Questions C4 (1162 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 C4 Q8
  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
  1. Show that
$$\int _ { 2 } ^ { 4 } x \left( x ^ { 2 } - 4 \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = 8 \sqrt { 3 }$$
OCR C4 Q2
  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
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
  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
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
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
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.
OCR C4 Q8
8. (i) Find $$\int x ^ { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } x } \mathrm {~d} x$$ (ii) Using the substitution \(u = \sin t\), evaluate $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { 2 } 2 t \cos t \mathrm {~d} t$$
OCR C4 Q2
2. (i) Find the binomial expansion of \(( 2 - 3 x ) ^ { - 3 }\) 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. (i) Express \(\frac { x + 11 } { ( x + 4 ) ( x - 3 ) }\) as a sum of partial fractions.
(ii) Evaluate $$\int _ { 0 } ^ { 2 } \frac { x + 11 } { ( x + 4 ) ( x - 3 ) } d x$$ giving your answer in the form \(\ln k\), where \(k\) is an exact simplified fraction.
OCR C4 Q4
4. A curve has the equation $$4 x ^ { 2 } - 2 x y - y ^ { 2 } + 11 = 0$$ Find an equation for the normal to the curve at the point with coordinates \(( - 1 , - 3 )\).
OCR C4 Q5
5.
\includegraphics[max width=\textwidth, alt={}, center]{5840974b-b08a-4818-9a59-97b2d3ce9890-1_469_809_1777_484} The diagram shows the curve with equation \(y = x \sqrt { 1 - x } , 0 \leq x \leq 1\).
Use the substitution \(u ^ { 2 } = 1 - x\) to show that the area of the region bounded by the curve and the \(x\)-axis is \(\frac { 4 } { 15 }\).
OCR C4 Q6
6. The number of people, \(n\), in a queue at a Post Office \(t\) minutes after it opens is modelled by the differential equation $$\frac { \mathrm { d } n } { \mathrm {~d} t } = \mathrm { e } ^ { 0.5 t } - 5 , \quad t \geq 0$$
  1. Find, to the nearest second, the time when the model predicts that there will be the least number of people in the queue.
  2. Given that there are 20 people in the queue when the Post Office opens, solve the differential equation.
  3. Explain why this model would not be appropriate for large values of \(t\).
OCR C4 Q7
7. (i) Show that ( \(2 x + 3\) ) is a factor of ( \(\left. 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 \right)\) and hence, simplify $$\frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } .$$ (ii) Show that $$\int _ { 2 } ^ { 5 } \frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } \mathrm {~d} x = \ln k$$ where \(k\) is an integer.
OCR C4 Q8
8. The points \(A\) and \(B\) have coordinates \(( 3,9 , - 7 )\) and \(( 13 , - 6 , - 2 )\) respectively.
  1. Find, in vector form, an equation for the line \(l\) which passes through \(A\) and \(B\).
  2. Show that the point \(C\) with coordinates \(( 9,0 , - 4 )\) lies on \(l\). The point \(D\) is the point on \(l\) closest to the origin, \(O\).
  3. Find the coordinates of \(D\).
  4. Find the area of triangle \(O A B\) to 3 significant figures.
OCR C4 Q9
9. A curve has parametric equations $$x = \sec \theta + \tan \theta , \quad y = \operatorname { cosec } \theta + \cot \theta , \quad 0 < \theta < \frac { \pi } { 2 }$$
  1. Show that \(x + \frac { 1 } { x } = 2 \sec \theta\). Given that \(y + \frac { 1 } { y } = 2 \operatorname { cosec } \theta\),
  2. find a cartesian equation for the curve.
  3. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} \theta } = \frac { 1 } { 2 } \left( x ^ { 2 } + 1 \right)\).
  4. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
OCR C4 Q1
  1. Express
$$\frac { 5 x } { ( x - 4 ) ( x + 1 ) } + \frac { 3 } { ( x - 2 ) ( x + 1 ) }$$ as a single fraction in its simplest form.
OCR C4 Q2
2. A curve has the equation $$x ^ { 2 } + 2 x y ^ { 2 } + y = 4$$ Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
OCR C4 Q3
3. Evaluate $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \sin 2 x \cos x d x$$
OCR C4 Q4
  1. A curve has parametric equations
$$x = \cos 2 t , \quad y = \operatorname { cosec } t , \quad 0 < t < \frac { \pi } { 2 }$$ The point \(P\) on the curve has \(x\)-coordinate \(\frac { 1 } { 2 }\).
  1. Find the value of the parameter \(t\) at \(P\).
  2. Show that the tangent to the curve at \(P\) has the equation $$y = 2 x + 1$$
OCR C4 Q5
  1. (i) Express \(\frac { 2 + 20 x } { 1 + 2 x - 8 x ^ { 2 } }\) as a sum of partial fractions.
    (ii) Hence find the series expansion of \(\frac { 2 + 20 x } { 1 + 2 x - 8 x ^ { 2 } } , | x | < \frac { 1 } { 4 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
  2. Use the substitution \(x = 2 \tan u\) to show that
$$\int _ { 0 } ^ { 2 } \frac { x ^ { 2 } } { x ^ { 2 } + 4 } d x = \frac { 1 } { 2 } ( 4 - \pi )$$
OCR C4 Q7
  1. A straight road passes through villages at the points \(A\) and \(B\) with position vectors \(( 9 \mathbf { i } - 8 \mathbf { j } + 2 \mathbf { k } )\) and ( \(4 \mathbf { j } + \mathbf { k }\) ) respectively, relative to a fixed origin.
The road ends at a junction at the point \(C\) with another straight road which lies along the line with equation $$\mathbf { r } = ( 2 \mathbf { i } + 16 \mathbf { j } - \mathbf { k } ) + t ( - 5 \mathbf { i } + 3 \mathbf { j } ) ,$$ where \(t\) is a scalar parameter.
  1. Find the position vector of \(C\). Given that 1 unit on each coordinate axis represents 200 metres,
  2. find the distance, in kilometres, from the village at \(A\) to the junction at \(C\).
OCR C4 Q8
8. (i) Find \(\int \tan ^ { 2 } x \mathrm {~d} x\).
(ii) Show that $$\int \tan x \mathrm {~d} x = \ln | \sec x | + c$$ where \(c\) is an arbitrary constant.
\includegraphics[max width=\textwidth, alt={}, center]{1e93a786-6105-4c69-a79a-a5f6e6c4aa0a-2_554_784_1484_507} The diagram shows part of the curve with equation \(y = x ^ { \frac { 1 } { 2 } } \tan x\).
The shaded region bounded by the curve, the \(x\)-axis and the line \(x = \frac { \pi } { 3 }\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
(iii) Show that the volume of the solid formed is \(\frac { 1 } { 18 } \pi ^ { 2 } ( 6 \sqrt { 3 } - \pi ) - \pi \ln 2\).
OCR C4 Q9
9. An entomologist is studying the population of insects in a colony. Initially there are 300 insects in the colony and in a model, the entomologist assumes that the population, \(P\), at time \(t\) weeks satisfies the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = k P$$ where \(k\) is a constant.
  1. Find an expression for \(P\) in terms of \(k\) and \(t\). Given that after one week there are 360 insects in the colony,
  2. find the value of \(k\) to 3 significant figures. Given also that after two and three weeks there are 440 and 600 insects respectively,
  3. comment on suitability of the modelling assumption. An alternative model assumes that $$\frac { \mathrm { d } P } { \mathrm {~d} t } = P ( 0.4 - 0.25 \cos 0.5 t )$$
  4. Using the initial data, \(P = 300\) when \(t = 0\), solve this differential equation.
  5. Compare the suitability of the two models.