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
June 2006 4 8 (i) Express 5 cos x + 12 sin x in the form R cos(x − α), whereR > 0and0 ◦ < α < 90 ◦ . [3] (ii) Hence give details of a pair of transformations which transforms the curve y = cos x to the curve y = 5cosx + 12 sin x. [3] (iii) Solve, for 0 ◦ < x < 360 ◦ , the equation 5 cos x + 12 sin x = 2, giving your answers correct to the nearest 0.1 ◦ . [5] 9 The diagram shows the curve with equation y = 2ln(x − 1). The point P has coordinates (0, p). The region R, shaded in the diagram, is bounded by the curve and the lines x = 0, y = 0andy = p. The units on the axes are centimetres. The region R is rotated completely about the y-axis to form a solid. (i) Show that the volume, V cm 3 , of the solid is given by V = π(e p + 4e 1 2 p + p − 5). [8] (ii) It is given that the point P is moving in the positive direction along the y-axis at a constant rate of 0.2 cm min −1 . Find the rate at which the volume of the solid is increasing at the instant when p = 4, giving your answer correct to 2 significant figures. [5] 4723/S06
2 Jan 2007 1 Find the equation of the tangent to the curve y = 2x + 1 3x − 1 at the point (1, 3 ), giving your answer in the 2 form ax + by + c = 0, where a, b and c are integers. [5] 2 It is given that θ is the acute angle such that sin θ = 12 . Find the exact value of 13 (i) cot θ, [2] (ii) cos 2θ. [3] 3 (a) It is given that a and b are positive constants. By sketching graphs of y = x 5 and y = a − bx on the same diagram, show that the equation x 5 + bx − a = 0 has exactly one real root. [3] (b) Use the iterative formula x n+1 = 5√ 53 − 2x n , with a suitable starting value, to find the real root of the equation x 5 + 2x − 53 = 0. Show the result of each iteration, and give the root correct to 3 decimal places. [4] 4 (i) Given that x =(4t + 9) 1 2 and y = 6e 1 2 x+1 , find expressions for dx dt (ii) Hencefindthevalueof dy dt and dy dx . [4] when t = 4, giving your answer correct to 3 significant figures. [3] 5 (i) Express 4 cos θ − sin θ in the form R cos(θ + α), whereR > 0and0 ◦ < α < 90 ◦ . [3] (ii) Hence solve the equation 4 cos θ − sin θ = 2, giving all solutions for which −180 ◦ < θ < 180 ◦ . [5] © 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: 6 June 2006 3 The diagram shows the
- 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 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
2<br />
Jan 2007<br />
1 Find the equation of the tangent to the curve y = 2x + 1<br />
3x − 1 at the point (1, 3 ), giving your answer in the<br />
2<br />
form ax + by + c = 0, where a, b and c are integers. [5]<br />
2 It is given that θ is the acute angle such that sin θ = 12 . Find the exact value of<br />
13<br />
(i) cot θ, [2]<br />
(ii) cos 2θ. [3]<br />
3 (a) It is given that a and b are positive constants. By sketching graphs of<br />
y = x 5 and y = a − bx<br />
on the same diagram, show that the equation<br />
x 5 + bx − a = 0<br />
has exactly one real root. [3]<br />
(b)<br />
Use the iterative formula x n+1<br />
= 5√ 53 − 2x n<br />
, with a suitable starting value, to find the real root<br />
of the equation x 5 + 2x − 53 = 0. Show the result of each iteration, and give the root correct to<br />
3 decimal places. [4]<br />
4 (i) Given that x =(4t + 9) 1 2<br />
and y = 6e 1 2 x+1 , find expressions for dx<br />
dt<br />
(ii) Hencefindthevalueof dy<br />
dt<br />
and<br />
dy<br />
dx . [4]<br />
when t = 4, giving your answer correct to 3 significant figures. [3]<br />
5 (i) Express 4 cos θ − sin θ in the form R cos(θ + α), whereR > 0and0 ◦ < α < 90 ◦ . [3]<br />
(ii) Hence solve the equation 4 cos θ − sin θ = 2, giving all solutions for which −180 ◦ < θ < 180 ◦ .<br />
[5]<br />
© <strong>OCR</strong> 2007 4723/01 Jan07
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