Questions — OCR FP1 AS (45 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
OCR FP1 AS 2018 March Q1
6 marks Moderate -0.8
1
  1. The complex number 3-4i is denoted by \(z _ { 1 }\). Write \(z _ { 1 }\) in modulus-argument form, giving your angle in radians to 3 significant figures.
  2. The complex number \(z _ { 2 }\) has modulus 6 and argument - 2.5 radians. Express \(z _ { 1 } z _ { 2 }\) in modulus-argument form with the angle in radians correct to 3 significant figures.
OCR FP1 AS 2018 March Q2
5 marks Moderate -0.5
2 In this question you must show detailed reasoning.
The quadratic equation \(3 x ^ { 2 } - 7 x + 5 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Hence find the values of the following expressions.
    1. \(\frac { 1 } { \alpha } + \frac { 1 } { \beta }\)
    2. \(\alpha ^ { 2 } + \beta ^ { 2 }\) \(3 \quad l _ { 1 }\) and \(l _ { 2 }\) are two intersecting straight lines with the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } 3 \\ 3 \\ - 5 \end{array} \right) \end{aligned}$$
OCR FP1 AS 2018 March Q3
6 marks Standard +0.3
$$\begin{aligned} l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } 4 \\ 2 \\ 5 \end{array} \right) + \lambda \left( \begin{array} { c } 1 \\ 3 \\ - 2 \end{array} \right) \\ & l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 1 \\ a \\ 1 \end{array} \right) + \mu \left( \begin{array} { c } 2 \\ 2 \\ - 3 \end{array} \right) \end{aligned}$$
  1. Find the position vector of the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
  2. Determine the value of \(a\).
OCR FP1 AS 2018 March Q4
6 marks Standard +0.8
4 Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(| z - 5 - 2 \mathrm { i } | \leqslant \sqrt { 32 }\) and \(\operatorname { Re } ( z ) \geqslant 9\).
OCR FP1 AS 2018 March Q5
7 marks Moderate -0.8
5 The matrix \(\mathbf { A }\) is given by \(\left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & a ^ { 2 } & 0 \\ 0 & 0 & 1 \end{array} \right)\) and the matrix \(\mathbf { B }\) is given by \(\left( \begin{array} { c c c } 0.6 & b & 0 \\ - b & 0.6 & 0 \\ 0 & 0 & 1 \end{array} \right)\).
  1. \(\mathbf { A }\) represents a reflection. Write down the value of \(\operatorname { det } \mathbf { A }\).
  2. Hence find the possible values of \(a\).
  3. \(\mathbf { r }\) is the position vector of a point \(R\). Given that \(\mathbf { A r } = \mathbf { r }\) describe the location of \(R\).
  4. \(\mathbf { B }\) represents a rotation. Write down the value of \(\operatorname { det } \mathbf { B }\).
  5. Hence find the possible values of \(b\).
OCR FP1 AS 2018 March Q6
10 marks Standard +0.3
6 The matrix \(\mathbf { A }\) is given by \(\left( \begin{array} { l l } 1 & 2 \\ 1 & a \end{array} \right)\) and the matrix \(\mathbf { B }\) is given by \(\left( \begin{array} { c c } 2 & 1 \\ - 1 & b \end{array} \right)\).
  1. Find the matrix \(\mathbf { A B }\).
  2. State the conditions on \(a\) and \(b\) for \(\mathbf { A B }\) to be a singular matrix. \(P Q R S\) is a quadrilateral and the vertices \(P , Q , R\) and \(S\) are in clockwise order. A transformation, T , is represented by the matrix \(\mathbf { A B }\).
  3. State the effect on both the area and also the orientation of the image of \(P Q R S\) under T in each of the following cases.
    1. \(\quad a = 1\) and \(b = 1\)
    2. \(\quad a = 2\) and \(b = 3\)
OCR FP1 AS 2018 March Q7
9 marks Standard +0.3
7 In this question you must show detailed reasoning.
  1. Find the square roots of the number \(528 + 46 \mathrm { i }\) giving your answers in the form \(a + b \mathrm { i }\).
  2. \(\quad 3 + 2 \mathrm { i }\) is a root of the equation \(x ^ { 3 } - a x + 78 = 0\), where \(a\) is a real number. Find the value of \(a\).
OCR FP1 AS 2018 March Q8
11 marks Challenging +1.3
8 In this question you must show detailed reasoning. A sequence of vectors \(\mathbf { a } _ { 1 } , \mathbf { a } _ { 2 } , \mathbf { a } _ { 3 } , \ldots\) is defined by
  • \(\mathbf { a } _ { 1 } = \left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\)
  • \(\quad \mathbf { a } _ { n + 1 } = \left( \mathbf { a } _ { n } \times \mathbf { b } \right) \times \mathbf { b }\), for integers \(n \geqslant 1\), where \(\mathbf { b }\) is the vector \(\frac { 1 } { 4 } \left( \begin{array} { c } - 3 \\ 1 \\ 2 \end{array} \right)\).
    1. Prove by induction that \(\mathbf { a } _ { n } = \left( - \frac { 7 } { 8 } \right) ^ { n - 1 } \left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\). for all integers \(n \geqslant 1\).
    2. Use an algebraic method to find the smallest value of \(n\) such that \(\left| \mathbf { a } _ { n } \right| < 0.001\).
\section*{END OF QUESTION PAPER}
OCR FP1 AS 2021 June Q2
12 marks Standard +0.3
2
The position vector of point \(A\) is \(\mathbf { a } = - 9 \mathbf { i } + 2 \mathbf { j } + 6 \mathbf { k }\).
The line \(l\) passes through \(A\) and is perpendicular to \(\mathbf { a }\).
  1. Determine the shortest distance between the origin, \(O\), and \(l\). \(l\) is also perpendicular to the vector \(\mathbf { b }\) where \(\mathbf { b } = - 2 \mathbf { i } + \mathbf { j } + \mathbf { k }\).
  2. Find a vector which is perpendicular to both \(\mathbf { a }\) and \(\mathbf { b }\).
  3. Write down an equation of \(l\) in vector form. \(P\) is a point on \(l\) such that \(P A = 2 O A\).
  4. Find angle \(P O A\) giving your answer to 3 significant figures. \(C\) is a point whose position vector, \(\mathbf { c }\), is given by \(\mathbf { c } = p \mathbf { a }\) for some constant \(p\). The line \(m\) passes through \(C\) and has equation \(\mathbf { r } = \mathbf { c } + \mu \mathbf { b }\). The point with position vector \(9 \mathbf { i } + 8 \mathbf { j } - 12 \mathbf { k }\) lies on \(m\).
  5. Find the value of \(p\). \section*{In this question you must show detailed reasoning.} You are given that \(\alpha , \beta\) and \(\gamma\) are the roots of the equation \(5 x ^ { 3 } - 2 x ^ { 2 } + 3 x + 1 = 0\).
    1. Find the value of \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\).
    2. Find a cubic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\) giving your answer in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d = 0\) where \(a , b , c\) and \(d\) are integers.
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]