Questions — OCR (4619 questions)

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OCR C4 Q6
11 marks Standard +0.3
6. Relative to a fixed origin, the points \(A , B\) and \(C\) have position vectors ( \(2 \mathbf { i } - \mathbf { j } + 6 \mathbf { k }\) ), \(( 5 \mathbf { i } - 4 \mathbf { j } )\) and \(( 7 \mathbf { i } - 6 \mathbf { j } - 4 \mathbf { k } )\) respectively.
  1. Show that \(A , B\) and \(C\) all lie on a single straight line.
  2. Write down the ratio \(A B : B C\) The point \(D\) has position vector \(( 3 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )\).
  3. Show that \(A D\) is perpendicular to \(B D\).
  4. Find the exact area of triangle \(A B D\).
OCR C4 Q7
11 marks Standard +0.3
7. A mathematician is selling goods at a car boot sale. She believes that the rate at which she makes sales depends on the length of time since the start of the sale, \(t\) hours, and the total value of sales she has made up to that time, \(\pounds x\). She uses the model $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { k ( 5 - t ) } { x }$$ where \(k\) is a constant.
Given that after two hours she has made sales of \(\pounds 96\) in total,
  1. solve the differential equation and show that she made \(\pounds 72\) in the first hour of the sale. The mathematician believes that is it not worth staying at the sale once she is making sales at a rate of less than \(\pounds 10\) per hour.
  2. Verify that at 3 hours and 5 minutes after the start of the sale, she should have already left.
OCR C4 Q8
12 marks Standard +0.3
8.
\includegraphics[max width=\textwidth, alt={}, center]{a86f277c-a2ec-4ba0-ab08-575cad2a5e53-3_424_698_246_479} The diagram shows the curve \(y = \mathrm { f } ( x )\) in the interval \(0 \leq x \leq 2 \pi\) where $$\mathrm { f } ( x ) = \frac { \cos x } { 2 - \sin x } , \quad x \in \mathbb { R }$$
  1. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 1 - 2 \sin x } { ( 2 - \sin x ) ^ { 2 } }\).
  2. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = \pi\).
  3. Find the minimum and maximum values of \(\mathrm { f } ( x )\) in the interval \(0 \leq x \leq 2 \pi\).
  4. Explain why your answers to part (c) are the minimum and maximum values of \(\mathrm { f } ( x )\) for all real values of \(x\).
OCR C4 Q1
4 marks Moderate -0.3
  1. Express
$$\frac { x - 10 } { ( x - 3 ) ( x + 4 ) } - \frac { x - 8 } { ( x - 3 ) ( 2 x - 1 ) }$$ as a single fraction in its simplest form.
OCR C4 Q2
5 marks Moderate -0.3
2. (i) Expand \(( 1 + 4 x ) ^ { \frac { 3 } { 2 } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
(ii) State the set of values of \(x\) for which your expansion is valid.
OCR C4 Q3
7 marks Standard +0.3
3. A curve has the equation $$3 x ^ { 2 } + x y - 2 y ^ { 2 } + 25 = 0$$ Find an equation for the normal to the curve at the point with coordinates \(( 1,4 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C4 Q4
9 marks Moderate -0.3
4. The line \(l _ { 1 }\) passes through the points \(P\) and \(Q\) with position vectors ( \(- \mathbf { i } - 8 \mathbf { j } + 3 \mathbf { k }\) ) and ( \(2 \mathbf { i } - 9 \mathbf { j } + \mathbf { k }\) ) respectively, relative to a fixed origin.
  1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has the equation $$\mathbf { r } = ( 6 \mathbf { i } + a \mathbf { j } + b \mathbf { k } ) + t ( \mathbf { i } + 4 \mathbf { j } - \mathbf { k } )$$ and also passes through the point \(Q\).
  2. Find the values of the constants \(a\) and \(b\).
  3. Find, in degrees to 1 decimal place, the acute angle between lines \(l _ { 1 }\) and \(l _ { 2 }\).
OCR C4 Q5
9 marks Standard +0.3
5. (i) Given that $$x = \sec \frac { y } { 2 } , \quad 0 \leq y < \pi$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { x \sqrt { x ^ { 2 } - 1 } }$$ (ii) Find an equation for the tangent to the curve \(y = \sqrt { 3 + 2 \cos x }\) at the point where \(x = \frac { \pi } { 3 }\).
OCR C4 Q6
11 marks Standard +0.3
6. A curve has parametric equations $$x = \frac { t } { 2 - t } , \quad y = \frac { 1 } { 1 + t } , \quad - 1 < t < 2$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 1 } { 2 } \left( \frac { 2 - t } { 1 + t } \right) ^ { 2 }\).
  2. Find an equation for the normal to the curve at the point where \(t = 1\).
  3. Show that the cartesian equation of the curve can be written in the form $$y = \frac { 1 + x } { 1 + 3 x }$$
OCR C4 Q7
11 marks Standard +0.3
  1. (i) Find
$$\int x ^ { 2 } \sin x \mathrm {~d} x$$ (ii) Use the substitution \(u = 1 + \sin x\) to find the value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos x ( 1 + \sin x ) ^ { 3 } d x$$
OCR C4 Q8
16 marks Challenging +1.2
8.
\includegraphics[max width=\textwidth, alt={}]{72221d03-8a4e-49d6-b5f9-cdcb4c9cbf1a-3_252_757_267_484}
The diagram shows a hemispherical bowl of radius 5 cm . The bowl is filled with water but the water leaks from a hole at the base of the bowl. At time \(t\) minutes, the depth of water is \(h \mathrm {~cm}\) and the volume of water in the bowl is \(V \mathrm {~cm} ^ { 3 }\), where $$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 15 - h ) .$$ In a model it is assumed that the rate at which the volume of water in the bowl decreases is proportional to \(V\).
  1. Show that $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - \frac { k h ( 15 - h ) } { 3 ( 10 - h ) } ,$$ where \(k\) is a positive constant.
  2. Express \(\frac { 3 ( 10 - h ) } { h ( 15 - h ) }\) in partial fractions. Given that when \(t = 0 , h = 5\),
  3. show that $$h ^ { 2 } ( 15 - h ) = 250 \mathrm { e } ^ { - k t } .$$ Given also that when \(t = 2 , h = 4\),
  4. find the value of \(k\) to 3 significant figures.
OCR C4 Q1
4 marks Standard +0.2
  1. Differentiate each of the following with respect to \(x\) and simplify your answers.
    1. \(\quad \ln ( \cos x )\)
    2. \(x ^ { 2 } \sin 3 x\)
    3. A curve has the equation
    $$x ^ { 2 } + 3 x y - 2 y ^ { 2 } + 17 = 0$$
  2. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  3. Find an equation for the normal to the curve at the point ( \(3 , - 2\) ).
OCR C4 Q3
9 marks Standard +0.3
3. $$f ( x ) = 3 - \frac { x - 1 } { x - 3 } + \frac { x + 11 } { 2 x ^ { 2 } - 5 x - 3 } , \quad | x | < \frac { 1 } { 2 }$$
  1. Show that $$f ( x ) = \frac { 4 x - 1 } { 2 x + 1 }$$
  2. Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
OCR C4 Q4
9 marks Standard +0.3
4. A curve has parametric equations $$x = t ^ { 3 } + 1 , \quad y = \frac { 2 } { t } , \quad t \neq 0$$
  1. Find an equation for the normal to the curve at the point where \(t = 1\), giving your answer in the form \(y = m x + c\).
  2. Find a cartesian equation for the curve in the form \(y = \mathrm { f } ( x )\).
OCR C4 Q5
10 marks Standard +0.3
5. $$f ( x ) = \frac { 15 - 17 x } { ( 2 + x ) ( 1 - 3 x ) ^ { 2 } } , \quad x \neq - 2 , \quad x \neq \frac { 1 } { 3 }$$
  1. Find the values of the constants \(A , B\) and \(C\) such that $$\mathrm { f } ( x ) = \frac { A } { 2 + x } + \frac { B } { 1 - 3 x } + \frac { C } { ( 1 - 3 x ) ^ { 2 } }$$
  2. Find the value of $$\int _ { - 1 } ^ { 0 } f ( x ) d x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers.
OCR C4 Q6
10 marks Standard +0.3
6. Relative to a fixed origin, \(O\), the line \(l\) has the equation $$\mathbf { r } = \left( \begin{array} { c } 1 \\ p \\ - 5 \end{array} \right) + \lambda \left( \begin{array} { c } 3 \\ - 1 \\ q \end{array} \right)$$ where \(p\) and \(q\) are constants and \(\lambda\) is a scalar parameter.
Given that the point \(A\) with coordinates \(( - 5,9 , - 9 )\) lies on \(l\),
  1. find the values of \(p\) and \(q\),
  2. show that the point \(B\) with coordinates \(( 25 , - 1,11 )\) also lies on \(l\). The point \(C\) lies on \(l\) and is such that \(O C\) is perpendicular to \(l\).
  3. Find the coordinates of \(C\).
  4. Find the ratio \(A C : C B\)
OCR C4 Q7
11 marks Standard +0.3
7. (i) Use the substitution \(x = 2 \sin u\) to evaluate $$\int _ { 0 } ^ { \sqrt { 3 } } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x$$ (ii) Evaluate $$\int _ { 0 } ^ { \frac { \pi } { 2 } } x \cos x \mathrm {~d} x$$
OCR C4 Q8
12 marks Moderate -0.8
  1. The rate of increase in the number of bacteria in a culture, \(N\), at time \(t\) hours is proportional to \(N\).
    1. Write down a differential equation connecting \(N\) and \(t\).
    Given that initially there are \(N _ { 0 }\) bacteria present in a culture,
  2. Show that \(N = N _ { 0 } \mathrm { e } ^ { k t }\), where \(k\) is a positive constant. Given also that the number of bacteria present doubles every six hours,
  3. find the value of \(k\),
  4. find how long it takes for the number of bacteria to increase by a factor of ten, giving your answer to the nearest minute.
OCR C4 Q1
5 marks Moderate -0.3
  1. Show that
$$\int _ { 2 } ^ { 4 } x \left( x ^ { 2 } - 4 \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = 8 \sqrt { 3 }$$
OCR C4 Q2
6 marks Moderate -0.3
  1. (i) Simplify
$$\frac { 2 x ^ { 2 } + 3 x - 9 } { 2 x ^ { 2 } - 7 x + 6 }$$ (ii) Find the quotient and remainder when ( \(2 x ^ { 4 } - 1\) ) is divided by ( \(x ^ { 2 } - 2\) ).
OCR C4 Q3
7 marks Standard +0.3
3. A curve has the equation $$2 \sin 2 x - \tan y = 0$$
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form in terms of \(x\) and \(y\).
  2. Show that the tangent to the curve at the point \(\left( \frac { \pi } { 6 } , \frac { \pi } { 3 } \right)\) has the equation $$y = \frac { 1 } { 2 } x + \frac { \pi } { 4 } .$$
OCR C4 Q4
8 marks Standard +0.3
  1. The gradient at any point \(( x , y )\) on a curve is proportional to \(\sqrt { y }\).
Given that the curve passes through the point with coordinates \(( 0,4 )\),
  1. show that the equation of the curve can be written in the form $$2 \sqrt { y } = k x + 4$$ where \(k\) is a positive constant. Given also that the curve passes through the point with coordinates ( 2,9 ),
  2. find the equation of the curve in the form \(y = \mathrm { f } ( x )\).
OCR C4 Q5
9 marks Standard +0.3
5.
\includegraphics[max width=\textwidth, alt={}, center]{00ad2596-cd76-425d-a373-a0deda11e3c0-2_444_702_246_516} The diagram shows the curve with parametric equations $$x = 2 - t ^ { 2 } , \quad y = t ( t + 1 ) , \quad t \geq 0$$
  1. Find the coordinates of the points where the curve meets the coordinate axes.
  2. Find an equation for the tangent to the curve at the point where \(t = 2\), giving your answer in the form \(a x + b y + c = 0\).
OCR C4 Q6
12 marks Standard +0.3
6. $$f ( x ) = \frac { 1 + 3 x } { ( 1 - x ) ( 1 - 3 x ) } , \quad | x | < \frac { 1 } { 3 }$$
  1. Find the values of the constants \(A\) and \(B\) such that $$\mathrm { f } ( x ) = \frac { A } { 1 - x } + \frac { B } { 1 - 3 x }$$
  2. Evaluate $$\int _ { 0 } ^ { \frac { 1 } { 4 } } f ( x ) d x$$ giving your answer as a single logarithm.
  3. Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
OCR C4 Q7
12 marks Standard +0.3
7. Relative to a fixed origin, two lines have the equations
and $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { c } 4 \\ 1 \\ 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 4 \\ 5 \end{array} \right) \\ & \mathbf { r } = \left( \begin{array} { c } - 3 \\ 1 \\ - 6 \end{array} \right) + t \left( \begin{array} { l } 3 \\ a \\ b \end{array} \right) , \end{aligned}$$ where \(a\) and \(b\) are constants and \(s\) and \(t\) are scalar parameters.
Given that the two lines are perpendicular,
  1. find a linear relationship between \(a\) and \(b\). Given also that the two lines intersect,
  2. find the values of \(a\) and \(b\),
  3. find the coordinates of the point where they intersect.