PURE

Question 1

i^2 = ?

Answer 1

-1

Question 2

i^3 = ?

Answer 2

-i

Question 3

i^4 = ?

Answer 3

1

Question 4

cos(-a) = ?

Answer 4

cos(a)

Question 5

sin(-a) = ?

Answer 5

-sin(a)

Question 6

tan(-a) = ?

Answer 6

-tan(a)

Question 7

What is modulus argument form?

Answer 7

z = r(cos(a)+isin(a))

Question 8

How do you invert a 3x3 matrix?

Answer 8

-Find the determinant
-Replace every element in the matrix by its minor
-Switch the sign of alternate elements
-Transpose the matrix
-Divide by the determinant

Question 9

If the coeffiecient matrix for a set of linear equations is singular then…

Answer 9

Either
-the equations have no solutions
Or
-the equations have an infinte number of solutions

Question 10

If the coeffiecient matrix for a set of linear equations is non-singular then…

Answer 10

The equations have a unique solution

Question 11

If there is a unique solution…

Answer 11

The three planes intersect at a single point

Question 12

If there are infinitely many solutions…

Answer 12

Either
-The three planes intersect on a common line (sheaf)
Or
-The planes are the same

Question 13

If there are no solutions…

Answer 13

Either
-The planes are parallel
Or
-The planes intersect to form a triangular prism

Question 14

What are the 4 types of linear transformation?

Answer 14

REFLECTION in the line…
ROTATION about the origin by … anti/clockwise
STRETCH parallel to the x/y axis by scale factor…
ENLARGEMENT with centre (0.0), by scale factor…

Question 15

What does the inverse of a linear transformation do?

Answer 15

Reverses original transformation

Question 15

What does the modulus of the determinant of the transformation matrix represent?

Answer 15

The area scale factor of the transformation

Question 15

What is an invariant point?

Answer 15

A point that has been transformed onto itself

Question 15

If the determinant of the transformation is negative, what does this tell you?

Answer 15

A reflection has taken place

Question 16

What is a line of invariant points?

Answer 16

A line on which every point on that line is invariant

Question 17

The point (x,y) is invariant if…

Answer 17

M(x y) = (x y)
Use this to find invariant lines

Question 18

What is an invariant line?

Answer 18

A line which is mapped onto itself

Question 19

If we are asked to find all invariant lines then we need to consider all possible lines. All possible lines are…

Answer 19

y = mx + c
x = k

Question 20

A general point on the line y = mx + c is…

Answer 20

(x,mx+c)

Question 21

A general point on the line x = k is…

Answer 21

(k,y)

Question 22

If matrix M represents a 2d transformation then...

Answer 22

-The first column is the image of the point (1,0) -The second column is the image of the point (0,1)

Question 23

If Matrix M represents of 3d matrix then...

Answer 23

-The first column is the image of the point (1,0,0) -The second column is the image of the point (0,1,0) -The third column is the image of the point (0,0,1)

Question 24

What is the convention of rotation?

Answer 24

The rotation will be clockwise or anticlockwise depending on how it appears if the axis of rotation is pointing straight at you

Question 25

What are the three possible reflections?

Answer 25

-In x = 0 (yz-plane) -In y = 0 (xz-plane) -In z = 0 (xy-plane)

Question 26

What is the matrix for a rotation about the x-axis?

Answer 26

1 0 0 0 cosA -sinA 0 sinA cosA

Question 27

What is the matrix for a rotation about the y-axis?

Answer 27

cosA o sinA 0 1 0 -sinA 0 cosA

Question 28

What is the matrix for a rotation about the z-axis?

Answer 28

cosA -sinA 0 sinA cosA 0 0 0 1

Question 29

What is the summation formula for 1?

Answer 29

=n

Question 30

What is the summation formula for n?

Answer 30

0.5n(n+1)

Question 31

What are the steps for proof by induction?

Answer 31

-Check the result is true for the base case -Assume result is true when n = k -Use this assumption to show that the result is true for n = k + 1 -Conclusion

Question 32

What is the conclusion for proof by induction?

Answer 32

We have shown that if the result is true for n=1 then it is true for n = k + 1 As the result is true for n = 1, it is therefore true for all positive integers

Question 33

What are the steps for stronger inudction?

Answer 33

-Check two base cases -Assume it is true for n = k and n = k + 1 -Show the result is true for n = k + 2 -Tweaked conclusion

Question 34

A + B =

Answer 34

-b/a

Question 35

AB =

Answer 35

c/a

Question 36

A^2 + B^2 =

Answer 36

(A + B)^2 - 2AB

Question 37

A^3 + B^3 =

Answer 37

(A + B)^3 - 3(AB)(A+B)

Question 38

A + B + C =

Answer 38

-b/a

Question 39

AB + AC + BC =

Answer 39

c/a

Question 40

ABC =

Answer 40

-d/a

Question 41

A^2 + B^2 + C^2 =

Answer 41

(A + B + C)^2 - 2(AB + AC + BC)

Question 42

A^3 + B^3 + C^3 =

Answer 42

(A + B + C)^3 -3(AB + AC + BC)(A + B + C) + 3ABC

Question 43

A + B + C + D =

Answer 43

-b/a

Question 44

AB + AC + AD + BC + BD + CD =

Answer 44

c/a

Question 45

ABC + ABD + ACD + BCD =

Answer 45

-d/a

Question 46

ABCD =

Answer 46

e/a

Question 47

A^2 + B^2 + C^2 + D^2 =

Answer 47

(A + B + C + D)^2 -2(AB + AC + AD + BC + BD + CD)