Questions FP1 (1385 questions)

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Edexcel FP1 2015 June Q3
3. (a) Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that $$\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r + 4 ) = \frac { n } { 3 } ( n + 4 ) ( n + 5 )$$ for all positive integers \(n\).
(b) Hence show that $$\sum _ { r = n + 1 } ^ { 2 n } ( r + 1 ) ( r + 4 ) = \frac { n } { 3 } ( n + 1 ) ( a n + b )$$ where \(a\) and \(b\) are integers to be found.
Edexcel FP1 2015 June Q4
4. $$z _ { 1 } = 3 \mathrm { i } \text { and } z _ { 2 } = \frac { 6 } { 1 + \mathrm { i } \sqrt { 3 } }$$
  1. Express \(z _ { 2 }\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real numbers.
  2. Find the modulus and the argument of \(z _ { 2 }\), giving the argument in radians in terms of \(\pi\).
  3. Show the three points representing \(z _ { 1 } , z _ { 2 }\) and \(\left( z _ { 1 } + z _ { 2 } \right)\) respectively, on a single Argand diagram.
Edexcel FP1 2015 June Q5
5. The rectangular hyperbola \(H\) has equation \(x y = 9\) The point \(A\) on \(H\) has coordinates \(\left( 6 , \frac { 3 } { 2 } \right)\).
  1. Show that the normal to \(H\) at the point \(A\) has equation $$2 y - 8 x + 45 = 0$$ The normal at \(A\) meets \(H\) again at the point \(B\).
  2. Find the coordinates of \(B\).
Edexcel FP1 2015 June Q6
  1. (i) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\),
$$\left( \begin{array} { r r } 1 & 0
- 1 & 5 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 & 0
- \frac { 1 } { 4 } \left( 5 ^ { n } - 1 \right) & 5 ^ { n } \end{array} \right)$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$
Edexcel FP1 2015 June Q7
$$\mathbf { A } = \left( \begin{array} { r r } 5 k & 3 k - 1
- 3 & k + 1 \end{array} \right) , \text { where } k \text { is a real constant. }$$ Given that \(\mathbf { A }\) is a singular matrix, find the possible values of \(k\).
(ii) $$\mathbf { B } = \left( \begin{array} { l l } 10 & 5
- 3 & 3 \end{array} \right)$$ A triangle \(T\) is transformed onto a triangle \(T ^ { \prime }\) by the transformation represented by the matrix \(\mathbf { B }\). The vertices of triangle \(T ^ { \prime }\) have coordinates \(( 0,0 ) , ( - 20,6 )\) and \(( 10 c , 6 c )\), where \(c\) is a positive constant. The area of triangle \(T ^ { \prime }\) is 135 square units.
  1. Find the matrix \(\mathbf { B } ^ { - 1 }\)
  2. Find the coordinates of the vertices of the triangle \(T\), in terms of \(c\) where necessary.
  3. Find the value of \(c\).
Edexcel FP1 2015 June Q8
  1. The point \(P \left( 3 p ^ { 2 } , 6 p \right)\) lies on the parabola with equation \(y ^ { 2 } = 12 x\) and the point \(S\) is the focus of this parabola.
    1. Prove that \(S P = 3 \left( 1 + p ^ { 2 } \right)\)
    The point \(Q \left( 3 q ^ { 2 } , 6 q \right) , p \neq q\), also lies on this parabola.
    The tangent to the parabola at the point \(P\) and the tangent to the parabola at the point \(Q\) meet at the point \(R\).
  2. Find the equations of these two tangents and hence find the coordinates of the point \(R\), giving the coordinates in their simplest form.
  3. Prove that \(S R ^ { 2 } = S P \cdot S Q\)
Edexcel FP1 2016 June Q1
  1. Given that \(k\) is a real number and that
$$\mathbf { A } = \left( \begin{array} { c c } 1 + k & k
k & 1 - k \end{array} \right)$$ find the exact values of \(k\) for which \(\mathbf { A }\) is a singular matrix. Give your answers in their simplest form.
(3)
Edexcel FP1 2016 June Q2
2. $$f ( x ) = 3 x ^ { \frac { 3 } { 2 } } - 25 x ^ { - \frac { 1 } { 2 } } - 125 , \quad x > 0$$
  1. Find \(f ^ { \prime } ( x )\). The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [12, 13].
  2. Using \(x _ { 0 } = 12.5\) 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 decimal places.
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Edexcel FP1 2016 June Q3
  1. (a) Using the formula for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) write down, in terms of \(n\) only, an expression for
$$\sum _ { r = 1 } ^ { 3 n } r ^ { 2 }$$ (b) Show that, for all integers \(n\), where \(n > 0\) $$\sum _ { r = 2 n + 1 } ^ { 3 n } r ^ { 2 } = \frac { n } { 6 } \left( a n ^ { 2 } + b n + c \right)$$ where the values of the constants \(a\), \(b\) and \(c\) are to be found.
Edexcel FP1 2016 June Q4
4. $$z = \frac { 4 } { 1 + \mathrm { i } }$$ Find, in the form \(a + \mathrm { i } b\) where \(a , b \in \mathbb { R }\)
  1. \(Z\)
  2. \(z ^ { 2 }\) Given that \(z\) is a complex root of the quadratic equation \(x ^ { 2 } + p x + q = 0\), where \(p\) and \(q\) are real integers,
  3. find the value of \(p\) and the value of \(q\).
Edexcel FP1 2016 June Q5
5. Points \(P \left( a p ^ { 2 } , 2 a p \right)\) and \(Q \left( a q ^ { 2 } , 2 a q \right)\), where \(p ^ { 2 } \neq q ^ { 2 }\), lie on the parabola \(y ^ { 2 } = 4 a x\).
  1. Show that the chord \(P Q\) has equation $$y ( p + q ) = 2 x + 2 a p q$$ Given that this chord passes through the focus of the parabola,
  2. show that \(p q = - 1\)
  3. Using calculus find the gradient of the tangent to the parabola at \(P\).
  4. Show that the tangent to the parabola at \(P\) and the tangent to the parabola at \(Q\) are perpendicular.
Edexcel FP1 2016 June Q6
6. $$\mathbf { P } = \left( \begin{array} { c c } - \frac { 1 } { \sqrt { } 2 } & - \frac { 1 } { \sqrt { } 2 }
\frac { 1 } { \sqrt { } 2 } & - \frac { 1 } { \sqrt { } 2 } \end{array} \right)$$
  1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { P }\). The transformation \(U\) maps the point \(A\), with coordinates \(( p , q )\), onto the point \(B\), with coordinates \(( 6 \sqrt { } 2,3 \sqrt { } 2 )\).
  2. Find the value of \(p\) and the value of \(q\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the line with equation \(y = x\).
  3. Write down the matrix \(\mathbf { Q }\). The transformation \(U\) followed by the transformation \(V\) is the transformation \(T\). The transformation \(T\) is represented by the matrix \(\mathbf { R }\).
  4. Find the matrix \(\mathbf { R }\).
  5. Deduce that the transformation \(T\) is self-inverse.
Edexcel FP1 2016 June Q7
7. A complex number \(z\) is given by $$z = a + 2 i$$ where \(a\) is a non-zero real number.
  1. Find \(z ^ { 2 } + 2 z\) in the form \(x +\) iy where \(x\) and \(y\) are real expressions in terms of \(a\). Given that \(z ^ { 2 } + 2 z\) is real,
  2. find the value of \(a\). Using this value for \(a\),
  3. find the values of the modulus and argument of \(z\), giving the argument in radians, and giving your answers to 3 significant figures.
  4. Show the points \(P , Q\) and \(R\), representing the complex numbers \(z , z ^ { 2 }\) and \(z ^ { 2 } + 2 z\) respectively, on a single Argand diagram with origin \(O\).
  5. Describe fully the geometrical relationship between the line segments \(O P\) and \(Q R\).
Edexcel FP1 2016 June Q8
8. (i) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$\sum _ { r = 1 } ^ { n } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } = 1 - \frac { 1 } { ( n + 1 ) ^ { 2 } }$$ (ii) A sequence of positive rational numbers is defined by $$\begin{aligned} u _ { 1 } & = 3
u _ { n + 1 } & = \frac { 1 } { 3 } u _ { n } + \frac { 8 } { 9 } , \quad n \in \mathbb { Z } ^ { + } \end{aligned}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$u _ { n } = 5 \times \left( \frac { 1 } { 3 } \right) ^ { n } + \frac { 4 } { 3 }$$
Edexcel FP1 2016 June Q9
9. The rectangular hyperbola, \(H\), has cartesian equation \(x y = 25\)
  1. Show that an equation of the normal to \(H\) at the point \(P \left( 5 p , \frac { 5 } { p } \right) , p \neq 0\), is $$y - p ^ { 2 } x = \frac { 5 } { p } - 5 p ^ { 3 }$$ This normal meets the line with equation \(y = - x\) at the point \(A\).
  2. Show that the coordinates of \(A\) are $$\left( - \frac { 5 } { p } + 5 p , \frac { 5 } { p } - 5 p \right)$$ The point \(M\) is the midpoint of the line segment \(A P\).
    Given that \(M\) lies on the positive \(x\)-axis,
  3. find the exact value of the \(x\) coordinate of point \(M\).
Edexcel FP1 2017 June Q2
2. $$\mathbf { A } = \left( \begin{array} { r r } 2 & - 1
4 & 3 \end{array} \right) , \quad \mathbf { P } = \left( \begin{array} { r r } 3 & 6
11 & - 8 \end{array} \right)$$
  1. Find \(\mathbf { A } ^ { - 1 }\)
    (2) The transformation represented by the matrix \(\mathbf { B }\) followed by the transformation represented by the matrix \(\mathbf { A }\) is equivalent to the transformation represented by the matrix \(\mathbf { P }\).
  2. Find \(\mathbf { B }\), giving your answer in its simplest form.
Edexcel FP1 2017 June Q3
3. The rectangular hyperbola \(H\) has parametric equations $$x = 4 t , \quad y = \frac { 4 } { t } \quad t \neq 0$$ The points \(P\) and \(Q\) on this hyperbola have parameters \(t = \frac { 1 } { 4 }\) and \(t = 2\) respectively.
The line \(l\) passes through the origin \(O\) and is perpendicular to the line \(P Q\).
  1. Find an equation for \(l\).
  2. Find a cartesian equation for \(H\).
  3. Find the exact coordinates of the two points where \(l\) intersects \(H\). Give your answers in their simplest form.
Edexcel FP1 2017 June Q4
4. (i) The complex number \(w\) is given by $$w = \frac { p - 4 \mathrm { i } } { 2 - 3 \mathrm { i } }$$ where \(p\) is a real constant.
  1. Express \(w\) in the form \(a + b i\), where \(a\) and \(b\) are real constants. Give your answer in its simplest form in terms of \(p\). Given that \(\arg w = \frac { \pi } { 4 }\)
  2. find the value of \(p\).
    (ii) The complex number \(z\) is given by $$z = ( 1 - \lambda i ) ( 4 + 3 i )$$ where \(\lambda\) is a real constant. Given that $$| z | = 45$$ find the possible values of \(\lambda\).
    Give your answers as exact values in their simplest form.
    II
Edexcel FP1 2017 June Q5
5. (i) $$\mathbf { A } = \left( \begin{array} { l l } p & 2
3 & p \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r } - 5 & 4
6 & - 5 \end{array} \right)$$ where \(p\) is a constant.
  1. Find, in terms of \(p\), the matrix \(\mathbf { A B }\) Given that $$\mathbf { A B } + 2 \mathbf { A } = k \mathbf { I }$$ where \(k\) is a constant and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
  2. find the value of \(p\) and the value of \(k\).
    (ii) $$\mathbf { M } = \left( \begin{array} { r r } a & - 9
    1 & 2 \end{array} \right) , \text { where } a \text { is a real constant }$$ Triangle \(T\) has an area of 15 square units.
    Triangle \(T\) is transformed to the triangle \(T ^ { \prime }\) by the transformation represented by the matrix M. Given that the area of triangle \(T ^ { \prime }\) is 270 square units, find the possible values of \(a\).
Edexcel FP1 2017 June Q6
6. Given that 4 and \(2 \mathrm { i } - 3\) are roots of the equation $$x ^ { 3 } + a x ^ { 2 } + b x - 52 = 0$$ where \(a\) and \(b\) are real constants,
  1. write down the third root of the equation,
  2. find the value of \(a\) and the value of \(b\).
Edexcel FP1 2017 June Q7
7. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a constant and \(a > 0\) The point \(Q \left( a q ^ { 2 } , 2 a q \right) , q > 0\), lies on the parabola \(C\).
  1. Show that an equation of the tangent to \(C\) at \(Q\) is $$q y = x + a q ^ { 2 }$$ The tangent to \(C\) at the point \(Q\) meets the \(x\)-axis at the point \(X \left( - \frac { 1 } { 4 } a , 0 \right)\) and meets the directrix of \(C\) at the point \(D\).
  2. Find, in terms of \(a\), the coordinates of \(D\). Given that the point \(F\) is the focus of the parabola \(C\),
  3. find the area, in terms of \(a\), of the triangle \(F X D\), giving your answer in its simplest form.
Edexcel FP1 2017 June Q8
8. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } + 8 r + 3 \right) = \frac { 1 } { 2 } n ( 2 n + 5 ) ( n + 3 )$$ for all positive integers \(n\). Given that $$\sum _ { r = 1 } ^ { 12 } \left( 3 r ^ { 2 } + 8 r + 3 + k \left( 2 ^ { r - 1 } \right) \right) = 3520$$ (b) find the exact value of the constant \(k\).
Edexcel FP1 2017 June Q9
9. (i) A sequence of numbers is defined by $$\begin{gathered} u _ { 1 } = 6 , \quad u _ { 2 } = 27
u _ { n + 2 } = 6 u _ { n + 1 } - 9 u _ { n } \quad n \geqslant 1 \end{gathered}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$u _ { n } = 3 ^ { n } ( n + 1 )$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 3 ^ { 3 n - 2 } + 2 ^ { 3 n + 1 } \text { is divisible by } 19$$ \includegraphics[max width=\textwidth, alt={}, center]{536d7ec7-91b0-4fda-a485-2ac4a72c7d59-29_56_20_109_1950}
Edexcel FP1 2018 June Q1
1. $$f ( z ) = 2 z ^ { 3 } - 4 z ^ { 2 } + 15 z - 13$$ Given that \(\mathrm { f } ( z ) \equiv ( z - 1 ) \left( 2 z ^ { 2 } + a z + b \right)\), where \(a\) and \(b\) are real constants,
  1. find the value of \(a\) and the value of \(b\).
  2. Hence use algebra to find the three roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
Edexcel FP1 2018 June Q2
2. $$f ( x ) = \frac { 3 } { 2 } x ^ { 2 } + \frac { 4 } { 3 x } + 2 x - 5 , \quad x < 0$$ The equation \(\mathrm { f } ( x ) = 0\) has a single root \(\alpha\).
  1. Show that \(\alpha\) lies in the interval \([ - 3 , - 2.5 ]\)
  2. Taking - 3 as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.
  3. Use linear interpolation once on the interval \([ - 3 , - 2.5 ]\) to find another approximation to \(\alpha\), giving your answer to 3 decimal places.