x ^ { 2 } + 9 x ^ { 3 }\).
    1. Show that \(\frac { \mathrm { d } V } { \mathrm {~d} x } = k \left( x ^ { 2 } - 4 x + 3 \right)\), and state the value of the integer \(k\).
    2. Hence find the two values of \(x\) for which \(\frac { \mathrm { d } V } { \mathrm {~d} x } = 0\).
  3. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\).
    1. Find the value of \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) for each of the two values of \(x\) found in part (b)(ii).
    2. Hence determine the value of \(x\) for which \(V\) has a maximum value.
    3. Find the maximum value of \(V\).
AQA C1 2008 June Q4
4
  1. Express \(x ^ { 2 } - 3 x + 4\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
    (2 marks)
  2. Hence write down the minimum value of the expression \(x ^ { 2 } - 3 x + 4\).
  3. Describe the geometrical transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } - 3 x + 4\).
AQA C1 2008 June Q5
5 The curve with equation \(y = 16 - x ^ { 4 }\) is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{fddf5016-a5bd-42db-b5c4-f4980b8d9d67-3_435_663_824_685} The points \(A ( - 2,0 ) , B ( 2,0 )\) and \(C ( 1,15 )\) lie on the curve.
  1. Find an equation of the straight line \(A C\).
    1. Find \(\int _ { - 2 } ^ { 1 } \left( 16 - x ^ { 4 } \right) \mathrm { d } x\).
    2. Hence calculate the area of the shaded region bounded by the curve and the line \(A C\).
AQA C1 2008 June Q6
6 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } - 8 x - 12\).
  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 \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
    2. Express \(\mathrm { p } ( x )\) as the product of linear factors.
    1. The curve with equation \(y = x ^ { 3 } + x ^ { 2 } - 8 x - 12\) passes through the point \(( 0 , k )\). State the value of \(k\).
    2. Sketch the graph of \(y = x ^ { 3 } + x ^ { 2 } - 8 x - 12\), indicating the values of \(x\) where the curve touches or crosses the \(x\)-axis.
AQA C1 2008 June Q7
7 The circle \(S\) has centre \(C ( 8,13 )\) and touches the \(x\)-axis, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{fddf5016-a5bd-42db-b5c4-f4980b8d9d67-4_444_755_356_641}
  1. Write down an equation for \(S\), giving your answer in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
  2. The point \(P\) with coordinates \(( 3,1 )\) lies on the circle.
    1. Find the gradient of the straight line passing through \(P\) and \(C\).
    2. Hence find an equation of the tangent to the circle \(S\) at the point \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
    3. The point \(Q\) also lies on the circle \(S\), and the length of \(P Q\) is 10 . Calculate the shortest distance from \(C\) to the chord \(P Q\).
AQA C1 2008 June Q8
8 The quadratic equation \(( k + 1 ) x ^ { 2 } + 4 k x + 9 = 0\) has real roots.
  1. Show that \(4 k ^ { 2 } - 9 k - 9 \geqslant 0\).
  2. Hence find the possible values of \(k\).
AQA C1 2009 June Q1
1 The line \(A B\) has equation \(3 x + 5 y = 11\).
    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 \(C\).
AQA C1 2009 June Q2
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 \(x\).