Questions FP1 (1385 questions)

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AQA FP1 2011 June Q6
6
  1. Expand \(( 5 + h ) ^ { 3 }\).
  2. A curve has equation \(y = x ^ { 3 } - x ^ { 2 }\).
    1. Find the gradient of the line passing through the point \(( 5,100 )\) and the point on the curve for which \(x = 5 + h\). Give your answer in the form $$p + q h + r h ^ { 2 }$$ where \(p , q\) and \(r\) are integers.
    2. Show how the answer to part (b)(i) can be used to find the gradient of the curve at the point \(( 5,100 )\). State the value of this gradient.
AQA FP1 2011 June Q7
7 The matrix \(\mathbf { A }\) is defined by $$\mathbf { A } = \left[ \begin{array} { c c } - 1 & - \sqrt { 3 }
\sqrt { 3 } & - 1 \end{array} \right]$$
    1. Calculate the matrix \(\mathbf { A } ^ { 2 }\).
    2. Show that \(\mathbf { A } ^ { 3 } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
  2. Describe the single geometrical transformation, or combination of two geometrical transformations, corresponding to each of the matrices:
    1. \(\mathrm { A } ^ { 3 }\);
    2. A.
AQA FP1 2011 June Q8
8 A curve has equation \(y = \frac { 1 } { x ^ { 2 } - 4 }\).
    1. Write down the equations of the three asymptotes of the curve.
    2. Sketch the curve, showing the coordinates of any points of intersection with the coordinate axes.
  2. Hence, or otherwise, solve the inequality $$\frac { 1 } { x ^ { 2 } - 4 } < - 2$$
AQA FP1 2011 June Q9
9 The diagram shows a parabola \(P\) which has equation \(y = \frac { 1 } { 8 } x ^ { 2 }\), and another parabola \(Q\) which is the image of \(P\) under a reflection in the line \(y = x\). The parabolas \(P\) and \(Q\) intersect at the origin and again at a point \(A\).
The line \(L\) is a tangent to both \(P\) and \(Q\).
\includegraphics[max width=\textwidth, alt={}, center]{7441c4e6-5448-483b-b100-f8076e7e6cd8-5_1015_1089_623_479}
    1. Find the coordinates of the point \(A\).
    2. Write down an equation for \(Q\).
    3. Give a reason why the gradient of \(L\) must be - 1 .
    1. Given that the line \(y = - x + c\) intersects the parabola \(P\) at two distinct points, show that $$c > - 2$$
    2. Find the coordinates of the points at which the line \(L\) touches the parabolas \(P\) and \(Q\).
      (No credit will be given for solutions based on differentiation.)
AQA FP1 2012 June Q1
1 The quadratic equation $$5 x ^ { 2 } - 7 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Show that \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = \frac { 39 } { 5 }\).
  3. Find a quadratic equation, with integer coefficients, which has roots $$\alpha + \frac { 1 } { \alpha } \quad \text { and } \quad \beta + \frac { 1 } { \beta }$$ (5 marks)
AQA FP1 2012 June Q2
2 A curve has equation \(y = x ^ { 4 } + x\).
  1. Find the gradient of the line passing through the point \(( - 2,14 )\) and the point on the curve for which \(x = - 2 + h\). Give your answer in the form $$p + q h + r h ^ { 2 } + h ^ { 3 }$$ where \(p , q\) and \(r\) are integers.
  2. Show how the answer to part (a) can be used to find the gradient of the curve at the point ( \(- 2,14\) ). State the value of this gradient.
AQA FP1 2012 June Q3
3 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.
  1. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$\mathrm { i } ( z + 7 ) + 3 \left( z ^ { * } - \mathrm { i } \right)$$
  2. Hence find the complex number \(z\) such that $$\mathrm { i } ( z + 7 ) + 3 \left( z ^ { * } - \mathrm { i } \right) = 0$$
AQA FP1 2012 June Q4
4 Find the general solution, in degrees, of the equation $$\sin \left( 70 ^ { \circ } - \frac { 2 } { 3 } x \right) = \cos 20 ^ { \circ }$$
AQA FP1 2012 June Q5
5 The curve \(C\) has equation \(y = \frac { x } { ( x + 1 ) ( x - 2 ) }\).
The line \(L\) has equation \(y = - \frac { 1 } { 2 }\).
  1. Write down the equations of the asymptotes of \(C\).
  2. The line \(L\) intersects the curve \(C\) at two points. Find the \(x\)-coordinates of these two points.
  3. Sketch \(C\) and \(L\) on the same axes.
    (You are given that the curve \(C\) has no stationary points.)
  4. Solve the inequality $$\frac { x } { ( x + 1 ) ( x - 2 ) } \leqslant - \frac { 1 } { 2 }$$
AQA FP1 2012 June Q6
6
  1. Using surd forms, find the matrix of a rotation about the origin through \(135 ^ { \circ }\) anticlockwise.
  2. The matrix \(\mathbf { M }\) is defined by \(\mathbf { M } = \left[ \begin{array} { r r } - 1 & - 1
    1 & - 1 \end{array} \right]\).
    1. Given that \(\mathbf { M }\) represents an enlargement followed by a rotation, find the scale factor of the enlargement and the angle of the rotation.
    2. The matrix \(\mathbf { M } ^ { 2 }\) also represents an enlargement followed by a rotation. State the scale factor of the enlargement and the angle of the rotation.
    3. Show that \(\mathbf { M } ^ { 4 } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
    4. Deduce that \(\mathbf { M } ^ { 2012 } = - 2 ^ { n } \mathbf { I }\) for some positive integer \(n\).
AQA FP1 2012 June Q7
7 The equation $$24 x ^ { 3 } + 36 x ^ { 2 } + 18 x - 5 = 0$$ has one real root, \(\alpha\).
  1. Show that \(\alpha\) lies in the interval \(0.1 < x < 0.2\).
  2. Starting from the interval \(0.1 < x < 0.2\), use interval bisection twice to obtain an interval of width 0.025 within which \(\alpha\) must lie.
  3. Taking \(x _ { 1 } = 0.2\) as a first approximation to \(\alpha\), use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to \(\alpha\). Give your answer to four decimal places.
    (4 marks)
AQA FP1 2012 June Q8
8 The diagram shows the ellipse \(E\) with equation $$\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 4 } = 1$$ and the straight line \(L\) with equation $$y = x + 4$$ \includegraphics[max width=\textwidth, alt={}, center]{9f8cd5ed-f5cf-4cf6-8c92-9fd0819238ca-5_675_1120_708_468}
  1. Write down the coordinates of the points where the ellipse \(E\) intersects the coordinate axes.
  2. The ellipse \(E\) is translated by the vector \(\left[ \begin{array} { c } p
    0 \end{array} \right]\), where \(p\) is a constant. Write down the equation of the translated ellipse.
  3. Show that, if the translated ellipse intersects the line \(L\), the \(x\)-coordinates of the points of intersection must satisfy the equation $$9 x ^ { 2 } - ( 8 p - 40 ) x + \left( 4 p ^ { 2 } + 60 \right) = 0$$
  4. Given that the line \(L\) is a tangent to the translated ellipse, find the coordinates of the two possible points of contact.
    (No credit will be given for solutions based on differentiation.)
AQA FP1 2013 June Q1
1 The equation $$x ^ { 3 } - x ^ { 2 } + 4 x - 900 = 0$$ has exactly one real root, \(\alpha\). Taking \(x _ { 1 } = 10\) as a first approximation to \(\alpha\), use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to \(\alpha\). Give your answer to four significant figures.
(3 marks)
\includegraphics[max width=\textwidth, alt={}]{d74d6295-d5b8-46da-8812-c5bf7c7a35f1-02_1659_1709_1048_153}
AQA FP1 2013 June Q2
2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { c c } p & 2
4 & p \end{array} \right] \quad \mathbf { B } = \left[ \begin{array} { l l } 3 & 1
2 & 3 \end{array} \right]$$
  1. Find, in terms of \(p\), the matrices:
    1. \(\mathbf { A } - \mathbf { B }\);
    2. AB .
  2. Show that there is a value of \(p\) for which \(\mathbf { A } - \mathbf { B } + \mathbf { A B } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix, and state the corresponding value of \(k\).
AQA FP1 2013 June Q3
3
  1. Find the general solution, in degrees, of the equation $$\cos \left( 5 x + 40 ^ { \circ } \right) = \cos 65 ^ { \circ }$$
  2. Given that $$\sin \frac { \pi } { 12 } = \frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$$ express \(\sin \frac { \pi } { 12 }\) in the form \(\left( \cos \frac { \pi } { 4 } \right) ( \cos ( a \pi ) + \cos ( b \pi ) )\), where \(a\) and \(b\) are rational.
    (3 marks)
AQA FP1 2013 June Q4
4
  1. It is given that \(z = x + y \mathrm { i }\), where \(x\) and \(y\) are real numbers.
    1. Write down, in terms of \(x\) and \(y\), an expression for \(( z - 2 \mathrm { i } ) ^ { * }\).
    2. Solve the equation $$( z - 2 \mathrm { i } ) ^ { * } = 4 \mathrm { i } z + 3$$ giving your answer in the form \(a + b \mathrm { i }\).
  2. It is given that \(p + q \mathrm { i }\), where \(p\) and \(q\) are real numbers, is a root of the equation \(z ^ { 2 } + 10 \mathrm { i } z - 29 = 0\). Without finding the values of \(p\) and \(q\), state why \(p - q\) i is not a root of the equation \(z ^ { 2 } + 10 \mathrm { i } z - 29 = 0\).
AQA FP1 2013 June Q5
5
  1. A curve has equation \(y = 2 x ^ { 2 } - 5 x\).
    The point \(P\) on the curve has coordinates \(( 1 , - 3 )\).
    The point \(Q\) on the curve has \(x\)-coordinate \(1 + h\).
    1. Show that the gradient of the line \(P Q\) is \(2 h - 1\).
    2. Explain how the result of part (a)(i) can be used to show that the tangent to the curve at the point \(P\) is parallel to the line \(x + y = 0\).
  2. For the improper integral \(\int _ { 1 } ^ { \infty } x ^ { - 4 } \left( 2 x ^ { 2 } - 5 x \right) \mathrm { d } x\), either show that the integral has a finite value and state its value, or explain why the integral does not have a finite value.
AQA FP1 2013 June Q6
6 The equation $$2 x ^ { 2 } + 3 x - 6 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
  2. Hence show that \(\alpha ^ { 3 } + \beta ^ { 3 } = - \frac { 135 } { 8 }\).
  3. Find a quadratic equation, with integer coefficients, whose roots are \(\alpha + \frac { \alpha } { \beta ^ { 2 } }\) and \(\beta + \frac { \beta } { \alpha ^ { 2 } }\).
AQA FP1 2013 June Q7
7
  1. Show that the equation \(4 x ^ { 3 } - x - 540000 = 0\) has a root, \(\alpha\), in the interval \(51 < \alpha < 52\).
  2. It is given that \(S _ { n } = \sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 }\).
    1. Use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that \(S _ { n } = \frac { n } { 3 } \left( k n ^ { 2 } - 1 \right)\), where \(k\) is an integer to be found.
    2. Hence show that \(6 S _ { n }\) can be written as the product of three consecutive integers.
  3. Find the smallest value of \(N\) for which the sum of the squares of the first \(N\) odd numbers is greater than 180000 .
AQA FP1 2013 June Q8
8 The diagram shows two triangles, \(T _ { 1 }\) and \(T _ { 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{d74d6295-d5b8-46da-8812-c5bf7c7a35f1-09_972_967_358_589}
  1. Find the matrix which represents the stretch that maps triangle \(T _ { 1 }\) onto triangle \(T _ { 2 }\).
  2. The triangle \(T _ { 2 }\) is reflected in the line \(y = \sqrt { 3 } x\) to give a third triangle, \(T _ { 3 }\). Find, using surd forms where appropriate:
    1. the matrix which represents the reflection that maps triangle \(T _ { 2 }\) onto triangle \(T _ { 3 }\);
    2. the matrix which represents the combined transformation that maps triangle \(T _ { 1 }\) onto triangle \(T _ { 3 }\).
      (2 marks)
AQA FP1 2013 June Q9
9 A curve has equation $$y = \frac { x ^ { 2 } - 2 x + 1 } { x ^ { 2 } - 2 x - 3 }$$
  1. Find the equations of the three asymptotes of the curve.
    1. Show that if the line \(y = k\) intersects the curve then $$( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0$$
    2. Given that the equation \(( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0\) has real roots, show that $$k ^ { 2 } - k \geqslant 0$$
    3. Hence show that the curve has only one stationary point and find its coordinates.
      (No credit will be given for solutions based on differentiation.)
  3. Sketch the curve and its asymptotes.
AQA FP1 2014 June Q1
5 marks
1 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.
[0pt] [5 marks]
AQA FP1 2014 June Q2
2 The quadratic equation $$2 x ^ { 2 } + 8 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
    1. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
    2. Hence, or otherwise, show that \(\alpha ^ { 4 } + \beta ^ { 4 } = \frac { 449 } { 2 }\).
  3. Find a quadratic equation, with integer coefficients, which has roots $$2 \alpha ^ { 4 } + \frac { 1 } { \beta ^ { 2 } } \text { and } 2 \beta ^ { 4 } + \frac { 1 } { \alpha ^ { 2 } }$$
    \includegraphics[max width=\textwidth, alt={}]{2eaee88a-9e08-4392-8a4c-79fc9861603e-04_2443_1707_260_153}
AQA FP1 2014 June Q4
8 marks
4 Find the complex number \(z\) such that $$5 \mathrm { i } z + 3 z ^ { * } + 16 = 8 \mathrm { i }$$ Give your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real.
[0pt] [6 marks] \(5 \quad\) 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.
  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.
    [0pt] [2 marks] \(6 \quad\) A curve \(C\) has equation \(y = \frac { 1 } { x ( x + 2 ) }\).
  3. Write down the equations of all the asymptotes of \(C\).
  4. 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.
    2. Sketch the curve \(C\).
  5. Solve the inequality $$\frac { 1 } { x ( x + 2 ) } \leqslant \frac { 1 } { 8 }$$
AQA FP1 2014 June Q7
5 marks
7
  1. Write down the \(2 \times 2\) matrix corresponding to each of the following transformations:
    1. a reflection in the line \(y = - x\);
    2. a stretch parallel to the \(y\)-axis of scale factor 7 .
  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 .
  3. The matrix \(\mathbf { A }\) is defined by \(\mathbf { A } = \left[ \begin{array} { c c } - 3 & - \sqrt { 3 }
    - \sqrt { 3 } & 3 \end{array} \right]\).
    1. Show that \(\mathbf { A } ^ { 2 } = k \mathbf { I }\), where \(k\) is a constant and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
    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 \(y = ( \tan \theta ) x\).
      [0pt] [5 marks]