Questions — OCR FP1 (210 questions)

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OCR FP1 2006 January Q1
5 marks Moderate -0.8
1
  1. Express \(( 1 + 8 i ) ( 2 - i )\) in the form \(x + i y\), showing clearly how you obtain your answer.
  2. Hence express \(\frac { 1 + 8 i } { 2 + i }\) in the form \(x + i y\).
OCR FP1 2006 January Q2
5 marks Standard +0.3
2 Prove by induction that, for \(n \geqslant 1 , \sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\).
OCR FP1 2006 January Q3
4 marks Moderate -0.8
3 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l l } 2 & 1 & 3 \\ 1 & 2 & 1 \\ 1 & 1 & 3 \end{array} \right)\).
  1. Find the value of the determinant of \(\mathbf { M }\).
  2. State, giving a brief reason, whether \(\mathbf { M }\) is singular or non-singular.
OCR FP1 2006 January Q4
5 marks Standard +0.3
4 Use the substitution \(x = u + 2\) to find the exact value of the real root of the equation $$x ^ { 3 } - 6 x ^ { 2 } + 12 x - 13 = 0$$
OCR FP1 2006 January Q5
6 marks Moderate -0.5
5 Use the standard results for \(\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \left( 8 r ^ { 3 } - 6 r ^ { 2 } + 2 r \right) = 2 n ^ { 3 } ( n + 1 )$$
OCR FP1 2006 January Q6
7 marks Standard +0.3
6 The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { l l } 1 & 2 \\ 3 & 8 \end{array} \right)\).
  1. Find \(\mathbf { C } ^ { - 1 }\).
  2. Given that \(\mathbf { C } = \mathbf { A B }\), where \(\mathbf { A } = \left( \begin{array} { l l } 2 & 1 \\ 1 & 3 \end{array} \right)\), find \(\mathbf { B } ^ { - 1 }\).
OCR FP1 2006 January Q7
10 marks Moderate -0.8
7
  1. The complex number \(3 + 2 \mathrm { i }\) is denoted by \(w\) and the complex conjugate of \(w\) is denoted by \(w ^ { * }\). Find
    1. the modulus of \(w\),
    2. the argument of \(w ^ { * }\), giving your answer in radians, correct to 2 decimal places.
  2. Find the complex number \(u\) given that \(u + 2 u ^ { * } = 3 + 2 \mathrm { i }\).
  3. Sketch, on an Argand diagram, the locus given by \(| z + 1 | = | z |\).
OCR FP1 2006 January Q8
9 marks Standard +0.3
8 The matrix \(\mathbf { T }\) is given by \(\mathbf { T } = \left( \begin{array} { r r } 2 & 0 \\ 0 & - 2 \end{array} \right)\).
  1. Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { T }\). [3]
  2. The transformation represented by matrix \(\mathbf { T }\) is equivalent to a transformation \(A\), followed by a transformation B. Give geometrical descriptions of possible transformations A and B, and state the matrices that represent them.
OCR FP1 2006 January Q9
10 marks Standard +0.3
9
  1. Show that \(\frac { 1 } { r } - \frac { 1 } { r + 2 } = \frac { 2 } { r ( r + 2 ) }\).
  2. Hence find an expression, in terms of \(n\), for $$\frac { 2 } { 1 \times 3 } + \frac { 2 } { 2 \times 4 } + \ldots + \frac { 2 } { n ( n + 2 ) }$$
  3. Hence find the value of
    1. \(\sum _ { r = 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) }\),
    2. \(\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) }\).
OCR FP1 2006 January Q10
11 marks Standard +0.3
10 The roots of the equation $$x ^ { 3 } - 9 x ^ { 2 } + 27 x - 29 = 0$$ are denoted by \(\alpha , \beta\) and \(\gamma\), where \(\alpha\) is real and \(\beta\) and \(\gamma\) are complex.
  1. Write down the value of \(\alpha + \beta + \gamma\).
  2. It is given that \(\beta = p + \mathrm { i } q\), where \(q > 0\). Find the value of \(p\), in terms of \(\alpha\).
  3. Write down the value of \(\alpha \beta \gamma\).
  4. Find the value of \(q\), in terms of \(\alpha\) only.
OCR FP1 2007 January Q1
3 marks Moderate -0.8
\(\mathbf { 1 }\) The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 1 \\ 3 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c } a & - 1 \\ - 3 & - 2 \end{array} \right)\).
  1. Given that \(2 \mathbf { A } + \mathbf { B } = \left( \begin{array} { l l } 1 & 1 \\ 3 & 2 \end{array} \right)\), write down the value of \(a\).
  2. Given instead that \(\mathbf { A B } = \left( \begin{array} { l l } 7 & - 4 \\ 9 & - 7 \end{array} \right)\), find the value of \(a\).
OCR FP1 2007 January Q2
6 marks Moderate -0.3
2 Use an algebraic method to find the square roots of the complex number \(15 + 8 \mathrm { i }\).
OCR FP1 2007 January Q3
6 marks Moderate -0.3
3 Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to find $$\sum _ { r = 1 } ^ { n } r ( r - 1 ) ( r + 1 ) ,$$ expressing your answer in a fully factorised form.
OCR FP1 2007 January Q4
6 marks Moderate -0.3
4
  1. Sketch, on an Argand diagram, the locus given by \(| z - 1 + \mathrm { i } | = \sqrt { 2 }\).
  2. Shade on your diagram the region given by \(1 \leqslant | z - 1 + \mathrm { i } | \leqslant \sqrt { 2 }\).
OCR FP1 2007 January Q5
7 marks Moderate -0.8
5
  1. Verify that \(z ^ { 3 } - 8 = ( z - 2 ) \left( z ^ { 2 } + 2 z + 4 \right)\).
  2. Solve the quadratic equation \(z ^ { 2 } + 2 z + 4 = 0\), giving your answers exactly in the form \(x + \mathrm { i } y\). Show clearly how you obtain your answers.
  3. Show on an Argand diagram the roots of the cubic equation \(z ^ { 3 } - 8 = 0\).
OCR FP1 2007 January Q6
8 marks Moderate -0.5
6 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = n ^ { 2 } + 3 n\), for all positive integers \(n\).
  1. Show that \(u _ { n + 1 } - u _ { n } = 2 n + 4\).
  2. Hence prove by induction that each term of the sequence is divisible by 2 .
OCR FP1 2007 January Q7
8 marks Standard +0.3
7 The quadratic equation \(x ^ { 2 } + 5 x + 10 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } = 5\).
  3. Hence find a quadratic equation which has roots \(\frac { \alpha } { \beta }\) and \(\frac { \beta } { \alpha }\).
OCR FP1 2007 January Q8
8 marks Standard +0.8
8
  1. Show that \(( r + 2 ) ! - ( r + 1 ) ! = ( r + 1 ) ^ { 2 } \times r !\).
  2. Hence find an expression, in terms of \(n\), for $$2 ^ { 2 } \times 1 ! + 3 ^ { 2 } \times 2 ! + 4 ^ { 2 } \times 3 ! + \ldots + ( n + 1 ) ^ { 2 } \times n ! .$$
  3. State, giving a brief reason, whether the series $$2 ^ { 2 } \times 1 ! + 3 ^ { 2 } \times 2 ! + 4 ^ { 2 } \times 3 ! + \ldots$$ converges.
OCR FP1 2007 January Q9
9 marks Standard +0.8
9 The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { r r } 0 & 3 \\ - 1 & 0 \end{array} \right)\).
  1. Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { C }\). The transformation represented by \(\mathbf { C }\) is equivalent to a rotation, R , followed by another transformation, S.
  2. Describe fully the rotation R and write down the matrix that represents R .
  3. Describe fully the transformation S and write down the matrix that represents S .
OCR FP1 2007 January Q10
11 marks Standard +0.3
10 The matrix \(\mathbf { D }\) is given by \(\mathbf { D } = \left( \begin{array} { r r r } a & 2 & 0 \\ 3 & 1 & 2 \\ 0 & - 1 & 1 \end{array} \right)\), where \(a \neq 2\).
  1. Find \(\mathbf { D } ^ { - 1 }\).
  2. Hence, or otherwise, solve the equations $$\begin{aligned} a x + 2 y & = 3 \\ 3 x + y + 2 z & = 4 \\ - y + z & = 1 \end{aligned}$$
OCR FP1 2008 January Q1
4 marks Easy -1.2
1 The transformation S is a shear with the \(y\)-axis invariant (i.e. a shear parallel to the \(y\)-axis). It is given that the image of the point \(( 1,1 )\) is the point \(( 1,0 )\).
  1. Draw a diagram showing the image of the unit square under the transformation S .
  2. Write down the matrix that represents S .
OCR FP1 2008 January Q2
5 marks Standard +0.3
2 Given that \(\sum _ { r = 1 } ^ { n } \left( a r ^ { 2 } + b \right) \equiv n \left( 2 n ^ { 2 } + 3 n - 2 \right)\), find the values of the constants \(a\) and \(b\).
OCR FP1 2008 January Q3
4 marks Standard +0.3
3 The cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + 24 x + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. Use the substitution \(x = \frac { 1 } { u }\) to find a cubic equation in \(u\) with integer coefficients.
  2. Hence, or otherwise, find the value of \(\frac { 1 } { \alpha \beta } + \frac { 1 } { \beta \gamma } + \frac { 1 } { \gamma \alpha }\).
OCR FP1 2008 January Q4
8 marks Moderate -0.8
4 The complex number \(3 - 4 \mathrm { i }\) is denoted by \(z\). Giving your answers in the form \(x + \mathrm { i } y\), and showing clearly how you obtain them, find
  1. \(2 z + 5 z ^ { * }\),
  2. \(( z - \mathrm { i } ) ^ { 2 }\),
  3. \(\frac { 3 } { z }\).
OCR FP1 2008 January Q5
8 marks Easy -1.2
5 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by \(\mathbf { A } = \left( \begin{array} { l } 3 \\ 1 \\ 2 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l } 4 \\ 0 \\ 3 \end{array} \right)\) and \(\mathbf { C } = \left( \begin{array} { l l l } 2 & 4 & - 1 \end{array} \right)\). Find
  1. \(\mathbf { A } - 4 \mathbf { B }\),
  2. BC ,
  3. CA .