Questions C1 (1442 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 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]
AQA C1 2014 June Q5
3 marks
5 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + c x ^ { 2 } + d x + 3$$ where \(c\) and \(d\) are integers.
  1. Given that \(x + 3\) is a factor of \(\mathrm { p } ( x )\), show that $$3 c - d = 8$$
  2. The remainder when \(\mathrm { p } ( x )\) is divided by \(x - 2\) is 65 . Obtain a further equation in \(c\) and \(d\).
  3. Use the equations from parts (a) and (b) to find the value of \(c\) and the value of \(d\). [3 marks]
AQA C1 2014 June Q6
7 marks
6 The diagram shows a curve and a line which intersect at the points \(A , B\) and \(C\).
\includegraphics[max width=\textwidth, alt={}, center]{f2124c89-79de-4758-b7b8-ff273345b9dd-7_574_844_349_609} The curve has equation \(y = x ^ { 3 } - x ^ { 2 } - 5 x + 7\) and the straight line has equation \(y = x + 7\). The point \(B\) has coordinates ( 0,7 ).
    1. Show that the \(x\)-coordinates of the points \(A\) and \(C\) satisfy the equation $$x ^ { 2 } - x - 6 = 0$$
    2. Find the coordinates of the points \(A\) and \(C\).
  2. Find \(\int \left( x ^ { 3 } - x ^ { 2 } - 5 x + 7 \right) \mathrm { d } x\).
  3. Find the area of the shaded region \(R\) bounded by the curve and the line segment \(A B\).
    [0pt] [4 marks]
    \(7 \quad\) A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 10 x + 12 y + 41 = 0\). The point \(A ( 3 , - 2 )\) lies on the circle.
AQA C1 2014 June Q8
4 marks
8 Solve the following inequalities:
  1. \(\quad 3 ( 1 - 2 x ) - 5 ( 3 x + 2 ) > 0\)
  2. \(\quad 6 x ^ { 2 } \leqslant x + 12\)
    [0pt] [4 marks]
AQA C1 2015 June Q1
1 The line \(A B\) has equation \(3 x + 5 y = 7\).
  1. Find the gradient of \(A B\).
  2. Find an equation of the line that is perpendicular to the line \(A B\) and which passes through the point \(( - 2 , - 3 )\). Express your answer in the form \(p x + q y + r = 0\), where \(p , q\) and \(r\) are integers.
  3. The line \(A C\) has equation \(2 x - 3 y = 30\). Find the coordinates of \(A\).
AQA C1 2015 June Q2
5 marks
2 The point \(P\) has coordinates \(( \sqrt { 3 } , 2 \sqrt { 3 } )\) and the point \(Q\) has coordinates \(( \sqrt { 5 } , 4 \sqrt { 5 } )\). Show that the gradient of \(P Q\) can be expressed as \(n + \sqrt { 15 }\), stating the value of the integer \(n\).
[0pt] [5 marks]
AQA C1 2015 June Q3
5 marks
3 The diagram shows a sketch of a curve and a line.
\includegraphics[max width=\textwidth, alt={}, center]{c7f38f7e-75aa-4b41-96fd-f38f968c225c-06_520_588_351_742} The curve has equation \(y = x ^ { 4 } + 3 x ^ { 2 } + 2\). The points \(A ( - 1,6 )\) and \(B ( 2,30 )\) lie on the curve.
  1. Find an equation of the tangent to the curve at the point \(A\).
    1. Find \(\int _ { - 1 } ^ { 2 } \left( x ^ { 4 } + 3 x ^ { 2 } + 2 \right) \mathrm { d } x\).
    2. Calculate the area of the shaded region bounded by the curve and the line \(A B\).
      [0pt] [3 marks] \(4 \quad\) A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 6 y - 40 = 0\).
AQA C1 2015 June Q5
5
  1. Express \(x ^ { 2 } + 3 x + 2\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
  2. A curve has equation \(y = x ^ { 2 } + 3 x + 2\).
    1. Use the result from part (a) to write down the coordinates of the vertex of the curve.
    2. State the equation of the line of symmetry of the curve.
  3. The curve with equation \(y = x ^ { 2 } + 3 x + 2\) is translated by the vector \(\left[ \begin{array} { l } 2
    4 \end{array} \right]\). Find the equation of the resulting curve in the form \(y = x ^ { 2 } + b x + c\).
AQA C1 2015 June Q6
4 marks
6 The diagram shows a cylindrical container of radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The container has an open top and a circular base.
\includegraphics[max width=\textwidth, alt={}, center]{c7f38f7e-75aa-4b41-96fd-f38f968c225c-12_389_426_404_751} The external surface area of the container's curved surface and base is \(48 \pi \mathrm {~cm} ^ { 2 }\).
When the radius of the base is \(r \mathrm {~cm}\), the volume of the container is \(V \mathrm {~cm} ^ { 3 }\).
    1. Find an expression for \(h\) in terms of \(r\).
    2. Show that \(V = 24 \pi r - \frac { \pi } { 2 } r ^ { 3 }\).
    1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\).
    2. Find the positive value of \(r\) for which \(V\) is stationary, and determine whether this stationary value is a maximum value or a minimum value.
      [0pt] [4 marks]
AQA C1 2015 June Q7
3 marks
7
  1. Sketch the curve with equation \(y = x ^ { 2 } ( x - 3 )\).
  2. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 2 } ( x - 3 ) + 20\).
    1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 4\).
    2. Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
    3. Express \(\mathrm { p } ( x )\) in the form \(( x + 2 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
    4. Hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root and state its value.
      [0pt] [3 marks]
AQA C1 2015 June Q8
8 A curve has equation \(y = x ^ { 2 } + ( 3 k - 4 ) x + 13\) and a line has equation \(y = 2 x + k\), where \(k\) is a constant.
  1. Show that the \(x\)-coordinate of any point of intersection of the line and curve satisfies the equation $$x ^ { 2 } + 3 ( k - 2 ) x + 13 - k = 0$$
  2. Given that the line and the curve do not intersect:
    1. show that \(9 k ^ { 2 } - 32 k - 16 < 0\);
    2. find the possible values of \(k\).
      \includegraphics[max width=\textwidth, alt={}]{c7f38f7e-75aa-4b41-96fd-f38f968c225c-18_1657_1714_1050_153}
AQA C1 2016 June Q1
2 marks
1 The line \(A B\) has equation \(5 x + 3 y + 3 = 0\).
  1. The line \(A B\) is parallel to the line with equation \(y = m x + 7\). Find the value of \(m\).
  2. The line \(A B\) intersects the line with equation \(3 x - 2 y + 17 = 0\) at the point \(B\). Find the coordinates of \(B\).
  3. The point with coordinates \(( 2 k + 3,4 - 3 k )\) lies on the line \(A B\). Find the value of \(k\).
    [0pt] [2 marks]
AQA C1 2016 June Q2
4 marks
2
  1. Simplify \(( 3 \sqrt { 5 } ) ^ { 2 }\).
  2. Express \(\frac { ( 3 \sqrt { 5 } ) ^ { 2 } + \sqrt { 5 } } { 7 + 3 \sqrt { 5 } }\) in the form \(m + n \sqrt { 5 }\), where \(m\) and \(n\) are integers.
    [0pt] [4 marks]
AQA C1 2016 June Q3
3 marks
3
    1. Express \(x ^ { 2 } - 7 x + 2\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
    2. Hence write down the minimum value of \(x ^ { 2 } - 7 x + 2\).
  2. Describe the geometrical transformation which maps the graph of \(y = x ^ { 2 } - 7 x + 2\) onto the graph of \(y = ( x - 4 ) ^ { 2 }\).
    [0pt] [3 marks]
AQA C1 2016 June Q4
3 marks
4 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 5 x ^ { 2 } - 8 x + 48\).
    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 three linear factors.
    1. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 2\).
    2. Express \(\mathrm { p } ( x )\) in the form \(( x - 2 ) \left( x ^ { 2 } + b x + c \right) + r\), where \(b , c\) and \(r\) are integers. [3 marks]
AQA C1 2016 June Q5
5 A circle with centre \(C ( 5 , - 3 )\) passes through the point \(A ( - 2,1 )\).
  1. Find the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
  2. Given that \(A B\) is a diameter of the circle, find the coordinates of the point \(B\).
  3. Find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(p x + q y + r = 0\), where \(p , q\) and \(r\) are integers.
  4. The point \(T\) lies on the tangent to the circle at \(A\) such that \(A T = 4\). Find the length of \(C T\).