Core 3 OCR Past Papers - The Grange School Blogs
Core 3 OCR Past Papers - The Grange School Blogs Core 3 OCR Past Papers - The Grange School Blogs
Jan 2007 6 3 The diagram shows the curve with equation y = and the lines x = 0, x = 2andy = 0. 1 √ 3x + 2 . The shaded region is bounded by the curve (i) Find the exact area of the shaded region. [4] (ii) The shaded region is rotated completely about the x-axis. Find the exact volume of the solid formed, simplifying your answer. [5] 7 The curve y = ln x is transformed to the curve y = ln( 1 x − a) bymeansofatranslationfollowedbya 2 stretch. It is given that a is a positive constant. (i) Give full details of the translation and stretch involved. [2] (ii) Sketch the graph of y = ln( 1 x − a). [2] 2 (iii) Sketch, on another diagram, the graph of y = ∣ ∣ ln(1 2 x − a)∣ ∣ . [2] (iv) State, in terms of a, thesetofvaluesofx for which ∣ ln(1 2 x − a)∣ ∣ =−ln(1 x − a). [2] 2 [Questions 8 and 9 are printed overleaf.] © OCR 2007 4723/01 Jan07 [Turn over
8 Jan 2007 4 The diagram shows the curve with equation y = x 8 e −x2 . The curve has maximum points at P and Q. The shaded region A is bounded by the curve, the line y = 0 and the line through Q parallel to the y-axis. The shaded region B is bounded by the curve and the line PQ. (i) Show by differentiation that the x-coordinate of Q is 2. [5] (ii) Use Simpson’s rule with 4 strips to find an approximation to the area of region A. Give your answer correct to 3 decimal places. [4] (iii) Deduce an approximation to the area of region B. [2] 9 Functions f and g are defined by f(x) =2sinx for − 1 2 π ≤ x ≤ 1 2 π, g(x) =4 − 2x 2 for x ∈. (i) State the range of f and the range of g. [2] (ii) Show that gf(0.5) =2.16,correctto3significantfigures,andexplainwhyfg(0.5) is not defined. [4] (iii) Find the set of values of x for which f −1 g(x) is not defined. [6] Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. © OCR 2007 4723/01 Jan07
- Page 1 and 2: The Grange School Maths Department
- Page 3 and 4: 3 June 2005 6 (a) Find the exact va
- Page 5 and 6: Jan 2006 1 Show that 2 8 2 3 dx =
- Page 7 and 8: 8 Jan 2006 4 The diagram shows part
- Page 9 and 10: 6 June 2006 3 The diagram shows the
- Page 11: 2 Jan 2007 1 Find the equation of t
- Page 15 and 16: June 2007 3 7 (i) Sketch the graph
- Page 17 and 18: Jan 2008 3 5 (a) Find (3x + 7) 9 d
- Page 19 and 20: June 2008 2 1 Find the exact soluti
- Page 21 and 22: June 2008 9 y 4 O x The function f
- Page 23 and 24: 6 Jan 2009 3 y y -1 = f ( x) P y =
- Page 25 and 26: 1 June 2009 y y y 2 x x x O O O Fig
- Page 27 and 28: June 2009 4 8 y y P y = ln x = 2 ln
- Page 29 and 30: Jan 2010 5 The equation of a curve
- Page 31 and 32: June 2010 1 Find dy in each of the
- Page 33: June 2010 4 9 The functions f and g
8<br />
Jan 2007<br />
4<br />
<strong>The</strong> diagram shows the curve with equation y = x 8 e −x2 . <strong>The</strong> curve has maximum points at P and Q.<br />
<strong>The</strong> shaded region A is bounded by the curve, the line y = 0 and the line through Q parallel to the<br />
y-axis. <strong>The</strong> shaded region B is bounded by the curve and the line PQ.<br />
(i) Show by differentiation that the x-coordinate of Q is 2. [5]<br />
(ii) Use Simpson’s rule with 4 strips to find an approximation to the area of region A. Give your<br />
answer correct to 3 decimal places. [4]<br />
(iii) Deduce an approximation to the area of region B. [2]<br />
9 Functions f and g are defined by<br />
f(x) =2sinx for − 1 2 π ≤ x ≤ 1 2 π,<br />
g(x) =4 − 2x 2 for x ∈.<br />
(i) State the range of f and the range of g. [2]<br />
(ii) Show that gf(0.5) =2.16,correctto3significantfigures,andexplainwhyfg(0.5) is not defined.<br />
[4]<br />
(iii) Find the set of values of x for which f −1 g(x) is not defined. [6]<br />
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable<br />
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will<br />
be pleased to make amends at the earliest possible opportunity.<br />
<strong>OCR</strong> is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES),<br />
which is itself a department of the University of Cambridge.<br />
© <strong>OCR</strong> 2007 4723/01 Jan07
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