Questions — Edexcel (9685 questions)

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Edexcel FP1 Q8
Standard +0.3
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
Challenging +1.2
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
5 marks Moderate -0.8
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
7 marks Moderate -0.3
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
4 marks Moderate -0.8
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
5 marks Standard +0.3
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
9 marks Moderate -0.3
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
5 marks Standard +0.3
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
6 marks Standard +0.3
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
10 marks Standard +0.8
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
10 marks Standard +0.8
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
14 marks Moderate -0.3
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
7 marks Moderate -0.8
  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
9 marks Moderate -0.3
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
4 marks Standard +0.3
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
6 marks Standard +0.3
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
8 marks Moderate -0.8
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 \(\mathbf { A } ^ { - 1 }\).
Edexcel FP1 2010 January Q6
8 marks Moderate -0.3
6. Given that 2 and \(5 + 2 \mathrm { i }\) are roots of the equation $$x ^ { 3 } - 12 x ^ { 2 } + c x + d = 0 , \quad c , d \in \mathbb { R }$$
  1. write down the other complex root of the equation.
  2. Find the value of \(c\) and the value of \(d\).
  3. Show the three roots of this equation on a single Argand diagram.
Edexcel FP1 2010 January Q7
9 marks Standard +0.8
7. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a constant. The point \(P \left( c t , \frac { c } { t } \right)\) is a general point on \(H\).
  1. Show that the tangent to \(H\) at \(P\) has equation $$t ^ { 2 } y + x = 2 c t$$ The tangents to \(H\) at the points \(A\) and \(B\) meet at the point \(( 15 c , - c )\).
  2. Find, in terms of \(c\), the coordinates of \(A\) and \(B\).
Edexcel FP1 2010 January Q8
12 marks Standard +0.3
8. (a) Prove by induction that, for any positive integer \(n\), $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$ (b) Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\), show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 3 } + 3 r + 2 \right) = \frac { 1 } { 4 } n ( n + 2 ) \left( n ^ { 2 } + 7 \right)$$ (c) Hence evaluate \(\sum _ { r = 15 } ^ { 25 } \left( r ^ { 3 } + 3 r + 2 \right)\)
Edexcel FP1 2010 January Q9
12 marks Standard +0.3
9. $$\mathbf { M } = \left( \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right)$$
  1. Describe fully the geometrical transformation represented by the matrix \(\mathbf { M }\). The transformation represented by \(\mathbf { M }\) maps the point \(A\) with coordinates \(( p , q )\) onto the point \(B\) with coordinates \(( 3 \sqrt { } 2,4 \sqrt { } 2 )\).
  2. Find the value of \(p\) and the value of \(q\).
  3. Find, in its simplest surd form, the length \(O A\), where \(O\) is the origin.
  4. Find \(\mathbf { M } ^ { 2 }\). The point \(B\) is mapped onto the point \(C\) by the transformation represented by \(\mathbf { M } ^ { 2 }\).
  5. Find the coordinates of \(C\).
Edexcel FP1 2011 January Q1
5 marks Moderate -0.8
1. $$z = 5 - 3 \mathrm { i } , \quad w = 2 + 2 \mathrm { i }$$ Express in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants,
  1. \(z ^ { 2 }\),
  2. \(\frac { z } { w }\).
Edexcel FP1 2011 January Q2
6 marks Easy -1.2
2. $$\mathbf { A } = \left( \begin{array} { l l } 2 & 0 \\ 5 & 3 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r } - 3 & - 1 \\ 5 & 2 \end{array} \right)$$
  1. Find \(\mathbf { A B }\). Given that $$\mathbf { C } = \left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right)$$
  2. describe fully the geometrical transformation represented by \(\mathbf { C }\),
  3. write down \(\mathbf { C } ^ { 100 }\). \includegraphics[max width=\textwidth, alt={}, center]{d20fa710-2d91-4ac2-adbc-46ccdcb93380-03_99_97_2631_1784}
Edexcel FP1 2011 January Q3
10 marks Standard +0.3
3. $$f ( x ) = 5 x ^ { 2 } - 4 x ^ { \frac { 3 } { 2 } } - 6 , \quad x \geqslant 0$$ The root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval \([ 1.6,1.8 ]\).
  1. Use linear interpolation once on the interval \([ 1.6,1.8 ]\) to find an approximation to \(\alpha\). Give your answer to 3 decimal places.
  2. Differentiate \(\mathrm { f } ( x )\) to find \(\mathrm { f } ^ { \prime } ( x )\).
  3. Taking 1.7 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.
Edexcel FP1 2011 January Q4
4 marks Moderate -0.8
4. Given that \(2 - 4 \mathrm { i }\) is a root of the equation $$z ^ { 2 } + p z + q = 0 ,$$ where \(p\) and \(q\) are real constants,
  1. write down the other root of the equation,
  2. find the value of \(p\) and the value of \(q\).