Questions — OCR C4 (317 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 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
Sort by: Default | Easiest first | Hardest first
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
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.
OCR C4 Q8
13 marks Standard +0.3
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
6 marks Moderate -0.3
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
7 marks Moderate -0.3
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
7 marks Standard +0.3
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
7 marks Standard +0.3
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
8 marks Moderate -0.3
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
9 marks Standard +0.3
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
10 marks Standard +0.8
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
13 marks Standard +0.8
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
6 marks Moderate -0.3
1 Express \(2 \sin \theta - 3 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants to be determined, and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Hence write down the greatest and least possible values of \(1 + 2 \sin \theta - 3 \cos \theta\).
OCR C4 Q2
7 marks Standard +0.3
2 Express \(4 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < { } _ { 2 } ^ { 1 } \pi\).-
Hence solve the equation \(4 \cos \theta - \sin \theta = 3\), for \(0 \leqslant \theta \leqslant 2 \pi\).
OCR C4 Q3
19 marks Challenging +1.2
3 Archimedes, about 2200 years ago, used regular polygons inside and outside circles to obtain approximations for \(\pi\).
  1. Fig. 8.1 shows a regular 12 -sided polygon inscribed in a circle of radius 1 unit, centre \(\mathrm { O } . \mathrm { AB }\) is one of the sides of the polygon. C is the midpoint of AB . Archimedes used the fact that the circumference of the circle is greater than the perimeter of this polygon. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c0fcd64b-8ca0-4309-9f58-c23cc4208f4d-2_457_422_457_936} \captionsetup{labelformat=empty} \caption{Fig. 8.1}
    \end{figure} (A) Show that \(\mathrm { AB } = 2 \sin 15 ^ { \circ }\).
    (B) Use a double angle formula to express \(\cos 30 ^ { \circ }\) in terms of \(\sin 15 ^ { \circ }\). Using the exact value of \(\cos 30 ^ { \circ }\), show that \(\sin 15 ^ { \circ } = \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }\).
    (C) Use this result to find an exact expression for the perimeter of the polygon. Hence show that \(\pi > 6 \sqrt { 2 - \sqrt { 3 } }\).
  2. In Fig. 8.2, a regular 12-sided polygon lies outside the circle of radius 1 unit, which touches each side of the polygon. F is the midpoint of DE. Archimedes used the fact that the circumference of the circle is less than the perimeter of this polygon. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c0fcd64b-8ca0-4309-9f58-c23cc4208f4d-2_450_416_1562_940} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
    \end{figure} (A) Show that \(\mathrm { DE } = 2 \tan 15 ^ { \circ }\).
    (B) Let \(t = \tan 15 ^ { \circ }\). Use a double angle formula to express \(\tan 30 ^ { \circ }\) in terms of \(t\). Hence show that \(t ^ { 2 } + 2 \sqrt { 3 } t - 1 = 0\).
    (C) Solve this equation, and hence show that \(\pi < 12 ( 2 - \sqrt { 3 } )\).
  3. Use the results in parts (i)( \(C\) ) and (ii)( \(C\) ) to establish upper and lower bounds for the value of \(\pi\), giving your answers in decimal form.
OCR C4 Q4
7 marks Moderate -0.3
4 Solve the equation \(\cos 2 \theta = \sin \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\), giving your answers in terms of \(\pi\).
OCR C4 Q5
6 marks Standard +0.3
5 Express \(\sqrt { 3 } \sin x - \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Express \(\alpha\) in the form \(k \pi\). Find the exact coordinates of the maximum point of the curve \(y = \sqrt { 3 } \sin x - \cos x\) for which \(0 < x < 2 \pi\).
OCR C4 Q6
7 marks Standard +0.3
6 Express \(\sin \theta - 3 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants to be determined, and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Hence solve the equation \(\sin \theta - 3 \cos \theta = 1\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
OCR C4 Q7
6 marks Moderate -0.3
7 Fig. 1 shows part of the graph of \(y = \sin x \quad \sqrt { 3 } \cos x\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c0fcd64b-8ca0-4309-9f58-c23cc4208f4d-3_452_613_1187_745} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Express \(\quad \sqrt { } \quad\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 \leqslant \alpha \leqslant \frac { 1 } { 2 } \pi\).
Hence write down the exact coordinates of the turning point P .