Questions F1 (197 questions)

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Edexcel F1 2024 January Q8
  1. (a) Use the standard results for summations to show that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } r \left( 2 r ^ { 2 } - 3 r - 1 \right) = \frac { 1 } { 2 } n ( n + 1 ) ^ { 2 } ( n - 2 )$$ (b) Hence show that, for all positive integers \(n\), $$\sum _ { r = n } ^ { 2 n } r \left( 2 r ^ { 2 } - 3 r - 1 \right) = \frac { 1 } { 2 } n ( n - 1 ) ( a n + b ) ( c n + d )$$ where \(a\), \(b\), \(c\) and \(d\) are integers to be determined.
Edexcel F1 2024 January Q9
  1. Given that
$$\frac { 3 z - 1 } { 2 } = \frac { \lambda + 5 i } { \lambda - 4 i }$$ where \(\lambda\) is a real constant,
  1. determine \(z\), giving your answer in the form \(x + y i\), where \(x\) and \(y\) are real and in terms of \(\lambda\). Given also that \(\arg z = \frac { \pi } { 4 }\)
  2. find the possible values of \(\lambda\).
Edexcel F1 2024 January Q10
  1. (i) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\)
$$\left( \begin{array} { r r } 5 & - 1
4 & 1 \end{array} \right) ^ { n } = 3 ^ { n - 1 } \left( \begin{array} { c c } 2 n + 3 & - n
4 n & 3 - 2 n \end{array} \right)$$ (ii) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 8 ^ { 2 n + 1 } + 6 ^ { 2 n - 1 }$$ is divisible by 7
Edexcel F1 2014 June Q1
  1. Find the value of
$$\sum _ { r = 1 } ^ { 200 } ( r + 1 ) ( r - 1 )$$
Edexcel F1 2014 June Q2
2. Given that \(- 2 + 3 \mathrm { i }\) 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 \(q\).
Edexcel F1 2014 June Q3
3. $$\mathbf { A } = \left( \begin{array} { l l } 4 & - 2
a & - 3 \end{array} \right)$$ where \(a\) is a real constant and \(a \neq 6\)
  1. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\). Given that \(\mathbf { A } + 2 \mathbf { A } ^ { - 1 } = \mathbf { I }\), where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
  2. find the value of \(a\).
Edexcel F1 2014 June Q4
4. $$\mathrm { f } ( x ) = x ^ { \frac { 3 } { 2 } } - 3 x ^ { \frac { 1 } { 2 } } - 3 , \quad x > 0$$ Given that \(\alpha\) is the only real root of the equation \(\mathrm { f } ( x ) = 0\),
  1. show that \(4 < \alpha < 5\)
  2. Taking 4.5 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.
    [0pt]
  3. Use linear interpolation once on the interval [4,5] to find another approximation to \(\alpha\), giving your answer to 3 decimal places.
Edexcel F1 2014 June Q5
  1. Given that \(z _ { 1 } = - 3 - 4 \mathrm { i }\) and \(z _ { 2 } = 4 - 3 \mathrm { i }\)
    1. show, on an Argand diagram, the point \(P\) representing \(z _ { 1 }\) and the point \(Q\) representing \(z _ { 2 }\)
    2. Given that \(O\) is the origin, show that \(O P\) is perpendicular to \(O Q\).
    3. Show the point \(R\) on your diagram, where \(R\) represents \(z _ { 1 } + z _ { 2 }\)
    4. Prove that \(O P R Q\) is a square.
Edexcel F1 2014 June Q6
6. It is given that \(\alpha\) and \(\beta\) are roots of the equation \(3 x ^ { 2 } + 5 x - 1 = 0\)
  1. Find the exact value of \(\alpha ^ { 3 } + \beta ^ { 3 }\)
  2. Find a quadratic equation which has roots \(\frac { \alpha ^ { 2 } } { \beta }\) and \(\frac { \beta ^ { 2 } } { \alpha }\), giving your answer in the form \(a x ^ { 2 } + b x + c = 0\), where \(a\), \(b\) and \(c\) are integers.
Edexcel F1 2014 June Q7
7. $$\mathbf { P } = \left( \begin{array} { c c } \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 }
\frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$
  1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { P }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the \(x\)-axis.
  2. Write down the matrix \(\mathbf { Q }\). Given that \(V\) followed by \(U\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\),
  3. find the matrix \(\mathbf { R }\).
  4. Show that there is a real number \(k\) for which the transformation \(T\) maps the point \(( 1 , k )\) onto itself. Give the exact value of \(k\) in its simplest form.
Edexcel F1 2014 June Q8
8. The hyperbola \(H\) has cartesian equation \(x y = 16\) The parabola \(P\) has parametric equations \(x = 8 t ^ { 2 } , y = 16 t\).
  1. Find, using algebra, the coordinates of the point \(A\) where \(H\) meets \(P\). Another point \(B ( 8,2 )\) lies on the hyperbola \(H\).
  2. Find the equation of the normal to \(H\) at the point (8, 2), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
  3. Find the coordinates of the points where this normal at \(B\) meets the parabola \(P\).
Edexcel F1 2014 June Q9
9. (i) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r + 2 ) = \frac { n ( n + 1 ) ( n + 2 ) ( n + 3 ) } { 4 }$$ (ii) Prove by induction that, $$4 ^ { n } + 6 n + 8 \text { is divisible by } 18$$ for all positive integers \(n\).
\includegraphics[max width=\textwidth, alt={}, center]{df5ab400-5cb1-4b51-8b0a-52dc3587f81a-16_62_44_2476_1889}
Edexcel F1 2015 June Q1
  1. Given that
$$2 z ^ { 3 } - 5 z ^ { 2 } + 7 z - 6 \equiv ( 2 z - 3 ) \left( z ^ { 2 } + a z + b \right)$$ where \(a\) and \(b\) are real constants,
  1. find the value of \(a\) and the value of \(b\).
  2. Given that \(z\) is a complex number, find the three exact roots of the equation $$2 z ^ { 3 } - 5 z ^ { 2 } + 7 z - 6 = 0$$
Edexcel F1 2015 June Q2
  1. Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that
$$\sum _ { r = 1 } ^ { n } ( 3 r - 2 ) ^ { 2 } = \frac { n } { 2 } \left( a n ^ { 2 } + b n + c \right)$$ where \(a , b\) and \(c\) are integers to be found.
Edexcel F1 2015 June Q3
3. It is given that \(\alpha\) and \(\beta\) are roots of the equation $$2 x ^ { 2 } - 7 x + 4 = 0$$
  1. Find the exact value of \(\alpha ^ { 2 } + \beta ^ { 2 }\)
  2. Find a quadratic equation which has roots \(\frac { \alpha } { \beta }\) and \(\frac { \beta } { \alpha }\), giving your answer in the form \(a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel F1 2015 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{04f06398-ff29-4690-a6fe-825d089fba39-05_663_665_228_644} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the parabola \(C\) with equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(S\) is the focus of \(C\) and the point \(Q\) lies on the directrix of \(C\). The point \(P\) lies on \(C\) where \(y > 0\) and the line segment \(Q P\) is parallel to the \(x\)-axis. Given that the length of \(P S\) is 13
  1. write down the length of \(P Q\). Given that the point \(P\) has \(x\) coordinate 9
    find
  2. the value of \(a\),
  3. the area of triangle \(P S Q\).
Edexcel F1 2015 June Q5
  1. In the interval \(2 < x < 3\), the equation
$$6 - x ^ { 2 } \cos \left( \frac { x } { 5 } \right) = 0 , \text { where } x \text { is measured in radians }$$ has exactly one root \(\alpha\).
[0pt]
  1. Starting with the interval [2,3], use interval bisection twice to find an interval of width 0.25 which contains \(\alpha\).
    [0pt]
  2. Use linear interpolation once on the interval [2,3] to find an approximation to \(\alpha\). Give your answer to 2 decimal places.
Edexcel F1 2015 June Q6
6. The rectangular hyperbola, \(H\), has cartesian equation $$x y = 36$$ The three points \(P \left( 6 p , \frac { 6 } { p } \right) , Q \left( 6 q , \frac { 6 } { q } \right)\) and \(R \left( 6 r , \frac { 6 } { r } \right)\), where \(p , q\) and \(r\) are distinct, non-zero values, lie on the hyperbola \(H\).
  1. Show that an equation of the line \(P Q\) is $$p q y + x = 6 ( p + q )$$ Given that \(P R\) is perpendicular to \(Q R\),
  2. show that the normal to the curve \(H\) at the point \(R\) is parallel to the line \(P Q\).
Edexcel F1 2015 June Q7
7. $$z = - 3 k - 2 k \mathrm { i } , \text { where } k \text { is a real, positive constant. }$$
  1. Find the modulus and the argument of \(z\), giving the argument in radians to 2 decimal places and giving the modulus as an exact answer in terms of \(k\).
  2. Express in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real and are given in terms of \(k\) where necessary,
    1. \(\frac { 4 } { z + 3 k }\)
    2. \(z ^ { 2 }\)
  3. Given that \(k = 1\), plot the points \(A , B , C\) and \(D\) representing \(z , z ^ { * } , \frac { 4 } { z + 3 k }\) and \(z ^ { 2 }\) respectively on a single Argand diagram.
Edexcel F1 2015 June Q8
8. $$\mathbf { P } = \left( \begin{array} { r r } 3 a & - 4 a
4 a & 3 a \end{array} \right) , \text { where } a \text { is a constant and } a > 0$$
  1. Find the matrix \(\mathbf { P } ^ { - 1 }\) in terms of \(a\).
    (3) The matrix \(\mathbf { P }\) represents the transformation \(U\) which transforms a triangle \(T _ { 1 }\) onto the triangle \(T _ { 2 }\).
    The triangle \(T _ { 2 }\) has vertices at the points ( \(- 3 a , - 4 a\) ), ( \(6 a , 8 a\) ), and ( \(- 20 a , 15 a\) ).
  2. Find the coordinates of the vertices of \(T _ { 1 }\)
  3. Hence, or otherwise, find the area of triangle \(T _ { 2 }\) in terms of \(a\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a rotation through an angle \(\alpha\) clockwise about the origin, where \(\tan \alpha = \frac { 4 } { 3 }\) and \(0 < \alpha < \frac { \pi } { 2 }\)
  4. Write down the matrix \(\mathbf { Q }\), giving each element as an exact value. The transformation \(U\) followed by the transformation \(V\) is the transformation \(W\). The matrix \(\mathbf { R }\) represents the transformation \(W\).
  5. Find the matrix \(\mathbf { R }\).
Edexcel F1 2015 June Q9
  1. (i) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\),
$$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 2 r - 1 ) = \frac { 1 } { 6 } n ( n + 1 ) \left( 3 n ^ { 2 } + n - 1 \right)$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\left( \begin{array} { c c } 7 & - 12
3 & - 5 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 6 n + 1 & - 12 n
3 n & 1 - 6 n \end{array} \right)$$
Edexcel F1 2016 June Q1
  1. Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } r \left( r ^ { 2 } - 3 \right) = \frac { n } { 4 } ( n + a ) ( n + b ) ( n + c )$$ where \(a\), \(b\) and \(c\) are integers to be found.
\includegraphics[max width=\textwidth, alt={}, center]{0b7ef4a1-51bf-4f0c-908a-7caf26a144dc-03_2673_1710_84_116}
\includegraphics[max width=\textwidth, alt={}, center]{0b7ef4a1-51bf-4f0c-908a-7caf26a144dc-03_24_21_109_2042}
Edexcel F1 2016 June Q2
2. A parabola \(P\) has cartesian equation \(y ^ { 2 } = 28 x\). The point \(S\) is the focus of the parabola \(P\).
  1. Write down the coordinates of the point \(S\). Points \(A\) and \(B\) lie on the parabola \(P\). The line \(A B\) is parallel to the directrix of \(P\) and cuts the \(x\)-axis at the midpoint of \(O S\), where \(O\) is the origin.
  2. Find the exact area of triangle \(A B S\).
    VILM SIHI NITIIIUMI ON OC
    VILV SIHI NI III HM ION OC
    VALV SIHI NI JIIIM ION OO \includegraphics[max width=\textwidth, alt={}, center]{0b7ef4a1-51bf-4f0c-908a-7caf26a144dc-05_2264_53_315_36}
Edexcel F1 2016 June Q3
3. $$\mathrm { f } ( x ) = x ^ { 2 } + \frac { 3 } { x } - 1 , \quad x < 0$$ The only real root, \(\alpha\), of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval \([ - 2 , - 1 ]\).
  1. Taking - 1.5 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 2 decimal places.
  2. Show that your answer to part (a) gives \(\alpha\) correct to 2 decimal places.
    tion 3continued -
Edexcel F1 2016 June Q4
4. Given that $$\mathbf { A } = \left( \begin{array} { c c } k & 3
- 1 & k + 2 \end{array} \right) \text {, where } k \text { is a constant }$$
  1. show that \(\operatorname { det } ( \mathbf { A } ) > 0\) for all real values of \(k\),
  2. find \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).