Questions — OCR MEI C2 (480 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 MEI C2 Q1
5 marks Moderate -0.8
An arithmetic progression has tenth term 11.1 and fiftieth term 7.1. Find the first term and the common difference. Find also the sum of the first fifty terms of the progression. [5]
OCR MEI C2 Q2
12 marks Moderate -0.3
Jill has 3 daughters and no sons. They are generation 1 of Jill's descendants. Each of her daughters has 3 daughters and no sons. Jill's 9 granddaughters are generation 2 of her descendants. Each of her granddaughters has 3 daughters and no sons; they are descendant generation 3. Jill decides to investigate what would happen if this pattern continues, with each descendant having 3 daughters and no sons.
  1. How many of Jill's descendants would there be in generation 8? [2]
  2. How many of Jill's descendants would there be altogether in the first 15 generations? [3]
  3. After \(n\) generations, Jill would have over a million descendants altogether. Show that \(n\) satisfies the inequality $$n > \frac{\log_{10}2000003}{\log_{10}3} - 1.$$ Hence find the least possible value of \(n\). [4]
  4. How many fewer descendants would Jill have altogether in 15 generations if instead of having 3 daughters, she and each subsequent descendant has 2 daughters? [3]
OCR MEI C2 Q3
5 marks Moderate -0.8
  1. Find \(\sum_{r=1}^{5} \frac{21}{r+2}\). [2]
  2. A sequence is defined by $$u_1 = a, \text{ where } a \text{ is an unknown constant,}$$ $$u_{n+1} = u_n + 5.$$ Find, in terms of \(a\), the tenth term and the sum of the first ten terms of this sequence. [3]
OCR MEI C2 Q4
5 marks Moderate -0.3
The second term of a geometric progression is 24. The sum to infinity of this progression is 150. Write down two equations in \(a\) and \(r\), where \(a\) is the first term and \(r\) is the common ratio. Solve your equations to find the possible values of \(a\) and \(r\). [5]
OCR MEI C2 Q5
3 marks Moderate -0.8
\(S\) is the sum to infinity of a geometric progression with first term \(a\) and common ratio \(r\).
  1. Another geometric progression has first term \(2a\) and common ratio \(r\). Express the sum to infinity of this progression in terms of \(S\). [1]
  2. A third geometric progression has first term \(a\) and common ratio \(r^2\). Express, in its simplest form, the sum to infinity of this progression in terms of \(S\) and \(r\). [2]
OCR MEI C2 Q6
4 marks Easy -1.8
Find the second and third terms in the sequence given by $$u_1 = 5,$$ $$u_{n+1} = u_n + 3.$$ Find also the sum of the first 50 terms of this sequence. [4]
OCR MEI C2 Q7
10 marks Standard +0.3
A geometric progression has first term \(a\) and common ratio \(r\). The second term is 6 and the sum to infinity is 25.
  1. Write down two equations in \(a\) and \(r\). Show that one possible value of \(a\) is 10 and find the other possible value of \(a\). Write down the corresponding values of \(r\). [7]
  2. Show that the ratio of the \(n\)th terms of the two geometric progressions found in part (i) can be written as \(2^{n-2} : 3^{n-2}\). [3]
OCR MEI C2 Q1
12 marks Standard +0.3
  1. An arithmetic progression has first term \(A\) and common difference \(D\). The sum of its first two terms is 25 and the sum of its first four terms is 250.
    1. Find the values of \(A\) and \(D\). [4]
    2. Find the sum of the 21st to 50th terms inclusive of this sequence. [3]
  2. A geometric progression has first term \(a\) and common ratio \(r\), with \(r \neq \pm 1\). The sum of its first two terms is 25 and the sum of its first four terms is 250. Use the formula for the sum of a geometric progression to show that \(\frac{r^4 - 1}{r^2 - 1} = 10\) and hence or otherwise find algebraically the possible values of \(r\) and the corresponding values of \(a\). [5]
OCR MEI C2 Q2
5 marks Moderate -0.8
A geometric progression has a positive common ratio. Its first three terms are 32, \(b\) and 12.5. Find the value of \(b\) and find also the sum of the first 15 terms of the progression. [5]
OCR MEI C2 Q3
5 marks Moderate -0.3
In an arithmetic progression, the second term is 11 and the sum of the first 40 terms is 3030. Find the first term and the common difference. [5]
OCR MEI C2 Q4
5 marks Moderate -0.3
The second term of a geometric sequence is 6 and the fifth term is \(-48\). Find the tenth term of the sequence. Find also, in simplified form, an expression for the sum of the first \(n\) terms of this sequence. [5]
OCR MEI C2 Q5
5 marks Moderate -0.3
The third term of an arithmetic progression is 24. The tenth term is 3. Find the first term and the common difference. Find also the sum of the 21st to 50th terms inclusive. [5] Simplify
OCR MEI C2 Q6
5 marks Easy -1.2
  1. Find the 51st term of the sequence given by $$u_1 = 5,$$ $$u_{n+1} = u_n + 4.$$ [3]
  2. Find the sum to infinity of the geometric progression which begins $$5 \quad 2 \quad 0.8 \quad \ldots$$ [2]
OCR MEI C2 Q7
5 marks Moderate -0.8
An arithmetic progression has first term 7 and third term 12.
  1. Find the 20th term of this progression. [2]
  2. Find the sum of the 21st to the 50th terms inclusive of this progression. [3]
OCR MEI C2 Q8
5 marks Moderate -0.3
The second term of a geometric progression is 18 and the fourth term is 2. The common ratio is positive. Find the sum to infinity of this progression. [5]
OCR MEI C2 Q9
3 marks Moderate -0.5
A geometric progression has 6 as its first term. Its sum to infinity is 5. Calculate its common ratio. [3]
OCR MEI C2 Q1
12 marks Moderate -0.8
Fig. 11.1 shows a village green which is bordered by 3 straight roads AB, BC and CA. The road AC runs due North and the measurements shown are in metres. \includegraphics{figure_1}
  1. Calculate the bearing of B from C, giving your answer to the nearest 0.1°. [4]
  2. Calculate the area of the village green. [2]
The road AB is replaced by a new road, as shown in Fig. 11.2. The village green is extended up to the new road. \includegraphics{figure_2} The new road is an arc of a circle with centre O and radius 130 m.
  1. (A) Show that angle AOB is 1.63 radians, correct to 3 significant figures. [2] (B) Show that the area of land added to the village green is 5300 m² correct to 2 significant figures. [4]
OCR MEI C2 Q2
5 marks Moderate -0.8
\includegraphics{figure_3} For triangle ABC shown in Fig. 4, calculate
  1. the length of BC, [3]
  2. the area of triangle ABC. [2]
OCR MEI C2 Q3
13 marks Moderate -0.3
  1. A boat travels from P to Q and then to R. As shown in Fig. 11.1, Q is 10.6 km from P on a bearing of 045°. R is 9.2 km from P on a bearing of 113°, so that angle QPR is 68°. \includegraphics{figure_4} Calculate the distance and bearing of R from Q. [5]
  2. Fig. 11.2 shows the cross-section, EBC, of the rudder of a boat. \includegraphics{figure_5} BC is an arc of a circle with centre A and radius 80 cm. Angle CAB = \(\frac{2\pi}{3}\) radians. EC is an arc of a circle with centre D and radius \(r\) cm. Angle CDE is a right angle.
    1. Calculate the area of sector ABC. [2]
    2. Show that \(r = 40\sqrt{3}\) and calculate the area of triangle CDA. [3]
    3. Hence calculate the area of cross-section of the rudder. [3]
OCR MEI C2 Q4
12 marks Standard +0.3
\emph{Arrowline Enterprises} is considering two possible logos: \includegraphics{figure_6}
  1. Fig. 10.1 shows the first logo ABCD. It is symmetrical about AC. Find the length of AB and hence find the area of this logo. [4]
  2. Fig. 10.2 shows a circle with centre O and radius 12.6 cm. ST and RT are tangents to the circle and angle SOR is 1.82 radians. The shaded region shows the second logo. Show that ST = 16.2 cm to 3 significant figures. Find the area and perimeter of this logo. [8]
OCR MEI C2 Q5
12 marks Moderate -0.3
  1. The course for a yacht race is a triangle, as shown in Fig. 11.1. The yachts start at A, then travel to B, then to C and finally back to A. \includegraphics{figure_7}
    1. Calculate the total length of the course for this race. [4]
    2. Given that the bearing of the first stage, AB, is 175°, calculate the bearing of the second stage, BC. [4]
  2. Fig. 11.2 shows the course of another yacht race. The course follows the arc of a circle from P to Q, then a straight line back to P. The circle has radius 120 m and centre O; angle POQ = 136°. \includegraphics{figure_8} Calculate the total length of the course for this race. [4]
OCR MEI C2 Q1
5 marks Moderate -0.8
  1. Starting with an equilateral triangle, prove that \(\cos 30° = \frac{\sqrt{3}}{2}\). [2]
  2. Solve the equation \(2 \sin \theta = -1\) for \(0 \leqslant \theta \leqslant 2\pi\), giving your answers in terms of \(\pi\). [3]
OCR MEI C2 Q2
2 marks Easy -1.2
Use an isosceles right-angled triangle to show that \(\cos 45° = \frac{1}{\sqrt{2}}\). [2]
OCR MEI C2 Q3
5 marks Easy -1.3
  1. On the same axes, sketch the graphs of \(y = \cos x\) and \(y = \cos 2x\) for values of \(x\) from \(0\) to \(2\pi\). [3]
  2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = 3 \cos x\). [2]
OCR MEI C2 Q4
3 marks Moderate -0.8
\(\theta\) is an acute angle and \(\sin \theta = \frac{1}{4}\). Find the exact value of \(\tan \theta\). [3]