More of the same, but different. Various pages were written on Series and binomial; updating and hence accessibility to other pages improved, I hope. This page should give exercises in series expansion at FPM and C1/C2. I’ll get around to the trigonometry - it’s in two forms, one working out angles between planes and lines as geometry but not as vectors (also to add) and trig as juggling the functions as identities and algebra.
DJS 20130213
Series expansions
1 Expand (1 - 3x)5/3 to four terms. [3]
2 Expand (3 - x)5/3 to four terms. [3]
3 Expand (1 - ⅔ x )5/3 to four terms. [3]
4 Expand (1 - 2 / 3x )5/3 to four terms.
Integrate this expansion to estimate the integral of (1 - 2 / 3x)5/3 [5]
5 Expand (2 - 3x/4) 5/3 to four terms. [3]
FPM question:
6 Expand (1 - x)-2 to five terms. [2]
Expand (1 - x)-1/2 to four terms. [2]
Use x=0.1 to estimate a value of √10. [4]
Find the % relative error. [very small] [1]
Find the coefficient of x7 in the expansion of (1 + x/√5)12 giving your answer in the form
a √5, where a is a rational number. [4]
[similar to 20120119 Q3]
7 Use sin(A+B) to show that sin(π/2- x) = cos x [2]
b Find the equivalent result for tan(π/2-x) [2]
c Solve cos x = 1/3 for 0<x<360º [3]
8 Use cos(A+B) = cosAcosB–sinAsinB to write an expression for cos2θ [1]
b Repeat this to show that cos 3θ = 4 cos³θ - 3 cos θ [3]
c Find all the solutions of cos 3θ = 0 for 0≤θ<360º [3]
9 sin(A + B) = sinAcosB + cosAsinB cos(A+B)=cosAcosB–sinAsinB
a Show that tan(A-B)= (tanA - tanB)/(1 + tan A tan B) [3]
b Hence write down an expression for tan 2θ in terms of tan θ [1]
Using the triangles 1:2:√3 and 1:1:√2. find an exact value for tan (π/3 - π/4), simplifying this to the form a + b√c, a,b,c∈Z [4]
10a Write down an expression for sin 2A in terms of sin A and cos A [1]
b Find an expression for cos² A in terms of sin A [2]
c Show that 2cos²A - sin3A = 4sin³A + 4sin³A - 4sin²A - 3sinA + 2 [4]
d Given that one factor of this is (2sinA-1) find the other factor [3]
e Hence solve 2cos²A - sin3A = 0 for 0≤A≤360º to the nearest degree [2]
[from 20110613 Q4]
Trigonometry
11 Triangle ABC has AB=7cm, AC=5cm, angle B is 20º and angle C is obtuse.
a Find, to the nearest degree, the size of angle C. [3]
The point D lies on BC produced and AD=5cm.
b Find to 3 sig fig, the length of CD. [3]
[similar to 20110621 Q3]
DJS 20130213
Bits of answers: index !!
10 4sin³A - 4sin²A - 3sinA + 2 = (2sinA-1) (2sin²A-sinA-2) => A=30º, 150º, 231º, 309º, (and reject one answer)
11 C = 123.4387=123º => ACD = 56.7º => CD=5.5
