Questions — OCR (4619 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 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 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 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 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 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 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics 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
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 .
OCR FP1 2008 January Q6
8 marks Standard +0.3
6 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by $$| z | = | z - 4 \mathbf { i } | \quad \text { and } \quad \arg z = \frac { 1 } { 6 } \pi$$ respectively.
  1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Hence find, in the form \(x +\) i \(y\), the complex number represented by the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
OCR FP1 2008 January Q7
7 marks Moderate -0.3
7 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c } a & 3 \\ - 2 & 1 \end{array} \right)\).
  1. Given that \(\mathbf { A }\) is singular, find \(a\).
  2. Given instead that \(\mathbf { A }\) is non-singular, find \(\mathbf { A } ^ { - 1 }\) and hence solve the simultaneous equations $$\begin{aligned} a x + 3 y & = 1 \\ - 2 x + y & = - 1 \end{aligned}$$
OCR FP1 2008 January Q8
7 marks Standard +0.8
8 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 1\) and \(u _ { n + 1 } = u _ { n } + 2 n + 1\).
  1. Show that \(u _ { 4 } = 16\).
  2. Hence suggest an expression for \(u _ { n }\).
  3. Use induction to prove that your answer to part (ii) is correct.
OCR FP1 2008 January Q9
8 marks Standard +0.8
9
  1. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } = ( \alpha + \beta ) ^ { 3 } - 3 \alpha \beta ( \alpha + \beta )\).
  2. The quadratic equation \(x ^ { 2 } - 5 x + 7 = 0\) has roots \(\alpha\) and \(\beta\). Find a quadratic equation with roots \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\).
  3. Show that \(\frac { 2 } { r } - \frac { 1 } { r + 1 } - \frac { 1 } { r + 2 } = \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }\).
  4. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 1 } ^ { n } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }$$
  5. Hence write down the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }\).
  6. Given that \(\sum _ { r = N + 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) } = \frac { 7 } { 10 }\), find the value of \(N\).
OCR FP1 2005 June Q1
6 marks Moderate -0.5
1 Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \left( 6 r ^ { 2 } + 2 r + 1 \right) = n \left( 2 n ^ { 2 } + 4 n + 3 \right)$$
OCR FP1 2005 June Q2
6 marks Moderate -0.3
2 The matrices \(\mathbf { A }\) and \(\mathbf { I }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & 2 \\ 1 & 3 \end{array} \right)\) and \(\mathbf { I } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right)\) respectively.
  1. Find \(\mathbf { A } ^ { 2 }\) and verify that \(\mathbf { A } ^ { 2 } = 4 \mathbf { A } - \mathbf { I }\).
  2. Hence, or otherwise, show that \(\mathbf { A } ^ { - 1 } = 4 \mathbf { I } - \mathbf { A }\).
OCR FP1 2005 June Q3
7 marks Easy -1.2
3 The complex numbers \(2 + 3 \mathrm { i }\) and \(4 - \mathrm { i }\) are denoted by \(z\) and \(w\) respectively. Express each of the following in the form \(x + \mathrm { i } y\), showing clearly how you obtain your answers.
  1. \(z + 5 w\),
  2. \(z ^ { * } w\), where \(z ^ { * }\) is the complex conjugate of \(z\),
  3. \(\frac { 1 } { w }\).
OCR FP1 2005 June Q4
6 marks Moderate -0.3
4 Use an algebraic method to find the square roots of the complex number 21-20i.
OCR FP1 2005 June Q5
7 marks Standard +0.3
5
  1. Show that $$\frac { r + 1 } { r + 2 } - \frac { r } { r + 1 } = \frac { 1 } { ( r + 1 ) ( r + 2 ) }$$
  2. Hence find an expression, in terms of \(n\), for $$\frac { 1 } { 6 } + \frac { 1 } { 12 } + \frac { 1 } { 20 } + \ldots + \frac { 1 } { ( n + 1 ) ( n + 2 ) }$$
  3. Hence write down the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( r + 1 ) ( r + 2 ) }\).
OCR FP1 2005 June Q6
7 marks Moderate -0.3
6 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by $$| z - 2 \mathrm { i } | = 2 \quad \text { and } \quad | z + 1 | = | z + \mathrm { i } |$$ respectively.
  1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Hence write down the complex numbers represented by the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
    \(7 \quad\) The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { r r r } a & 1 & 3 \\ 2 & 1 & - 1 \\ 0 & 1 & 2 \end{array} \right)\).
  3. Given that \(\mathbf { B }\) is singular, show that \(a = - \frac { 2 } { 3 }\).
  4. Given instead that \(\mathbf { B }\) is non-singular, find the inverse matrix \(\mathbf { B } ^ { - 1 }\).
  5. Hence, or otherwise, solve the equations $$\begin{aligned} - x + y + 3 z & = 1 \\ 2 x + y - z & = 4 \\ y + 2 z & = - 1 \end{aligned}$$
OCR FP1 2005 June Q8
11 marks Standard +0.3
8
  1. The quadratic equation \(x ^ { 2 } - 2 x + 4 = 0\) has roots \(\alpha\) and \(\beta\).
    1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
    2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } = - 4\).
    3. Hence find a quadratic equation which has roots \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).
  2. The cubic equation \(x ^ { 3 } - 12 x ^ { 2 } + a x - 48 = 0\) has roots \(p , 2 p\) and \(3 p\).
    1. Find the value of \(p\).
    2. Hence find the value of \(a\).
OCR FP1 2005 June Q9
12 marks Standard +0.3
9
  1. Write down the matrix \(\mathbf { C }\) which represents a stretch, scale factor 2 , in the \(x\)-direction.
  2. The matrix \(\mathbf { D }\) is given by \(\mathbf { D } = \left( \begin{array} { l l } 1 & 3 \\ 0 & 1 \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { D }\).
  3. The matrix \(\mathbf { M }\) represents the combined effect of the transformation represented by \(\mathbf { C }\) followed by the transformation represented by \(\mathbf { D }\). Show that $$\mathbf { M } = \left( \begin{array} { l l } 2 & 3 \\ 0 & 1 \end{array} \right)$$
  4. Prove by induction that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 3 \left( 2 ^ { n } - 1 \right) \\ 0 & 1 \end{array} \right)\), for all positive integers \(n\).
OCR FP1 2006 June Q1
4 marks Easy -1.3
\(\mathbf { 1 }\) The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 4 & 1 \\ 0 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 1 & 1 \\ 0 & - 1 \end{array} \right)\).
  1. Find \(\mathbf { A } + 3 \mathbf { B }\).
  2. Show that \(\mathbf { A } - \mathbf { B } = k \mathbf { I }\), where \(\mathbf { I }\) is the identity matrix and \(k\) is a constant whose value should be stated. \end{itemize}