Print exam correction: Section B, Q6h.

,
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Victorian Certificate of Education Formula Sheet:
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TO ATTACH 2., Data LABEL
PROCESSING analysis,
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Letter
STUDENT NUMBER

SPECIALIST MATHEMATICS
Written examination 2
Monday 6 November 2023
Reading time: 11.45 am to 12.00 pm (15 minutes)
Writing time: 12.00 pm to 2.00 pm (2 hours)

QUESTION AND ANSWER BOOK

Structure of book
Section Number of Number of questions Number of
questions to be answered marks
A 20 20 20
B 6 6 60
Total 80

• Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers,
sharpeners, rulers, a protractor, set squares, aids for curve sketching, one bound reference, one
approved technology (calculator or software) and, if desired, one scientific calculator. Calculator
memory DOES NOT need to be cleared. For approved computer-based CAS, full functionality may
be used.
• Students are NOT permitted to bring into the examination room: blank sheets of paper and/or
correction fluid/tape.
Materials supplied
• Question and answer book of 23 pages
• Formula sheet
• Answer sheet for multiple-choice questions
Instructions
• Write your student number in the space provided above on this page.
• Check that your name and student number as printed on your answer sheet for multiple-choice
questions are correct, and sign your name in the space provided to verify this.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
• All written responses must be in English.
At the end of the examination
• Place the answer sheet for multiple-choice questions inside the front cover of this book.
• You may keep the formula sheet.

Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic
devices into the examination room.
© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2023
2023 SPECMATH EXAM 2 2

SECTION A – Multiple-choice questions

Instructions for Section A

Answer all questions in pencil on the answer sheet provided for multiple-choice questions.
Choose the response that is correct for the question.
A correct answer scores 1; an incorrect answer scores 0.
Marks will not be deducted for incorrect answers.
No marks will be given if more than one answer is completed for any question.
Unless otherwise indicated, the diagrams in this book are not drawn to scale.
Take the acceleration due to gravity to have magnitude g m s–2, where g = 9.8

Question 1

do not write in this area

Consider the following statement.
‘If my football team plays badly, then they are not training enough.’
Which one of the following statements is the contrapositive of the statement above?
A. If they are not training enough, then my football team plays badly.
B. If my football team plays badly, then they need more training.
C. If they are training enough, then my football team does not play badly.
D. If my football team doesn’t play badly, then they are training enough.
E. If they are training enough, then my football team will most likely win.

Question 2
x3
The graph of y  has asymptotes given by y = 2x + 1 and x = 1. The values of a, b and c are,
ax 2  bx  c
respectively
A. 2, −4, 2

1 1 1
B. , − , −
2 4 4
1 1 3
C. , , −
2 4 4
1 1 3
D. , − , −
2 4 4

E. 2, −4, −8

Question 3
In the interval −π ≤ x ≤ π, the graph of y = a + sec(x), where a ∈ R, has two x-intercepts when
A. 0 ≤ a ≤ 1
B. −1 < a < 1
C. a ≤ −1 or a > 1
D. −1 ≤ a < 0
E. a < −1 or a ≥ 1

SECTION A – continued
 3 2023 SPECMATH EXAM 2

Question 4
4a
If z = − (2a + 1) + 2ai, where a is a non-zero real constant, then is equal to
1+ z
 
A. 2cis  
4

 3 
B. 2cis  
 4 

 
C. cis  
4

 3 
D. 2cis   
 4 

 
do not write in this area

E. cis   
 4

Question 5
Let z be a complex number where Re(z) > 0 and Im(z) > 0.
Given z = 4 and arg(z3) = −π , then z2 is equivalent to
A.   4z
B. −2z
C.   3z
D.   z 2
E. −4z

Question 6
Consider the following pseudocode.
define f(x,y)=exy
x0
y0
h0.5
n0
while n≥0
yy+h×f(x,y)
xx+h
nn+1
print y
end while
After how many iterations will the pseudocode print 2.709?
A. 1
B. 2
C. 3
D. 4
E. 5

SECTION A – continued
TURN OVER
2023 SPECMATH EXAM 2 4

Question 7

x
O

do not write in this area

−2 −1 1 2

−1

−2

The direction field for a differential equation is shown above. On a certain solution curve of this differential
equation, y = 2 when x = −1.
The value of y on the same solution curve when x = 1.5 is closest to
A. −0.5
B.  0
C.   0.5
D.   1.0
E.   1.5

SECTION A – continued
 5 2023 SPECMATH EXAM 2

Question 8
Initially a spa pool is filled with 8000 litres of water that contains a quantity of dissolved chemical. It is
discovered that too much chemical is contained in the spa pool water. To correct this situation, 20 litres of
well-mixed spa pool water is pumped out every minute while 15 litres of fresh water is pumped in each
minute.
Let Q be the number of kilograms of chemical that remains dissolved in the spa pool after t minutes.
The differential equation relating Q to t is
dQ 4Q
A. 
dt t  1600

dQ Q
B. 
dt 400

dQ 3Q
C. 
dt t  1600
do not write in this area

dQ 3Q
D. 
dt 1600  t

dQ 4Q
E. 
dt 1600  t

Question 9
The position of a particle moving in the Cartesian plane, at time t, is given by the parametric equations
6t 8
x(t )  and y (t )  2 , where t ≥ 0.
t 1 t 4
What is the slope of the tangent to the path of the particle when t = 2?

1
A. −
3
1
B. −
4
1
C.   
3
3
D.   
4
4
E.   
3

SECTION A – continued
TURN OVER
2023 SPECMATH EXAM 2 6

Question 10

  (1  x) e  dx, where n ∈ N, then for n ≥ 1, I

1
n x
If I n  n equals
0

A. −1 + nIn − 1
B. nIn − 1
C. −1 − nIn − 1
D. −nIn − 1
E. (1 − x)ne x + nIn − 1

Question 11
The area of the curved surface generated by revolving part of the curve with equation y = cos−1(x) from
 

do not write in this area

 0,  to (1, 0) about the y-axis can be found by evaluating
 2

 1 

1
A. 2 2
 cos ( x) 1  2  dx
0  x  1 

1 
  cos
1 1
B. 2 ( x) 1   dx
0 x  1 
2

C. 2
 cos( y ) 1  sin 2 ( y ) dy
2
0


D. 2
0
2
1  u 2 du, where u = sin(y)

1
E. 2
0
1  u 2 du, where u = sin (y)

Question 12
The acceleration, a m s−2, of a particle that starts from rest and moves in a straight line is described by
a = 1 + v, where v m s−1 is its velocity after t seconds.
The velocity of the particle after loge(e + 1) seconds is
A. e
B. e + 1
C. e2 + 1
D. loge (1 + e) + 1
E. log e  log e (1  e)  1

SECTION A – continued
 7 2023 SPECMATH EXAM 2

Question 13
A tourist in a hot air balloon, which is rising vertically at 2.5 m s−1, accidentally drops a phone over the side
when the phone is 80 metres above the ground.
Assuming air resistance is negligible, how long in seconds, correct to two decimal places, does it take for the
phone to hit the ground?
A. 2.86
B. 2.98
C. 3.79
D. 4.04
E. 4.30

Question 14
Let a  i  j , b  i  j and c  i  2 j  3k .
         
do not write in this area

If n is a unit vector such that a.n = 0 and b.n = 0, then c.n is equal to
    
A. 2
B. 3
C. 4
D. 5
E. 6

Question 15
If the sum of two unit vectors is a unit vector, then the magnitude of the difference of the two vectors is
A. 0
1
B.
2

C. 2

D. 3

E. 5

Question 16
A student throws a ball for his dog to retrieve. The position vector of the ball, relative to an origin O at
ground level t seconds after release, is given by r B (t )  5t i  7t j  (15t  4.9t 2  1.5)k . Displacement
   
components are measured in metres, where i is a unit vector to the east, j is a unit vector to the north and
k is a unit vector vertically up.  

The total vertical distance, in metres, travelled by the ball before it hits the ground is closest to
A.   1.5
B. 11.5
C. 13.0
D. 24.5
E. 26.0

SECTION A – continued
TURN OVER
2023 SPECMATH EXAM 2 8

Question 17
Consider the vectors a   i  j  k , b  3i   j  4k and c  2 i  7 j   k , where α, β, γ ∈ R. If a  b  c , then
           
A. α = −2, β = −1, γ = −5   
B. α = −1, β = 2, γ = −1
C. α = 1, β = −2, γ = −5
D. α = −2, β = −1, γ = −1
E. α = 1, β = −2, γ = 5

Question 18
What value of k, where k ∈ R, will make the following planes perpendicular?
Π1 : 2x − ky + 3z = 1
Π2 : 2kx + 3y − 2z = 4

do not write in this area

A.  2
B.   4
C.  6
D.  8
E. 10

Question 19
A company accountant knows that the amount owed on any individual unpaid invoice is normally distributed
with a mean of $800 and a standard deviation of $200.
What is the probability, correct to three decimal places, that in a random sample of 16 unpaid invoices the
total amount owed is more than $13 500?
A. 0.087
B. 0.191
C. 0.413
D. 0.587
E. 0.809

Question 20
The lifespan of a certain electronic component is normally distributed with a mean of μ hours and a standard
deviation of σ hours.
Given that a 99% confidence interval, based on a random sample of 100 such components, is
(10 500, 15 500), the value of σ is closest to
A.   9710
B. 10 750
C. 12 750
D. 15 190
E. 19 390

END OF SECTION A
 9 2023 SPECMATH EXAM 2

SECTION B

Instructions for Section B

Answer all questions in the spaces provided.
Unless otherwise specified, an exact answer is required to a question.
In questions where more than one mark is available, appropriate working must be shown.
Unless otherwise indicated, the diagrams in this book are not drawn to scale.
Take the acceleration due to gravity to have magnitude g m s–2, where g = 9.8
do not write in this area

SECTION B – continued
TURN OVER
2023 SPECMATH EXAM 2 10

Question 1 (10 marks)

Viewed from above, a scenic walking track from point O to point D is shown below. Its shape is
given by

 x( x  a ) 2 , 0  x 1
f ( x)  
x 1
e  x  b, 1 x  2 .

The minimum turning point of section OABC occurs at point A. Point B is a point of inflection and
the curves meet at point C(1, 0). Distances are measured in kilometres.

do not write in this area

D

0.5

C
x
O 1 2
B
A

a. Show that a = −1 and b = 0. 1 mark

b. Verify that the two curves meet smoothly at point C. 2 marks

SECTION B – Question 1 – continued

 11 2023 SPECMATH EXAM 2

c. i. Find the coordinates of point A. 1 mark

ii. Find the coordinates of point B. 1 mark

do not write in this area

The return track from point D to point O follows an elliptical path given by

 
x = 2cos(t) + 2, y = (e − 2)sin(t), where t   ,   .
2 
d. Find the Cartesian equation of the elliptical path. 2 marks

e. Sketch the elliptical path from D to O on the diagram on page 10. 1 mark

f. i. Write down a definite integral in terms of t that gives the length of the elliptical path
from D to O. 1 mark

ii. Find the length of the elliptical path from D to O.

Give your answer in kilometres correct to three decimal places. 1 mark

SECTION B – continued
TURN OVER
2023 SPECMATH EXAM 2 12

Question 2 (10 marks)

 2 
Let w  cis  .
 7 
a. Verify that w is a root of z7 − 1 = 0. 1 mark

b. List the other roots of z7 − 1 = 0 in polar form. 1 mark

do not write in this area

c. On the Argand diagram below, plot and label the points that represent all the roots of
z7 − 1 = 0. 2 marks

Im(z)

Re(z)
−1 O 1

−1

SECTION B – Question 2 – continued

 13 2023 SPECMATH EXAM 2

d. i. On the Argand diagram below, sketch the ray that originates at the real root of
 2 
z7 − 1 = 0 and passes through the point represented by cis  . 1 mark
 7 

Im(z)

Re(z)
−1 O 1
do not write in this area

−1

ii. Find the equation of this ray in the form Arg(z − z0) = θ, where z0 ∈ C, and θ is measured
in radians in terms of π. 1 mark

SECTION B – Question 2 – continued

TURN OVER
2023 SPECMATH EXAM 2 14

e. Verify that the equation z7 − 1 = 0 can be expressed in the form

(z − 1)(z6 + z5 + z4 + z3 + z2 + z + 1) = 0. 1 mark

 2   12  +
f. i. Express cis    cis   in the form Acos (Bπ), where A, B ∈ R . 1 mark
 7   7 

do not write in this area

 2  6 5 4 3 2
ii.
Given that w  cis   satisfies (z − 1)(z + z + z + z + z + z + 1) = 0, use
 7 
 2   4   6  1

De Moivre’s theorem to show that cos    cos    cos   . 2 marks
 7   7   7  2

SECTION B – continued
 15 2023 SPECMATH EXAM 2

Question 3 (10 marks)

The curve given by y2 = x − 1, where 2 ≤ x ≤ 5, is rotated about the x-axis to form a solid of
revolution.
a. i. Write down the definite integral, in terms of x, for the volume of this solid of revolution. 1 mark

ii. Find the volume of the solid of revolution. 1 mark

do not write in this area

b
b. i. Express the curved surface area of the solid in the form 
are all positive integers.

a
Ax  B dx , where a, b, A, B
2 marks

ii. Hence or otherwise, find the curved surface area of the solid correct to three decimal
places. 1 mark

The total surface area of the solid consists of the curved surface area plus the areas of the two
circular discs at each end.
The ‘efficiency ratio’ of a body is defined as its total surface area divided by the enclosed volume.
c. Find the efficiency ratio of the solid of revolution correct to two decimal places. 2 marks

SECTION B – Question 3 – continued

TURN OVER
2023 SPECMATH EXAM 2 16

d. Another solid of revolution is formed by rotating the curve given by y2 = x − 1 about the
x-axis for 2 ≤ x ≤ k, where k ∈ R. This solid has a volume of 24π.
Find the efficiency ratio for this solid, giving your answer correct to two decimal places. 3 marks

do not write in this area

SECTION B – continued
 17 2023 SPECMATH EXAM 2

Question 4 (10 marks)

A fish farmer releases 200 fish into a pond that originally contained no fish. The fish population,
dP  P 
P, grows according to the logistic model,  P 1   , where t is the time in years after the
release of the 200 fish. dt  1000 

a. The above logistic differential equation can be expressed as

A B
 P  1 P 
dP  dt , where A, B ∈ R.

1000
Find the values of A and B. 1 mark
do not write in this area

1000
One form of the solution for P is P  , where D is a real constant.
1  De t
b. Find the value of D. 1 mark

The farmer releases a batch of n fish into a second pond, pond 2, which originally contained no
1000
fish. The population, Q, of fish in pond 2 can be modelled by Q  , where t is the time in
years after the n fish are released. 1  9e 1.1t

c. Find the value of n. 1 mark

d. Find the value of Q when t = 6.

Give your answer correct to the nearest integer. 1 mark

SECTION B – Question 4 – continued

TURN OVER
2023 SPECMATH EXAM 2 18

dQ 11  Q  d 2Q
e. i. Given that  Q 1   , express 2 in terms of Q. 1 mark
dt 10  1000  dt

ii. Hence or otherwise, find the size of the fish population in pond 2 and the value of t when
the rate of growth of the population is a maximum. Give your answer for t correct to the
nearest year. 2 marks

do not write in this area

SECTION B – Question 4 – continued

 19 2023 SPECMATH EXAM 2

f. Sketch the graph of Q versus t on the set of axes below. Label any axis intercepts and any
asymptotes with their equations. 2 marks

Q
1000

800

600

400
do not write in this area

200

t
O 1 2 3 4 5 6

The farmer wishes to take 5.5% of the fish from pond 2 each year. The modified logistic differential
equation that would model the fish population, Q, in pond 2 after t years in this situation is

dQ 11  Q 
 Q 1    0.055Q .
dt 10  1000 

g. Find the maximum number of fish that could be supported in pond 2 in this situation. 1 mark

SECTION B – continued
TURN OVER
2023 SPECMATH EXAM 2 20

Question 5 (11 marks)

The points with coordinates A(1, 1, 2), B(1, 2, 3) and C(3, 2, 4) all lie in a plane Π.
→ →
a. Find the vectors AB and AC, and hence show that the area of triangle ABC is 1.5 square units. 2 marks

do not write in this area

b. Find the shortest distance from point B to the line segment AC. 2 marks

A second plane, ψ, has the Cartesian equation 2x − 2y − z = −18.

c.


 to the nearest
 

At what acute angle does the line given by r (t )  3i  2 j  4k  t i  2 j  2k , t  R ,
  degree.
intersect the plane ψ? Give your answer in degrees correct 2 marks

A line L passes through the origin and is normal to the plane ψ. The line L intersects ψ at a point D.
d. Write down an equation of the line L in parametric form. 1 mark

e. Find the shortest distance from the origin to the plane ψ. 2 marks

SECTION B – Question 5 – continued

 21 2023 SPECMATH EXAM 2

f. Find the coordinates of point D. 2 marks

do not write in this area

SECTION B – continued
TURN OVER
2023 SPECMATH EXAM 2 22

Question 6 (9 marks)
A forest ranger wishes to investigate the mass of adult male koalas in a Victorian forest. A random
sample of 20 such koalas has a sample mean of 11.39 kg.
It is known that the mass of adult male koalas in the forest is normally distributed with a standard
deviation of 1 kg.
a. Find a 95% confidence interval for the population mean (the mean mass of all adult male
koalas in the forest). Give your values correct to two decimal places. 1 mark

do not write in this area

b. Sixty such random samples are taken and their confidence intervals are calculated.
In how many of these confidence intervals would the actual mean mass of all adult male
koalas in the forest be expected to lie? 1 mark

The ranger wants to decrease the width of the 95% confidence interval by 60% to get a better
estimate of the population mean.
c. How many adult male koalas should be sampled to achieve this? 1 mark

It is thought that the mean mass of adult male koalas in the forest is 12 kg. The ranger thinks that
the true mean mass is less than this and decides to apply a one-tailed statistical test. A random
sample of 40 adult male koalas is taken and the sample mean is found to be 11.6 kg.
d. Write down the null hypothesis, H0 , and the alternative hypothesis, H1 , for the test. 1 mark

SECTION B – Question 6 – continued

 23 2023 SPECMATH EXAM 2

The ranger decides to apply the one-tailed test at the 1% level of significance and assumes the
mass of adult male koalas in the forest is normally distributed with a mean of 12 kg and a standard
deviation of 1 kg.
e. i. Find the p value for the test correct to four decimal places. 1 mark

ii. Draw a conclusion about the null hypothesis in part d. from the p value found above,
giving a reason for your conclusion. 1 mark
do not write in this area

f. What is the critical sample mean (the smallest sample mean for H0 not to be rejected) in this
test? Give your answer in kilograms correct to three decimal places. 1 mark

Suppose that the true mean mass of adult male koalas in the forest is 11.4 kg, and the standard
deviation is 1 kg. The level of significance of the test is still 1%.
g. What is the probability, correct to three decimal places, of the ranger making a type II error in
the statistical test? 1 mark

h. The frequency curves for the sampling distributions associated with H0 and H1 are shown
below.
Label the critical sample mean on the diagram and shade the region that represents the
type II error. 1 mark

H1 H0

x
11 11.5 12 12.5

END OF QUESTION AND ANSWER BOOK

Victorian Certificate of Education
2023

SPECIALIST MATHEMATICS
Written examination 2

FORMULA SHEET

Instructions

This formula sheet is provided for your reference.

A question and answer book is provided with this formula sheet.

Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic
devices into the examination room.

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2023

SPECMATH EXAM 2

Mensuration Calculus

area of a r2 volume of d n
circle segment 2
  sin ( )  a sphere
4 3
3
πr
dx
 x   n xn 1
volume of area of d ax
a cylinder
π r 2h
a triangle
1
2
bc sin ( A )
dx
 e   a eax
volume of a b c d
a cone
1 2
3
πr h sine rule = =
sin ( A) sin ( B) sin (C ) dx
 log e ( x)  
1
x

volume of 1 d
a pyramid
Ah cosine rule c2 = a2 + b2 − 2ab cos (C)  sin (ax)   a cos (ax)
3 dx

d
Algebra, number and structure (complex numbers)  cos (ax)   a sin (ax)
dx
z  x  iy  r  cos ( )  i sin ( )   r cis ( ) z  x2  y 2  r
d
 tan (ax)   a sec2 (ax)
−π < Arg (z) ≤ π z1 z2  r1r2 cis 1   2  dx

z1 r1 d
 cis 1   2  de Moivre’s
z n = r n cis (n θ)  cot (ax)   a cosec2 (ax)
z2 r2 theorem dx

d
Data analysis, probability and statistics  sec (ax)   a sec (ax) tan (ax)
dx
E  aX1  b   a E  X1   b d
 cosec( ax)   a cosec (ax) cot (ax)
E  a1 X1  a2 X 2    an X n  dx

for independent  a1E  X1   a2 E  X 2     an E  X n  d a

random variables dx
 sin 1 (ax)  
Var  aX1  b   a 2 Var  X1  1  (ax) 2
X1 , X2 … Xn
Var  a1 X1  a2 X 2    an X n  d a
 cos 1 (ax)  
 a12 Var  X1   a2 2 Var  X 2     an 2 Var  X n  dx 1  (ax) 2

for independent E  X 1  X 2    X n   n d a
identically distributed dx
 tan 1 (ax)  
1  (ax) 2
variables
X1 , X2 … Xn Var  X1  X 2    X n   n 2

approximate confidence  s s 
x  z ,xz 
interval for µ  n n 

mean E X µ
distribution of sample
mean X 2
variance Var  X  
n
 3 SPECMATH EXAM

Calculus – continued

d dv du
 x dx  n  1 x
n 1 n 1  c, n  1 product rule (uv)  u  v
dx dx dx

1 ax du dv
 e ax dx 
a
e c quotient rule d u
 
dx  v 
v
dx
u
v2
dx


1
dx  log e x  c dy dy du
x chain rule  
dx du dx


1
sin (ax) dx   cos (ax)  c dv du
a integration by parts
 u dx dx  u v   v dx dx
 cos (ax) dx  a sin (ax)  c dy
1
If = f ( x, y ) , x0 = a and y0 = b,
dx
Euler’s method then xn +1 = xn + h and

1
sec 2 (ax) dx  tan (ax)  c yn  1  yn  h  f  xn , yn .
a
t2 2 2
 dx   dy 
 
1
cosec 2 (ax)dx   cot (ax)  c arc length parametric      dt
a t1  dt   dt 

 sec (ax) tan (ax) dx  a sec (ax)  c

1
surface area Cartesian x2 2
 dy 
about x-axis  x1
2 y 1    dx
 dx 

 cosec (ax) cot (ax) dx   a cosec (ax)  c

1
) surface area Cartesian y2 2
 dx 
about y-axis  y1
2 x 1    dy
 dy 
x

1
dx  sin 1    c, a  0
a x
2 2 a surface area parametric t2 2
 dx   dy 
2

about x-axis t1

2 y      dt
 dt   dt 
1 x
 a2  x2
dx  cos 1    c, a  0
a surface area parametric t2 2
 dx   dy 
2

about y-axis t1

2 x      dt
 dt   dt 
a x
 a x
2 2
dx  tan 1    c
a Kinematics

 (ax  b) n 1
dx  (ax  b) n  1  c, n  1 d 2 x dv dv d  1 
a (n  1) acceleration a   v   v2 
dt 2 dt dx dx  2 

 ax  b dx  a log
1 1
e ax  b  c 1 2
v = u + at s  ut  at
constant acceleration 2
formulas
1
v2 = u2 + 2as s  (u  v) t
2

TURN OVER
SPECMATH EXAM 4

Vectors in two and three dimensions

r (t )  x (t ) i + y (t ) j  z (t )k r (t )  x(t ) 2 + y (t ) 2  z (t ) 2
    
d r dx dy dz
r (t )    i + j k
 dt dt  dt  dt 

vector scalar product

r 1 . r 2  r 1 r 2 cos ( )  x1 x2 + y1 y2  z1 z2
   
for r1  x1 i + y1 j  z1k
    vector cross product
and r 2  x2 i + y2 j  z2 k
    i

j k


r 1  r 2  x1 y1 z1   y1 z2  y2 z1  i +  x2 z1  x1 z2  j   x1 y2  x2 y1  k
    
x2 y2 z2

vector equation of a line r (t )  r1  t r 2   x1  x2t  i +  y1  y2t  j   z1  z2t  k

     
parametric equation of a line x (t) = x1 + x2t  y (t) = y1 + y2t  z (t) = z1 + z2t

r ( s, t )  r 0  s r 1  t r 2
vector equation of a plane    
  x0 + x1s + x2t  i   y0 + y1s + y2t  j   z0 + z1s + z2t  k
  
parametric equation of a plane x (s, t) = x0 + x1s + x2t, y (s, t) = y0 + y1s + y2t, z (s, t) = z0 + z1s + z2t

Cartesian equation of a plane ax + by + cz = d

Circular functions

cos2 (x) + sin2 (x) = 1

1 + tan2 (x) = sec2 (x) cot2 (x) + 1 = cosec2 (x)

sin (x + y) = sin (x) cos (y) + cos (x) sin (y) sin (x − y) = sin (x) cos (y) − cos (x) sin (y)

cos (x + y) = cos (x) cos (y) − sin (x) sin (y) cos (x − y) = cos (x) cos (y) + sin (x) sin (y)

tan ( x)  tan ( y ) tan ( x)  tan ( y )

tan ( x  y )  tan ( x  y ) 
1  tan ( x) tan ( y ) 1  tan ( x) tan ( y )

sin (2x) = 2 sin (x) cos (x)

2 tan ( x)
cos (2x) = cos2 (x) − sin2 (x) = 2 cos2 (x) − 1 = 1 − 2 sin2 (x) tan (2 x) 
1  tan 2 ( x)

1 1
sin 2 (ax)  1  cos (2ax)  cos 2 (ax)  1  cos (2ax) 
2 2

END OF FORMULA SHEET