Questions — Edexcel FP1 (269 questions)

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Edexcel FP1 2012 June Q8
8. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a positive constant. The point \(P \left( c t , \frac { c } { t } \right) , t \neq 0\), is a general point on \(H\).
  1. Show that an equation for the tangent to \(H\) at \(P\) is $$x + t ^ { 2 } y = 2 c t$$ The tangent to \(H\) at the point \(P\) meets the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\). Given that the area of the triangle \(O A B\), where \(O\) is the origin, is 36 ,
  2. find the exact value of \(c\), expressing your answer in the form \(k \sqrt { } 2\), where \(k\) is an integer.
Edexcel FP1 2012 June Q9
9. $$\mathbf { M } = \left( \begin{array} { r r } 3 & 4
2 & - 5 \end{array} \right)$$
  1. Find \(\operatorname { det } \mathbf { M }\). The transformation represented by \(\mathbf { M }\) maps the point \(S ( 2 a - 7 , a - 1 )\), where \(a\) is a constant, onto the point \(S ^ { \prime } ( 25 , - 14 )\).
  2. Find the value of \(a\). The point \(R\) has coordinates \(( 6,0 )\). Given that \(O\) is the origin,
  3. find the area of triangle \(O R S\). Triangle \(O R S\) is mapped onto triangle \(O R ^ { \prime } S ^ { \prime }\) by the transformation represented by \(\mathbf { M }\).
  4. Find the area of triangle \(O R ^ { \prime } S ^ { \prime }\). Given that $$\mathbf { A } = \left( \begin{array} { r r } 0 & - 1
    1 & 0 \end{array} \right)$$
  5. describe fully the single geometrical transformation represented by \(\mathbf { A }\). The transformation represented by \(\mathbf { A }\) followed by the transformation represented by \(\mathbf { B }\) is equivalent to the transformation represented by \(\mathbf { M }\).
  6. Find B.
Edexcel FP1 2012 June Q10
10. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$f ( n ) = 2 ^ { 2 n - 1 } + 3 ^ { 2 n - 1 } \text { is divisible by } 5 .$$
Edexcel FP1 2013 June Q1
  1. The complex numbers \(z\) and \(w\) are given by
$$z = 8 + 3 \mathrm { i } , \quad w = - 2 \mathrm { i }$$ Express in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants,
  1. \(z - w\),
  2. \(z w\).
Edexcel FP1 2013 June Q2
2. (i) $$\mathbf { A } = \left( \begin{array} { c c } 2 k + 1 & k
- 3 & - 5 \end{array} \right) , \quad \text { where } k \text { is a constant }$$ Given that $$\mathbf { B } = \mathbf { A } + 3 \mathbf { I }$$ where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix, find
  1. \(\mathbf { B }\) in terms of \(k\),
  2. the value of \(k\) for which \(\mathbf { B }\) is singular.
    (ii) Given that $$\mathbf { C } = \left( \begin{array} { r } 2
    - 3
Edexcel FP1 2013 June Q4
4 \end{array} \right) , \quad \mathbf { D } = \left( \begin{array} { l l l } 2 & - 1 & 5 \end{array} \right)$$ and $$\mathbf { E } = \mathbf { C D }$$ find \(\mathbf { E }\).
3. $$f ( x ) = \frac { 1 } { 2 } x ^ { 4 } - x ^ { 3 } + x - 3$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) between \(x = 2\) and \(x = 2.5\)
    [0pt]
  2. Starting with the interval [2,2.5] use interval bisection twice to find an interval of width 0.125 which contains \(\alpha\). The equation \(\mathrm { f } ( x ) = 0\) has a root \(\beta\) in the interval \([ - 2 , - 1 ]\).
  3. Taking - 1.5 as a first approximation to \(\beta\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\beta\). Give your answer to 2 decimal places.
    4. $$f ( x ) = \left( 4 x ^ { 2 } + 9 \right) \left( x ^ { 2 } - 2 x + 5 \right)$$
  4. Find the four roots of \(\mathrm { f } ( x ) = 0\)
  5. Show the four roots of \(\mathrm { f } ( x ) = 0\) on a single Argand diagram.
Edexcel FP1 2013 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3f8492ef-c576-4642-b75f-1735387e11ba-06_828_1091_228_422} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a rectangular hyperbola \(H\) with parametric equations $$x = 3 t , \quad y = \frac { 3 } { t } , \quad t \neq 0$$ The line \(L\) with equation \(6 y = 4 x - 15\) intersects \(H\) at the point \(P\) and at the point \(Q\) as shown in Figure 1.
  1. Show that \(L\) intersects \(H\) where \(4 t ^ { 2 } - 5 t - 6 = 0\)
  2. Hence, or otherwise, find the coordinates of points \(P\) and \(Q\).
Edexcel FP1 2013 June Q6
6. $$\mathbf { A } = \left( \begin{array} { r r } 0 & 1
- 1 & 0 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { l l } 2 & 3
1 & 4 \end{array} \right)$$ The transformation represented by \(\mathbf { B }\) followed by the transformation represented by \(\mathbf { A }\) is equivalent to the transformation represented by \(\mathbf { P }\).
  1. Find the matrix \(\mathbf { P }\). Triangle \(T\) is transformed to the triangle \(T ^ { \prime }\) by the transformation represented by \(\mathbf { P }\). Given that the area of triangle \(T ^ { \prime }\) is 24 square units,
  2. find the area of triangle \(T\). Triangle \(T ^ { \prime }\) is transformed to the original triangle \(T\) by the matrix represented by \(\mathbf { Q }\).
  3. Find the matrix \(\mathbf { Q }\).
Edexcel FP1 2013 June Q7
7. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(P \left( a t ^ { 2 } , 2 a t \right)\) is a general point on \(C\).
  1. Show that the equation of the tangent to \(C\) at \(P \left( a t ^ { 2 } , 2 a t \right)\) is $$t y = x + a t ^ { 2 }$$ The tangent to \(C\) at \(P\) meets the \(y\)-axis at a point \(Q\).
  2. Find the coordinates of \(Q\). Given that the point \(S\) is the focus of \(C\),
  3. show that \(P Q\) is perpendicular to \(S Q\).
Edexcel FP1 2013 June Q8
8. (a) Prove by induction, that for \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } r ( 2 r - 1 ) = \frac { 1 } { 6 } n ( n + 1 ) ( 4 n - 1 )$$ (b) Hence, show that $$\sum _ { r = n + 1 } ^ { 3 n } r ( 2 r - 1 ) = \frac { 1 } { 3 } n \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be found.
Edexcel FP1 2013 June Q9
9. The complex number \(w\) is given by $$w = 10 - 5 \mathrm { i }$$
  1. Find \(| w |\).
  2. Find arg \(w\), giving your answer in radians to 2 decimal places. The complex numbers \(z\) and \(w\) satisfy the equation $$( 2 + i ) ( z + 3 i ) = w$$
  3. Use algebra to find \(z\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. Given that $$\arg ( \lambda + 9 i + w ) = \frac { \pi } { 4 }$$ where \(\lambda\) is a real constant,
  4. find the value of \(\lambda\).
Edexcel FP1 2013 June Q10
10. (i) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) and \(\sum _ { r = 1 } ^ { n } r\) to evaluate $$\sum _ { r = 1 } ^ { 24 } \left( r ^ { 3 } - 4 r \right)$$ (ii) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that $$\sum _ { r = 0 } ^ { n } \left( r ^ { 2 } - 2 r + 2 n + 1 \right) = \frac { 1 } { 6 } ( n + 1 ) ( n + a ) ( b n + c )$$ for all integers \(n \geqslant 0\), where \(a , b\) and \(c\) are constant integers to be found.
Edexcel FP1 2013 June Q1
  1. \(\mathbf { M } = \left( \begin{array} { c c } a & 1
    1 & 2 - a \end{array} \right)\), where \(a\) is a constant.
    1. Find det M in terms of \(a\).
      (2)
    A triangle \(T\) is transformed to \(T ^ { \prime }\) by the matrix M .
    Given that the area of \(T ^ { \prime }\) is 0 ,
  2. find the value of \(a\).
    (3)
Edexcel FP1 2013 June Q2
2. $$f ( z ) = z ^ { 3 } + 5 z ^ { 2 } + 11 z + 15$$ Given that \(z = 2 i - 1\) is a solution of the equation \(f ( z ) = 0\), use algebra to solve \(f ( z ) = 0\) completely.
(5)
Edexcel FP1 2013 June Q3
3. $$z _ { 1 } = \frac { 1 } { 2 } ( 1 + \mathrm { i } \sqrt { } 3 ) , z _ { 2 } = - \sqrt { } 3 + \mathrm { i }$$
  1. Express \(z _ { 1 }\) and \(z _ { 2 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) giving exact values of \(r\) and \(\theta\).
    (4)
  2. Find \(\left| z _ { 1 } z _ { 2 } \right|\).
  3. Show and label \(z _ { 1 }\) and \(z _ { 2 }\) on a single Argand diagram.
    (2)
Edexcel FP1 2013 June Q4
4. The hyperbola \(H\) has equation $$x y = 3$$ The point \(Q ( 1,3 )\) is on \(H\).
  1. Find the equation of the normal to \(H\) at \(Q\) in the form \(y = a x + b\), where \(a\) and \(b\) are constants.
    (5) The normal at \(Q\) intersects \(H\) again at the point \(R\).
  2. Find the coordinates of \(R\).
    (5)
Edexcel FP1 2013 June Q5
5. Prove, by induction, that \(3 ^ { 2 n } + 7\) is divisible by 8 for all positive integers \(n\).
Edexcel FP1 2013 June Q6
6. A curve \(C\) is in the form of a parabola with equation \(y ^ { 2 } = 4 x\).
\(P \left( p ^ { 2 } , 2 p \right)\) and \(Q \left( q ^ { 2 } , 2 q \right)\) are points on \(C\) where \(p > q\).
  1. Find an equation of the tangent to \(C\) at \(P\).
    (5)
  2. The tangent at \(P\) and the tangent at \(Q\) are perpendicular and intersect at the point \(R ( - 1,2 )\).
    1. Find the exact value of \(p\) and the exact value of \(q\).
    2. Find the area of the triangle \(P Q R\).
Edexcel FP1 2013 June Q7
7. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r - 1 ) = \frac { n ( n + 1 ) ( 3 n + 2 ) ( n - 1 ) } { 12 }$$ for all positive integers \(n\).
(b) Hence find the sum of the series $$10 ^ { 2 } \times 9 + 11 ^ { 2 } \times 10 + 12 ^ { 2 } \times 11 + \ldots + 50 ^ { 2 } \times 49$$
Edexcel FP1 2013 June Q8
8. $$f ( x ) = x ^ { 3 } - 2 x - 3$$
  1. Show that \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), in the interval \([ 1,2 ]\).
  2. Starting with the interval \([ 1,2 ]\), use interval bisection twice to find an interval of width 0.25 which contains \(\alpha\).
  3. Using \(x _ { 0 } = 1.8\) 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 2013 June Q9
9. With reference to a fixed origin \(O\) and coordinate axes \(O x\) and \(O y\), a transformation from \(\mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix \(A\) where $$A = \left( \begin{array} { c c } 3 & 1
1 & - 2 \end{array} \right)$$
  1. Find \(\mathrm { A } ^ { 2 }\).
  2. Show that the matrix A is non-singular.
  3. Find \(\mathrm { A } ^ { - 1 }\). The transformation represented by matrix A maps the point \(P\) onto the point \(Q\).
    Given that \(Q\) has coordinates \(( k - 1,2 - k )\), where \(k\) is a constant,
  4. show that \(P\) lies on the line with equation \(y = 4 x - 1\)
Edexcel FP1 2013 June Q1
1. $$\mathbf { M } = \left( \begin{array} { c c } x & x - 2
3 x - 6 & 4 x - 11 \end{array} \right)$$ Given that the matrix \(\mathbf { M }\) is singular, find the possible values of \(x\).
Edexcel FP1 2013 June Q2
2. $$\mathrm { f } ( x ) = \cos \left( x ^ { 2 } \right) - x + 3 , \quad 0 < x < \pi$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 2.5,3 ]\).
    [0pt]
  2. Use linear interpolation once on the interval [2.5,3] to find an approximation for \(\alpha\), giving your answer to 2 decimal places.
Edexcel FP1 2013 June Q3
3. Given that \(x = \frac { 1 } { 2 }\) is a root of the equation $$2 x ^ { 3 } - 9 x ^ { 2 } + k x - 13 = 0 , \quad k \in \mathbb { R }$$ find
  1. the value of \(k\),
  2. the other 2 roots of the equation.
Edexcel FP1 2013 June Q4
4. The rectangular hyperbola \(H\) has Cartesian equation \(x y = 4\) The point \(P \left( 2 t , \frac { 2 } { t } \right)\) lies on \(H\), where \(t \neq 0\)
  1. Show that an equation of the normal to \(H\) at the point \(P\) is $$t y - t ^ { 3 } x = 2 - 2 t ^ { 4 }$$ The normal to \(H\) at the point where \(t = - \frac { 1 } { 2 }\) meets \(H\) again at the point \(Q\).
  2. Find the coordinates of the point \(Q\).