Questions — AQA (3548 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 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 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
Sort by: Default | Easiest first | Hardest first
AQA FP1 2007 January Q1
10 marks Easy -1.2
1
  1. Solve the following equations, giving each root in the form \(a + b \mathrm { i }\) :
    1. \(x ^ { 2 } + 16 = 0\);
    2. \(x ^ { 2 } - 2 x + 17 = 0\).
    1. Expand \(( 1 + x ) ^ { 3 }\).
    2. Express \(( 1 + \mathrm { i } ) ^ { 3 }\) in the form \(a + b \mathrm { i }\).
    3. Hence, or otherwise, verify that \(x = 1 + \mathrm { i }\) satisfies the equation $$x ^ { 3 } + 2 x - 4 \mathrm { i } = 0$$
AQA FP1 2007 January Q2
11 marks Moderate -0.3
2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left[ \begin{array} { c c } \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\ \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right] , \mathbf { B } = \left[ \begin{array} { c c } \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \\ \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \end{array} \right]$$
  1. Calculate:
    1. \(\mathbf { A } + \mathbf { B }\);
    2. \(\mathbf { B A }\).
  2. Describe fully the geometrical transformation represented by each of the following matrices:
    1. \(\mathbf { A }\);
    2. \(\mathbf { B }\);
    3. \(\mathbf { B A }\).
AQA FP1 2007 January Q3
8 marks Moderate -0.3
3 The quadratic equation $$2 x ^ { 2 } + 4 x + 3 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } = 1\).
  3. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 }\).
AQA FP1 2007 January Q4
6 marks Moderate -0.5
4 The variables \(x\) and \(y\) are related by an equation of the form $$y = a x ^ { b }$$ where \(a\) and \(b\) are constants.
  1. Using logarithms to base 10 , reduce the relation \(y = a x ^ { b }\) to a linear law connecting \(\log _ { 10 } x\) and \(\log _ { 10 } y\).
  2. The diagram shows the linear graph that results from plotting \(\log _ { 10 } y\) against \(\log _ { 10 } x\). \includegraphics[max width=\textwidth, alt={}, center]{49539feb-f842-49f4-b809-72e8147072e7-3_711_1223_1503_411} Find the values of \(a\) and \(b\).
AQA FP1 2007 January Q5
10 marks Standard +0.3
5 A curve has equation $$y = \frac { x } { x ^ { 2 } - 1 }$$
  1. Write down the equations of the three asymptotes to the curve.
  2. Sketch the curve.
    (You are given that the curve has no stationary points.)
  3. Solve the inequality $$\frac { x } { x ^ { 2 } - 1 } > 0$$
AQA FP1 2007 January Q6
10 marks Moderate -0.5
6
    1. Expand \(( 2 r - 1 ) ^ { 2 }\).
    2. Hence show that $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$
  2. Hence find the sum of the squares of the odd numbers between 100 and 200 .
AQA FP1 2007 January Q7
8 marks Moderate -0.3
7 The function f is defined for all real numbers by $$f ( x ) = \sin \left( x + \frac { \pi } { 6 } \right)$$
  1. Find the general solution of the equation \(\mathrm { f } ( x ) = 0\).
  2. The quadratic function g is defined for all real numbers by $$\mathrm { g } ( x ) = \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } x - \frac { 1 } { 4 } x ^ { 2 }$$ It can be shown that \(\mathrm { g } ( x )\) gives a good approximation to \(\mathrm { f } ( x )\) for small values of \(x\).
    1. Show that \(\mathrm { g } ( 0.05 )\) and \(\mathrm { f } ( 0.05 )\) are identical when rounded to four decimal places.
    2. A chord joins the points on the curve \(y = \mathrm { g } ( x )\) for which \(x = 0\) and \(x = h\). Find an expression in terms of \(h\) for the gradient of this chord.
    3. Using your answer to part (b)(ii), find the value of \(\mathrm { g } ^ { \prime } ( 0 )\).
AQA FP1 2007 January Q8
12 marks Standard +0.3
8 A curve \(C\) has equation $$\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 9 } = 1$$
  1. Find the \(y\)-coordinates of the points on \(C\) for which \(x = 10\), giving each answer in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
  2. Sketch the curve \(C\), indicating the coordinates of any points where the curve intersects the coordinate axes.
  3. Write down the equation of the tangent to \(C\) at the point where \(C\) intersects the positive \(x\)-axis.
    1. Show that, if the line \(y = x - 4\) intersects \(C\), the \(x\)-coordinates of the points of intersection must satisfy the equation $$16 x ^ { 2 } - 200 x + 625 = 0$$
    2. Solve this equation and hence state the relationship between the line \(y = x - 4\) and the curve \(C\).
AQA FP1 2009 January Q1
5 marks Moderate -0.5
1 A curve passes through the point \(( 0,1 )\) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { 1 + x ^ { 2 } }$$ Starting at the point \(( 0,1 )\), use a step-by-step method with a step length of 0.2 to estimate the value of \(y\) at \(x = 0.4\). Give your answer to five decimal places.
AQA FP1 2009 January Q2
5 marks Moderate -0.8
2 The complex number \(2 + 3 \mathrm { i }\) is a root of the quadratic equation $$x ^ { 2 } + b x + c = 0$$ where \(b\) and \(c\) are real numbers.
  1. Write down the other root of this equation.
  2. Find the values of \(b\) and \(c\).
AQA FP1 2009 January Q3
5 marks Moderate -0.5
3 Find the general solution of the equation $$\tan \left( \frac { \pi } { 2 } - 3 x \right) = \sqrt { 3 }$$
AQA FP1 2009 January Q4
7 marks Standard +0.3
4 It is given that $$S _ { n } = \sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 3 r + 1 \right)$$
  1. Use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that \(S _ { n } = n ^ { 3 }\).
  2. Hence show that \(\sum _ { r = n + 1 } ^ { 2 n } \left( 3 r ^ { 2 } - 3 r + 1 \right) = k n ^ { 3 }\) for some integer \(k\).
AQA FP1 2009 January Q5
12 marks Standard +0.3
5 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { c c } k & k \\ k & - k \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { c c } - k & k \\ k & k \end{array} \right]$$ where \(k\) is a constant.
  1. Find, in terms of \(k\) :
    1. \(\mathbf { A } + \mathbf { B }\);
    2. \(\mathbf { A } ^ { 2 }\).
  2. Show that \(( \mathbf { A } + \mathbf { B } ) ^ { 2 } = \mathbf { A } ^ { 2 } + \mathbf { B } ^ { 2 }\).
  3. It is now given that \(k = 1\).
    1. Describe the geometrical transformation represented by the matrix \(\mathbf { A } ^ { 2 }\).
    2. The matrix \(\mathbf { A }\) represents a combination of an enlargement and a reflection. Find the scale factor of the enlargement and the equation of the mirror line of the reflection.
AQA FP1 2009 January Q6
10 marks Moderate -0.3
6 A curve has equation $$y = \frac { ( x - 1 ) ( x - 3 ) } { x ( x - 2 ) }$$
    1. Write down the equations of the three asymptotes of this curve.
    2. State the coordinates of the points at which the curve intersects the \(x\)-axis.
    3. Sketch the curve.
      (You are given that the curve has no stationary points.)
  2. Hence, or otherwise, solve the inequality $$\frac { ( x - 1 ) ( x - 3 ) } { x ( x - 2 ) } < 0$$
AQA FP1 2009 January Q7
10 marks Standard +0.3
7 The points \(P ( a , c )\) and \(Q ( b , d )\) lie on the curve with equation \(y = \mathrm { f } ( x )\). The straight line \(P Q\) intersects the \(x\)-axis at the point \(R ( r , 0 )\). The curve \(y = \mathrm { f } ( x )\) intersects the \(x\)-axis at the point \(S ( \beta , 0 )\). \includegraphics[max width=\textwidth, alt={}, center]{38c2a2c8-84cc-4bd2-b3ad-f9dee59763ba-4_951_971_470_539}
  1. Show that $$r = a + c \left( \frac { b - a } { c - d } \right)$$
  2. Given that $$a = 2 , b = 3 \text { and } \mathrm { f } ( x ) = 20 x - x ^ { 4 }$$
    1. find the value of \(r\);
    2. show that \(\beta - r \approx 0.18\).
AQA FP1 2009 January Q8
7 marks Standard +0.3
8 For each of the following improper integrals, find the value of the integral or explain why it does not have a value:
  1. \(\int _ { 1 } ^ { \infty } x ^ { - \frac { 3 } { 4 } } \mathrm {~d} x\);
  2. \(\int _ { 1 } ^ { \infty } x ^ { - \frac { 5 } { 4 } } \mathrm {~d} x\);
  3. \(\quad \int _ { 1 } ^ { \infty } \left( x ^ { - \frac { 3 } { 4 } } - x ^ { - \frac { 5 } { 4 } } \right) \mathrm { d } x\).
AQA FP1 2009 January Q9
14 marks Standard +0.3
9 A hyperbola \(H\) has equation $$x ^ { 2 } - \frac { y ^ { 2 } } { 2 } = 1$$
  1. Find the equations of the two asymptotes of \(H\), giving each answer in the form \(y = m x\).
  2. Draw a sketch of the two asymptotes of \(H\), using roughly equal scales on the two coordinate axes. Using the same axes, sketch the hyperbola \(H\).
    1. Show that, if the line \(y = x + c\) intersects \(H\), the \(x\)-coordinates of the points of intersection must satisfy the equation $$x ^ { 2 } - 2 c x - \left( c ^ { 2 } + 2 \right) = 0$$
    2. Hence show that the line \(y = x + c\) intersects \(H\) in two distinct points, whatever the value of \(c\).
    3. Find, in terms of \(c\), the \(y\)-coordinates of these two points.
AQA FP1 2011 January Q1
7 marks Standard +0.3
1 The quadratic equation \(x ^ { 2 } - 6 x + 18 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Find a quadratic equation, with integer coefficients, which has roots \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).
  3. Hence find the values of \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).
AQA FP1 2011 January Q2
6 marks Standard +0.3
2
  1. Find, in terms of \(p\) and \(q\), the value of the integral \(\int _ { p } ^ { q } \frac { 2 } { x ^ { 3 } } \mathrm {~d} x\).
  2. Show that only one of the following improper integrals has a finite value, and find that value:
    1. \(\int _ { 0 } ^ { 2 } \frac { 2 } { x ^ { 3 } } \mathrm {~d} x\);
    2. \(\int _ { 2 } ^ { \infty } \frac { 2 } { x ^ { 3 } } \mathrm {~d} x\).
AQA FP1 2011 January Q3
13 marks Moderate -0.3
3
  1. Write down the \(2 \times 2\) matrix corresponding to each of the following transformations:
    1. a rotation about the origin through \(90 ^ { \circ }\) clockwise;
    2. a rotation about the origin through \(180 ^ { \circ }\).
  2. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { r r } 2 & 4 \\ - 1 & - 3 \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { l l } - 2 & 1 \\ - 4 & 3 \end{array} \right]$$
    1. Calculate the matrix \(\mathbf { A B }\).
    2. Show that \(( \mathbf { A } + \mathbf { B } ) ^ { 2 } = k \mathbf { I }\), where \(\mathbf { I }\) is the identity matrix, for some integer \(k\).
  3. Describe the single geometrical transformation, or combination of two geometrical transformations, represented by each of the following matrices:
    1. \(\mathbf { A } + \mathbf { B }\);
    2. \(( \mathbf { A } + \mathbf { B } ) ^ { 2 }\);
    3. \(( \mathbf { A } + \mathbf { B } ) ^ { 4 }\).
AQA FP1 2011 January Q4
6 marks Standard +0.3
4 Find the general solution of the equation $$\sin \left( 4 x - \frac { 2 \pi } { 3 } \right) = - \frac { 1 } { 2 }$$ giving your answer in terms of \(\pi\).
(6 marks)
AQA FP1 2011 January Q5
8 marks Moderate -0.3
5
  1. It is given that \(z _ { 1 } = \frac { 1 } { 2 } - \mathrm { i }\).
    1. Calculate the value of \(z _ { 1 } ^ { 2 }\), giving your answer in the form \(a + b \mathrm { i }\).
    2. Hence verify that \(z _ { 1 }\) is a root of the equation $$z ^ { 2 } + z ^ { * } + \frac { 1 } { 4 } = 0$$
  2. Show that \(z _ { 2 } = \frac { 1 } { 2 } + \mathrm { i }\) also satisfies the equation in part (a)(ii).
  3. Show that the equation in part (a)(ii) has two equal real roots.
AQA FP1 2011 January Q6
8 marks Standard +0.3
6 The diagram shows a circle \(C\) and a line \(L\), which is the tangent to \(C\) at the point \(( 1,1 )\). The equations of \(C\) and \(L\) are $$x ^ { 2 } + y ^ { 2 } = 2 \text { and } x + y = 2$$ respectively. \includegraphics[max width=\textwidth, alt={}, center]{a4c5d61d-1af9-449e-b27a-d1e656dcd75a-4_760_1301_552_395} The circle \(C\) is now transformed by a stretch with scale factor 2 parallel to the \(x\)-axis. The image of \(C\) under this stretch is an ellipse \(E\).
  1. On the diagram below, sketch the ellipse \(E\), indicating the coordinates of the points where it intersects the coordinate axes.
  2. Find equations of:
    1. the ellipse \(E\);
    2. the tangent to \(E\) at the point \(( 2,1 )\). \includegraphics[max width=\textwidth, alt={}, center]{a4c5d61d-1af9-449e-b27a-d1e656dcd75a-4_743_1301_1921_420}
AQA FP1 2011 January Q7
15 marks Standard +0.8
7 A graph has equation $$y = \frac { x - 4 } { x ^ { 2 } + 9 }$$
  1. Explain why the graph has no vertical asymptote and give the equation of the horizontal asymptote.
  2. Show that, if the line \(y = k\) intersects the graph, the \(x\)-coordinates of the points of intersection of the line with the graph must satisfy the equation $$k x ^ { 2 } - x + ( 9 k + 4 ) = 0$$
  3. Show that this equation has real roots if \(- \frac { 1 } { 2 } \leqslant k \leqslant \frac { 1 } { 18 }\).
  4. Hence find the coordinates of the two stationary points on the curve.
    (No credit will be given for methods involving differentiation.)
AQA FP1 2011 January Q8
12 marks Standard +0.3
8
  1. The equation $$x ^ { 3 } + 2 x ^ { 2 } + x - 100000 = 0$$ has one real root. Taking \(x _ { 1 } = 50\) as a first approximation to this root, use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to the root.
    1. Given that \(S _ { n } = \sum _ { r = 1 } ^ { n } r ( 3 r + 1 )\), use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that $$S _ { n } = n ( n + 1 ) ^ { 2 }$$
    2. The lowest integer \(n\) for which \(S _ { n } > 100000\) is denoted by \(N\). Show that $$N ^ { 3 } + 2 N ^ { 2 } + N - 100000 > 0$$
  3. Find the value of \(N\), justifying your answer.