Questions — Edexcel (9685 questions)

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Edexcel F1 2018 Specimen Q2
5 marks Standard +0.8
  1. 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 \(A B S\).
Edexcel F1 2018 Specimen Q3
7 marks Standard +0.3
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. \includegraphics[max width=\textwidth, alt={}, center]{38217fcb-8f26-49ac-9bb1-61c2f304006e-06_2250_51_317_1980}
Edexcel F1 2018 Specimen Q4
5 marks Moderate -0.5
  1. 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 \(k\).
Edexcel F1 2018 Specimen Q5
5 marks Moderate -0.3
5. $$2 z + z ^ { * } = \frac { 3 + 4 \mathrm { i } } { 7 + \mathrm { i } }$$ Find \(z\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants. You must show all your working.
Edexcel F1 2018 Specimen Q6
10 marks Standard +0.3
  1. The rectangular hyperbola \(H\) has equation \(x y = 25\)
    1. Verify that, for \(t \neq 0\), the point \(P \left( 5 t , \frac { 5 } { t } \right)\) is a general point on \(H\).
    The point \(A\) on \(H\) has parameter \(t = \frac { 1 } { 2 }\)
  2. Show that the normal to \(H\) at the point \(A\) has equation $$8 y - 2 x - 75 = 0$$ This normal at \(A\) meets \(H\) again at the point \(B\).
  3. Find the coordinates of \(B\). \includegraphics[max width=\textwidth, alt={}, center]{38217fcb-8f26-49ac-9bb1-61c2f304006e-13_2261_50_312_36}
    VIAV SIHI NI BIIIM ION OCVGHV SIHI NI GHIYM ION OCVJ4V SIHI NI JIIYM ION OC
Edexcel F1 2018 Specimen Q7
10 marks Moderate -0.3
7. $$\mathbf { P } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 } \\ \frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)$$
  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 line with equation \(y = x\)
  2. Write down the matrix \(\mathbf { Q }\). Given that the transformation \(V\) followed by the transformation \(U\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\),
  3. find the matrix \(\mathbf { R }\).
  4. Show that there is a value of \(k\) for which the transformation \(T\) maps each point on the straight line \(y = k x\) onto itself, and state the value of \(k\).
    \includegraphics[max width=\textwidth, alt={}, center]{38217fcb-8f26-49ac-9bb1-61c2f304006e-17_2261_54_312_34}
    VIAV SIHI NI BIIIM ION OCVGHV SIHI NI GHIYM ION OCVJ4V SIHI NI JIIYM ION OC
Edexcel F1 2018 Specimen Q8
9 marks Moderate -0.3
8. $$\mathrm { f } ( z ) = z ^ { 4 } + 6 z ^ { 3 } + 76 z ^ { 2 } + a z + b$$ where \(a\) and \(b\) are real constants.
Given that \(- 3 + 8 \mathrm { i }\) is a complex root of the equation \(\mathrm { f } ( z ) = 0\)
  1. write down another complex root of this equation.
  2. Hence, or otherwise, find the other roots of the equation \(\mathrm { f } ( z ) = 0\)
  3. Show on a single Argand diagram all four roots of the equation \(\mathrm { f } ( z ) = 0\)
    VIAV SIHI NI BIIIM ION OCVGHV SIHI NI GHIYM ION OCVJ4V SIHI NI JIIYM ION OC
Edexcel F1 2018 Specimen Q9
9 marks Standard +0.8
  1. The quadratic equation
$$2 x ^ { 2 } + 4 x - 3 = 0$$ has roots \(\alpha\) and \(\beta\).
Without solving the quadratic equation,
  1. find the exact value of
    1. \(\alpha ^ { 2 } + \beta ^ { 2 }\)
    2. \(\alpha ^ { 3 } + \beta ^ { 3 }\)
  2. Find a quadratic equation which has roots ( \(\alpha ^ { 2 } + \beta\) ) and ( \(\beta ^ { 2 } + \alpha\) ), giving your answer in the form \(a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are integers.
    VIAV SIHI NI BIIIM ION OCVGHV SIHI NI GHIYM ION OCVJ4V SIHI NI JIIYM ION OC
Edexcel F1 2018 Specimen Q10
11 marks Standard +0.3
  1. (i) A sequence of positive numbers is defined by
$$\begin{aligned} u _ { 1 } & = 5 \\ u _ { n + 1 } & = 3 u _ { n } + 2 , \quad n \geqslant 1 \end{aligned}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$u _ { n } = 2 \times ( 3 ) ^ { n } - 1$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } \frac { 4 r } { 3 ^ { r } } = 3 - \frac { ( 3 + 2 n ) } { 3 ^ { n } }$$
\(\_\_\_\_\) VJYV SIHI NI JIIYM ION OO \(\quad\) VJYV SIHI NI JIIYM ION OC \(\quad\) VJYV SIHI NI JLIVM ION OC \includegraphics[max width=\textwidth, alt={}, center]{38217fcb-8f26-49ac-9bb1-61c2f304006e-31_2255_51_316_36}
Edexcel F1 Specimen Q1
7 marks Moderate -0.8
  1. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by
$$z _ { 1 } = 2 + 8 \mathrm { i } \quad \text { and } \quad z _ { 2 } = 1 - \mathrm { i }$$ 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 F1 Specimen Q2
10 marks Standard +0.3
2. $$f ( x ) = 5 x ^ { 2 } - 4 x ^ { \frac { 3 } { 2 } } - 6 , \quad x \geqslant 0$$ 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. 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 F1 Specimen Q3
7 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa5a23b5-d52c-4bae-97c7-2eb7220a3dc4-04_736_659_299_660} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the parabola \(C\) with equation \(y ^ { 2 } = 8 x\).
The point \(P\) lies on \(C\), where \(y > 0\), and the point \(Q\) lies on \(C\), where \(y < 0\) The line segment \(P Q\) is parallel to the \(y\)-axis. Given that the distance \(P Q\) is 12 ,
  1. write down the \(y\) coordinate of \(P\),
  2. find the \(x\) coordinate of \(P\). Figure 1 shows the point \(S\) which is the focus of \(C\). The line \(l\) passes through the point \(P\) and the point \(S\).
  3. Find an equation for \(l\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
Edexcel F1 Specimen Q4
12 marks Standard +0.3
  1. The quadratic equation
$$5 x ^ { 2 } - 4 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
  2. Show that \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = \frac { 6 } { 5 }\)
  3. Find a quadratic equation with integer coefficients, which has roots $$\alpha + \frac { 1 } { \alpha } \text { and } \beta + \frac { 1 } { \beta }$$
Edexcel F1 Specimen Q5
6 marks Standard +0.3
  1. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\),
$$\mathrm { f } ( n ) = 5 ^ { n } + 8 n + 3 \text { is divisible by } 4$$
Edexcel F1 Specimen Q6
10 marks Standard +0.8
6. (a) Using the formulae for \(\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\), show that $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r + 3 ) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( 3 n + k )$$ where \(k\) is a constant to be found.
(b) Hence evaluate \(\sum _ { r = 21 } ^ { 40 } r ( r + 1 ) ( r + 3 )\)
Edexcel F1 Specimen Q7
12 marks Challenging +1.2
  1. The point \(\mathrm { P } \left( 6 \mathrm { t } , \frac { 6 } { \mathrm { t } } \right) , t \neq 0\), lies on the rectangular hyperbola \(H\) with equation \(x y = 36\) (a) Show that an equation for the tangent to \(H\) at \(P\) is
$$y = - \frac { 1 } { t ^ { 2 } } x + \frac { 12 } { t }$$ The tangent to \(H\) at the point \(A\) and the tangent to \(H\) at the point \(B\) meet at the point \(( - 9,12 )\).
(b) Find the coordinates of \(A\) and \(B\).
Edexcel F1 Specimen Q8
11 marks Standard +0.3
8. (i) The transformation \(U\) is represented by the matrix \(\mathbf { P }\) where, $$P = \left( \begin{array} { r r } - \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \\ \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \end{array} \right)$$
  1. Describe fully the transformation \(U\). The transformation \(V\), represented by the matrix \(\mathbf { Q }\), is a stretch scale factor 3 parallel to the \(x\)-axis.
  2. Write down the matrix \(\mathbf { Q }\). Transformation \(U\) followed by transformation \(V\) is a transformation which is represented by matrix \(\mathbf { R }\).
  3. Find the matrix \(\mathbf { R }\).
    (ii) $$S = \left( \begin{array} { r r } 1 & - 3 \\ 3 & 1 \end{array} \right)$$ Given that the matrix \(\mathbf { S }\) represents an enlargement, with a positive scale factor and centre \(( 0,0 )\), followed by a rotation with centre \(( 0,0 )\),
  4. find the scale factor of the enlargement,
  5. find the angle and direction of rotation, giving your answer in degrees to 1 decimal place.
Edexcel FP1 Q1
5 marks Moderate -0.8
1. $$\mathrm { f } ( x ) = 2 x ^ { 3 } - 8 x ^ { 2 } + 7 x - 3$$ Given that \(x = 3\) is a solution of the equation \(\mathrm { f } ( x ) = 0\), solve \(\mathrm { f } ( x ) = 0\) completely.
Edexcel FP1 Q5
7 marks Standard +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 Q6
5 marks Moderate -0.3
6. A series of positive integers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 6 \text { and } u _ { n + 1 } = 6 u _ { n } - 5 , \text { for } n \geq 1 .$$ Prove by induction that \(u _ { n } = 5 \times 6 ^ { n - 1 } + 1\), for \(n \geq 1\).
Edexcel FP1 Q7
5 marks Standard +0.3
7. Given that \(\mathbf { X } = \left( \begin{array} { r r } 2 & a \\ - 1 & - 1 \end{array} \right)\), 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 Q8
12 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 Q9
10 marks Standard +0.8
9. Given that \(z _ { 1 } = 3 + 2 \mathrm { i }\) and \(z _ { 2 } = \frac { 12 - 5 \mathrm { i } } { z _ { 1 } }\),
  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 Q1
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.
(5)
Edexcel FP1 Q6
Standard +0.3
6. A series of positive integers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 6 \text { and } u _ { n + 1 } = 6 u _ { n } - 5 , \text { for } n \geqslant 1 .$$ Prove by induction that \(u _ { n } = 5 \times 6 ^ { n - 1 } + 1\), for \(n \geqslant 1\).