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Date May 2017 Marks available 6 Reference code 17M.1.SL.TZ1.S_10
Level Standard Level Paper Paper 1 (without calculator) Time zone Time zone 1
Command term Find Question number S_10 Adapted from N/A

Question

The following table shows the probability distribution of a discrete random variable A, in terms of an angle θ.

M17/5/MATME/SP1/ENG/TZ1/10

Show that cosθ=34.

[6]
a.

Given that tanθ>0, find tanθ.

[3]
b.

Let y=1cosx, for 0<x<π2. The graph of ybetween x=θ and x=π4 is rotated 360° about the x-axis. Find the volume of the solid formed.

[6]
c.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

evidence of summing to 1     (M1)

egp=1

correct equation     A1

egcosθ+2cos2θ=1

correct equation in cosθ     A1

egcosθ+2(2cos2θ1)=1, 4cos2θ+cosθ3=0

evidence of valid approach to solve quadratic     (M1)

egfactorizing equation set equal to 0, 1±14×4×(3)8

correct working, clearly leading to required answer     A1

eg(4cosθ3)(cosθ+1), 1±78

correct reason for rejecting cosθ1     R1

egcosθ is a probability (value must lie between 0 and 1), cosθ>0

 

Note:     Award R0 for cosθ1 without a reason.

 

cosθ=34    AG  N0

a.

valid approach     (M1)

egsketch of right triangle with sides 3 and 4, sin2x+cos2x=1

correct working     

(A1)

egmissing side =7, 7434

tanθ=73     A1     N2

[3 marks]

b.

attempt to substitute either limits or the function into formula involving f2     (M1)

egππ4θf2, (1cosx)2

correct substitution of both limits and function     (A1)

egππ4θ(1cosx)2dx

correct integration     (A1)

egtanx

substituting their limits into their integrated function and subtracting     (M1)

egtanπ4tanθ

 

Note:     Award M0 if they substitute into original or differentiated function.

 

tanπ4=1    (A1)

eg1tanθ

V=ππ73     A1     N3

 

[6 marks]

c.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.

Syllabus sections

Topic 5 —Calculus » SL 5.5—Integration introduction, areas between curve and x axis
Show 162 related questions
Topic 3— Geometry and trigonometry » SL 3.6—Pythagorean identity, double angles
Topic 4—Statistics and probability » SL 4.7—Discrete random variables
Topic 2—Functions » SL 2.7—Solutions of quadratic equations and inequalities, discriminant and nature of roots
Topic 2—Functions » SL 2.10—Solving equations graphically and analytically
Topic 5 —Calculus » SL 5.11—Definite integrals, areas under curve onto x-axis and areas between curves
Topic 2—Functions
Topic 3— Geometry and trigonometry
Topic 4—Statistics and probability
Topic 5 —Calculus

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