Mathematics SL review material

Topic 1—Algebra............................................................................................................................2Topic 2—Functions and equations..................................................................................................5Topic 3—Circular functions and trigonometry.............................................................................12Topic 4—Matrices.........................................................................................................................18Topic 5—Vectors...........................................................................................................................20Topic 6—Statistics and probability...............................................................................................23Topic 7—Calculus.........................................................................................................................31

NC = non calculator question

1

Topic 1—Algebra

1. In an arithmetic sequence, the first term is 8 and the 78th term is 1009. Find the common difference d.

2. Find the sum of the arithmetic series 100 + 97 + 94 + ... + (– 1967).

3. Find the general term of the geometric sequence which has: u3=45and u7=3645

4. Find the sum of the geometric series 3 + (–6) + 12 + (–24) + ... + 49152.

5. Find the sum of the infinite geometric series2+ 2

3+ 2

9+ 2

27+. . .

6. Calculate

a) ∑n=2

8

(2n−1) b)

∑n=0

4

3n

7. 50 000 € is invested for 20 years. The rate of interest is fixed at 2.75% per annum. Interest is compounded annually. Calculate, giving your answer to the nearest euro, the total value of the investment at the end of the 20 years.

8. A town has a population of 4500 people. If the population of the town is growing by 3% p.a., how long will it take for the town to double in size?

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Answers:

1. 132. – 644 115

3. un=5×3n−1

4. 32 7695. 36. a) 63 b) 1217. 86 021 €8. 24 years

2

1. (NC) Simplify

a) (3 x2 y )3

b)

10 a7 b4

5 a3 b2

c) 2723

2. (NC) Simplifya) ln 2+ ln3

b) log 10−log 5c) 3 log 2

3. (NC) If log a=x and log b= y

find the expression in x and y for the following

a) log a3

b) log ( ab)

c)log a

b

d)log a2

b3

4. Calculate to 3 s.f.a) log 211

b) log5103

________________________

Answers:

1. a) 27 x6 y3 b) 2 a4 b2

c) 9

3

2. a) ln 6 b) log 2 c) log 83. a) 3x b) x + y c) x – y d) 2x - 3y4. a) 3.46 b) 2.88

1. Use your calculator to find the value of the following

a)(62 )

b)(75 )

2. Expand ( x+2 )4

3. Find the coefficient of x4

in the expansion of (3−x )10

4. Find the constant term in the expansion of (x2− 1

x2 )6

_______________________

Answers:

1. a) 15 b) 212. x4+8 x3+24 x2+32 x+163. 1530904. -20

4

Topic 2—Functions and equations

1. (NC) State the domain and range of the following functions:a) f ( x )= ln x

b) f ( x )=−2 x+3

c) f ( x )=√ x−1

d)f ( x )= 2

x−1

2. (NC) f ( x )=2 x+1 and g( x )=x2+x , find

a) f (−3 )

b) g( x )=0

c) ( f ∘g )(x )

d) ( g∘ f )(x )

3. (NC) a) Find the inverse of f ( x )=4 x+2

b) Find ( f ∘ f−1 )(x ) and ( f −1∘ f )(x ). What do you notice?

________________________

Answers:

1. a) x>0 , y∈R b) x∈ R , y∈R c)

x≥1 , y≥0 d) x≠1 , y≠0

2. a) -5 b) x = 0 or x = -1 c) 2 x2+2 x+1 d) 4 x2+6 x+2

3. a) f−1( x )= x−2

4 b) the answer is an identity function f(x) = x5

1. The function is defined by f ( x )= 2

√4−x2 for −2<x<2 .

a) Sketch the graph of f(x).b) Write down the equation of each vertical asymptote.

2. Solve the equation using GDC x3+2 x=2x

.

________________________

Answers:

1. a) b) x = 2, x = -22. x = 0.647

6

1. (NC) Given the function y=x2

translate it horizontally 2 units to the right reflect in the x-axis translate it vertically 3 units up

Write down the equation of the transformed graph.

2. (NC) If y=2

x under goes a translation of (−4

3 )write down the equation of the translated

graph. What are the asymptotes of the translated graph?

3. (NC) Sketch the graph of y= 1

x+3−2

. What are the asymptotes?

________________________

Answers:

1. y=−( x−2 )2+3

2.y= 2

x+4+3

asymptotes: x=−4

y=3

7

3. x=−3

y=−2

1. (NC) f ( x )=2 x2+8 x−4

, find

a) coordinates of the y-intercept

b) the axis of symmetry

c) coordinates of the vertex

2. (NC) The graph below has an equation in the form y=a ( x−h )2+k , find a, h and k.

x-10 -5 5 10

y

-10

-5

5

10

3. (NC) Write y=x2+6 x+8 in the form y=a ( x−h )2+k . What are the coordinates of the

vertex?

4. (NC) The graph has an equation in the form y=a ( x−p )( x−q ) , find a, p and q.

________________________

Answers:

1. a) (0,-4) b) x = -2 c) (-2,-12) 8

x-10 -5 5 10

y

-10

-5

5

10

2. a = 2, h = 2, k = -3

3. y= (x+3 )2−1 , (−3 ,−1)4. a = -1/2 p = -1 and q = 4

1. Solve the following equations using quadratic formulaa) x2+2 x−2=0

b) −5 x2=−x−7

2. (NC) Solve x2−3 x−10=0 using factorization.

3. (NC) Find the value for k so that equation a) x2−6 x+k=0 has exactly one real root.b) kx 2−4 x+1=0has two real solutions.c) x2−kx+2=0 has no real roots.

________________________

Answers:

1. a) -2.73 or 0.732 b) 1.29 or -1.092. 5 or -2

3. a) k = 9 b) k < 4 c) -2√2<k <2√2

9

1. (NC) Evaluate

a) log3 9

b)log 2

18

c) 10log 4

2. (NC) Find the inverses of the following functions

a) f ( x )=102 x+3

b) f ( x )=2x−1c) f ( x )=log4( x+1 )

3. Sketch the graphs off ( x )=3x and g( x )=log3 x in the same set of axis. What do you

notice?

4. Solve to 3 s.f.a) 10x=3

b) 4×23 x=13

c) log x=1 .2

________________________

Answers:

1. a) 2 b) -3 c) 4

2. a) f−1( x )=1

2log( x−3 )

b) f−1( x )=log2( x+1) c) f

−1( x )=4x−1

10

3. The graph of f(x) is the reflection in the line y = x of the graph g(x). Functions are inverse functions.

4. a) 0.477 b) 0.567 c) 15.8

1. Solve to 3 s.f.a) 2 ex=10

b) ln (x+1 )=8

2. (NC )Find the inverse of the following functions

a) f ( x )=e x+1

b) f ( x )= ln x+3

3. Find the value of the given investment after the indicated years at the given ratea) $6 000 after 3 years at 7% p.a.b) $10 000 after 10 years at 5% p.a.

4. A person has $15 000 to invest at 5% p.a., how long does it take for his investment to exceed $22 000?

5. The number of bacteria present in a culture at t (hours) is given by the equation N=400 e0.4 t

a) Find the number of bacteria present initially.b) Find the number of bacteria present after 8 hours.c) How long does it take the number of bacteria to double?

6. A radioactive substance is decaying so that its weight W (g) after t days is given by the equationW =100 e−0 . 2t

.a) What is the weight of the substance after 15 days?b) How long does it take to get to 25% of its original weight?

________________________

Answers:

11

1. a) 1.61 b) 29802. a) f

−1( x )=ln x−1 b) f−1( x )=ex−3

3. a) $7350 b) $162894. 8 years5. a) 400 b) 9813 c) 1.73 hrs (1 hr 44 min)6. a) 4.98 g b) 7 days

Topic 3—Circular functions and trigonometry

1. Convert into radiansa) 60∘

b) 20∘

c) 111∘

2. Convert into degrees

a)π6

b) 3 π

c) 2.13

3. Calculate the perimeter of the sector, if radius is 2.7 cm and angle is2π3 .

4. Calculate the area of the sector if radius is 3.1 cm and angle is 3.8 rad.

_______________________

Answers:

1. a) π3 b)

π9 c)

37 π60

12

2. a) 30∘ b) 540∘

c) 122∘

3. 11.0 cm4. 18.3 cm2

1. (NC) Find the exact values of the following using the unit circle.

a)sin 3 π

2

b) cos1500

c) sin 2250

2. (NC) sin 2π

3=√3

2 andcos 2 π

3=−1

2 . Find exact value oftan 2π

3 .

3. (NC) Make cosθ the subject cos2θ+sin2θ=1 .

4. (NC) If sin θ=3

4 and π2<θ<π

find cosθ .

5. (NC) Find the equation of the straight line which makes an angle of 60 degrees with the x-axis and cuts the y-axis at (0,-2).

_______________________

13

Answers:

1. a) -1 b) −√3

2 c)

− 1√2

2. −√3

3. cosθ=±√1−sin2 θ

4.−√7

4

5. y=√3 x−2

1. (NC) Factorize sin 2 x−sin x .

2. (NC) Write cos2 θ+3sin θ=2 in the form a sin2θ+b sin θ+c , where a , b , c∈ R .

_______________________

Answers:

1. sin x (2cos x−1)

2. 2 sin2 θ−3 sinθ+1

14

1. (NC) State the domain, range and period of the following functionsa) f ( x )=sin x

b) f ( x )=cos x

c) f ( x )= tan x

2. (NC)f ( x )=2 cos (3 ( x−4 ) )+1

, state thea) amplitudeb) periodc) vertical translationd) horizontal translation

3. The model for the height of tide (in meters above mean sea level) is

h( t )=4 sin ( πt

6 ) (t is number of hours after midnight) a) What was the height at 1 pm?b) When was the high tide and what was was the maximum height?

4. The model for the height (in meters above the ground) of a light on a Ferris wheel is

h( t )=−18 sin( 2 πt3 )+22

(t is time in minutes) a) Where is the light at time t = 0 min?b) How long does the wheel take to complete one revolution?c) At what time was the light at its highest in the first revolution of the wheel?

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Answers:

15

1. a) x∈ R , -1≤ y≤1 b)

x∈ R , -1≤ y≤1 c)

x≠π2+nπ , y∈R

2. a) 2 b) 2 π3

c) 1 step up d) 4 steps to the right3. a) 2 m above the mean b) 4 m above the mean4. a) 22.0 m b) 3 min c) 2 ¼ min

1. Solve a) 2sin x=3cos x , 0≤x≤2 π

b) 2 sin 2x=3cos x , 0∘≤x≤180∘

c) 2 sin x=cos2 x , -π≤x≤π

2. (NC) Solve 4 sin(2(x+ π

3 ) )=2, 0≤x≤2 π

3. (NC) Solve

a) 2 sin2 x+sin x−1=0 , 0< x <3 π

b)cos22 x= 1

4, - π<x<π

_______________________

16

Answers:

1. a) 0.983 rad, 4.12 rad b) 90 degrees, 48.6 degrees, 131.4 degrees c) 0.375 rad, 2.77 rad

2.π

12, 13 π12

, 3 π4

, 7 π4

3. a) π2

, 5 π2 b)

± π6

,± π3

,±2 π3

,±5 π6

1. A triangle has sides 5 cm, 7 cm and 8 cm. Find the size of the biggest angle in degrees.

2. Find the two values for the angle A in the triangle ABC where angle C = 35 degrees, AB = 9 cm and BC = 14 cm.

3. Find the area of the triangle with sides AB = 10 cm, AC = 9 cm and the angle BAC = 54 degrees.

4. Three towns are positioned so that A is 41 km from B and C is 35 km from B. If the angle ABC = 46 degrees, how far apart are A and C?

_______________________

17

Answers:

1. 81.8 degrees2. 63.2 or 116.8 degrees3. 36.4 cm2

4. 30.2 km

Topic 4—Matrices

1. (NC) a) What is the order of the following matrices

i)

A=(1 4 20 5 3 )

ii)

B=( 3−5

4 )b) Calculate AB and BA.

2. (NC) Solve the equations, where A, B and X are matrices.a) 2X = Ab) AX = Bc) XA = B

18

_______________________

Answers:

1. a) i) 2x3 ii) 3x1 b) AB=(−17

−25) , BA is not defined

2. a) X=1

2A

b) X=A−1 B c) X=BA−1

1. (NC) Calculate the determinant

a)A=(1 2

3 4 ) b)

B=( 2 −2 0−1 5 13 4 5 )

2. (NC) Find the inverse of A=(4 −3

2 −2 )

3. Find the inverse of

B=(1 3 31 4 31 3 4 )

4. (NC) A=(−4 2

k 1 ) state k when A-1 exists.

5. Solve using matrices

19

a)

x+2 y=10x−2 y=6 b)

3 x+2 y− z=12 x−2 y+4 z=−2−2 x+ y−2 z=0

_______________________

Answers:

1. a) -2 b) 26

2. A−1=− 1

2 (−2 3−2 4 )

3.

B−1=( 7 −3 −3−1 1 0−1 0 1 )

4. k≠25. a) x = 8, y = 1 b) x = 1, y = -2, z = -2

Topic 5—Vectors

1. (NC) A = (2,3) and B = (5,7). Find

a) A⃗B

b) magnitude of A⃗B

2. Calculate the distance between the points (2, 0, 3) and (1, 6, 4).

20

3. (NC) Find the unit vector in the direction of 2 i+ j

_______________________

Answers:

1. a) A⃗B=(34 )

b)

|⃗AB|=5

2. 6.16

3.

1√5

(2i+ j )

1. (NC) Consider the vectors a=5 i+ j and b=−3 i+2 j . Calculate the scalar producta⋅b .

2. (NC) Find k so that vectors a=2 i+3 j and b=6 i+k j are

a) parallelb) perpendicular

3. Calculate the angle between the vectors

a=(421 )

and

b=(−22

−2).

21

_______________________

Answers:

1. -132. a) 9 b) -43. 112 degrees

1. (NC) Write down the vector equation of the line parallel to the line

r=(123 )+ t( 0−25 )

and

passing through the point (3,7,8).

2. Find the angle between the two lines with equations(x

y)=(22 )+ t(46 )

and

(xy)=(−1

4 )+ t(−22 )

3. An object is moving with constant velocity(23 )

km/h along a straight line. At the time t = 0 it is at the point (-1,5). a) Write down the vector equation representing the position of the object.b) Find the speed of the object.

22

4. Find the point of intersection of the following lines

r=(512 )+t( 2−1−1)

and

s=(125)+k (−110 )

_______________________

Answers:

1.

r=(378 )+t ( 0−25 )

2. 78.7 degrees

3. a) r=(−1

5 )+ t(23 )

b) 3.61 km/h4. (-1,4,5)

Topic 6—Statistics and probability

1. What is the difference between discrete and continuous data?

2. (NC) Using the box and whisker plot given, finda) the medianb) the lower quartilec) the upper quartiled) the range.

23

______________________

Answers:

1. Discrete data can be counted and is usually recorded as whole numbers with a limited number of values. Continuous data can be measured.

2. a) 86 b) 72 c) 90 d) 27

1. (NC) Find the mean, median and mode of the following dataa) 2, 5, 2b) 2, 1, 8, 1

2. (NC) Find thei) rangeii) lower quartile iii) upper quartileiv) interquartile range (IQR) of the following

a) 1, 3, 3, 5, 5, 4, 4, 4, 2, 2, 2, 2b) 1, 2, 3, 3, 4, 4

3. Calculate the standard deviation and variancea) 167, 171, 173, 176, 180b)

24

score frequency1 32 13 84 5

______________________

Answers:

1. a) mean: 3, median: 2, mode: 2 b) mean: 3, median: 1.5, mode: 2 2. a) i) 4 ii) 2 iii) 4 iv) 2 b) i) 3 ii) 1 iii) 4 iv) 3 3. a) var: 19.4 sd: 4.41 b) var: 1.04 sd: 1.02

1. Use the cumulative frequency graph to estimate thea) medianb) upper quartilec) lover quartiled) 90th percentile

25

______________________

Answers:

1. a) 27.5 b) 31.5 c) 23.5 d) 36.5

1. (NC) List the sample space fora) rolling a dieb) tossing two coins

2. (NC) P(A) = 0.5 , P(B) = 0.3 andP( A∪B )=0 .6 . Find P( A∩B ).

3. (NC) From a bag containing 5 white balls, 2 black balls, and 11 red balls, 1 ball is drawn. What is the probability that it is either black or red?

______________________

Answers:26

1. a) {1,2,3,4,5,6} b) {HT, TH, TT, HH}2. 0.23. 13/18

1. Two machined are operating independently. Machine A operates for 70 % of the time and machine B operated for 80 % of the time. Draw a tree diagram and

a) Calculate the probability that both machines are workingb) Calculate the probability that only one machine is workingc) Given that only one machine is working, what is the probability that it is machine B?

2. 70 % of the people shop at market A and 60 % at market B. 50 % shop at both markets. Represent this on a Venn diagram and

a) Calculate probability that a person doesn’t shop at either market

b) Find P( A|B )and P( B|A ) .

3. A pair of dice is rolled.  What is the probability that the sum of the numbers rolled isa) either 7 or 11?b) either an even number or a multiple of 3?

4. A jar contains 3 red, 5 green, 2 blue and 6 orange marbles. A marble is chosen at random from the jar. After replacing it, a second marble is chosen. What is the probability of choosing a green and an orange marble?

5. A bag of sweets contains 10 red and 12 yellow sweets. Two sweets are taken (without replacement) what is the probability that

a) both are yellowb) only one is yellow

27

c) at least one is yellow?

______________________

Answers:

1. a) 0.56 b) 0.38 c) 0.4212. a) 0.2 b) 0.833 and 0.7143. a) 2/9 b) 2/34. 15/1285. a) 0.286 b) 0.519 c) 0.805

1. a) Show that P( X=x )= 1

18(4+ x ) , x=1,2,3

is a probability distribution function.

b) Calculate the expected value.

2. A coin is tossed once. It a tail appears you win 3 euros and if head appears you lose 2 euros. How much would you expect to win when playing the game four times?

28

______________________

Answers:

1. a) - b) 2.112. 2 euros

1. (NC) Which ones are binomial random variables?a) 2 % of bolts are defective. A random sample of 50 bolts is taken with replacement.

The variable is the number of faulty bolts.b) A die is tossed 60 times. The variable is the number of sixes.c) A bag contains 3 brown and 5 white sweets. I eat 2 sweets. The variable is the

number of white sweets eaten.d) There are 30 true-false questions in the test. Student guesses all the answers. The

variable is the number of correct answers.

2. If X ~ B(10 , 0. 2) . Calculate

a) P( X=3)

b) P( X≤3)

c) P( X≥5)

d) the mean.

3. A student is taking a multiple-choice exam in which each question has four choices. If there are five multiple-choice questions on the exam, what is the probability that student will get

a) five questions correct?b) at least four questions correct?

29

______________________

Answers:

1. a, b and d2. a) 0.201 b) 0.879 c) 0.0328 d) 23. a) 0.0098 b) 0.0156

1. If Z is a standard normal variable, find

a) P( Z≤1. 2)

b) P( Z≥0 .63 )

c) P(−0 .21≤Z≤0 .57 )

2. If Z is a standard normal variable, find k

a) P( Z≤k )=0 .79

b) P( Z≥k )=0 . 18

3. X is a random variable and distributed normally with mean 50 and standard deviation 4 find

a) P( X≤62)

b) P( X≥40 )

c) P(20≤X≤59 )

4. The height of the students are normally distributed with mean 171 cm and standard deviation 9 cm. Find the probability that a randomly selected student is

a) At least 173 cm tallb) Between 165 cm and 175 cm

5. 85% of the students will pass the exam. Exam results are expected to be normally distributed with the mean 53 and standard deviation 9. Find the lowest score needed to pass the exam.

30

______________________

Answers:

1. a) 0.885 b) 0.264 c) 0.2992. a) 0.806 b) 1.923. a) 0.999 b) 0.994 c) 0.9884. a) 0.412 b) 0.4195. 62

31

Topic 7—Calculus

1. (NC) Find the derivative of f ( x )=2 x2+3 using the first principles method (=definition of

the derivative).

2. (NC) Find the gradient of the tangent to the curve y=sin x atx=π .

3. (NC) Find the equation of the tangent to the curve y=x2at the point x = 3.

4. (NC) Find the gradient of the normal to the curve y=cos x atx=π

2 .

5. (NC) Find the equation of the normal to the curve y=exat the point x = 1.

6. (NC) Find the range of values of x where the function

a) f ( x )=x3−9 x is increasing

b)f ( x )= 1

3x3−2 x2

is decreasing.

______________________

Answers:

1. f ' ( x )=4 x

2. -13. y=6 x−94. -1

5.y=−1

ex+e+1

e32

6. a) x≤−√3 , x≥√3

b)

0≤x≤4

1. (NC) Differentiate

a) y=sin x+cos x b)

y=3 tan x−5ex+2 ln x

2. (NC) Find the derivative (using the chain rule)

a) (3 x2+6 x )10

b) sin 5 x

c) cos x2

d) cos 2 x

e) tan 4 x

f) e3 x

g) ln (3 x2+x )

3. (NC) Find the gradient function

a) f ( x )=5 x2 sin x

b) f ( x )= e x

4 x3+ x

4. (NC) Find d2 ydx2

if y=sin2 x

_____________________

Answers:

1. a)

dydx

=cos x−sin x b)

dydx

= 3cos2 x

−5 ex+ 2x

33

2. a) 10 (3 x2+6 x )9 (6 x+6 )

b) 5 cos5 x

c) −2 x sin x2

d) −2sin xcos x

e)

4cos2 4 x

f) 3 e3 x

g)

6 x+13 x2+ x

3. a) f ' ( x )=10 x sin x+5 x2cos x b) f ' ( x )=

ex (4 x3+x )−(12 x2+1)ex

( 4 x3+x )2

4.d2 ydx2 =−4 sin 2x

1. (NC) Find the local maximum and minimum value of

a)f ( x )=1

3x3−x2−2

b) f ( x )=x4+2 x2+1

2. (NC) A manufacturer produces toys at a cost of f ( x )=x2−6 x+30 cents each. Each toy is

sold for 45 cents.a) Write down the expression for the profit made in making and selling x toys.b) Find the number of toys that should be made to maximize the profit.

3. (NC) A rectangular field is formed using 1200 m of fencing. Find the dimensions of the field if the area is a maximum.

4. A container is made by cutting out the squares from the corners of a 12 cm by 12 cm square sheet of metal and folding the remaining sheet to form the container. What size squares must be cut out in order to maximize the volume of the container? What is the volume in this case?

______________________

34

Answers:

1. a) max f (0)=−2

, minf (2)=−3 1

3 b) min

f (0)=1, no max

2. a) −x3+6 x2+15 x b) 5 toys3. both sides are 300 m 4. 2 cm by 2 cm, 128 cm3

1. (NC) Integrate

a) y=x6

b) y=sin xc) y=cos x

d)y=1

x

e) y=ex

2. (NC) Integrate

a) y=(2 x+1 )6

b) y=sin3 x

c) y=cos( 4 x+1 )

d)y= 1

5 x+1

e) y=e−x+3

35

______________________

Answers:

1. a) 17

x7+C b) −cos x+C c) sin x+C d) ln x+C e)

e x+C

2. a)

114

(2 x+1)7+C b)

−13

cos3 x+C c)

14

sin( 4 x+1)+C d)

15

ln (5 x+1 )+C

e) −e− x+3+C

1. (NC) Find f(x) given that f ' ( x )=3 x2+x and f(0) = 10.

2. Evaluate

a)∫2

45x

dx

b)∫0

π6

cos 3xdx

3. (NC) Find the area between the curve y=−x2+4 and the x-axis .

4. (NC) Find the area between the curves y=−x2+4 and y=x+2

.

36

5. (NC) Find the volume of revolution generated by revolving the region between y=x2 and

x=2 about the x-axis.

______________________

Answers:

1.f ( x )=x3+ 1

2x2+10

2. a) 3.47 b) 1/3

3.10 2

3 units2

4.3 1

4 units2

5.32 π

5 units3

37

1. (NC) An object moves so that its position from the origin (in meters) is given by the

functions( t )=3t3−3 t2

, t in seconds. Find itsa) positionb) velocityc) acceleration when t =2 sec

2. (NC) Find the distance traveled by an object moving with a velocityv (t )=t2+2 t in the first 4 seconds.

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Answers:

1. a) 12 m b) 24 m/s c) 30 m/s2

2. 37.3 units

38

1. (NC) Find the asymptotes

a)f ( x )=5 x−2

x+3

b)f ( x )= 6 x

x2−4

2. (NC) For f ( x )=x4−4 x3+2 find the intervals where the function is

a) concave upb) concave down

3. Find the points of inflexion of

a) y=x3

b) y=x3−6 x2−12 x+2

c) y=x 4

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Answers:

1. a) x = -3, y = 5 b) x = 2, x = -2, y = 02. a) x≤0 , x≥2 b)

0≤x≤2

3. a) (0,0) b) (2,−38) c) no inflexion points

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