Questions — OCR MEI (4301 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 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 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 MEI C3 Q1
5 marks Standard +0.3
1 Evaluate \(\int _ { 1 } ^ { 2 } x ^ { 2 } \ln x \mathrm {~d} x\), giving your answer in an exact form.
OCR MEI C3 Q2
16 marks Standard +0.3
2 Fig. 7 shows the curve \(y = \frac { x ^ { 2 } } { 1 + 2 x ^ { 3 } }\). It is undefined at \(x = a\); the line \(x = a\) is a vertical asymptote. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00c12cc4-f7ee-4219-8d34-a1854284f65d-1_647_1027_832_534} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Calculate the value of \(a\), giving your answer correct to 3 significant figures.
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x - 2 x ^ { 4 } } { \left( 1 + 2 x ^ { 3 } \right) ^ { 2 } }\). Hence determine the coordinates of the turning points of the curve.
  3. Show that the area of the region between the curve and the \(x\)-axis from \(x = 0\) to \(x = 1\) is \(\frac { 1 } { 6 } \ln 3\).
OCR MEI C3 Q3
20 marks Standard +0.3
3 Fig. 8 shows part of the curve \(y = x \cos 2 x\), together with a point P at which the curve crosses the \(x\)-axis. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00c12cc4-f7ee-4219-8d34-a1854284f65d-2_425_974_478_591} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find the exact coordinates of P .
  2. Show algebraically that \(x \cos 2 x\) is an odd function, and interpret this result graphically.
  3. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  4. Show that turning points occur on the curve for values of \(x\) which satisfy the equation \(x \tan 2 x = \frac { 1 } { 2 }\).
  5. Find the gradient of the curve at the origin. Show that the second derivative of \(x \cos 2 x\) is zero when \(x = 0\).
  6. Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x \cos 2 x \mathrm {~d} x\), giving your answer in terms of \(\pi\). Interpret this result graphically.
OCR MEI C3 Q1
18 marks Standard +0.3
1 Fig. 7 shows the curve $$y = 2 x - x \ln x , \text { where } x > 0 .$$ The curve crosses the \(x\)-axis at A , and has a turning point at B . The point C on the curve has \(x\)-coordinate 1 . Lines CD and BE are drawn parallel to the \(y\)-axis. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{74cc215f-bd55-489d-aa4b-0f67c2c8de52-1_529_1259_657_602} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Find the \(x\)-coordinate of A , giving your answer in terms of e .
  2. Find the exact coordinates of B .
  3. Show that the tangents at A and C are perpendicular to each other.
  4. Using integration by parts, show that $$\int x \ln x \mathrm {~d} x = \frac { 1 } { 2 } x ^ { 2 } \ln x - \frac { 1 } { 4 } x ^ { 2 } + c$$ Hence find the exact area of the region enclosed by the curve, the \(x\)-axis and the lines CD and BE .
OCR MEI C3 Q2
6 marks Standard +0.3
2 Show that \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } x \sin 2 x \mathrm {~d} x = \frac { 3 \sqrt { 3 } \pi } { 24 }\).
OCR MEI C3 Q3
17 marks Standard +0.3
3 Fig. 8 shows part of the curve \(y = x \sin 3 x\). It crosses the \(x\)-axis at P . The point on the curve with \(x\)-coordinate \(\frac { 1 } { 6 } \pi\) is Q . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{74cc215f-bd55-489d-aa4b-0f67c2c8de52-2_420_780_549_655} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find the \(x\)-coordinate of P .
  2. Show that Q lies on the line \(y = x\).
  3. Differentiate \(x \sin 3 x\). Hence prove that the line \(y = x\) touches the curve at Q .
  4. Show that the area of the region bounded by the curve and the line \(y = x\) is \(\frac { 1 } { 72 } \left( \pi ^ { 2 } - 8 \right)\).
  5. Differentiate \(x \cos 2 x\) with respect to \(x\).
  6. Integrate \(x \cos 2 x\) with respect to \(x\).
OCR MEI C3 Q1
4 marks Moderate -0.5
1 Either prove or disprove each of the following statements.
  1. 'If \(m\) and \(n\) are consecutive odd numbers, then at least one of \(m\) and \(n\) is a prime number.'
  2. 'If \(m\) and \(n\) are consecutive even numbers, then \(m n\) is divisible by 8 .'
OCR MEI C3 Q2
4 marks Moderate -0.8
2
  1. Disprove the following statement: $$3 ^ { n } + 2 \text { is prime for all integers } n \geqslant 0 .$$
  2. Prove that no number of the form \(3 ^ { n }\) (where \(n\) is a positive integer) has 5 as its final digit.
OCR MEI C3 Q3
4 marks Moderate -0.5
3
  1. Factorise fully \(n ^ { 3 } - n\).
  2. Hence prove that, if \(n\) is an integer, \(n ^ { 3 } - n\) is divisible by 6 .
OCR MEI C3 Q4
2 marks Moderate -0.5
4 Prove or disprove the following statement:
'No cube of an integer has 2 as its units digit.'
OCR MEI C3 Q5
3 marks Easy -1.2
5 Use the triangle in Fig. 4 to prove that \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta = 1\). For what values of \(\theta\) is this proof valid? \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{64cb5e15-35b0-43a3-9f6d-f8e1c04b80b8-1_357_595_1831_798} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
OCR MEI C3 Q6
4 marks Standard +0.3
6
  1. Multiply out \(\left( 3 ^ { n } + 1 \right) \left( 3 ^ { n } - 1 \right)\).
  2. Hence prove that if \(n\) is a positive integer then \(3 ^ { 2 n } - 1\) is divisible by 8 .
OCR MEI C3 Q7
3 marks Moderate -0.8
7 State whether the following statements are true or false; if false, provide a counterexample.
  1. If \(a\) is rational and \(b\) is rational, then \(a + b\) is rational.
  2. If \(a\) is rational and \(b\) is irrational, then \(a + b\) is irrational.
  3. If \(a\) is irrational and \(b\) is irrational, then \(a + b\) is irrational.
OCR MEI C3 Q8
3 marks Moderate -0.8
8
  1. Disprove the following statement.
    'If \(p > q\), then \(\frac { 1 } { p } < \frac { 1 } { q }\).
  2. State a condition on \(p\) and \(q\) so that the statement is true.
  3. Show that
OCR MEI C3 Q11
3 marks Moderate -0.8
11 Use the method of exhaustion to prove the following result.
No 1 - or 2 -digit perfect square ends in \(2,3,7\) or 8
State a generalisation of this result.
OCR MEI C3 Q12
2 marks Easy -1.2
12 Prove that the following statement is false.
For all integers \(n\) greater than or equal to \(1 , n ^ { 2 } + 3 n + 1\) is a prime number.
OCR MEI C3 Q13
6 marks Standard +0.3
13 Positive integers \(a , b\) and \(c\) are said to form a Pythagorean triple if \(a ^ { 2 } + b ^ { 2 } = c ^ { 2 }\).
  1. Given that \(t\) is an integer greater than 1 , show that \(2 t , t ^ { 2 } - 1\) and \(t ^ { 2 } + 1\) form a Pythagorean triple.
  2. The two smallest integers of a Pythagorean triple are 20 and 21. Find the third integer. Use this triple to show that not all Pythagorean triples can be expressed in the form \(2 t , t ^ { 2 } - 1\) and \(t ^ { 2 } + 1\).
OCR MEI C4 Q4
5 marks Standard +0.3
4 Fig. 4 shows a sketch of the region enclosed by the curve \(\sqrt { 1 + \mathrm { e } ^ { - 2 x } }\), the \(x\)-axis, the \(y\)-axis and the line \(x = 1\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{88eadcf5-4335-4016-baf5-d3f74513bbb8-02_517_755_1576_649} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Find the volume of the solid generated when this region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Give your answer in an exact form.
OCR MEI C4 2007 January Q1
4 marks Moderate -0.8
1 Solve the equation \(\frac { 1 } { x } + \frac { x } { x + 2 } = 1\).
OCR MEI C4 2007 January Q2
5 marks Moderate -0.8
2 Fig. 2 shows part of the curve \(y = \sqrt { 1 + x ^ { 3 } }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5dcd4f44-4c61-4384-be1b-a8d63cb6b5aa-2_540_648_662_712} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. Use the trapezium rule with 4 strips to estimate \(\int _ { 0 } ^ { 2 } \sqrt { 1 + x ^ { 3 } } \mathrm {~d} x\), giving your answer correct to 3 significant figures.
  2. Chris and Dave each estimate the value of this integral using the trapezium rule with 8 strips. Chris gets a result of 3.25, and Dave gets 3.30. One of these results is correct. Without performing the calculation, state with a reason which is correct.
    [0pt] [2]
OCR MEI C4 2007 January Q3
7 marks Moderate -0.8
3
  1. Use the formula for \(\sin ( \theta + \phi )\), with \(\theta = 45 ^ { \circ }\) and \(\phi = 60 ^ { \circ }\), to show that \(\sin 105 ^ { \circ } = \frac { \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }\).
  2. In triangle ABC , angle \(\mathrm { BAC } = 45 ^ { \circ }\), angle \(\mathrm { ACB } = 30 ^ { \circ }\) and \(\mathrm { AB } = 1\) unit (see Fig. 3). Fig. 3 Using the sine rule, together with the result in part (i), show that \(\mathrm { AC } = \frac { \sqrt { 3 } + 1 } { \sqrt { 2 } }\).
OCR MEI C4 2007 January Q4
7 marks Standard +0.3
4 Show that \(\frac { 1 + \tan ^ { 2 } \theta } { 1 - \tan ^ { 2 } \theta } = \sec 2 \theta\).
Hence, or otherwise, solve the equation \(\frac { 1 + \tan ^ { 2 } \theta } { 1 - \tan ^ { 2 } \theta } = 2\), for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
OCR MEI C4 2007 January Q5
5 marks Moderate -0.5
5 Find the first four terms in the binomial expansion of \(( 1 + 3 x ) ^ { \frac { 1 } { 3 } }\).
State the range of values of \(x\) for which the expansion is valid.
OCR MEI C4 2007 January Q6
8 marks Standard +0.3
6
  1. Express \(\frac { 1 } { ( 2 x + 1 ) ( x + 1 ) }\) in partial fractions.
  2. A curve passes through the point \(( 0,2 )\) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y } { ( 2 x + 1 ) ( x + 1 ) }$$ Show by integration that \(y = \frac { 4 x + 2 } { x + 1 }\). Section B (36 marks)
OCR MEI C4 2007 January Q7
20 marks Standard +0.3
7 Fig. 7 shows the curve with parametric equations $$x = \cos \theta , y = \sin \theta - \frac { 1 } { 8 } \sin 2 \theta , 0 \leqslant \theta < 2 \pi$$ The curve crosses the \(x\)-axis at points \(\mathrm { A } ( 1,0 )\) and \(\mathrm { B } ( - 1,0 )\), and the positive \(y\)-axis at C . D is the maximum point of the curve, and E is the minimum point. The solid of revolution formed when this curve is rotated through \(360 ^ { \circ }\) about the \(x\)-axis is used to model the shape of an egg. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5dcd4f44-4c61-4384-be1b-a8d63cb6b5aa-4_744_1207_776_431} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Show that, at the point \(\mathrm { A } , \theta = 0\). Write down the value of \(\theta\) at the point B , and find the coordinates of C .
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). Hence show that, at the point D, $$2 \cos ^ { 2 } \theta - 4 \cos \theta - 1 = 0 .$$
  3. Solve this equation, and hence find the \(y\)-coordinate of D , giving your answer correct to 2 decimal places. The cartesian equation of the curve (for \(0 \leqslant \theta \leqslant \pi\) ) is $$y = \frac { 1 } { 4 } ( 4 - x ) \sqrt { 1 - x ^ { 2 } } .$$
  4. Show that the volume of the solid of revolution of this curve about the \(x\)-axis is given by $$\frac { 1 } { 16 } \pi \int _ { - 1 } ^ { 1 } \left( 16 - 8 x - 15 x ^ { 2 } + 8 x ^ { 3 } - x ^ { 4 } \right) \mathrm { d } x .$$ Evaluate this integral.