Questions FP1 AS (86 questions)

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OCR FP1 AS 2021 June Q4
6 marks Standard +0.3
4 In this question you must show detailed reasoning. \(\mathbf { M }\) is the matrix \(\left( \begin{array} { l l } 1 & 6 \\ 0 & 2 \end{array} \right)\).
Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & 3 \left( 2 ^ { n + 1 } - 2 \right) \\ 0 & 2 ^ { n } \end{array} \right)\), for any positive integer \(n\).
OCR FP1 AS 2021 June Q1
4 marks Easy -1.3
1 Matrices \(\mathbf { P }\) and \(\mathbf { Q }\) are given by \(\mathbf { P } = \left( \begin{array} { c c c } 1 & k & 0 \\ - 2 & 1 & 3 \end{array} \right)\) and \(\mathbf { Q } = ( ( 1 + k ) - 1 )\) where \(k\) is a constant.
Exactly one of statements A and B is true.
Statement A: \(\quad \mathbf { P }\) and \(\mathbf { Q }\) (in that order) are conformable for multiplication.
Statement B: \(\quad \mathbf { Q }\) and \(\mathbf { P }\) (in that order) are conformable for multiplication.
  1. State, with a reason, which one of A and B is true.
  2. Find either \(\mathbf { P Q }\) or \(\mathbf { Q P }\) in terms of \(k\).
OCR FP1 AS 2021 June Q2
14 marks Standard +0.3
2 In this question you must show detailed reasoning. You are given that \(\mathrm { f } ( z ) = 4 z ^ { 4 } - 12 z ^ { 3 } + 41 z ^ { 2 } - 128 z + 185\) and that \(2 + \mathrm { i }\) is a root of the equation \(\mathrm { f } ( z ) = 0\).
  1. Express \(\mathrm { f } ( z )\) as the product of two quadratic factors with integer coefficients.
  2. Solve \(\mathrm { f } ( z ) = 0\). Two loci on an Argand diagram are defined by \(C _ { 1 } = \left\{ z : | z | = r _ { 1 } \right\}\) and \(C _ { 2 } = \left\{ z : | z | = r _ { 2 } \right\}\) where \(r _ { 1 } > r _ { 2 }\). You are given that two of the points representing the roots of \(\mathrm { f } ( z ) = 0\) are on \(C _ { 1 }\) and two are on \(C _ { 2 } \cdot R\) is the region on the Argand diagram between \(C _ { 1 }\) and \(C _ { 2 }\).
  3. Find the exact area of \(R\).
  4. \(\omega\) is the sum of all the roots of \(\mathrm { f } ( z ) = 0\). Determine whether or not the point on the Argand diagram which represents \(\omega\) lies in \(R\).
OCR FP1 AS 2021 June Q3
5 marks Standard +0.8
3 A transformation T is represented by the matrix \(\mathbf { T }\) where \(\mathbf { T } = \left( \begin{array} { c c } x ^ { 2 } + 1 & - 4 \\ 3 - 2 x ^ { 2 } & x ^ { 2 } + 5 \end{array} \right)\). A quadrilateral \(Q\), whose area is 12 units, is transformed by T to \(Q ^ { \prime }\). Find the smallest possible value of the area of \(Q ^ { \prime }\).
OCR FP1 AS 2021 June Q4
7 marks Standard +0.3
4 A transformation A is represented by the matrix \(\mathbf { A }\) where \(\mathbf { A } = \left( \begin{array} { c c c } - 1 & x & 2 \\ 7 - x & - 6 & 1 \\ 5 & - 5 x & 2 x \end{array} \right)\).
The tetrahedron \(H\) has vertices at \(O , P , Q\) and \(R\). The volume of \(H\) is 6 units. \(P ^ { \prime } , Q ^ { \prime } , R ^ { \prime }\) and \(H ^ { \prime }\) are the images of \(P , Q , R\) and \(H\) under A .
  1. In the case where \(x = 5\)
OCR FP1 AS 2021 June Q1
3 marks Standard +0.3
1 In this question you must show detailed reasoning.
The cubic equation \(2 x ^ { 3 } + 3 x ^ { 2 } - 5 x + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). By making an appropriate substitution, or otherwise, find a cubic equation with integer coefficients whose roots are \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\).
OCR FP1 AS 2021 June Q2
7 marks Moderate -0.8
2 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r c } 2 & 1 & 2 \\ 1 & - 1 & 1 \\ 2 & 2 & a \end{array} \right)\).
  1. Show that \(\operatorname { det } \mathbf { A } = 6 - 3 a\).
  2. State the value of \(a\) for which \(\mathbf { A }\) is singular.
  3. Given that \(\mathbf { A }\) is non-singular find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\).
OCR FP1 AS 2021 June Q3
7 marks Standard +0.3
3 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } t & 6 \\ t & - 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 2 t & 4 \\ t & - 2 \end{array} \right)\) where \(t\) is a constant.
  1. Show that \(| \mathrm { A } | = | \mathrm { B } |\).
  2. Verify that \(| \mathrm { AB } | = | \mathrm { A } | | \mathrm { B } |\).
  3. Given that \(| \mathbf { A B } | = - 1\) explain what this means about the constant \(t\). The \(2 \times 2\) matrix \(A\) represents a transformation \(T\) which has the following properties.
    The transformation \(S\) is represented by the matrix \(B\) where \(B = \left( \begin{array} { l l } 3 & 1 \\ 2 & 2 \end{array} \right)\).
    (b) Find the equation of the line of invariant points of S .
    (c) Show that any line of the form \(y = x + c\) is an invariant line of S .
OCR FP1 AS 2021 June Q1
6 marks Moderate -0.3
1 In this question you must show detailed reasoning.
Use an algebraic method to find the square roots of \(- 77 - 36 \mathrm { i }\).
OCR FP1 AS 2021 June Q2
13 marks Moderate -0.3
2 In this question you must show detailed reasoning.
The complex number \(7 - 4 \mathrm { i }\) is denoted by \(z\).
  1. Giving your answers in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are rational numbers, find the following.
    1. \(3 z - 4 z ^ { * }\)
    2. \(( z + 1 - 3 \mathrm { i } ) ^ { 2 }\)
    3. \(\frac { z + 1 } { z - 1 }\)
  2. Express \(z\) in modulus-argument form giving the modulus exactly and the argument correct to 3 significant figures.
  3. The complex number \(\omega\) is such that \(z \omega = \sqrt { 585 } ( \cos ( 0.5 ) + \mathrm { i } \sin ( 0.5 ) )\). Find the following.
OCR FP1 AS 2021 June Q3
7 marks Challenging +1.8
3 In this question you must show detailed reasoning.
The cubic equation \(5 x ^ { 3 } + 3 x ^ { 2 } - 4 x + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation with integer coefficients whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).
OCR FP1 AS 2021 June Q4
5 marks Standard +0.3
4 Prove that \(n ! > 2 ^ { 2 n }\) for all integers \(n \geqslant 9\).
OCR FP1 AS 2021 June Q2
6 marks Standard +0.3
2 You are given the system of equations $$\begin{array} { r } a ^ { 2 } x - 2 y = 1 \\ x + b ^ { 2 } y = 3 \end{array}$$ where \(a\) and \(b\) are real numbers.
  1. Use a matrix method to find \(x\) and \(y\) in terms of \(a\) and \(b\).
  2. Explain why the method used in part (a) works for all values of \(a\) and \(b\).
OCR FP1 AS 2021 June Q3
6 marks Standard +0.3
3 The equations of two intersecting lines are \(\mathbf { r } = \left( \begin{array} { c } - 12 \\ a \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ 2 \\ 1 \end{array} \right) \quad \mathbf { r } = \left( \begin{array} { l } 2 \\ 0 \\ 5 \end{array} \right) + \mu \left( \begin{array} { c } - 3 \\ 1 \\ - 1 \end{array} \right)\) where \(a\) is a constant.
  1. Find a vector, \(\mathbf { b }\), which is perpendicular to both lines.
  2. Show that b. \(\left( \begin{array} { c } - 12 \\ a \\ - 1 \end{array} \right) =\) b. \(\left( \begin{array} { l } 2 \\ 0 \\ 5 \end{array} \right)\).
  3. Hence, or otherwise, find the value of \(a\).
OCR FP1 AS 2021 June Q4
8 marks Standard +0.8
4 Two loci, \(C _ { 1 }\) and \(C _ { 2 }\), are defined by $$\begin{aligned} & C _ { 1 } = \left\{ z : | z | = \left| z - 4 d ^ { 2 } - 36 \right| \right\} \\ & C _ { 2 } = \left\{ z : \arg ( z - 12 d - 3 \mathrm { i } ) = \frac { 1 } { 4 } \pi \right\} \end{aligned}$$ where \(d\) is a real number.
  1. Find, in terms of \(d\), the complex number which is represented on an Argand diagram by the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
    [0pt] [You may assume that \(C _ { 1 } \cap C _ { 2 } \neq \varnothing\).]
  2. Explain why the solution found in part (a) is not valid when \(d = 3\).
OCR FP1 AS 2021 June Q1
5 marks Moderate -0.8
  1. Find a vector which is perpendicular to both \(\begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix}\) and \(\begin{pmatrix} -3 \\ -6 \\ 4 \end{pmatrix}\). [2]
  2. The cartesian equation of a line is \(\frac{x}{2} = y - 3 = \frac{z + 4}{4}\). Express the equation of this line in vector form. [3]
OCR FP1 AS 2021 June Q2
9 marks Standard +0.3
In this question you must show detailed reasoning. The complex numbers \(z_1\) and \(z_2\) are given by \(z_1 = 2 - 3i\) and \(z_2 = a + 4i\) where \(a\) is a real number.
  1. Express \(z_1\) in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures. [3]
  2. Find \(z_1z_2\) in terms of \(a\), writing your answer in the form \(c + id\). [2]
  3. The real and imaginary parts of a complex number on an Argand diagram are \(x\) and \(y\) respectively. Given that the point representing \(z_1z_2\) lies on the line \(y = x\), find the value of \(a\). [2]
  4. Given instead that \(z_1z_2 = (z_1z_2)^*\) find the value of \(a\). [2]
OCR FP1 AS 2021 June Q3
10 marks Moderate -0.3
In this question you must show detailed reasoning.
  1. Express \((2 + 3i)^3\) in the form \(a + ib\). [3]
  2. Hence verify that \(2 + 3i\) is a root of the equation \(3z^3 - 8z^2 + 23z + 52 = 0\). [3]
  3. Express \(3z^3 - 8z^2 + 23z + 52\) as the product of a linear factor and a quadratic factor with real coefficients. [4]
OCR FP1 AS 2021 June Q4
6 marks Standard +0.3
Prove by induction that \(2^{n+1} + 5 \times 9^n\) is divisible by 7 for all integers \(n \geq 1\). [6]
OCR FP1 AS 2017 December Q1
4 marks Standard +0.3
The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} -3 & 3 & 2 \\ 5 & -4 & -3 \\ -1 & 1 & 1 \end{pmatrix}\).
  1. Find \(\mathbf{A}^{-1}\). [1]
  2. Solve the simultaneous equations $$-3x + 3y + 2z = 12a$$ $$5x - 4y - 3z = -6$$ $$-x + y + z = 7$$ giving your solution in terms of \(a\). [3]
OCR FP1 AS 2017 December Q2
9 marks Standard +0.3
The loci \(C_1\) and \(C_2\) are given by \(|z - (3 + 2i)| = 2\) and \(\arg(z - (3 + 2i)) = \frac{5\pi}{6}\) respectively.
  1. Sketch \(C_1\) and \(C_2\) on a single Argand diagram. [4]
  2. Find, in surd form, the number represented by the point of intersection of \(C_1\) and \(C_2\). [3]
  3. Indicate, by shading, the region of the Argand diagram for which $$|z - (3 + 2i)| \leq 2 \text{ and } \frac{5\pi}{6} \leq \arg(z - (3 + 2i)) \leq \pi.$$ [2]
OCR FP1 AS 2017 December Q3
8 marks Standard +0.3
Two lines, \(l_1\) and \(l_2\), have the following equations. $$l_1: \mathbf{r} = \begin{pmatrix} -11 \\ 10 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$$ \(P\) is the point of intersection of \(l_1\) and \(l_2\).
  1. Find the position vector of \(P\). [3]
  2. Find, correct to 1 decimal place, the acute angle between \(l_1\) and \(l_2\). [3]
\(Q\) is a point on \(l_1\) which is 12 metres away from \(P\). \(R\) is the point on \(l_2\) such that \(QR\) is perpendicular to \(l_1\).
  1. Determine the length \(QR\). [2]
OCR FP1 AS 2017 December Q4
7 marks Standard +0.8
In this question you must show detailed reasoning. The distinct numbers \(\omega_1\) and \(\omega_2\) both satisfy the quadratic equation \(4x^2 + 4x + 17 = 0\).
  1. Write down the value of \(\omega_1 \omega_2\). [1]
  2. \(A\), \(B\) and \(C\) are the points on an Argand diagram which represent \(\omega_1\), \(\omega_2\) and \(\omega_1 \omega_2\). Find the area of triangle \(ABC\). [6]
OCR FP1 AS 2017 December Q5
7 marks Challenging +1.3
In this question you must show detailed reasoning. The equation \(x^3 + 3x^2 - 2x + 4 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).
  1. Using the identity \(\alpha^3 + \beta^3 + \gamma^3 \equiv (\alpha + \beta + \gamma)^3 - 3(\alpha\beta + \beta\gamma + \gamma\alpha)(\alpha + \beta + \gamma) + 3\alpha\beta\gamma\) find the value of \(\alpha^3 + \beta^3 + \gamma^3\). [3]
  2. Given that \(\alpha^2\beta^3 + \beta^3\gamma^3 + \gamma^3\alpha^3 = 112\) find a cubic equation whose roots are \(\alpha^2\), \(\beta^3\) and \(\gamma^3\). [4]
OCR FP1 AS 2017 December Q6
5 marks Standard +0.3
Prove by induction that \(n! \geq 6n\) for \(n \geq 4\). [5]