CH159 – Part b Test
Answer sheet for practice tests
The time allowed for the test is 40 minutes.
Attempt every question giving your answers clearly in the space provided in black or blue ink.
Graphical calculators are not permitted.
The formula for solving a quadratic equation is:
2
⎛
⎞
⎜ x = − b ± (b − 4ac) ⎟
⎜
⎟
2a
⎝
⎠
Standard Derivatives
d ax
d x
e = ae ax
e = ex
dx
dx
d
(sin x ) = cos x
dx
d
( cos x ) = − sin x
dx
d
1
d
a
tan x =
tan ax =
2
dx
cos x dx
cos 2 ax
d
1
d ax
ln x =
( e ) = aeax
dx
x
dx
d
sin ( ax ) = a cos ( ax )
dx
d
cos ( ax ) = − a sin ( ax )
dx
Standard Integrals
1
∫ x dx = n + 1 x
n
n +1
+c
∫x
(n ≠ −1)
1
ax
dx = ln x + c
1
∫ cos ( ax ) dx = a sin ( ax ) + c
∫e
−1
∫ sin ( ax ) dx = − a cos ( ax ) + c
1
dx = e ax + c
a
∫ ln x dx = x ln x − x + c
Taylor Series
(
) ()
f (x) = f (a) + x − a f ! a +
1
x−a
2!
(
) f !!(a) + 3!1 ( x − a)
2
3
()
f !!! a +…
Practice Test 1
1. Evaluate the following, given the matrices
⎛ 0 1 −1⎞
⎛1 2 0⎞
⎛1 2 ⎞
⎜
⎟
⎜
⎟
⎜
⎟
A = ⎜ 2 2 1 ⎟ , B= ⎜ 0 −3 1 ⎟ , C= ⎜1 1 ⎟ ,
⎜ 1 3 −1⎟
⎜1 3 1⎟
⎜1 3 ⎟
⎝
⎠
⎝
⎠
⎝
⎠
(a)
⎛ 1 −4 ⎞
D= ⎜
⎟
⎝2 3 ⎠
A+B
[2]
⎛ 1 3 −1 ⎞
⎜ 2 −1 2 ⎟
⎜
⎟
⎝ 2 6 0 ⎠
(b)
AC
[3]
⎛ 0 −2 ⎞
⎜ 5 9 ⎟
⎜
⎟
⎝ 3 2 ⎠
(c)
|D|
(d)
BT
[2]
11
[1]
⎛ 1 0 1
⎜ 2 −3 3
⎜
⎝ 0 1 1
⎞
⎟
⎟
⎠
2. All parts of this question relate to the following simultaneous equations:
(a)
(b)
(c)
2x − 3y = 1
.
x + 2y = 3
write these simultaneous equations in matrix form
⎛ 2 −3 ⎞ ⎛ x ⎞ ⎛ 1 ⎞
⎜⎝ 1 2 ⎟⎠ ⎜ y ⎟ = ⎜⎝ 3 ⎟⎠
⎝
⎠
Calculate the inverse of the matrix of coefficients
1⎛ 2 3 ⎞
7 ⎜⎝ −1 2 ⎟⎠
[2]
Use the inverse matrix to solve the simultaneous equations
⎛ x ⎞ 1⎛ 2 3 ⎞⎛ 1 ⎞
⎜ y ⎟ = 7 ⎜ −1 2 ⎟ ⎜ 3 ⎟
⎝
⎠⎝
⎠
⎝
⎠
[3]
⎛
⎜
=⎜
⎜
⎜⎝
11
7
5
7
⎞
⎟
⎟
⎟
⎟⎠
( also accept answer in the form x = 11 / 7, y = 5 / 7, but should show use of inverse matrix)
[3]
⎛ 4 2⎞
3. Given the matrix M = ⎜
⎟:
3
3
⎝
⎠
(a) calculate the eigenvalues of M
[4]
4−λ
3
2
=0
3− λ
( 4 − λ )( 3 − λ ) − 3 × 2 = λ 2 − 7λ + 6
= ( λ − 6 ) ( λ − 1) = 0
so two solutions (eigenvalues) are: λ = 6 and λ = 1
(b) hence find the two eigenvectors of M
for λ = 6
⎛ x ⎞
⎛ 4 2 ⎞⎛ x ⎞
⎜⎝ 3 3 ⎟⎠ ⎜ y ⎟ = 6 ⎜ y ⎟
⎝
⎠
⎝
⎠
⎛ 4x + 2y ⎞ ⎛ 6x ⎞
⎜
⎟ =⎜
⎟
⎝ 3x + 3y ⎠ ⎝ 6y ⎠
⎛ a ⎞
both imply that x=y, so acceptable eigenvectors are anything of the form ⎜
⎝ a ⎟⎠
⎛ 1 ⎞
⎛ 2 ⎞
where a is any number, e.g. ⎜
or
⎜⎝ 2 ⎟⎠ ...
⎝ 1 ⎟⎠
for λ = 1
⎛ 4 2 ⎞⎛ x ⎞ ⎛ x ⎞
⎜⎝ 3 3 ⎟⎠ ⎜ y ⎟ = ⎜ y ⎟
⎝
⎠ ⎝
⎠
⎛ 4x + 2y ⎞ ⎛ x ⎞
⎜
⎟ =⎜
⎟
⎝ 3x + 3y ⎠ ⎝ y ⎠
⎛ 2a ⎞
both imply that 3x = -2y, so acceptable eigenvectors are anything of the form ⎜
⎝ −3a ⎟⎠
⎛ 1 ⎞
⎛ −2 ⎞
where a is any number, e.g. ⎜
or ⎜
...
⎟
⎝ −1.5 ⎠
⎝ 3 ⎟⎠
[4]
4. (a) Draw a vector that corresponds to the sum of the two vectors shown in the diagram below.
[2]
To add them, align head to tail and then draw vector from first tail to last head (in black)
(b)
Given the vectors v = 2i + 3j – k and u = i – 2j + 4k, evaluate the following:
(i) u – v
– i – 5j + 5k
(ii) 3v
6i + 9j – 3k
(iii) u ⋅ v
–8
(iv) u × v
–10i + 9j + 7k
[2]
[2]
[2]
[3]
5. (a) Write out the Taylor series expansion for ln x about the point x=1, giving the first four
terms.
[5]
evaluate at x = 1( = a )
function and derivatives
f =0
f (x) = ln x
df 1
=
dx x
d 2 f −1
=
dx 2 x 2
d3 f
2
= 3
3
dx
x
df
=1
dx
d2 f
= −1
dx 2
d3 f
=2
dx 3
So the Taylor series is
1
1
2
3
f (x) = 0 + ( x − 1)1+ ( x − 1) ( −1) + ( x − 1) ( 2 )
2
6
1
1
2
3
= ( x − 1) − ( x − 1) + ( x − 1)
2
3
(b)
Hence evaluate ln(0.9) to four decimal places without using your calculator.
use Taylor series; if x = 0.9, then ( x − 1) = −0.1
f (x) = −0.1000 − 0.0050 − 0.00033
= −0.1053
[3]
6. All parts of this question refer to the complex numbers c = 2 + 3i ; d = 1 − i ; k = 2i ; m = 4
(a)
plot c, d and m on an Argand diagram
[3]
c
m
d
(b)
(c)
(d)
(e)
evaluate c + 2d
4+i
evaluate k × d
2 + 2i
evaluate c × d
5+i
evaluate c × c*
13
[1]
[2]
[2]
[2]
Practice Test 2
1. Evaluate the following, given the matrices
⎛ 1 2 −1⎞
⎛ 3 1 2⎞
⎛0 2 1 ⎞
⎜
⎟
⎜
⎟
A = ⎜ 2 0 1 ⎟ , B= ⎜ 1 −3 0 ⎟ , C= ⎜
⎟,
1
1
−
1
⎝
⎠
⎜ 4 3 −1⎟
⎜2 2 1⎟
⎝
⎠
⎝
⎠
(a)
(b)
(c)
(d)
⎛ 1 −2 ⎞
D= ⎜
⎟
⎝ −1 2 ⎠
A–B
CB
|D|
BT
[2]
[3]
[2]
[1]
2. All parts of this question relate to the following simultaneous equations:
(a)
(b)
(c)
2 x − 2 y = −4
.
x + 3y = 2
write these simultaneous equations in matrix form
Calculate the inverse of the matrix of coefficients
Use the inverse matrix to solve the simultaneous equations
⎛ 4 2⎞
3. Given the matrix M = ⎜
⎟:
⎝ −1 1 ⎠
(a) calculate the eigenvalues of M
(b) hence find the two eigenvectors of M
[2]
[3]
[3]
[4]
[4]
4. (a) Given the vectors a and b below, draw a vector that corresponds b – a.
a
b
[2]
(b)
Given the vectors v = i + 3j – 2k and u = 3i – j – k, evaluate the following:
(i) u + 2v
(ii) 2u
(iii) u ⋅ v
(iv) u × v
[2]
[2]
[2]
[3]
5. (a) Write out the Taylor series expansion for x cos x about the point x=0, giving the first four
terms.
[5]
(b)
Hence evaluate xcosx when x = 0.1 to three decimal places without using your calculator.
[3]
6. All parts of this question refer to the complex numbers c = 1 + 4i ; d = 2 − 3i ; k = i ; m = 2
(a)
plot c, d and m on an Argand diagram
[3]
c
m
d
(b)
(c)
(d)
(e)
evaluate 3c – d
evaluate k×c
evaluate c×d
evaluate d×d*
[1]
[2]
[2]
[2]
Practice Test 3
1. Evaluate the following, given the matrices
⎛ 1 4 10 ⎞
⎛ −2 3 1 ⎞
⎛ 2 0 −2 ⎞
⎜
⎟
⎜
⎟
A = ⎜ 3 2 −2 ⎟ , B= ⎜ 0 0 2 ⎟ , C= ⎜
⎟,
4
1
−
1
⎝
⎠
⎜2 0 1 ⎟
⎜ 2 −1 3 ⎟
⎝
⎠
⎝
⎠
(a)
⎛ 1 −1⎞
D= ⎜
⎟
⎝2 3 ⎠
A–B
[2]
⎛ 3 1 9 ⎞
⎜ 3 2 −4 ⎟
⎜
⎟
⎝ 0 1 −2 ⎠
(b)
DC
[3]
⎛ −2 −1 −1 ⎞
⎜⎝ 18 3 −7 ⎟⎠
(c)
|D|
(d)
BT
[2]
5
[1]
⎛ −2 0 2 ⎞
⎜ 3 0 −1 ⎟
⎜
⎟
⎝ 1 2 3 ⎠
2. All parts of this question relate to the following simultaneous equations:
(a)
(b)
(c)
x − 3y = 8
.
2x + y = 2
write these simultaneous equations in matrix form
⎛ 1 −3 ⎞ ⎛ x ⎞ ⎛ 8 ⎞
⎜⎝ 2 1 ⎟⎠ ⎜ y ⎟ = ⎜⎝ 2 ⎟⎠
⎝
⎠
Calculate the inverse of the matrix of coefficients
1⎛ 1 3 ⎞
7 ⎜⎝ −2 1 ⎟⎠
Use the inverse matrix to solve the simultaneous equations
⎛ x ⎞ 1⎛ 1 3 ⎞⎛ 8 ⎞
⎜ y ⎟ = 7 ⎜ −2 1 ⎟ ⎜ 2 ⎟
⎝
⎠⎝
⎠
⎝
⎠
⎛ 2 ⎞
=⎜
⎝ −2 ⎟⎠
( also accept answer in the form x = 2, y = −2, but should show use of inverse matrix)
[2]
[3]
[3]
⎛3 2⎞
3. Given the matrix M = ⎜
⎟:
3
4
⎝
⎠
(a) calculate the eigenvalues of M
[4]
3− λ
3
2
=0
4−λ
( 4 − λ )( 3 − λ ) − 3 × 2 = λ 2 − 7λ + 6
= ( λ − 6 ) ( λ − 1) = 0
so two solutions (eigenvalues) are: λ = 6 and λ = 1
(b) hence find the two eigenvectors of M
for λ = 6
[4]
⎛ x ⎞
⎛ 3 2 ⎞⎛ x ⎞
⎜⎝ 3 4 ⎟⎠ ⎜ y ⎟ = 6 ⎜ y ⎟
⎝
⎠
⎝
⎠
⎛ 3x + 2y ⎞ ⎛ 6x ⎞
⎜
⎟ =⎜
⎟
⎝ 3x + 4y ⎠ ⎝ 6y ⎠
⎛ 2a ⎞
both imply that 3x = 2y, so acceptable eigenvectors are anything of the form ⎜
⎝ 3a ⎟⎠
⎛ 2 ⎞
⎛ 3 ⎞
where a is any number, e.g. ⎜
or
⎜⎝ 4.5 ⎟⎠ ...
⎝ 3 ⎟⎠
for λ = 1
⎛ 3 2 ⎞⎛ x ⎞ ⎛ x ⎞
⎜⎝ 3 4 ⎟⎠ ⎜ y ⎟ = ⎜ y ⎟
⎝
⎠ ⎝
⎠
⎛ 3x + 2y ⎞ ⎛ x ⎞
⎜
⎟ =⎜
⎟
⎝ 3x + 4y ⎠ ⎝ y ⎠
⎛ a ⎞
both imply that x = -y, so acceptable eigenvectors are anything of the form ⎜
⎝ −a ⎟⎠
⎛ 1 ⎞
⎛ −2 ⎞
where a is any number, e.g. ⎜
or ⎜
...
⎟
⎝ −1 ⎠
⎝ 2 ⎟⎠
(b)
4. (a) Given the vectors a and b below, draw a vector that corresponds 2a – b.
[2]
(b)
Given the vectors v = i + 4j – 2k and u = 2i – 3j + k, evaluate the following:
(i) 3u
6i – 9j + 3k
(ii) u + 2v
4i + 5j – 3k
(iii) v ⋅ u
–12
(iv) u × v
2i + 5j + 11k
[2]
[2]
[2]
[3]
5. (a) Write out the Taylor series expansion for x ln x about the point x=1, giving the first four
terms.
[5]
(b)
Hence evaluate ln(1.1) to two decimal places without using your calculator.
[3]
function and derivatives
evaluate at x = 1
f (x) = x ln x
df
= ln x + 1
dx
d2 f 1
=
dx 2 x
d 3 f −1
=
dx 3 x 2
(b)
f (x) = 0
df
=1
dx
d2 f
=1
dx 2
d3 f
= −1
dx 3
So the Taylor series is
1
1
2
3
f (x) = 0 + ( x − 1)1+ ( x − 1) (1) + ( x − 1) ( −1)
2
6
1
1
2
3
= ( x − 1) + ( x − 1) − ( x − 1)
2
6
Hence evaluate ln(0.9) to four decimal places without using your calculator.
use Taylor series; if x = 1.1, then ( x − 1) = 0.1
f (x) = 0.1000 + 0.0050 − 0.00017
= 0.1033
[3]
6. All parts of this question refer to the complex numbers c = 2 + 3i ; d = 1 + 2i ; k = 5i ; m = −2
(a)
plot c, d and m on an Argand diagram
[3]
c
d
m
(b)
evaluate 2c – d
(c)
evaluate k×c
(d)
evaluate c×d
[1]
3 + 4i
[2]
–15 + 10i
(e)
[2]
*
–4 + 7i
evaluate d×d
[2]
5