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May 2025 Section II

0/54 marks

2025 · 6 questions · 24 parts · 54 marks

0Correct0Partial0Wrong24Unattempted
1(A)1(B)1(C)1(D)2(A)2(B)2(C)2(D)3(A)3(B)3(C)3(D)4(A)4(B)4(C)4(D)5(A)5(B)5(C)5(D)6(A)6(B)6(C)6(D)
Context for question 2
The shaded region R is bounded by the graphs of the functions f and g, where f(x)=x2−2x and g(x)=x+sin(πx), as shown in the figure.
Graphs of y = f(x), a parabola through the origin and the point (2, 0), and y = g(x), an oscillating curve through the origin. The two curves meet at the origin and at the point (3, 3), and the region R between them from x = 0 to x = 3 is shaded. The x-axis is marked at 1, 2 and 3 and the y-axis at -2, -1, 1, 2, 3 and 4.
(Note: Your calculator should be in radian mode.)
2(A)Areas and Accumulated ChangeEasy2 marks
Find the area of R. Show the setup for your calculations.
2(B)Volumes, Arc Length and Surface AreaEasy2 marks
Region R is the base of a solid. For this solid, at each x the cross section perpendicular to the x-axis is a rectangle with height x and base in region R. Find the volume of the solid. Show the setup for your calculations.
2(C)Volumes, Arc Length and Surface AreaHard3 marks
Write, but do not evaluate, an integral expression for the volume of the solid generated when the region R is rotated about the horizontal line y=−2.
2(D)Tangents, Normals and Linear ApproximationEasy2 marks
It can be shown that g′(x)=1+πcos(πx). Find the value of x, for 0<x<1, at which the line tangent to the graph of f is parallel to the line tangent to the graph of g. Show the work that leads to your answer.
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