Questions — OCR C4 (310 questions)

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OCR C4 2007 June Q8
8 The height, \(h\) metres, of a shrub \(t\) years after planting is given by the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 6 - h } { 20 }$$ A shrub is planted when its height is 1 m .
  1. Show by integration that \(t = 20 \ln \left( \frac { 5 } { 6 - h } \right)\).
  2. How long after planting will the shrub reach a height of 2 m ?
  3. Find the height of the shrub 10 years after planting.
  4. State the maximum possible height of the shrub.
OCR C4 2007 June Q9
9 Lines \(L _ { 1 } , L _ { 2 }\) and \(L _ { 3 }\) have vector equations $$\begin{aligned} & L _ { 1 } : \mathbf { r } = ( 5 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) + s ( - 6 \mathbf { i } + 8 \mathbf { j } - 2 \mathbf { k } ) ,
& L _ { 2 } : \mathbf { r } = ( 3 \mathbf { i } - 8 \mathbf { j } ) + t ( \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } ) ,
& L _ { 3 } : \mathbf { r } = ( 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) + u ( 3 \mathbf { i } + c \mathbf { j } + \mathbf { k } ) . \end{aligned}$$
  1. Calculate the acute angle between \(L _ { 1 }\) and \(L _ { 2 }\).
  2. Given that \(L _ { 1 }\) and \(L _ { 3 }\) are parallel, find the value of \(c\).
  3. Given instead that \(L _ { 2 }\) and \(L _ { 3 }\) intersect, find the value of \(c\). 4
OCR C4 2008 June Q1
1
  1. Simplify \(\frac { \left( 2 x ^ { 2 } - 7 x - 4 \right) ( x + 1 ) } { \left( 3 x ^ { 2 } + x - 2 \right) ( x - 4 ) }\).
  2. Find the quotient and remainder when \(x ^ { 3 } + 2 x ^ { 2 } - 6 x - 5\) is divided by \(x ^ { 2 } + 4 x + 1\).
OCR C4 2008 June Q2
2 Find the exact value of \(\int _ { 1 } ^ { \mathrm { e } } x ^ { 4 } \ln x \mathrm {~d} x\).
OCR C4 2008 June Q3
3 The equation of a curve is \(x ^ { 2 } y - x y ^ { 2 } = 2\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } - 2 x y } { x ^ { 2 } - 2 x y }\).
  2. (a) Show that, if \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\), then \(y = 2 x\).
    (b) Hence find the coordinates of the point on the curve where the tangent is parallel to the \(x\)-axis.
OCR C4 2008 June Q4
4 Relative to an origin \(O\), the points \(A\) and \(B\) have position vectors \(3 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\) and \(\mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }\) respectively.
  1. Find a vector equation of the line passing through \(A\) and \(B\).
  2. Find the position vector of the point \(P\) on \(A B\) such that \(O P\) is perpendicular to \(A B\).
OCR C4 2008 June Q5
5
  1. Show that \(\sqrt { \frac { 1 - x } { 1 + x } } \approx 1 - x + \frac { 1 } { 2 } x ^ { 2 }\), for \(| x | < 1\).
  2. By taking \(x = \frac { 2 } { 7 }\), show that \(\sqrt { 5 } \approx \frac { 111 } { 49 }\).
OCR C4 2008 June Q6
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
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
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
1 Find the quotient and remainder when \(x ^ { 4 } + 1\) is divided by \(x ^ { 2 } + 1\).
OCR C4 Specimen Q2
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
3 Find \(\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\), giving your answer in terms of e.
OCR C4 Specimen Q4
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
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
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
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
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
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 C4 Q1
  1. Express
$$\frac { 2 x } { 2 x ^ { 2 } + 3 x - 5 } \div \frac { x ^ { 3 } } { x ^ { 2 } - x }$$ as a single fraction in its simplest form.
OCR C4 Q2
2. A curve has the equation $$2 x ^ { 2 } + x y - y ^ { 2 } + 18 = 0$$ Find the coordinates of the points where the tangent to the curve is parallel to the \(x\)-axis.
OCR C4 Q3
3. The first four terms in the series expansion of \(( 1 + a x ) ^ { n }\) in ascending powers of \(x\) are $$1 - 4 x + 24 x ^ { 2 } + k x ^ { 3 }$$ where \(a , n\) and \(k\) are constants and \(| a x | < 1\).
  1. Find the values of \(a\) and \(n\).
  2. Show that \(k = - 160\).
OCR C4 Q4
4. Relative to a fixed origin, \(O\), the points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } 1
5
- 1 \end{array} \right)\) and \(\left( \begin{array} { c } 6
3
- 6 \end{array} \right)\) respectively. Find, in exact, simplified form,
  1. the cosine of \(\angle A O B\),
  2. the area of triangle \(O A B\),
  3. the shortest distance from \(A\) to the line \(O B\).
OCR C4 Q5
5. (i) Use the derivatives of \(\sin x\) and \(\cos x\) to prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \tan x ) = \sec ^ { 2 } x$$ The tangent to the curve \(y = 2 x \tan x\) at the point where \(x = \frac { \pi } { 4 }\) meets the \(y\)-axis at the point \(P\).
(ii) Find the \(y\)-coordinate of \(P\) in the form \(k \pi ^ { 2 }\) where \(k\) is a rational constant.
OCR C4 Q6
6. (i) Find $$\int \cot ^ { 2 } 2 x \mathrm {~d} x$$ (ii) Use the substitution \(u ^ { 2 } = x + 1\) to evaluate $$\int _ { 0 } ^ { 3 } \frac { x ^ { 2 } } { \sqrt { x + 1 } } \mathrm {~d} x$$