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 2009 1 Find 2 (i) 8e −2x dx, (ii) (4x + 5) 6 dx. [5] 2 (i) Use Simpson’s rule with four strips to find an approximation to 12 ln x dx, 4 giving your answer correct to 2 decimal places. [4] 12 (ii) Deduce an approximation to ln(x 10 ) dx. [1] 4 3 (i) Express 2 tan 2 θ − 1 in terms of sec θ. [3] cos θ (ii) Hence solve, for 0 ◦ < θ < 360 ◦ , the equation 2 tan 2 θ − 1 = 4. [4] cos θ 4 For each of the following curves, find dy and determine the exact x-coordinate of the stationary point: dx (i) y = (4x 2 + 1) 5 , [3] (ii) y = x2 ln x . [4] 5 The mass, M grams, of a certain substance is increasing exponentially so that, at time t hours, the mass is given by M = 40e kt , where k is a constant. The following table shows certain values of t and M. t 0 21 63 M 80 (i) In either order, (a) find the values missing from the table, [3] (b) determine the value of k. [2] (ii) Find the rate at which the mass is increasing when t = 21. [3] © OCR 2009 4723 Jan09
6 Jan 2009 3 y y –1 = f ( x) P y = f( x) O Q x The function f is defined for all real values of x by f(x) = 3√ 1 2 x + 2. The graphs of y = f(x) and y = f −1 (x) meet at the point P, and the graph of y = f −1 (x) meets the x-axis at Q (see diagram). (i) Find an expression for f −1 (x) and determine the x-coordinate of the point Q. [3] (ii) State how the graphs of y = f(x) and y = f −1 (x) are related geometrically, and hence show that the x-coordinate of the point P is the root of the equation x = 3√ 1 x + 2. [2] 2 (iii) Use an iterative process, based on the equation x = 3√ 1 x + 2, to find the x-coordinate of P, giving 2 your answer correct to 2 decimal places. [4] 7 y O x The diagram shows the curve y = e kx − a, where k and a are constants. (i) Give details of the pair of transformations which transforms the curve y = e x to the curve y = e kx − a. [3] (ii) Sketch the curve y = ∣ ∣ ekx − a ∣ ∣ . [2] (iii) Given that the curve y = ∣ ∣ e kx − a∣ ∣ passes through the points (0, 13) and (ln 3, 13), find the values of k and a. [4] © OCR 2009 4723 Jan09 Turn over
- 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 and 12: 2 Jan 2007 1 Find the equation of t
- Page 13 and 14: 8 Jan 2007 4 The diagram shows the
- 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: June 2008 9 y 4 O x The function f
- 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
Jan 2009<br />
1 Find<br />
2<br />
(i) 8e −2x dx,<br />
(ii) (4x + 5) 6 dx.<br />
[5]<br />
2 (i) Use Simpson’s rule with four strips to find an approximation to<br />
12<br />
ln x dx,<br />
4<br />
giving your answer correct to 2 decimal places. [4]<br />
12<br />
(ii) Deduce an approximation to ln(x 10 ) dx. [1]<br />
4<br />
3 (i) Express 2 tan 2 θ − 1 in terms of sec θ. [3]<br />
cos θ<br />
(ii) Hence solve, for 0 ◦ < θ < 360 ◦ , the equation<br />
2 tan 2 θ − 1 = 4. [4]<br />
cos θ<br />
4 For each of the following curves, find dy and determine the exact x-coordinate of the stationary point:<br />
dx<br />
(i) y = (4x 2 + 1) 5 , [3]<br />
(ii) y = x2<br />
ln x . [4]<br />
5 <strong>The</strong> mass, M grams, of a certain substance is increasing exponentially so that, at time t hours, the<br />
mass is given by<br />
M = 40e kt ,<br />
where k is a constant. <strong>The</strong> following table shows certain values of t and M.<br />
t 0 21 63<br />
M 80<br />
(i) In either order,<br />
(a) find the values missing from the table, [3]<br />
(b) determine the value of k. [2]<br />
(ii) Find the rate at which the mass is increasing when t = 21. [3]<br />
© <strong>OCR</strong> 2009 4723 Jan09
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