Integration Past Papers Unit 2 Outcome 2

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1 Integration Past Papers Unit 2 utcome 2 Multiple Choice Questions Each correct answer in this section is worth two marks.. Evaluate A. 2 B. 7 6 C. 2 D. 2 4 /2 d. 2. The diagram shows the area bounded b the curves = and = 2 2 between = and = 2. = = Represent the shaded area as an integral. A ) d B. C. D ) d ) d ) d [END F MULTIPLE CHICE QUESTINS] Written Questions 3. Find ) d ) d Find hsn.uk.net Page Questions marked c SQA

2 5. Evaluate 2 + ) 2 d Find 2 5 d Find 2 2) 2 + 2) 2 d, = Find the value of u u 2 du a) Find the value of 4 2) d. 3 b) Sketch a graph and shade the area represented b the integral in a) Evaluate 9 + d. 5. Find the value of 4 d Find 7 3) 2 d Evaluate ) 2 d. 4 ) 4. Evaluate d and draw a sketch to illustrate the area represented b this integral. 5 Questions marked c SQA hsn.uk.net Page 2

3 5. a) Find the coordinates of the points of intersection of the curves with equations = 2 2 and = b) Find the area completel enclosed between these two curves For all points on the curve = f ), f ) = 2. If the curve passes through the point 2, ), find the equation of the curve A curve for which d d = 32 + passes through the point, 2). Epress in terms of A curve for which d d = 62 2 passes through the point, 2). Epress in terms of A point moves in a straight line such that its acceleration a is given b a = 24 t) 2, 0 t 4. If it starts at rest, find an epression for the velocit v where a = dv dt A curve with equation = f ) passes through the point 2, ) and is such that f ) = 4 3. Epress f ) in terms of The graph of = g) passes through the point, 2). If d d = 3 + 2, epress in terms of rag hsn.uk.net Page 3 Questions marked c SQA

4 23. The parabola shown crosses the -ais at 0, 0) and 4, 0), and has a maimum at 2, 4). 4 The shaded area is bounded b the parabola, the -ais and the lines = 2 and = k. a) Find the equation of the parabola. 2 k 4 2 b) Hence show that the shaded area, A, is given b A = 3 k3 + 2k rag hsn.uk.net Page 4 Questions marked c SQA

5 25. rag 26. rag hsn.uk.net Page 5 Questions marked c SQA

6 27. rag hsn.uk.net Page 6 Questions marked c SQA

7 28. The diagram shows a sketch of the graphs of = and = The two curves intersect at A and touch at B, i.e. at B the curves have a common tangent. = A B = a) i) Find the -coordinates of the point of the curves where the gradients are equal. 4 ii) B considering the corresponding -coordinates, or otherwise, distinguish geometricall between the two cases found in part i). b) The point A is, 2) and B is 3, 8). Find the area enclosed between the two curves. 5 hsn.uk.net Page 7 Questions marked c SQA

8 29. A firm asked for a logo to be designed involving the letters A and U. Their initial sketch is shown in the heagon. A mathematical representation of the final logo is shown PSfrag in the coordinate diagram. The curve has equation = + ) ) 3) and the straight line has equation = 5 5. The point, 0) is the centre of half-turn smmetr. Calculate the total shaded area , 5) 2, 5) 30. Calculate the shaded area enclosed between the parabolas with equations = = and = = hsn.uk.net Page 8 Questions marked c SQA

9 3. rag hsn.uk.net Page 9 Questions marked c SQA

10 32. rag 33. rag hsn.uk.net Page 0 Questions marked c SQA

11 34. rag 35. rag 36. Functions f and g are defined on the set of real numbers b f ) = and g) = 2. a) Find formulae for i) f g) ) ii) g f ) ). 4 b) The function h is defined b h) = f g) ) + g f ) ). Show that h) = and sketch the graph of h. 3 c) Find the area enclosed between this graph and the -ais. 4 hsn.uk.net Page Questions marked c SQA

12 37. A function f is defined b the formula f ) = 4 2 3) where R. a) Write down the coordinates of the points where the curve with equation = f ) meets the - and -aes. 2 b) Find the stationar points of = f ) and determine the nature of each. 6 c) Sketch the curve = f ). 2 d) Find the area completel enclosed b the curve = f ) and the -ais. 4 [END F WRITTEN QUESTINS] hsn.uk.net Page 2 Questions marked c SQA

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