Processing math: 100%

User interface language: English | Español

Date May 2011 Marks available 17 Reference code 11M.2.hl.TZ2.13
Level HL only Paper 2 Time zone TZ2
Command term Show that and Solve Question number 13 Adapted from N/A

Question

Prove by mathematical induction that, for nZ+,

1+2(12)+3(12)2+4(12)3+...+n(12)n1=4n+22n1.

[8]
A.

(a)     Using integration by parts, show that e2xsinxdx=15e2x(2sinxcosx)+C .

(b)     Solve the differential equation dydx=1y2e2xsinx, given that y = 0 when x = 0,

writing your answer in the form y=f(x) .

(c)     (i)     Sketch the graph of y=f(x) , found in part (b), for 0x1.5 .

Determine the coordinates of the point P, the first positive intercept on the x-axis, and mark it on your sketch.

(ii)     The region bounded by the graph of y=f(x) and the x-axis, between the origin and P, is rotated 360° about the x-axis to form a solid of revolution.

Calculate the volume of this solid.

[17]
B.

Markscheme

prove that 1+2(12)+3(12)2+4(12)3+...+n(12)n1=4n+22n1

for n = 1

LHS=1, RHS=41+220=43=1

so true for n = 1     R1

assume true for n = k     M1

so 1+2(12)+3(12)2+4(12)3+...+k(12)k1=4k+22k1

now for n = k +1

LHS: 1+2(12)+3(12)2+4(12)3+...+k(12)k1+(k+1)(12)k     A1

=4k+22k1+(k+1)(12)k     M1A1

=42(k+2)2k+k+12k(or equivalent)     A1

=4(k+1)+22(k+1)1(accept 4k+32k)     A1

Therefore if it is true for n = k it is true for n = k + 1. It has been shown to be true for n = 1 so it is true for all n (Z+).     R1

Note: To obtain the final R mark, a reasonable attempt at induction must have been made.

 

[8 marks]

A.

(a)

METHOD 1

e2xsinxdx=cosxe2x+2e2xcosxdx     M1A1A1

=cosxe2x+2e2xsinx4e2xsinxdx     A1A1

5e2xsinxdx=cosxe2x+2e2xsinx     M1

e2xsinxdx=15e2x(2sinxcosx)+C     AG 

METHOD 2

sinxe2xdx=sinxe2x2cosxe2x2dx     M1A1A1

=sinxe2x2cosxe2x4sinxe2x4dx     A1A1

54e2xsinxdx=e2xsinx2cosxe2x4     M1

e2xsinxdx=15e2x(2sinxcosx)+C     AG

[6 marks]

 

(b)

dy1y2=e2xsinxdx     M1A1

arcsiny=15e2x(2sinxcosx)(+C)     A1

when x=0, y=0C=15     M1

y=sin(15e2x(2sinxcosx)+15)     A1

[5 marks]

 

(c)

(i)         A1

P is (1.16, 0)     A1

Note: Award A1 for 1.16 seen anywhere, A1 for complete sketch.

 

Note: Allow FT on their answer from (b)

 

(ii)     V=1.162...0πy2dx     M1A1

=1.05     A2

Note: Allow FT on their answers from (b) and (c)(i).

 

[6 marks]

B.

Examiners report

Part A: Given that this question is at the easier end of the ‘proof by induction’ spectrum, it was disappointing that so many candidates failed to score full marks. The n = 1 case was generally well done. The whole point of the method is that it involves logic, so ‘let n = k’ or ‘put n = k’, instead of ‘assume ... to be true for n = k’, gains no marks. The algebraic steps need to be more convincing than some candidates were able to show. It is astonishing that the R1 mark for the final statement was so often not awarded.

A.

Part B: Part (a) was often well done, although some faltered after the first integration. Part (b) was also generally well done, although there were some errors with the constant of integration. In (c) the graph was often attempted, but errors in (b) usually led to manifestly incorrect plots. Many attempted the volume of integration and some obtained the correct value.

B.

Syllabus sections

Topic 9 - Option: Calculus » 9.5 » First-order differential equations.
Show 43 related questions

View options