Question 1
The transformation `T_1` is represented by the matrix `[[0,-1], [-1,0]]`and the transformation `T_2` is represented by the matrix `[[0,1],[-1,0]]`
(a) Calculate the matrix `(T_1T_2)^-1`.
(b) Describe the transformation represented by the matrix `(T_1T_2)^-1`.
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Question 2
The matrix `M = [[0.2, 0.7],[0.,0.3]]`has eigenvalues -0.5 and 1
(a) Find an eigenvector corresponding to the eigenvalue of . Give your answer in the form `((a),(b))`where a, b`in`Z.
A switch has two states, A and B. Each second it either remains in the same state or moves according to the following rule: If it is in state A it will move to state B with a probability of 0.8 and if it is in state B it will move to state A with a probability of 0.7.
(b) Using your answer to (a), or otherwise, find the long-term probability of the switch being in state A. Give your answer in the form `c/d`, where `c,d in Z^+` .
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Question 3
A particle moves such that its displacement, metres, from a point at time seconds is given by the differential equation
`(d^2x)/(dt^2) + 5(dx)/(dt) + 6x = 0`
(a)
(i) Use the substitution `y = dx/dt` to show that this equation can be written as `((dx/dt),(dy/dt)) = ((0, 1),(-6, -5))((x),(y)).`
(ii) Find the eigenvalues for the matrix `[[0,1],[-6,-5]]`
(iii) Hence state the long-term velocity of the particle.
(b)
(i) Use the substitution `y = dx/dt` to write the differential equation as a system of coupled, first order differential equations.
When t = 0 the particle is stationary at O.
(ii) Use Euler's method with a step length of 0.1 to find the displacement of the particle when t = 1.
(iii) Find the long-term velocity of the particle.
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Question 4
The matrix `((1, -2, -3),(1, -k, -13),(-3, 5, k))`is singular. Find the values of k.
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Question 5
A flying drone is programmed to complete a series of movements in a horizontal plane relative to an origin O and a set of x-y-axes.
In each case, the drone moves to a new position represented by the following transformations:
All the movements are performed in the listed order.
(a)
(i) Write down each of the transformations in matrix form, clearly stating which matrix represents each transformation.
(ii) Find a single matrix P that defines a transformation that represents the overall change in position.
(iii) Find `P^2`
(iv) Hence state what the value of `P^2` indicates for the possible movement of the drone.
(b)
Three drones are initially positioned at the points A, B and C. After performing the movements listed above, the drones are positioned at points A', B' and C' respectively.
(c) Find a single transformation that is equivalent to the three transformations represented by matrix P.
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Question 6
Given that `A = ((3, -2),(-3, 4)) " and " I = ((1, 0),(0, 1))` , find the values of `lambda` for which `(A - lambdaI)` is a singular matrix.
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Question 7
A shock absorber on a car contains a spring surrounded by a fluid. When the car travels over uneven ground the spring is compressed and then returns to an equilibrium position.
The displacement, `x`, of the spring is measured, in centimetres, from the equilibrium position of `x=0`. The value of can be modelled by the following second order differential equation, where t is the time, measured in seconds, after the initial displacement.
`ddot(x) + 3dot(x) + 1.25x = 0`
(a) Given that `y = dot(x)`, show that `dot(y) = -1.25x - 3y`
The differential equation can be expressed in the form `((dot(x)),(dot(y))) = (A)((x),(y)),`,where is a 2 x 2 matrix.
(b) Write down the matrix A
(c)
(i) Find the eigenvalues of matrix A
(ii) Find the eigenvectors of matrix A.
(d) Given that when t = 0 the shock absorber is displaced 8cm and its velocity is zero, find an expression for x in terms of t.
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Question 8
Let `A = ((5, 1),(6, 2)) " and " B = ((2, -1),(-6, 5)).`
(a)
(i) Find AB
(ii) Write down the inverse of A.
Let `X = ((x),(y)) " and " C = ((8),(-4)).`
(b) Solve the matrix equation AX = C.
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Question 9
The drivers of a delivery company can park their vans overnight either at its headquarters or at home.
Urvashi is a driver for the company. If Urvashi has parked her van overnight at headquarters on a given day, the probability that she parks her van at headquarters on the following day is 0.88. If Urvashi has parked her van overnight at home on a given day, the probability that she parks her van at home on the following day is 0.92.
(a) Write down a transition matrix T, that shows the movement of Urvashi's van between headquarters and home.
On Monday morning she collected her van from headquarters where it was parked overnight.
(b) Find the probability that Urvashi's van will be parked at home at the end of the week on Friday evening.
(c) Write down the characteristic polynomial for the matrix T. Give your answer in the form `lambda^2 + blambda + c`
(d) Calculate eigenvectors for the matrix .
(e) Write down matrices P and D such that `T = PDP^-1`, where D is a diagonal matrix.
(f) Hence find the long-term probability that Urvashi's van is parked at home.
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Question 10
The equation of the line `y = mx + c`, can be expressed in vector form `r = a + lambda b`
(a) Find the vectors a and b in terms of m and or c.
The matrix M is defined by `[[6,3],[4,2]]`
(b) Find the value of `detM`.
The line `y = mx +c`(where `m !=2`) is transformed into a new line using the transformation described by matrix M.
(c) Show that the equation of the resulting line does not depend on m or c.
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Question 1
The transformation `T_1` is represented by the matrix `[[0,-1], [-1,0]]`and the transformation `T_2` is represented by the matrix `[[0,1],[-1,0]]`
(a) Calculate the matrix `(T_1T_2)^-1`.
(b) Describe the transformation represented by the matrix `(T_1T_2)^-1`.
(a)
`(T_1 T_2)^(-1) = ([[0, -1], [-1, 0]] [[0, 1], [-1, 0]]) ^(-1) = [[1, 0], [0, -1]] ^(-1) = [[1, 0], [0, -1]]`
(b)
This matrix represents a reflection in the `x-`axis.
Question 2
The matrix `M = [[0.2, 0.7],[0.,0.3]]`has eigenvalues -0.5 and 1
(a) Find an eigenvector corresponding to the eigenvalue of . Give your answer in the form `((a),(b))`where a, b`in`Z.
A switch has two states, A and B. Each second it either remains in the same state or moves according to the following rule: If it is in state A it will move to state B with a probability of 0.8 and if it is in state B it will move to state A with a probability of 0.7.
(b) Using your answer to (a), or otherwise, find the long-term probability of the switch being in state A. Give your answer in the form `c/d`, where `c,d in Z^+` .
(a)
`lambda = 1`
`((-0.8, 0.7),(0.8, -0.7))((x),(y)) = ((0),(0))`
OR
`((0.2, 0.7),(0.8, 0.3))((x),(y)) = ((x),(y))`
`0.8x=0.7y`
an eigenvector `((7),(8))`(or equivalent with integer values)
(b)
EITHER (the long-term probability matrix is given by the eigenvector corresponding to the eigenvalue equal to 1, scaled so that the sum of the entries is 1)
8 + 7 = 15
THEN probability of being in state A is `7/15`
Question 3
A particle moves such that its displacement, metres, from a point at time seconds is given by the differential equation
`(d^2x)/(dt^2) + 5(dx)/(dt) + 6x = 0`
(a)
(i) Use the substitution `y = dx/dt` to show that this equation can be written as `((dx/dt),(dy/dt)) = ((0, 1),(-6, -5))((x),(y)).`
(ii) Find the eigenvalues for the matrix `[[0,1],[-6,-5]]`
(iii) Hence state the long-term velocity of the particle.
(b)
(i) Use the substitution `y = dx/dt` to write the differential equation as a system of coupled, first order differential equations.
When t = 0 the particle is stationary at O.
(ii) Use Euler's method with a step length of 0.1 to find the displacement of the particle when t = 1.
(iii) Find the long-term velocity of the particle.
(a)
(i)
`y = (dx)/(dt) (dy)/(dt) + 5(dx)/(dt) + 6x = 0` OR `(dy)/(dt) + 5y + 6x = 0`
`((dx/dt),(dy/dt)) = ((0, 1),(-6, -5))((x),(y))`
(ii)
`det((-lambda, 1),(-6, -5-lambda)) = 0`
`-lambda(-5 - lambda) + 6 = 0`
(iii)
on a phase portrait the particle approaches (0,0) as t increases so long term velocity (y) is 0.
(b)
(i)
`y = (dx)/(dt) \ \ (d^2x)/(dt^2) = (dy)/(dt) \ \ (dy)/(dt) + 5y + 6x = 3t + 4`
(ii)
recognition that h = 0.1 in any recurrence formula (when t = 1)
`t_(n+1) = t_n + 0.1 \ \ x_(n+1) = x_n + 0.1y_n \ \ y_(n+1) = y_n + 0.1(3t_n + 4 - 5y_n - 6x_n)`
`x approx 0.64402... approx 0.644 " m"`
(iii)
recognizing that y is the velocity `0.5 ms^-1`
Question 4
The matrix `((1, -2, -3),(1, -k, -13),(-3, 5, k))`is singular. Find the values of k.
The matrix is singular if its determinant is zero. Then,
`((1, -2, -3),(1, -k, -13),(-3, 5, k))`
`= |((-k, -13),(5, k))| + 2|((1, -13),(-3, k))| - 3|((1, -k),(-3, 5))|`
`= k^2 + 65 + 2k - 78 - 15 + 9k`
`= -(k^2 - 11k + 28)`
`= -(k-4)(k-7)`
Therefore, the matrix is singular if k = 4 or k = 7.
Question 5
A flying drone is programmed to complete a series of movements in a horizontal plane relative to an origin O and a set of x-y-axes.
In each case, the drone moves to a new position represented by the following transformations:
All the movements are performed in the listed order.
(a)
(i) Write down each of the transformations in matrix form, clearly stating which matrix represents each transformation.
(ii) Find a single matrix P that defines a transformation that represents the overall change in position.
(iii) Find `P^2`
(iv) Hence state what the value of `P^2` indicates for the possible movement of the drone.
(b)
Three drones are initially positioned at the points A, B and C. After performing the movements listed above, the drones are positioned at points A', B' and C' respectively.
(c) Find a single transformation that is equivalent to the three transformations represented by matrix P.
(a)
(i) rotation anticlockwise `pi/6`is
`((0.866, -0.5),(0.5, 0.866))`
(ii) an attempt to multiply three matrices
`P = ((1/2, sqrt(3)/2),(-sqrt(3)/2, 1/2)) ((1/2, sqrt(3)/2),(sqrt(3)/2, -1/2)) ((sqrt(3)/2, -1/2),(1/2, sqrt(3)/2))`
`P = ((sqrt(3)/2, -1/2),(-1/2, -sqrt(3)/2))`OR `((0.866, -0.5),(-0.5, -0.866))`
(iii) `P^2 = ((sqrt(3)/2, -1/2),(-1/2, -sqrt(3)/2))((sqrt(3)/2, -1/2),(-1/2, -sqrt(3)/2)) = ((1, 0),(0, 1))`
(iv) if the overall movement of the drone is repeated the drone would return to its original position
(b) METHOD statement of fact that rotation leaves area unchanged statement of fact that reflection leaves area unchanged area triangle ABC = area of triangle A'B'C'
(c) attempt to find angles associated with values of elements in matrix
`P((sqrt(3)/2, -1/2),(-1/2, -sqrt(3)/2)) = ((cos(-pi/6), sin(-pi/6)),(sin(-pi/6), -cos(-pi/6)))`
reflection (in `y = (tantheta)x`)
where `2theta = - pi/6`
reflection in `y = tan (-pi/12)x (= - 0,268x)`
Question 6
Given that `A = ((3, -2),(-3, 4)) " and " I = ((1, 0),(0, 1))` , find the values of `lambda` for which `(A - lambdaI)` is a singular matrix.
singular matrix `=> det = 0`
`A - lambda I = |(3 - lambda, -2),(-3, 4 - lambda)|`
`(3 - lambda) ( 4-lambda)-6=`
`=> lambda^2 - 7lambda + 6 = 0`
`lambda = 1 or lambda =6`
Question 7
A shock absorber on a car contains a spring surrounded by a fluid. When the car travels over uneven ground the spring is compressed and then returns to an equilibrium position.
The displacement, `x`, of the spring is measured, in centimetres, from the equilibrium position of `x=0`. The value of can be modelled by the following second order differential equation, where t is the time, measured in seconds, after the initial displacement.
`ddot(x) + 3dot(x) + 1.25x = 0`
(a) Given that `y = dot(x)`, show that `dot(y) = -1.25x - 3y`
The differential equation can be expressed in the form `((dot(x)),(dot(y))) = (A)((x),(y)),`,where is a 2 x 2 matrix.
(b) Write down the matrix A
(c)
(i) Find the eigenvalues of matrix A
(ii) Find the eigenvectors of matrix A.
(d) Given that when t = 0 the shock absorber is displaced 8cm and its velocity is zero, find an expression for x in terms of t.
(a)
`dot(y) = ddot(x) \ \ dot(y) + 3(dot(x)) + 1.25x = 0`
`dot(y) = -3y - 1.25x`
(b) `A = [[0,1],[-1.25, -3]]`
(c)
(i)
`|(-lambda, 1),(-1.25, -3 - lambda)| = 0`
`lambda(lambda + 3) + 1.25 = 0`
`lambda = -2.5 and lambda = -0.`
(ii)
For `lambda = -2.5`:
`((2.5, 1),(-1.25, -0.5))((a),(b)) = ((0),(0)) \ \ 2.5a + b = 0 \ \ bf(v)_1 = ((-2),(5))`
For `lambda = -0.5`:
`((0.5, 1),(-1.25, -2.5))((a),(b)) = ((0),(0)) \ \ 0.5a + b = 0 \ \ bf(v)_2 = ((-2),(1))`
(d)
`((x),(y)) = A e^(-2.5t) ((-2),(5)) + B e^(-0.5t) ((-2),(1))`
`t = 0 => x = 8, dot(x) = y = 0`
`-2A - 2B = 8`
`5A + B = 0`
`A = 1, B = -5`
`x = -2e^(-2.5t) + 10e^(-0.5t)`
Question 8
Let `A = ((5, 1),(6, 2)) " and " B = ((2, -1),(-6, 5)).`
(a)
(i) Find AB
(ii) Write down the inverse of A.
Let `X = ((x),(y)) " and " C = ((8),(-4)).`
(b) Solve the matrix equation AX = C.
(a)
(i)
`AB = ((4, 0),(0, 4)) (= 4I)`
(ii)
`A^-1 = 1/4 ((2, -1),(-6, 5)), 1/4 B, ((1/2, -1/4),(-3/2, 5/4))`
(b)
`= A^-1C = 1/4 ((2, -1),(-6, 5))((8),(-4)) ( ((1/2, -1/4),(-3/2, 5/4))((8),(-4)) )`
`((x),(y)) = ((5),(-17))`
Question 9
The drivers of a delivery company can park their vans overnight either at its headquarters or at home.
Urvashi is a driver for the company. If Urvashi has parked her van overnight at headquarters on a given day, the probability that she parks her van at headquarters on the following day is 0.88. If Urvashi has parked her van overnight at home on a given day, the probability that she parks her van at home on the following day is 0.92.
(a) Write down a transition matrix T, that shows the movement of Urvashi's van between headquarters and home.
On Monday morning she collected her van from headquarters where it was parked overnight.
(b) Find the probability that Urvashi's van will be parked at home at the end of the week on Friday evening.
(c) Write down the characteristic polynomial for the matrix T. Give your answer in the form `lambda^2 + blambda + c`
(d) Calculate eigenvectors for the matrix .
(e) Write down matrices P and D such that `T = PDP^-1`, where D is a diagonal matrix.
(f) Hence find the long-term probability that Urvashi's van is parked at home.
(a)
`((0.88, 0.08),(0.12, 0.92))`
(b)
5 (seen)
`((0.88, 0.08),(0.12, 0.92))^5 ((1),(0)) = ((0.596608),(0.403392)) OP("Friday evening") = 0.403 (0.403392)`
(c)
attempt to find `det(A - lambdaI)`
`|(0.88 - lambda, 0.08),(0.12, 0.92 - lambda)|; lambda^2 - 1.8lambda + 0.8`
(d)
eigenvalues are 0.8 and 1
`((0.88, 0.08),(0.12, 0.92))((x),(y)) = 1((x),(y))`
`0.88x + 0.08y = x \ =>\ 0.08y = 0.12x`
THEN eigenvector eg. `((2),(3))`
(e)
`D = ((1, 0),(0, 0.8)), P = ((2, 1),(3, -1))`
(f)
EITHER attempt to use `T^n = (PDP^-1)^n = PD^nP^-1`
limit of Dn calculated
`((2, 1),(3, -1))((1, 0),(0, 0))((3, -2),(-1, 1))`
THEN 0.6
Question 10
The equation of the line `y = mx + c`, can be expressed in vector form `r = a + lambda b`
(a) Find the vectors a and b in terms of m and or c.
The matrix M is defined by `[[6,3],[4,2]]`
(b) Find the value of `detM`.
The line `y = mx +c`(where `m !=2`) is transformed into a new line using the transformation described by matrix M.
(c) Show that the equation of the resulting line does not depend on m or c.
(a)
one vector to the line is `((0),(c))`herefore
a = `((0),(c))`
the line goes m up for every 1 across
(so the direction vector is)
b = `((1),(m))`
(b) (from GDC OR 6 x 2 - 4 x 3)
|M| = 0
(c)
`((X),(Y)) = ((6, 3),(4, 2))((x),(mx + c)) = ((6x + 3(mx + c)),(4x + 2(mx + c))) = ((3(2x + mx + c)),(2(2x + mx + c)))`
therefore the new line has equation `3Y = 2X`
which is independent of m or c.
Question 1
The transformation `T_1` is represented by the matrix `[[0,-1], [-1,0]]`and the transformation `T_2` is represented by the matrix `[[0,1],[-1,0]]`
(a) Calculate the matrix `(T_1T_2)^-1`.
(b) Describe the transformation represented by the matrix `(T_1T_2)^-1`.
Question 2
The matrix `M = [[0.2, 0.7],[0.,0.3]]`has eigenvalues -0.5 and 1
(a) Find an eigenvector corresponding to the eigenvalue of . Give your answer in the form `((a),(b))`where a, b`in`Z.
A switch has two states, A and B. Each second it either remains in the same state or moves according to the following rule: If it is in state A it will move to state B with a probability of 0.8 and if it is in state B it will move to state A with a probability of 0.7.
(b) Using your answer to (a), or otherwise, find the long-term probability of the switch being in state A. Give your answer in the form `c/d`, where `c,d in Z^+` .
Question 3
A particle moves such that its displacement, metres, from a point at time seconds is given by the differential equation
`(d^2x)/(dt^2) + 5(dx)/(dt) + 6x = 0`
(a)
(i) Use the substitution `y = dx/dt` to show that this equation can be written as `((dx/dt),(dy/dt)) = ((0, 1),(-6, -5))((x),(y)).`
(ii) Find the eigenvalues for the matrix `[[0,1],[-6,-5]]`
(iii) Hence state the long-term velocity of the particle.
(b)
(i) Use the substitution `y = dx/dt` to write the differential equation as a system of coupled, first order differential equations.
When t = 0 the particle is stationary at O.
(ii) Use Euler's method with a step length of 0.1 to find the displacement of the particle when t = 1.
(iii) Find the long-term velocity of the particle.
Question 4
The matrix `((1, -2, -3),(1, -k, -13),(-3, 5, k))`is singular. Find the values of k.
Question 5
A flying drone is programmed to complete a series of movements in a horizontal plane relative to an origin O and a set of x-y-axes.
In each case, the drone moves to a new position represented by the following transformations:
All the movements are performed in the listed order.
(a)
(i) Write down each of the transformations in matrix form, clearly stating which matrix represents each transformation.
(ii) Find a single matrix P that defines a transformation that represents the overall change in position.
(iii) Find `P^2`
(iv) Hence state what the value of `P^2` indicates for the possible movement of the drone.
(b)
Three drones are initially positioned at the points A, B and C. After performing the movements listed above, the drones are positioned at points A', B' and C' respectively.
(c) Find a single transformation that is equivalent to the three transformations represented by matrix P.
Question 6
Given that `A = ((3, -2),(-3, 4)) " and " I = ((1, 0),(0, 1))` , find the values of `lambda` for which `(A - lambdaI)` is a singular matrix.
Question 7
A shock absorber on a car contains a spring surrounded by a fluid. When the car travels over uneven ground the spring is compressed and then returns to an equilibrium position.
The displacement, `x`, of the spring is measured, in centimetres, from the equilibrium position of `x=0`. The value of can be modelled by the following second order differential equation, where t is the time, measured in seconds, after the initial displacement.
`ddot(x) + 3dot(x) + 1.25x = 0`
(a) Given that `y = dot(x)`, show that `dot(y) = -1.25x - 3y`
The differential equation can be expressed in the form `((dot(x)),(dot(y))) = (A)((x),(y)),`,where is a 2 x 2 matrix.
(b) Write down the matrix A
(c)
(i) Find the eigenvalues of matrix A
(ii) Find the eigenvectors of matrix A.
(d) Given that when t = 0 the shock absorber is displaced 8cm and its velocity is zero, find an expression for x in terms of t.
Question 8
Let `A = ((5, 1),(6, 2)) " and " B = ((2, -1),(-6, 5)).`
(a)
(i) Find AB
(ii) Write down the inverse of A.
Let `X = ((x),(y)) " and " C = ((8),(-4)).`
(b) Solve the matrix equation AX = C.
Question 9
The drivers of a delivery company can park their vans overnight either at its headquarters or at home.
Urvashi is a driver for the company. If Urvashi has parked her van overnight at headquarters on a given day, the probability that she parks her van at headquarters on the following day is 0.88. If Urvashi has parked her van overnight at home on a given day, the probability that she parks her van at home on the following day is 0.92.
(a) Write down a transition matrix T, that shows the movement of Urvashi's van between headquarters and home.
On Monday morning she collected her van from headquarters where it was parked overnight.
(b) Find the probability that Urvashi's van will be parked at home at the end of the week on Friday evening.
(c) Write down the characteristic polynomial for the matrix T. Give your answer in the form `lambda^2 + blambda + c`
(d) Calculate eigenvectors for the matrix .
(e) Write down matrices P and D such that `T = PDP^-1`, where D is a diagonal matrix.
(f) Hence find the long-term probability that Urvashi's van is parked at home.
Question 10
The equation of the line `y = mx + c`, can be expressed in vector form `r = a + lambda b`
(a) Find the vectors a and b in terms of m and or c.
The matrix M is defined by `[[6,3],[4,2]]`
(b) Find the value of `detM`.
The line `y = mx +c`(where `m !=2`) is transformed into a new line using the transformation described by matrix M.
(c) Show that the equation of the resulting line does not depend on m or c.
