Questions — AQA FP1 (182 questions)

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AQA FP1 2005 June Q5
7 marks Moderate -0.3
5 Find the general solutions of the following equations, giving your answers in terms of \(\pi\) :
  1. \(\quad \tan 3 x = \sqrt { 3 }\);
  2. \(\quad \tan \left( 3 x - \frac { \pi } { 3 } \right) = - \sqrt { 3 }\).
AQA FP1 2005 June Q6
11 marks Standard +0.3
6 The equation $$x ^ { 2 } - 4 x + 13 = 0$$ has roots \(\alpha\) and \(\beta\).
    1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
    2. Deduce that \(\alpha ^ { 2 } + \beta ^ { 2 } = - 10\).
    3. Explain why the statement \(\alpha ^ { 2 } + \beta ^ { 2 } = - 10\) implies that \(\alpha\) and \(\beta\) cannot both be real.
  2. Find in the form \(p + \mathrm { i } q\) the values of:
    1. \(( \alpha + \mathrm { i } ) + ( \beta + \mathrm { i } )\);
    2. \(( \alpha + \mathrm { i } ) ( \beta + \mathrm { i } )\).
  3. Hence find a quadratic equation with roots \(( \alpha + \mathrm { i } )\) and \(( \beta + \mathrm { i } )\).
AQA FP1 2005 June Q7
11 marks Standard +0.3
7 [Figure 1, printed on the insert, is provided for use in this question.]
The diagram shows a triangle with vertices \(A ( 1,1 ) , B ( 3,1 )\) and \(C ( 3,2 )\). \includegraphics[max width=\textwidth, alt={}, center]{5bfb4d19-8772-43d7-b667-bd124d2504a8-04_1114_1141_552_360}
  1. The triangle \(D E F\) is obtained by applying to triangle \(A B C\) the transformation T represented by the matrix $$\left[ \begin{array} { r r } 2 & 2 \\ - 2 & 2 \end{array} \right]$$
    1. Calculate the coordinates of \(D , E\) and \(F\).
    2. Draw the triangle \(D E F\) on Figure 1.
  2. Given that T is a combination of an enlargement and a rotation, find the exact value of:
    1. the scale factor of the enlargement;
    2. the magnitude of the angle of the rotation.
AQA FP1 2005 June Q8
8 marks Standard +0.3
8 The diagram shows a part of the curve $$\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 6 } = 1$$ and a chord \(P Q\) of the curve, where \(P\) lies on the \(x\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{5bfb4d19-8772-43d7-b667-bd124d2504a8-05_751_1072_680_459}
  1. Write down the coordinates of \(P\).
  2. The gradient of the chord \(P Q\) is 2 . Find the coordinates of \(Q\).
AQA FP1 2005 June Q9
13 marks Standard +0.3
9 The function f is defined by $$f ( x ) = \frac { x ^ { 2 } + 4 x } { x ^ { 2 } + 9 }$$
    1. The graph of \(y = \mathrm { f } ( x )\) has an asymptote which is parallel to the \(x\)-axis. Find the equation of this asymptote.
    2. Explain why the graph of \(y = \mathrm { f } ( x )\) has no asymptotes parallel to the \(y\)-axis.
  2. Show that the equation \(\mathrm { f } ( x ) = k\) has two equal roots if \(9 k ^ { 2 } - 9 k - 4 = 0\).
  3. Hence find the coordinates of the two stationary points on the graph of \(y = \mathrm { f } ( x )\).
    SurnameOther Names
    Centre NumberCandidate Number
    Candidate Signature
    General Certificate of Education
    June 2005
    Advanced Subsidiary Examination MATHEMATICS
    MFP1
    Unit Further Pure 1 ASSESSMENT and
    QUALIFICATIONS
    ALLIANCE Wednesday 22 June 2005 Afternoon Session Insert for use in Question 7.
    Fill in the boxes at the top of this page.
    Fasten this insert securely to your answer book.
AQA FP1 2006 June Q1
9 marks Standard +0.3
1 The quadratic equation $$3 x ^ { 2 } - 6 x + 2 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the numerical values of \(\alpha + \beta\) and \(\alpha \beta\).
    1. Expand \(( \alpha + \beta ) ^ { 3 }\).
    2. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } = 4\).
  3. Find a quadratic equation with roots \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\), giving your answer in the form \(p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers.
AQA FP1 2006 June Q2
6 marks Moderate -0.3
2 A curve satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \log _ { 10 } x$$ Starting at the point \(( 2,3 )\) on the curve, use a step-by-step method with a step length of 0.2 to estimate the value of \(y\) at \(x = 2.4\). Give your answer to three decimal places.
AQA FP1 2006 June Q3
4 marks Moderate -0.8
3 Show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r \right) = k n ( n + 1 ) ( n - 1 )$$ where \(k\) is a rational number.
AQA FP1 2006 June Q4
5 marks Moderate -0.5
4 Find, in radians, the general solution of the equation $$\cos 3 x = \frac { \sqrt { 3 } } { 2 }$$ giving your answers in terms of \(\pi\).
AQA FP1 2006 June Q5
9 marks Moderate -0.3
5 The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\ - \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right]$$
  1. Find the matrix:
    1. \(\mathbf { M } ^ { 2 }\);
    2. \(\mathbf { M } ^ { 4 }\).
  2. Describe fully the geometrical transformation represented by \(\mathbf { M }\).
  3. Find the matrix \(\mathbf { M } ^ { 2006 }\).
AQA FP1 2006 June Q6
7 marks Moderate -0.3
6 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.
  1. Write down, in terms of \(x\) and \(y\), an expression for $$( z + \mathrm { i } ) ^ { * }$$ where \(( z + \mathrm { i } ) ^ { * }\) denotes the complex conjugate of \(( z + \mathrm { i } )\).
  2. Solve the equation $$( z + \mathrm { i } ) ^ { * } = 2 \mathrm { i } z + 1$$ giving your answer in the form \(a + b \mathrm { i }\).
AQA FP1 2006 June Q7
6 marks Standard +0.3
7
  1. Describe a geometrical transformation by which the hyperbola $$x ^ { 2 } - 4 y ^ { 2 } = 1$$ can be obtained from the hyperbola \(x ^ { 2 } - y ^ { 2 } = 1\).
  2. The diagram shows the hyperbola \(H\) with equation $$x ^ { 2 } - y ^ { 2 } - 4 x + 3 = 0$$
    \includegraphics[max width=\textwidth, alt={}]{e44987a7-2cdf-442a-aecb-abd3e889ecd4-4_951_1216_824_402}
    By completing the square, describe a geometrical transformation by which the hyperbola \(H\) can be obtained from the hyperbola \(x ^ { 2 } - y ^ { 2 } = 1\).
AQA FP1 2006 June Q8
10 marks Standard +0.3
8
  1. The function f is defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 1$$
    1. Express \(\mathrm { f } ( 1 + h ) - \mathrm { f } ( 1 )\) in the form $$p h + q h ^ { 2 } + r h ^ { 3 }$$ where \(p , q\) and \(r\) are integers.
    2. Use your answer to part (a)(i) to find the value of \(f ^ { \prime } ( 1 )\).
  2. The diagram shows the graphs of $$y = \frac { 1 } { x ^ { 2 } } \quad \text { and } \quad y = x + 1 \quad \text { for } \quad x > 0$$
    \includegraphics[max width=\textwidth, alt={}]{e44987a7-2cdf-442a-aecb-abd3e889ecd4-5_643_791_1160_596}
    The graphs intersect at the point \(P\).
    1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(\mathrm { f } ( x ) = 0\), where f is the function defined in part (a).
    2. Taking \(x _ { 1 } = 1\) as a first approximation to the root of the equation \(\mathrm { f } ( x ) = 0\), use the Newton-Raphson method to find a second approximation \(x _ { 2 }\) to the root.
      (3 marks)
  3. The region enclosed by the curve \(y = \frac { 1 } { x ^ { 2 } }\), the line \(x = 1\) and the \(x\)-axis is shaded on the diagram. By evaluating an improper integral, find the area of this region.
    (3 marks)
AQA FP1 2006 June Q9
16 marks Standard +0.8
9 A curve \(C\) has equation $$y = \frac { ( x + 1 ) ( x - 3 ) } { x ( x - 2 ) }$$
    1. Write down the coordinates of the points where \(C\) intersects the \(x\)-axis. (2 marks)
    2. Write down the equations of all the asymptotes of \(C\).
    1. Show that, if the line \(y = k\) intersects \(C\), then $$( k - 1 ) ( k - 4 ) \geqslant 0$$
    2. Given that there is only one stationary point on \(C\), find the coordinates of this stationary point.
      (No credit will be given for solutions based on differentiation.)
  3. Sketch the curve \(C\).
AQA FP1 2014 June Q1
5 marks Moderate -0.8
A curve passes through the point \((9, 6)\) and satisfies the differential equation $$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{2 + \sqrt{x}}$$ Use a step-by-step method with a step length of \(0.25\) to estimate the value of \(y\) at \(x = 9.5\). Give your answer to four decimal places. [5 marks]
AQA FP1 2014 June Q2
11 marks Standard +0.3
The quadratic equation $$2x^2 + 8x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\). [2 marks]
    1. Find the value of \(\alpha^2 + \beta^2\). [2 marks]
    2. Hence, or otherwise, show that \(\alpha^4 + \beta^4 = \frac{449}{2}\). [2 marks]
  3. Find a quadratic equation, with integer coefficients, which has roots $$2\alpha^4 + \frac{1}{\beta^2} \text{ and } 2\beta^4 + \frac{1}{\alpha^2}$$ [5 marks]
AQA FP1 2014 June Q3
4 marks Standard +0.3
Use the formulae for \(\sum_{r=1}^{n} r^3\) and \(\sum_{r=1}^{n} r^2\) to find the value of $$\sum_{r=3}^{60} r^2(r - 6)$$ [4 marks]
AQA FP1 2014 June Q4
6 marks Standard +0.3
Find the complex number \(z\) such that $$5iz + 3z^* + 16 = 8i$$ Give your answer in the form \(a + bi\), where \(a\) and \(b\) are real. [6 marks]
AQA FP1 2014 June Q5
5 marks Moderate -0.8
A curve \(C\) has equation \(y = x(x + 3)\).
  1. Find the gradient of the line passing through the point \((-5, 10)\) and the point on \(C\) with \(x\)-coordinate \(-5 + h\). Give your answer in its simplest form. [3 marks]
  2. Show how the answer to part (a) can be used to find the gradient of the curve \(C\) at the point \((-5, 10)\). State the value of this gradient. [2 marks]
AQA FP1 2014 June Q6
10 marks Standard +0.3
A curve \(C\) has equation \(y = \frac{1}{x(x + 2)}\).
  1. Write down the equations of all the asymptotes of \(C\). [2 marks]
  2. The curve \(C\) has exactly one stationary point. The \(x\)-coordinate of the stationary point is \(-1\).
    1. Find the \(y\)-coordinate of the stationary point. [1 mark]
    2. Sketch the curve \(C\). [2 marks]
  3. Solve the inequality $$\frac{1}{x(x + 2)} \leqslant \frac{1}{8}$$ [5 marks]
AQA FP1 2014 June Q7
10 marks Moderate -0.3
  1. Write down the \(2 \times 2\) matrix corresponding to each of the following transformations:
    1. a reflection in the line \(y = -x\); [1 mark]
    2. a stretch parallel to the \(y\)-axis of scale factor \(7\). [1 mark]
  2. Hence find the matrix corresponding to the combined transformation of a reflection in the line \(y = -x\) followed by a stretch parallel to the \(y\)-axis of scale factor \(7\). [2 marks]
  3. The matrix \(\mathbf{A}\) is defined by \(\mathbf{A} = \begin{bmatrix} -3 & -\sqrt{3} \\ -\sqrt{3} & 3 \end{bmatrix}\).
    1. Show that \(\mathbf{A}^2 = k\mathbf{I}\), where \(k\) is a constant and \(\mathbf{I}\) is the \(2 \times 2\) identity matrix. [1 mark]
    2. Show that the matrix \(\mathbf{A}\) corresponds to a combination of an enlargement and a reflection. State the scale factor of the enlargement and state the equation of the line of reflection in the form \(y = (\tan \theta)x\). [5 marks]
AQA FP1 2014 June Q8
9 marks Standard +0.3
  1. Find the general solution of the equation $$\cos\left(\frac{5}{4}x - \frac{\pi}{3}\right) = \frac{\sqrt{2}}{2}$$ giving your answer for \(x\) in terms of \(\pi\). [5 marks]
  2. Use your general solution to find the sum of all the solutions of the equation $$\cos\left(\frac{5}{4}x - \frac{\pi}{3}\right) = \frac{\sqrt{2}}{2}$$ that lie in the interval \(0 \leqslant x \leqslant 20\pi\). Give your answer in the form \(k\pi\), stating the exact value of \(k\). [4 marks]
AQA FP1 2014 June Q9
15 marks Standard +0.8
An ellipse \(E\) has equation $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$
  1. Sketch the ellipse \(E\), showing the values of the intercepts on the coordinate axes. [2 marks]
  2. Given that the line with equation \(y = x + k\) intersects the ellipse \(E\) at two distinct points, show that \(-5 < k < 5\). [5 marks]
  3. The ellipse \(E\) is translated by the vector \(\begin{bmatrix} a \\ b \end{bmatrix}\) to form another ellipse whose equation is \(9x^2 + 16y^2 + 18x - 64y = c\). Find the values of the constants \(a\), \(b\) and \(c\). [5 marks]
  4. Hence find an equation for each of the two tangents to the ellipse \(9x^2 + 16y^2 + 18x - 64y = c\) that are parallel to the line \(y = x\). [3 marks]
AQA FP1 2016 June Q1
7 marks Moderate -0.3
The quadratic equation \(x^2 - 6x + 14 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\). [2 marks]
  2. Find a quadratic equation, with integer coefficients, which has roots \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\). [5 marks]
AQA FP1 2016 June Q2
5 marks Moderate -0.8
A curve \(C\) has equation \(y = (2 - x)(1 + x) + 3\).
  1. A line passes through the point \((2, 3)\) and the point on \(C\) with \(x\)-coordinate \(2 + h\). Find the gradient of the line, giving your answer in its simplest form. [3 marks]
  2. Show how your answer to part (a) can be used to find the gradient of the curve \(C\) at the point \((2, 3)\). State the value of this gradient. [2 marks]