Questions — OCR (4619 questions)

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OCR C4 2008 June Q6
8 marks Standard +0.3
6 Two lines have equations $$\mathbf { r } = \left( \begin{array} { r } 1 \\ 0 \\ - 5 \end{array} \right) + t \left( \begin{array} { l } 2 \\ 3 \\ 4 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } 12 \\ 0 \\ 5 \end{array} \right) + s \left( \begin{array} { r } 1 \\ - 4 \\ - 2 \end{array} \right) .$$
  1. Show that the lines intersect.
  2. Find the angle between the lines.
OCR C4 2008 June Q8
11 marks Standard +0.3
8
  1. Given that \(\frac { 2 t } { ( t + 1 ) ^ { 2 } }\) can be expressed in the form \(\frac { A } { t + 1 } + \frac { B } { ( t + 1 ) ^ { 2 } }\), find the values of the constants \(A\) and \(B\).
  2. Show that the substitution \(t = \sqrt { 2 x - 1 }\) transforms \(\int \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x\) to \(\int \frac { 2 t } { ( t + 1 ) ^ { 2 } } \mathrm {~d} t\).
  3. Hence find the exact value of \(\int _ { 1 } ^ { 5 } \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x\).
OCR C4 2008 June Q12
Moderate -0.3
12
0
5 \end{array} \right) + s \left( \begin{array} { r } 1
- 4
- 2 \end{array} \right) .$$
  1. Show that the lines intersect.
  2. Find the angle between the lines.
  3. Show that, if \(y = \operatorname { cosec } x\), then \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed as \(- \operatorname { cosec } x \cot x\).
  4. Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \sin x \tan x \cot t$$ given that \(x = \frac { 1 } { 6 } \pi\) when \(t = \frac { 1 } { 2 } \pi\). 8
  5. Given that \(\frac { 2 t } { ( t + 1 ) ^ { 2 } }\) can be expressed in the form \(\frac { A } { t + 1 } + \frac { B } { ( t + 1 ) ^ { 2 } }\), find the values of the constants \(A\) and \(B\).
  6. Show that the substitution \(t = \sqrt { 2 x - 1 }\) transforms \(\int \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x\) to \(\int \frac { 2 t } { ( t + 1 ) ^ { 2 } } \mathrm {~d} t\).
  7. Hence find the exact value of \(\int _ { 1 } ^ { 5 } \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x\). 9 The parametric equations of a curve are $$x = 2 \theta + \sin 2 \theta , \quad y = 4 \sin \theta$$ and part of its graph is shown below.
    \includegraphics[max width=\textwidth, alt={}, center]{b8ba126f-c5fa-4828-9439-e5162a03ca5b-3_646_1150_1050_500}
  8. Find the value of \(\theta\) at \(A\) and the value of \(\theta\) at \(B\).
  9. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec \theta\).
  10. At the point \(C\) on the curve, the gradient is 2 . Find the coordinates of \(C\), giving your answer in an exact form.
OCR C4 Specimen Q1
4 marks Moderate -0.8
1 Find the quotient and remainder when \(x ^ { 4 } + 1\) is divided by \(x ^ { 2 } + 1\).
OCR C4 Specimen Q2
5 marks Moderate -0.8
2
  1. Expand \(( 1 - 2 x ) ^ { - \frac { 1 } { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
  2. State the set of values for which the expansion in part (i) is valid.
OCR C4 Specimen Q3
5 marks Moderate -0.3
3 Find \(\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\), giving your answer in terms of e.
OCR C4 Specimen Q4
7 marks Moderate -0.5
4
\includegraphics[max width=\textwidth, alt={}, center]{798da17d-0af5-4aa6-b731-564642dc28d5-2_428_572_861_760} As shown in the diagram the points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) with respect to the origin \(O\).
  1. Make a sketch of the diagram, and mark the points \(C , D\) and \(E\) such that \(\overrightarrow { O C } = 2 \mathbf { a } , \overrightarrow { O D } = 2 \mathbf { a } + \mathbf { b }\) and \(\overrightarrow { O E } = \frac { 1 } { 3 } \overrightarrow { O D }\).
  2. By expressing suitable vectors in terms of \(\mathbf { a }\) and \(\mathbf { b }\), prove that \(E\) lies on the line joining \(A\) and \(B\).
OCR C4 Specimen Q5
8 marks Standard +0.3
5
  1. For the curve \(2 x ^ { 2 } + x y + y ^ { 2 } = 14\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Deduce that there are two points on the curve \(2 x ^ { 2 } + x y + y ^ { 2 } = 14\) at which the tangents are parallel to the \(x\)-axis, and find their coordinates.
OCR C4 Specimen Q6
9 marks Moderate -0.3
6
\includegraphics[max width=\textwidth, alt={}, center]{798da17d-0af5-4aa6-b731-564642dc28d5-3_766_611_251_703} The diagram shows the curve with parametric equations $$x = a \sin \theta , \quad y = a \theta \cos \theta$$ where \(a\) is a positive constant and \(- \pi \leqslant \theta \leqslant \pi\). The curve meets the positive \(y\)-axis at \(A\) and the positive \(x\)-axis at \(B\).
  1. Write down the value of \(\theta\) corresponding to the origin, and state the coordinates of \(A\) and \(B\).
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - \theta \tan \theta\), and hence find the equation of the tangent to the curve at the origin.
OCR C4 Specimen Q7
11 marks Standard +0.3
7 The line \(L _ { 1 }\) passes through the point \(( 3,6,1 )\) and is parallel to the vector \(2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k }\). The line \(L _ { 2 }\) passes through the point ( \(3 , - 1,4\) ) and is parallel to the vector \(\mathbf { i } - 2 \mathbf { j } + \mathbf { k }\).
  1. Write down vector equations for the lines \(L _ { 1 }\) and \(L _ { 2 }\).
  2. Prove that \(L _ { 1 }\) and \(L _ { 2 }\) intersect, and find the coordinates of their point of intersection.
  3. Calculate the acute angle between the lines.
OCR C4 Specimen Q8
12 marks Standard +0.3
8 Let \(I = \int \frac { 1 } { x ( 1 + \sqrt { } x ) ^ { 2 } } \mathrm {~d} x\).
  1. Show that the substitution \(u = \sqrt { } x\) transforms \(I\) to \(\int \frac { 2 } { u ( 1 + u ) ^ { 2 } } \mathrm {~d} u\).
  2. Express \(\frac { 2 } { u ( 1 + u ) ^ { 2 } }\) in the form \(\frac { A } { u } + \frac { B } { 1 + u } + \frac { C } { ( 1 + u ) ^ { 2 } }\).
  3. Hence find \(I\).
OCR C4 Specimen Q9
11 marks Standard +0.3
9
\includegraphics[max width=\textwidth, alt={}, center]{798da17d-0af5-4aa6-b731-564642dc28d5-4_572_917_294_607} A cylindrical container has a height of 200 cm . The container was initially full of a chemical but there is a leak from a hole in the base. When the leak is noticed, the container is half-full and the level of the chemical is dropping at a rate of 1 cm per minute. It is required to find for how many minutes the container has been leaking. To model the situation it is assumed that, when the depth of the chemical remaining is \(x \mathrm {~cm}\), the rate at which the level is dropping is proportional to \(\sqrt { } x\). Set up and solve an appropriate differential equation, and hence show that the container has been leaking for about 80 minutes.
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
    (a) \(\sum _ { r = 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) }\),
    (b) \(\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) }\).
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 \(a\). \end{itemize}
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.