Questions FP1 (1385 questions)

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Edexcel FP1 Q1
1. $$\mathrm { f } ( x ) = 2 x ^ { 3 } - 8 x ^ { 2 } + 7 x - 3$$ Given that \(x = 3\) is a solution of the equation \(\mathrm { f } ( x ) = 0\), solve \(\mathrm { f } ( x ) = 0\) completely.
Edexcel FP1 Q5
5. $$f ( x ) = 3 \sqrt { } x + \frac { 18 } { \sqrt { } x } - 20$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 1.1,1.2 ]\).
  2. Find \(\mathrm { f } ^ { \prime } ( x )\).
  3. Using \(x _ { 0 } = 1.1\) as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to find a second approximation to \(\alpha\), giving your answer to 3 significant figures.
Edexcel FP1 Q6
6. A series of positive integers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 6 \text { and } u _ { n + 1 } = 6 u _ { n } - 5 , \text { for } n \geq 1 .$$ Prove by induction that \(u _ { n } = 5 \times 6 ^ { n - 1 } + 1\), for \(n \geq 1\).
Edexcel FP1 Q7
7. Given that \(\mathbf { X } = \left( \begin{array} { r r } 2 & a
- 1 & - 1 \end{array} \right)\), where \(a\) is a constant, and \(a \neq 2\),
  1. find \(\mathbf { X } ^ { - 1 }\) in terms of \(a\). Given that \(\mathbf { X } + \mathbf { X } ^ { - 1 } = \mathbf { I }\), where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
  2. find the value of \(a\).
Edexcel FP1 Q8
8. A parabola has equation \(y ^ { 2 } = 4 a x , a > 0\). The point \(Q \left( a q ^ { 2 } , 2 a q \right)\) lies on the parabola.
  1. Show that an equation of the tangent to the parabola at \(Q\) is $$y q = x + a q ^ { 2 }$$ This tangent meets the \(y\)-axis at the point \(R\).
  2. Find an equation of the line \(l\) which passes through \(R\) and is perpendicular to the tangent at \(Q\).
  3. Show that \(l\) passes through the focus of the parabola.
  4. Find the coordinates of the point where \(l\) meets the directrix of the parabola.
Edexcel FP1 Q9
9. Given that \(z _ { 1 } = 3 + 2 \mathrm { i }\) and \(z _ { 2 } = \frac { 12 - 5 \mathrm { i } } { z _ { 1 } }\),
  1. find \(z _ { 2 }\) in the form \(a + i b\), where \(a\) and \(b\) are real.
  2. Show, on an Argand diagram, the point \(P\) representing \(z _ { 1 }\) and the point \(Q\) representing \(z _ { 2 }\).
  3. Given that \(O\) is the origin, show that \(\angle P O Q = \frac { \pi } { 2 }\). The circle passing through the points \(O , P\) and \(Q\) has centre \(C\). Find
  4. the complex number represented by \(C\),
  5. the exact value of the radius of the circle.
Edexcel FP1 Q1
1. $$f ( x ) = 2 x ^ { 3 } - 8 x ^ { 2 } + 7 x - 3$$ Given that \(x = 3\) is a solution of the equation \(\mathrm { f } ( x ) = 0\), solve \(\mathrm { f } ( x ) = 0\) completely.
(5)
Edexcel FP1 Q6
6. A series of positive integers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 6 \text { and } u _ { n + 1 } = 6 u _ { n } - 5 , \text { for } n \geqslant 1 .$$ Prove by induction that \(u _ { n } = 5 \times 6 ^ { n - 1 } + 1\), for \(n \geqslant 1\).
Edexcel FP1 Q8
8. A parabola has equation \(y ^ { 2 } = 4 a x , a > 0\). The point \(Q \left( a q ^ { 2 } , 2 a q \right)\) lies on the parabola.
  1. Show that an equation of the tangent to the parabola at \(Q\) is $$y q = x + a q ^ { 2 }$$ This tangent meets the \(y\)-axis at the point \(R\).
  2. Find an equation of the line \(l\) which passes through \(R\) and is perpendicular to the tangent at \(Q\).
  3. Show that \(l\) passes through the focus of the parabola.
  4. Find the coordinates of the point where \(I\) meets the directrix of the parabola.
Edexcel FP1 Q9
9. Given that \(z _ { 1 } = 3 + 2 i\) and \(z _ { 2 } = \frac { 12 - 5 i } { z _ { 1 } }\),
  1. find \(z _ { 2 }\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real.
  2. Show on an Argand diagram the point \(P\) representing \(z _ { 1 }\) and the point \(Q\) representing \(z _ { 2 }\).
  3. Given that \(O\) is the origin, show that \(\angle P O Q = \frac { \pi } { 2 }\). The circle passing through the points \(O , P\) and \(Q\) has centre \(C\). Find
  4. the complex number represented by C,
  5. the exact value of the radius of the circle.
Edexcel FP1 2009 January Q1
1. $$f ( x ) = 2 x ^ { 3 } - 8 x ^ { 2 } + 7 x - 3$$ Given that \(x = 3\) is a solution of the equation \(\mathrm { f } ( x ) = 0\), solve \(\mathrm { f } ( x ) = 0\) completely.
Edexcel FP1 2009 January Q2
2. (a) Show, using the formulae for \(\sum r\) and \(\sum r ^ { 2 }\), that $$\sum _ { r = 1 } ^ { n } \left( 6 r ^ { 2 } + 4 r - 1 \right) = n ( n + 2 ) ( 2 n + 1 )$$ (b) Hence, or otherwise, find the value of \(\sum _ { r = 11 } ^ { 20 } \left( 6 r ^ { 2 } + 4 r - 1 \right)\).
Edexcel FP1 2009 January Q3
3. The rectangular hyperbola, \(H\), has parametric equations \(x = 5 t , y = \frac { 5 } { t } , t \neq 0\).
  1. Write the cartesian equation of \(H\) in the form \(x y = c ^ { 2 }\). Points \(A\) and \(B\) on the hyperbola have parameters \(t = 1\) and \(t = 5\) respectively.
  2. Find the coordinates of the mid-point of \(A B\).
Edexcel FP1 2009 January Q4
4. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) } = \frac { n } { n + 1 }$$
Edexcel FP1 2009 January Q5
5. $$f ( x ) = 3 \sqrt { } x + \frac { 18 } { \sqrt { } x } - 20$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [1.1,1.2].
  2. Find \(\mathrm { f } ^ { \prime } ( x )\).
  3. Using \(x _ { 0 } = 1.1\) as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to find a second approximation to \(\alpha\), giving your answer to 3 significant figures.
Edexcel FP1 2009 January Q6
6. A series of positive integers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 6 \text { and } u _ { n + 1 } = 6 u _ { n } - 5 \text {, for } n \geqslant 1 \text {. }$$ Prove by induction that \(u _ { n } = 5 \times 6 ^ { n - 1 } + 1\), for \(n \geqslant 1\).
Edexcel FP1 2009 January Q7
7. Given that \(\mathbf { X } = \left( \begin{array} { c c } 2 & a
- 1 & - 1 \end{array} \right)\), where \(a\) is a constant, and \(a \neq 2\),
  1. find \(\mathbf { X } ^ { - 1 }\) in terms of \(a\). Given that \(\mathbf { X } + \mathbf { X } ^ { - 1 } = \mathbf { I }\), where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
  2. find the value of \(a\).
Edexcel FP1 2009 January Q8
8. A parabola has equation \(y ^ { 2 } = 4 a x , a > 0\). The point \(Q \left( a q ^ { 2 } , 2 a q \right)\) lies on the parabola.
  1. Show that an equation of the tangent to the parabola at \(Q\) is $$y q = x + a q ^ { 2 } .$$ This tangent meets the \(y\)-axis at the point \(R\).
  2. Find an equation of the line \(l\) which passes through \(R\) and is perpendicular to the tangent at \(Q\).
  3. Show that \(l\) passes through the focus of the parabola.
  4. Find the coordinates of the point where \(l\) meets the directrix of the parabola.
Edexcel FP1 2009 January Q9
9. Given that \(z _ { 1 } = 3 + 2 i\) and \(z _ { 2 } = \frac { 12 - 5 i } { z _ { 1 } }\),
  1. find \(z _ { 2 }\) in the form \(a + i b\), where \(a\) and \(b\) are real.
  2. Show on an Argand diagram the point \(P\) representing \(z _ { 1 }\) and the point \(Q\) representing \(z _ { 2 }\).
  3. Given that \(O\) is the origin, show that \(\angle P O Q = \frac { \pi } { 2 }\). The circle passing through the points \(O , P\) and \(Q\) has centre \(C\). Find
  4. the complex number represented by C,
  5. the exact value of the radius of the circle.
Edexcel FP1 2009 January Q10
10. $$\mathbf { A } = \left( \begin{array} { c c } 3 \sqrt { } 2 & 0
0 & 3 \sqrt { } 2 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { c c } 0 & 1
1 & 0 \end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { c c } \frac { 1 } { \sqrt { } 2 } & - \frac { 1 } { \sqrt { } 2 }
\frac { 1 } { \sqrt { } 2 } & \frac { 1 } { \sqrt { } 2 } \end{array} \right)$$
  1. Describe fully the transformations described by each of the matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\). It is given that the matrix \(\mathbf { D } = \mathbf { C A }\), and that the matrix \(\mathbf { E } = \mathbf { D B }\).
  2. Find \(\mathbf { D }\).
  3. Show that \(\mathbf { E } = \left( \begin{array} { c c } - 3 & 3
    3 & 3 \end{array} \right)\). The triangle \(O R S\) has vertices at the points with coordinates \(( 0,0 ) , ( - 15,15 )\) and \(( 4,21 )\). This triangle is transformed onto the triangle \(O R ^ { \prime } S ^ { \prime }\) by the transformation described by \(\mathbf { E }\).
  4. Find the coordinates of the vertices of triangle \(O R ^ { \prime } S ^ { \prime }\).
  5. Find the area of triangle \(O R ^ { \prime } S ^ { \prime }\) and deduce the area of triangle \(O R S\).
Edexcel FP1 2010 January Q1
  1. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by
$$z _ { 1 } = 2 + 8 i \quad \text { and } \quad z _ { 2 } = 1 - i$$ Find, showing your working,
  1. \(\frac { Z _ { 1 } } { Z _ { 2 } }\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real,
  2. the value of \(\left| \frac { z _ { 1 } } { z _ { 2 } } \right|\),
  3. the value of \(\arg \frac { Z _ { 1 } } { Z _ { 2 } }\), giving your answer in radians to 2 decimal places.
Edexcel FP1 2010 January Q2
2. $$f ( x ) = 3 x ^ { 2 } - \frac { 11 } { x ^ { 2 } }$$
  1. Write down, to 3 decimal places, the value of \(\mathrm { f } ( 1.3 )\) and the value of \(\mathrm { f } ( 1.4 )\). The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) between 1.3 and 1.4
    [0pt]
  2. Starting with the interval [1.3, 1.4], use interval bisection to find an interval of width 0.025 which contains \(\alpha\).
  3. Taking 1.4 as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\), giving your answer to 3 decimal places.
Edexcel FP1 2010 January Q3
3. A sequence of numbers is defined by $$\begin{aligned} u _ { 1 } & = 2
u _ { n + 1 } & = 5 u _ { n } - 4 , \quad n \geqslant 1 . \end{aligned}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + } , u _ { n } = 5 ^ { n - 1 } + 1\).
Edexcel FP1 2010 January Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cfad960a-f56a-4471-b4ad-92ab670d8121-05_791_874_265_518} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the parabola with equation \(y ^ { 2 } = 12 x\).
The point \(P\) on the parabola has \(x\)-coordinate \(\frac { 1 } { 3 }\).
The point \(S\) is the focus of the parabola.
  1. Write down the coordinates of \(S\). The points \(A\) and \(B\) lie on the directrix of the parabola.
    The point \(A\) is on the \(x\)-axis and the \(y\)-coordinate of \(B\) is positive. Given that \(A B P S\) is a trapezium,
  2. calculate the perimeter of \(A B P S\).
Edexcel FP1 2010 January Q5
5. \(\mathbf { A } = \left( \begin{array} { c c } a & - 5
2 & a + 4 \end{array} \right)\), where \(a\) is real.
  1. Find \(\operatorname { det } \mathbf { A }\) in terms of \(a\).
  2. Show that the matrix \(\mathbf { A }\) is non-singular for all values of \(a\). Given that \(a = 0\),
  3. find \(\mathbf { A } ^ { - 1 }\).