Questions FP1 (1491 questions)

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AQA FP1 2013 June Q2
7 marks Moderate -0.8
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
8 marks Standard +0.3
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
7 marks Standard +0.3
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
8 marks Standard +0.3
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
11 marks Standard +0.8
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
11 marks Standard +0.3
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
6 marks Standard +0.3
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
14 marks Challenging +1.2
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 2015 June Q1
9 marks Standard +0.3
1 The quadratic equation \(2 x ^ { 2 } + 6 x + 7 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
  2. Find a quadratic equation, with integer coefficients, which has roots \(\alpha ^ { 2 } - 1\) and \(\beta ^ { 2 } - 1\).
  3. Hence find the values of \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).
AQA FP1 2015 June Q2
5 marks Challenging +1.2
2
  1. Explain why \(\int _ { 0 } ^ { 4 } \frac { x - 4 } { x ^ { 1.5 } } \mathrm {~d} x\) is an improper integral.
  2. Either find the value of the integral \(\int _ { 0 } ^ { 4 } \frac { x - 4 } { x ^ { 1.5 } } \mathrm {~d} x\) or explain why it does not have a finite value.
    [0pt] [4 marks]
    \includegraphics[max width=\textwidth, alt={}]{e45b07a3-e303-4caf-8f3a-5341bad7560a-04_1970_1712_737_150}
AQA FP1 2015 June Q3
11 marks Standard +0.3
3
  1. Show that \(( 2 + \mathrm { i } ) ^ { 3 }\) can be expressed in the form \(2 + b \mathrm { i }\), where \(b\) is an integer.
  2. It is given that \(2 + \mathrm { i }\) is a root of the equation $$z ^ { 3 } + p z + q = 0$$ where \(p\) and \(q\) are real numbers.
    1. Show that \(p = - 11\) and find the value of \(q\).
    2. Given that \(2 - \mathrm { i }\) is also a root of \(z ^ { 3 } + p z + q = 0\), find a quadratic factor of \(z ^ { 3 } + p z + q\) with real coefficients.
    3. Find the real root of the equation \(z ^ { 3 } + p z + q = 0\).
      \includegraphics[max width=\textwidth, alt={}]{e45b07a3-e303-4caf-8f3a-5341bad7560a-06_1568_1707_1139_155}
AQA FP1 2015 June Q4
6 marks Moderate -0.3
4
  1. Find the general solution, in degrees, of the equation $$2 \sin \left( 3 x + 45 ^ { \circ } \right) = 1$$
  2. Use your general solution to find the solution of \(2 \sin \left( 3 x + 45 ^ { \circ } \right) = 1\) that is closest to \(200 ^ { \circ }\).
    [0pt] [1 mark]
AQA FP1 2015 June Q5
13 marks Moderate -0.3
5
  1. The matrix \(\mathbf { A }\) is defined by \(\mathbf { A } = \left[ \begin{array} { c c } - 2 & c \\ d & 3 \end{array} \right]\).
    Given that the image of the point \(( 5,2 )\) under the transformation represented by \(\mathbf { A }\) is \(( - 2,1 )\), find the value of \(c\) and the value of \(d\).
    [0pt] [4 marks]
  2. The matrix \(\mathbf { B }\) is defined by \(\mathbf { B } = \left[ \begin{array} { c c } \sqrt { 2 } & \sqrt { 2 } \\ - \sqrt { 2 } & \sqrt { 2 } \end{array} \right]\).
    1. Show that \(\mathbf { B } ^ { 4 } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
    2. Describe the transformation represented by the matrix \(\mathbf { B }\) as a combination of two geometrical transformations.
    3. Find the matrix \(\mathbf { B } ^ { 17 }\). \(6 \quad \mathrm {~A}\) curve \(C _ { 1 }\) has equation $$\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$$
AQA FP1 2015 June Q7
15 marks Moderate -0.3
7
  1. The equation \(2 x ^ { 3 } + 5 x ^ { 2 } + 3 x - 132000 = 0\) has exactly one real root \(\alpha\).
    1. Show that \(\alpha\) lies in the interval \(39 < \alpha < 40\).
    2. Taking \(x _ { 1 } = 40\) as a first approximation to \(\alpha\), use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to \(\alpha\). Give your answer to two decimal places.
  2. Use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that $$\sum _ { r = 1 } ^ { n } 2 r ( 3 r + 2 ) = n ( n + p ) ( 2 n + q )$$ where \(p\) and \(q\) are integers.
    1. Express \(\log _ { 8 } 4 ^ { r }\) in the form \(\lambda r\), where \(\lambda\) is a rational number.
    2. By first finding a suitable cubic inequality for \(k\), find the greatest value of \(k\) for which \(\sum _ { r = k + 1 } ^ { 60 } ( 3 r + 2 ) \log _ { 8 } 4 ^ { r }\) is greater than 106060.
      [0pt] [4 marks]
AQA FP1 2015 June Q8
11 marks Challenging +1.2
8 A curve \(C\) has equation $$y = \frac { x ( x - 3 ) } { x ^ { 2 } + 3 }$$
  1. State the equation of the asymptote of \(C\).
  2. The line \(y = k\) intersects the curve \(C\). Show that \(4 k ^ { 2 } - 4 k - 3 \leqslant 0\).
  3. Hence find the coordinates of the stationary points of the curve \(C\). (No credit will be given for solutions based on differentiation.) \includegraphics[max width=\textwidth, alt={}, center]{e45b07a3-e303-4caf-8f3a-5341bad7560a-24_2488_1728_219_141}
Edexcel FP1 2019 June Q1
5 marks Moderate -0.8
  1. Use Simpson's rule with 4 intervals to estimate
$$\int _ { 0.4 } ^ { 2 } e ^ { x ^ { 2 } } d x$$
Edexcel FP1 2019 June Q2
4 marks Challenging +1.8
  1. Given that \(k\) is a real non-zero constant and that
$$y = x ^ { 3 } \sin k x$$ use Leibnitz's theorem to show that $$\frac { \mathrm { d } ^ { 5 } y } { \mathrm {~d} x ^ { 5 } } = \left( k ^ { 2 } x ^ { 2 } + A \right) k ^ { 3 } x \cos k x + B \left( k ^ { 2 } x ^ { 2 } + C \right) k ^ { 2 } \sin k x$$ where \(A\), \(B\) and \(C\) are integers to be determined.
Edexcel FP1 2019 June Q3
9 marks Challenging +1.2
3. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x - y ^ { 2 }$$
  1. Show that $$\frac { \mathrm { d } ^ { 5 } y } { \mathrm {~d} x ^ { 5 } } = a y \frac { \mathrm {~d} ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } + b \frac { \mathrm {~d} y } { \mathrm {~d} x } \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } + c \left( \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \right) ^ { 2 }$$ where \(a\), \(b\) and \(c\) are integers to be determined.
  2. Hence find a series solution, in ascending powers of \(x\) as far as the term in \(x ^ { 5 }\), of the differential equation (I), given that \(y = 1\) at \(x = 0\)
Edexcel FP1 2019 June Q4
8 marks Challenging +1.2
  1. The parabola \(C\) has equation
$$y ^ { 2 } = 16 x$$ The distinct points \(P \left( p ^ { 2 } , 4 p \right)\) and \(Q \left( q ^ { 2 } , 4 q \right)\) lie on \(C\), where \(p \neq 0 , q \neq 0\) The tangent to \(C\) at \(P\) and the tangent to \(C\) at \(Q\) meet at the point \(R ( - 28,6 )\).
Show that the area of triangle \(P Q R\) is 1331
Edexcel FP1 2019 June Q5
8 marks Challenging +1.2
5. $$I = \int \frac { 1 } { 4 \cos x - 3 \sin x } \mathrm {~d} x \quad 0 < x < \frac { \pi } { 4 }$$ Use the substitution \(t = \tan \left( \frac { x } { 2 } \right)\) to show that $$I = \frac { 1 } { 5 } \ln \left( \frac { 2 + \tan \left( \frac { x } { 2 } \right) } { 1 - 2 \tan \left( \frac { x } { 2 } \right) } \right) + k$$ where \(k\) is an arbitrary constant.
Edexcel FP1 2019 June Q6
17 marks Challenging +1.2
  1. The concentration of a drug in the bloodstream of a patient, \(t\) hours after the drug has been administered, where \(t \leqslant 6\), is modelled by the differential equation
$$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } C } { \mathrm {~d} t ^ { 2 } } - 5 t \frac { \mathrm {~d} C } { \mathrm {~d} t } + 8 C = t ^ { 3 }$$ where \(C\) is measured in micrograms per litre.
  1. Show that the transformation \(t = \mathrm { e } ^ { x }\) transforms equation (I) into the equation $$\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} C } { \mathrm {~d} x } + 8 C = \mathrm { e } ^ { 3 x }$$
  2. Hence find the general solution for the concentration \(C\) at time \(t\) hours. Given that when \(t = 6 , C = 0\) and \(\frac { \mathrm { d } C } { \mathrm {~d} t } = - 36\)
  3. find the maximum concentration of the drug in the bloodstream of the patient.
Edexcel FP1 2019 June Q7
10 marks Challenging +1.2
  1. With respect to a fixed origin \(O\), the points \(A\), \(B\) and \(C\) have coordinates \(( 3,4,5 ) , ( 10 , - 1,5 )\) and ( \(4,7 , - 9\) ) respectively.
The plane \(\Pi\) has equation \(4 x - 8 y + z = 2\) The line segment \(A B\) meets the plane \(\Pi\) at the point \(P\) and the line segment \(B C\) meets the plane \(\Pi\) at the point \(Q\).
  1. Show that, to 3 significant figures, the area of quadrilateral \(A P Q C\) is 38.5 The point \(D\) has coordinates \(( k , 4 , - 1 )\), where \(k\) is a constant.
    Given that the vectors \(\overrightarrow { A B } , \overrightarrow { A C }\) and \(\overrightarrow { A D }\) form three edges of a parallelepiped of volume 226
  2. find the possible values of the constant \(k\).
Edexcel FP1 2019 June Q8
14 marks Challenging +1.8
  1. The hyperbola \(H\) has equation
$$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$ The line \(l _ { 1 }\) is the tangent to \(H\) at the point \(P ( 4 \cosh \theta , 3 \sinh \theta )\).
The line \(l _ { 1 }\) meets the \(x\)-axis at the point \(A\).
The line \(l _ { 2 }\) is the tangent to \(H\) at the point \(( 4,0 )\).
The lines \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(B\) and the midpoint of \(A B\) is the point \(M\).
  1. Show that, as \(\theta\) varies, a Cartesian equation for the locus of \(M\) is $$y ^ { 2 } = \frac { 9 ( 4 - x ) } { 4 x } \quad p < x < q$$ where \(p\) and \(q\) are values to be determined. Let \(S\) be the focus of \(H\) that lies on the positive \(x\)-axis.
  2. Show that the distance from \(M\) to \(S\) is greater than 1
Edexcel FP1 2020 June Q1
5 marks Standard +0.3
  1. Use l'Hospital's Rule to show that
$$\lim _ { x \rightarrow \frac { \pi } { 2 } } \frac { \left( e ^ { \sin x } - \cos ( 3 x ) - e \right) } { \tan ( 2 x ) } = - \frac { 3 } { 2 }$$
Edexcel FP1 2020 June Q2
6 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9f127ab1-0e03-4f9f-87c2-01c553c54ee9-04_807_649_251_708} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the vertical cross-section of the entrance to a tunnel. The width at the base of the tunnel entrance is 2 metres and its maximum height is 3 metres. The shape of the cross-section can be modelled by the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 3 \cos \left( \frac { \pi } { 2 } x ^ { 2 } \right) \quad x \in [ - 1,1 ]$$ A wooden door of uniform thickness 85 mm is to be made to seal the tunnel entrance.
Use Simpson's rule with 6 intervals to estimate the volume of wood required for this door, giving your answer in \(\mathrm { m } ^ { 3 }\) to 4 significant figures.