Questions — Edexcel (9685 questions)

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Edexcel FP1 2013 June Q7
8 marks Standard +0.8
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
10 marks Standard +0.3
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 marks Standard +0.3
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
8 marks Moderate -0.3
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
5 marks Moderate -0.8
  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
5 marks Standard +0.3
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
8 marks Moderate -0.5
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
10 marks Challenging +1.2
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
6 marks Standard +0.3
5. Prove, by induction, that \(3 ^ { 2 n } + 7\) is divisible by 8 for all positive integers \(n\).
Edexcel FP1 2013 June Q6
13 marks Standard +0.8
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
8 marks Challenging +1.2
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
11 marks Standard +0.3
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 marks Moderate -0.5
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
4 marks Moderate -0.5
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
5 marks Moderate -0.5
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
7 marks Moderate -0.3
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
9 marks Standard +0.8
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 \(Q\).
Edexcel FP1 2013 June Q5
10 marks Standard +0.3
5. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } ( r + 2 ) ( r + 3 ) = \frac { 1 } { 3 } n \left( n ^ { 2 } + 9 n + 26 \right)$$ for all positive integers \(n\).
(b) Hence show that $$\sum _ { r = n + 1 } ^ { 3 n } ( r + 2 ) ( r + 3 ) = \frac { 2 } { 3 } n \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be found.
Edexcel FP1 2013 June Q6
11 marks Standard +0.8
6. A parabola \(C\) has equation \(y ^ { 2 } = 4 a x , \quad a > 0\) The points \(P \left( a p ^ { 2 } , 2 a p \right)\) and \(Q \left( a q ^ { 2 } , 2 a q \right)\) lie on \(C\), where \(p \neq 0 , q \neq 0 , p \neq q\).
  1. Show that an equation of the tangent to the parabola at \(P\) is $$p y - x = a p ^ { 2 }$$
  2. Write down the equation of the tangent at \(Q\). The tangent at \(P\) meets the tangent at \(Q\) at the point \(R\).
  3. Find, in terms of \(p\) and \(q\), the coordinates of \(R\), giving your answers in their simplest form. Given that \(R\) lies on the directrix of \(C\),
  4. find the value of \(p q\).
Edexcel FP1 2013 June Q7
11 marks Moderate -0.3
7. $$z _ { 1 } = 2 + 3 \mathrm { i } , \quad z _ { 2 } = 3 + 2 \mathrm { i } , \quad z _ { 3 } = a + b \mathrm { i } , \quad a , b \in \mathbb { R }$$
  1. Find the exact value of \(\left| z _ { 1 } + z _ { 2 } \right|\). Given that \(w = \frac { z _ { 1 } z _ { 3 } } { z _ { 2 } }\),
  2. find \(w\) in terms of \(a\) and \(b\), giving your answer in the form \(x + \mathrm { i } y , \quad x , y \in \mathbb { R }\) Given also that \(w = \frac { 17 } { 13 } - \frac { 7 } { 13 } \mathrm { i }\),
  3. find the value of \(a\) and the value of \(b\),
  4. find \(\arg w\), giving your answer in radians to 3 decimal places.
Edexcel FP1 2013 June Q8
8 marks Standard +0.3
8. $$\mathbf { A } = \left( \begin{array} { c c } 6 & - 2 \\ - 4 & 1 \end{array} \right)$$ and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
  1. Prove that $$\mathbf { A } ^ { 2 } = 7 \mathbf { A } + 2 \mathbf { I }$$
  2. Hence show that $$\mathbf { A } ^ { - 1 } = \frac { 1 } { 2 } ( \mathbf { A } - 7 \mathbf { I } )$$ The transformation represented by \(\mathbf { A }\) maps the point \(P\) onto the point \(Q\).
    Given that \(Q\) has coordinates \(( 2 k + 8 , - 2 k - 5 )\), where \(k\) is a constant,
  3. find, in terms of \(k\), the coordinates of \(P\).
Edexcel FP1 2013 June Q9
10 marks Standard +0.8
9. (a) A sequence of numbers is defined by $$\begin{aligned} & u _ { 1 } = 8 \\ & u _ { n + 1 } = 4 u _ { n } - 9 n , \quad n \geqslant 1 \end{aligned}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$u _ { n } = 4 ^ { n } + 3 n + 1$$ (b) Prove by induction that, for \(m \in \mathbb { Z } ^ { + }\), $$\left( \begin{array} { l l } 3 & - 4 \\ 1 & - 1 \end{array} \right) ^ { m } = \left( \begin{array} { c c } 2 m + 1 & - 4 m \\ m & 1 - 2 m \end{array} \right)$$
Edexcel FP1 2014 June Q1
5 marks Moderate -0.3
  1. The roots of the equation
$$2 z ^ { 3 } - 3 z ^ { 2 } + 8 z + 5 = 0$$ are \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) Given that \(z _ { 1 } = 1 + 2 i\), find \(z _ { 2 }\) and \(z _ { 3 }\)
Edexcel FP1 2014 June Q2
9 marks Standard +0.3
2. $$\mathrm { f } ( x ) = 3 \cos 2 x + x - 2 , \quad - \pi \leqslant x < \pi$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [2,3].
    [0pt]
  2. Use linear interpolation once on the interval [2,3] to find an approximation to \(\alpha\). Give your answer to 3 decimal places.
  3. The equation \(\mathrm { f } ( x ) = 0\) has another root \(\beta\) in the interval \([ - 1,0 ]\). Starting with this interval, use interval bisection to find an interval of width 0.25 which contains \(\beta\).
Edexcel FP1 2014 June Q3
6 marks Moderate -0.3
3. (i) $$\mathbf { A } = \left( \begin{array} { c c } \frac { 1 } { \sqrt { } 2 } & \frac { - 1 } { \sqrt { } 2 } \\ \frac { 1 } { \sqrt { } 2 } & \frac { 1 } { \sqrt { } 2 } \end{array} \right)$$
  1. Describe fully the single transformation represented by the matrix \(\mathbf { A }\). The matrix \(\mathbf { B }\) represents an enlargement, scale factor - 2 , with centre the origin.
  2. Write down the matrix \(\mathbf { B }\).
    (ii) $$\mathbf { M } = \left( \begin{array} { c c } 3 & k \\ - 2 & 3 \end{array} \right) , \quad \text { where } k \text { is a positive constant. }$$ Triangle \(T\) has an area of 16 square units. Triangle \(T\) is transformed onto the triangle \(T ^ { \prime }\) by the transformation represented by the matrix M. Given that the area of the triangle \(T ^ { \prime }\) is 224 square units, find the value of \(k\).