Questions C4 (1162 questions)

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OCR C4 Q7
7. Relative to a fixed origin, two lines have the equations
and $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { c } 7
0
- 3 \end{array} \right) + s \left( \begin{array} { c } 5
4
- 2 \end{array} \right)
& \mathbf { r } = \left( \begin{array} { l } a
6
3 \end{array} \right) + t \left( \begin{array} { c } - 5
14
2 \end{array} \right) , \end{aligned}$$ where \(a\) is a constant and \(s\) and \(t\) are scalar parameters.
Given that the two lines intersect,
  1. find the position vector of their point of intersection,
  2. find the value of \(a\). Given also that \(\theta\) is the acute angle between the lines,
  3. find the value of \(\cos \theta\) in the form \(k \sqrt { 5 }\) where \(k\) is rational.
OCR C4 Q8
8. A small town had a population of 9000 in the year 2001. In a model, it is assumed that the population of the town, \(P\), at time \(t\) years after 2001 satisfies the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = 0.05 P \mathrm { e } ^ { - 0.05 t }$$
  1. Show that, according to the model, the population of the town in 2011 will be 13300 to 3 significant figures.
  2. Find the value which the population of the town will approach in the long term, according to the model.
OCR C4 Q14
14
2 \end{array} \right) , \end{aligned}$$ where \(a\) is a constant and \(s\) and \(t\) are scalar parameters.
Given that the two lines intersect,
  1. find the position vector of their point of intersection,
  2. find the value of \(a\). Given also that \(\theta\) is the acute angle between the lines,
  3. find the value of \(\cos \theta\) in the form \(k \sqrt { 5 }\) where \(k\) is rational.
    8. A small town had a population of 9000 in the year 2001. In a model, it is assumed that the population of the town, \(P\), at time \(t\) years after 2001 satisfies the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = 0.05 P \mathrm { e } ^ { - 0.05 t }$$
  4. Show that, according to the model, the population of the town in 2011 will be 13300 to 3 significant figures.
  5. Find the value which the population of the town will approach in the long term, according to the model.
    9. A curve has parametric equations $$x = t ( t - 1 ) , \quad y = \frac { 4 t } { 1 - t } , \quad t \neq 1$$
  6. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). The point \(P\) on the curve has parameter \(t = - 1\).
  7. Show that the tangent to the curve at \(P\) has the equation $$x + 3 y + 4 = 0$$ The tangent to the curve at \(P\) meets the curve again at the point \(Q\).
  8. Find the coordinates of \(Q\).
OCR C4 Q1
  1. Express
$$\frac { 2 x ^ { 3 } + x ^ { 2 } } { x ^ { 2 } - 4 } \times \frac { x - 2 } { 2 x ^ { 2 } - 5 x - 3 }$$ as a single fraction in its simplest form.
OCR C4 Q2
2. A curve has the equation $$x ^ { 3 } + 2 x y - y ^ { 2 } + 24 = 0$$ Show that the normal to the curve at the point \(( 2 , - 4 )\) has the equation \(y = 3 x - 10\).
OCR C4 Q3
3. Using the substitution \(u = \mathrm { e } ^ { x } - 1\), show that $$\int _ { \ln 2 } ^ { \ln 5 } \frac { \mathrm { e } ^ { 2 x } } { \sqrt { \mathrm { e } ^ { x } - 1 } } \mathrm {~d} x = \frac { 20 } { 3 }$$
OCR C4 Q4
  1. (i) Expand \(( 1 + a x ) ^ { - 3 } , | a x | < 1\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\). Give each coefficient as simply as possible in terms of the constant \(a\).
Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { 6 - x } { ( 1 + a x ) ^ { 3 } } , | a x | < 1\), is 3 ,
(ii) find the two possible values of \(a\). Given also that \(a < 0\),
(iii) show that the coefficient of \(x ^ { 3 }\) in the expansion of \(\frac { 6 - x } { ( 1 + a x ) ^ { 3 } }\) is \(\frac { 14 } { 9 }\).
OCR C4 Q5
5. $$f ( x ) = \frac { 7 + 3 x + 2 x ^ { 2 } } { ( 1 - 2 x ) ( 1 + x ) ^ { 2 } } , \quad | x | > \frac { 1 } { 2 }$$
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that $$\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = p - \ln q$$ where \(p\) is rational and \(q\) is an integer.
OCR C4 Q6
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
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
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
  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
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
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
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
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
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
  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
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
  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
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
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
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
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
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 $$\int _ { 0 } ^ { \frac { \pi } { 2 } } x \cos x \mathrm {~d} x$$