Questions C2 (1550 questions)

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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
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OCR MEI C2 Q1
3 marks Easy -1.8
Find \(\int 7x^2 dx\). [3]
OCR MEI C2 Q2
5 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = \frac{18}{x} + 2\). The curve passes through the point \((3, 6)\). Find the equation of the curve. [5]
OCR MEI C2 Q3
5 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = 6x^{\frac{1}{2}} - 5\). Given also that the curve passes through the point \((4, 20)\), find the equation of the curve. [5]
OCR MEI C2 Q4
3 marks Moderate -0.8
Find \(\int_2^5 (2x^3 + 3) dx\). [3]
OCR MEI C2 Q5
5 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = 6\sqrt{x} - 2\). Given also that the curve passes through the point \((9, 4)\), find the equation of the curve. [5]
OCR MEI C2 Q6
4 marks Moderate -0.3
Find \(\int_2^5 \left(1 - \frac{6}{x^3}\right) dx\). [4]
OCR MEI C2 Q7
4 marks Easy -1.2
Find \(\int_1^2 (12x^5 + 5) dx\). [4]
OCR MEI C2 Q8
5 marks Moderate -0.8
The gradient of a curve is \(3\sqrt{x} - 5\). The curve passes through the point \((4, 6)\). Find the equation of the curve. [5]
OCR MEI C2 Q9
4 marks Moderate -0.8
A curve has gradient given by \(\frac{dy}{dx} = 6\sqrt{x}\). Find the equation of the curve, given that it passes through the point \((9, 105)\). [4]
OCR MEI C2 Q10
5 marks Moderate -0.8
Find \(\int_1^2 \left(x^4 - \frac{3}{x^2} + 1\right) dx\), showing your working. [5]
OCR MEI C2 Q11
3 marks Easy -1.8
Find \(\int 30x^2 dx\). [3]
OCR MEI C2 Q12
4 marks Easy -1.2
Find \(\int (x^5 + 10x^3) dx\). [4]
OCR MEI C2 Q1
4 marks Easy -1.2
Find \(\int (3x^5 + 2x^{-\frac{1}{2}}) dx\). [4]
OCR MEI C2 Q2
11 marks Moderate -0.3
Fig. 11 shows the curve \(y = x^3 - 3x^2 - x + 3\). \includegraphics{figure_11}
  1. Use calculus to find \(\int_{-1}^{3} (x^3 - 3x^2 - x + 3) dx\) and state what this represents. [6]
  2. Find the \(x\)-coordinates of the turning points of the curve \(y = x^3 - 3x^2 - x + 3\), giving your answers in surd form. Hence state the set of values of \(x\) for which \(y = x^3 - 3x^2 - x + 3\) is a decreasing function. [5]
OCR MEI C2 Q3
3 marks Easy -1.2
Find \(\int \left(x - \frac{3}{x^2}\right) dx\). [3]
OCR MEI C2 Q4
4 marks Easy -1.2
Find \(\int (20x^4 + 6x^{-\frac{2}{3}}) dx\). [4]
OCR MEI C2 Q5
10 marks Easy -1.2
Find \(\int (12x^5 + \sqrt[5]{x} + 7) dx\). [5]
OCR MEI C2 Q6
5 marks Moderate -0.8
Find \(\int \left(x^{\frac{1}{2}} + \frac{6}{x^3}\right) dx\). [5]
OCR MEI C2 Q7
4 marks Easy -1.2
Find \(\int \left(x^4 + \frac{1}{x^3}\right) dx\). [4]
OCR MEI C2 Q8
5 marks Easy -1.3
  1. Differentiate \(12\sqrt{x}\). [2]
  2. Integrate \(\frac{6}{x^5}\). [3]
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]