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 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 \(x\).
AQA C1 2009 June Q3
3 The curve with equation \(y = x ^ { 5 } + 20 x ^ { 2 } - 8\) passes through the point \(P\), where \(x = - 2\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Verify that the point \(P\) is a stationary point of the curve.
    1. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(P\).
    2. Hence, or otherwise, determine whether \(P\) is a maximum point or a minimum point.
  4. Find an equation of the tangent to the curve at the point where \(x = 1\).
AQA C1 2009 June Q4
4
  1. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - x + 6\).
    1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
    2. Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
    3. Express \(\mathrm { p } ( x ) = x ^ { 3 } - x + 6\) in the form \(( x + 2 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
    4. The equation \(\mathrm { p } ( x ) = 0\) has one root equal to - 2 . Show that the equation has no other real roots.
  2. The curve with equation \(y = x ^ { 3 } - x + 6\) is sketched below.
    \includegraphics[max width=\textwidth, alt={}, center]{5f1ff5fa-b6e8-4c4f-aef7-63eb947b299f-3_529_702_945_667} The curve cuts the \(x\)-axis at the point \(A ( - 2,0 )\) and the \(y\)-axis at the point \(B\).
    1. State the \(y\)-coordinate of the point \(B\).
    2. Find \(\int _ { - 2 } ^ { 0 } \left( x ^ { 3 } - x + 6 \right) \mathrm { d } x\).
    3. Hence find the area of the shaded region bounded by the curve \(y = x ^ { 3 } - x + 6\) and the line \(A B\).
AQA C1 2009 June Q5
5 A circle with centre \(C\) has equation $$( x - 5 ) ^ { 2 } + ( y + 12 ) ^ { 2 } = 169$$
  1. Write down:
    1. the coordinates of \(C\);
    2. the radius of the circle.
    1. Verify that the circle passes through the origin \(O\).
    2. Given that the circle also passes through the points \(( 10,0 )\) and \(( 0 , p )\), sketch the circle and find the value of \(p\).
  3. The point \(A ( - 7 , - 7 )\) lies on the circle.
    1. Find the gradient of \(A C\).
    2. Hence find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
AQA C1 2009 June Q6
6
    1. Express \(x ^ { 2 } - 8 x + 17\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
    2. Hence write down the minimum value of \(x ^ { 2 } - 8 x + 17\).
    3. State the value of \(x\) for which the minimum value of \(x ^ { 2 } - 8 x + 17\) occurs.
      (1 mark)
  2. The point \(A\) has coordinates (5,4) and the point \(B\) has coordinates ( \(x , 7 - x\) ).
    1. Expand \(( x - 5 ) ^ { 2 }\).
    2. Show that \(A B ^ { 2 } = 2 \left( x ^ { 2 } - 8 x + 17 \right)\).
    3. Use your results from part (a) to find the minimum value of the distance \(A B\) as \(x\) varies.
AQA C1 2009 June Q7
7 The curve \(C\) has equation \(y = k \left( x ^ { 2 } + 3 \right)\), where \(k\) is a constant.
The line \(L\) has equation \(y = 2 x + 2\).
  1. Show that the \(x\)-coordinates of any points of intersection of the curve \(C\) with the line \(L\) satisfy the equation $$k x ^ { 2 } - 2 x + 3 k - 2 = 0$$
  2. The curve \(C\) and the line \(L\) intersect in two distinct points.
    1. Show that $$3 k ^ { 2 } - 2 k - 1 < 0$$
    2. Hence find the possible values of \(k\).