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Q1[11 marks]hardCh1 · Matrices and transformations· Working with matrices
A company has two factories, Factory 1 and Factory 2, that produce and sell items of type A, B, and C. The number of items produced is represented by matrix M and the number of items sold is represented by matrix N.
M=(1−10324) (row 1 for Factory 1, row 2 for Factory 2; columns for A, B, C)
N=(021−2−13) (row 1 for Factory 1, row 2 for Factory 2; columns for A, B, C)
(a) Calculate the resultant matrix R=3M−2N. [6]
(b) Interpret the meaning of the elements of R in the context of the problem. [5]
Q2[10 marks]hardCh1 · Matrices and transformations· Invariant points
A transformation is represented by the matrix T=(31−20).
(a) Find the equations of any lines of invariant points for this transformation. [6]
(b) Explain the difference between an invariant point and a point on an invariant line, using specific examples from this transformation if possible. [4]
Q3[8 marks]mediumCh1 · Matrices and transformations· Invariance
Fig 1.1 shows a unit square (solid lines) with vertices (0,0), (1,0), (1,1), (0,1) transformed into a parallelogram (dashed lines) with vertices (0,0), (1,0), (1,2), (0,2).
(a) Identify any invariant lines for this transformation from Fig 1.1. [3]
(b) Sketch the image of the line y=x+1 under the transformation shown in Fig 1.1, and hence determine if y=x+1 is an invariant line. Clearly label your sketch. [5]
Q4[6 marks]easyCh1 · Matrices and transformations· Invariant lines
A linear transformation is represented by the matrix M=(34−1−2).
(a) Verify that the line y = 2x is an invariant line for this transformation. [4]
(b) Explain why an invariant line does not necessarily consist of invariant points. [2]
Q5[9 marks]mediumCh1 · Matrices and transformations· Finding the matrix that represents a given transformation
A shear transformation is applied to points in the Cartesian plane. The x-axis remains fixed, and the point (0, 1) is mapped to (4, 1).
(a) Find the matrix that represents this shear. [5]
(b) Show the effect of this shear on a rectangle with vertices (0,0), (2,0), (2,3), (0,3) by drawing the object and its image on a coordinate grid. [4]
Q6[8 marks]mediumCh1 · Matrices and transformations· Working with matrices
Consider matrix equations involving unknown matrices and elements.
(a) Find the matrix X such that X+(1−230)=(5417). [4]
(b) Solve for x and y in the matrix equation (2yx3)+(12−54)=(3007). [4]
Q7[7 marks]mediumCh1 · Matrices and transformations· Special matrices
The zero matrix plays a crucial role in matrix algebra, similar to the number zero in scalar arithmetic, having specific properties related to addition.
(a) Describe the properties of a zero matrix. [3]
(b) Give an example of two matrices C and D such that C+D=O, where O is the zero matrix. [4]
Q8[9 marks]mediumCh1 · Matrices and transformations· Successive transformations
A triangle with vertices at the origin O(0,0), A(1,0), and B(0,1) is defined. This triangle undergoes a transformation.
(a) Plot the image of the triangle with vertices (0,0), (1,0), (0,1) under the transformation represented by M = (1201). [3]
(b) Determine the single matrix that represents a reflection in the x-axis followed by the transformation M. [3]
(c) Describe the combined transformation from part (b). [3]
Q9[11 marks]hardCh1 · Matrices and transformations· Summary of transformations in two dimensions
A geometric transformation maps points in the Cartesian plane.
(a) Determine the matrix that represents a reflection in the line y = -x. [4]
(b) Find the image of the line y = 2x + 1 under this transformation. [4]
(c) Show that the line y = -x is an invariant line for this transformation. [3]
Q10[5 marks]easyCh1 · Matrices and transformations· Successive transformations
Consider a point in a two-dimensional plane that undergoes two successive transformations.
(a) Find the single matrix that represents a rotation of 90 degrees anticlockwise about the origin followed by a reflection in the x-axis. [5]
Q11[10 marks]hardCh1 · Matrices and transformations· Special matrices
The identity matrix, often denoted by I, is a fundamental concept in linear algebra, acting as a multiplicative identity. Its structure is very specific.
(a) Explain why an identity matrix must always be a square matrix. [5]
(b) Determine the values of a and b if (a+100b−2) is an identity matrix of order 2. [5]
Q12[7 marks]mediumCh1 · Matrices and transformations· Multiplication of matrices
A point (2, 3) is transformed by the matrix T=(32−10).
(a) Calculate the image of the point (2, 3) under this transformation. [4]
(b) Determine the coordinates of the image point and plot both the object and image point on a coordinate grid. [3]
Q13[5 marks]easyCh1 · Matrices and transformations· Matrices
Matrices are fundamental mathematical structures used to represent data and transformations.
(a) Define the term 'order' when referring to a matrix. [2]
(b) State the order of the matrix A=(2−35017) and the element in the second row and first column. [3]
Q14[4 marks]easyCh1 · Matrices and transformations· Invariance
Transformations can map points and lines to new positions. However, some lines may remain in their original position.
(a) Define what is meant by an invariant line under a transformation. [2]
(b) Give an example of a simple transformation that has at least one invariant line. [2]
Q15[8 marks]mediumCh1 · Matrices and transformations· Properties of matrix multiplication
Matrix multiplication, while not commutative, does obey other algebraic properties. Consider the following matrices:
A=(1023)B=(4−112)C=(025−3)
(a) Show that for these matrices, the distributive property A(B+C) = AB + AC holds. [5]
(b) Deduce the order of the resulting matrix from the expression (B+C). [3]
Q16[10 marks]hardCh1 · Matrices and transformations· Proving results in trigonometry
Rotation matrices can be used to derive trigonometric identities. Consider two successive rotations about the origin.
(a) Use matrix multiplication of two rotation matrices, one for an angle A and another for an angle A, to derive the double angle formula for cos(2A). [6]
(b) Deduce the corresponding double angle formula for sin(2A) using a similar method. [4]
Q17[5 marks]easyCh1 · Matrices and transformations· Associativity and commutativity
Matrix operations have specific properties that govern how they can be manipulated. Understanding these properties is crucial for accurate matrix algebra.
(a) State the definition of associativity for matrix multiplication. [2]
(b) Explain why matrix multiplication is generally not commutative. [3]
Q18[8 marks]mediumCh1 · Matrices and transformations· Transformations
Rotations are a common type of transformation that can be represented by a matrix.
(a) Describe the transformation represented by the matrix (cos(90∘)sin(90∘)−sin(90∘)cos(90∘)). [4]
(b) Show the effect of this transformation on the unit square by drawing the object and its image on a coordinate grid. [4]
Q19[12 marks]hardCh1 · Matrices and transformations· Finding the matrix that represents a given transformation
A geometric transformation reflects points in a line passing through the origin. Consider the line y=(tanθ)x, which makes an angle θ with the positive x-axis.
(a) Derive the matrix that represents a reflection in the line y=(tanθ)x. [5]
(b) Show that for θ=45∘, your derived matrix matches the standard reflection matrix in y=x. [4]
(c) Calculate the image of the point (1, 0) under a reflection in the line y=3x. [3]
Q20[9 marks]mediumCh1 · Matrices and transformations· Multiplication of matrices
Two matrices, M1 and M2, are given as M1=(2−314) and M2=(05−12).
(a) Find the matrix product M1M2. [6]
(b) Verify if M1M2=M2M1 for the given matrices. [3]
Q21[5 marks]easyCh1 · Matrices and transformations· Proving results in trigonometry
Matrix transformations can be used to represent geometric operations like rotations. Consider a rotation about the origin.
(a) State the matrix for a rotation of 90 degrees anticlockwise about the origin. [2]
(b) Show that applying this rotation twice is equivalent to a rotation of 180 degrees. [3]
Q22[7 marks]mediumCh1 · Matrices and transformations· Summary of transformations in two dimensions
A unit square has vertices at (0,0), (1,0), (1,1) and (0,1).
(a) Draw the image of the unit square under the transformation represented by \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. [4]
(b) Describe this transformation fully. [3]
Q23[8 marks]mediumCh1 · Matrices and transformations· Finding the transformation represented by a given matrix
A unit square with vertices (0,0), (1,0), (1,1), (0,1) is transformed by a matrix M to produce an image. Fig 1.1 shows the object (unit square) and its image (a rectangle).
(a) Identify the type of transformation represented by the matrix M shown in Fig 1.1. [3]
(b) Determine the matrix M. [3]
(c) Calculate the image of the point (3, 2) under this transformation. [2]
Q24[12 marks]hardCh1 · Matrices and transformations· Multiplication of matrices
Geometric transformations can be represented by matrix multiplication, mapping an object to its image. Fig 1.1 shows a triangle PQR and its image P'Q'R' after a transformation.
(a) Analyse the transformation applied to the triangle PQR to obtain P'Q'R' as shown in Fig 1.1, and determine the transformation matrix M. [6]
(b) Compare the area of triangle PQR with the area of triangle P'Q'R' and deduce the scale factor of the transformation from the determinant of M. [6]
Q25[11 marks]hardCh1 · Matrices and transformations· Finding the transformation represented by a given matrix
A triangle with vertices A(1,1), B(3,1), C(1,3) is transformed by a matrix M to produce an image triangle with vertices A'( -1, 1), B'( -1, 3), C'( -3, 1). Fig 1.2 shows the object triangle and its image triangle.
(a) Describe the transformation represented by the matrix M shown in Fig 1.2. [4]
(b) Find the matrix M. [4]
(c) Determine the area of the image of a square of side length 2 units under this transformation. [3]
Q26[5 marks]easyCh1 · Matrices and transformations· Shears
Shears are a type of linear transformation that distort a shape in a specific way while preserving its area.
(a) State the matrix for a shear with the x-axis fixed and shear factor k. [2]
(b) Describe the effect of a shear on the area of a shape. [3]
Q27[7 marks]mediumCh1 · Matrices and transformations· Finding the matrix that represents a given transformation
Transformations can be applied to points and shapes using matrix multiplication. Consider an enlargement transformation.
(a) Find the matrix that represents an enlargement, centre the origin, with scale factor 3. [3]
(b) Calculate the image of the point (2, -1) under this enlargement. [4]
Q28[10 marks]hardCh1 · Matrices and transformations· Transformations
Consider the transformation represented by the matrix M=(1k01), where k is a non-zero constant.
(a) Analyse the effect of applying this transformation to a general point (x,y). [4]
(b) Explain why this transformation is classified as a shear. [3]
(c) Derive the equation of the fixed line for this shear. [3]
Q29[9 marks]mediumCh1 · Matrices and transformations· Transformations
Transformations can be represented by matrices, and their effects can be observed by looking at how they map points or shapes.
(a) Calculate the image of the point (2, 3) under a rotation of 60∘ anticlockwise about the origin. [3]
(b) Describe the transformation if the image of (1, 0) is (2, 0) and the image of (0, 1) is (0, 1). [3]
(c) Draw the object and image for the transformation described in part (b) using the unit square as the object. [3]
Q30[8 marks]mediumCh1 · Matrices and transformations· Shears
A shear transformation, S, has the y-axis as its fixed line. This means that points on the y-axis remain unchanged, and other points move parallel to the y-axis.
(a) Find the matrix for a shear with the y-axis fixed such that the point (2, 1) maps to (2, 7). [4]
(b) Calculate the image of the point (5, -3) under this shear. [4]
Q31[8 marks]mediumCh1 · Matrices and transformations· Successive transformations
A point in the Cartesian plane undergoes a sequence of transformations.
(a) Calculate the single matrix representing a stretch of factor 3 parallel to the x-axis followed by a shear with the y-axis fixed and shear factor 2. [4]
(b) Describe the transformation represented by this single matrix. [4]
Q32[10 marks]hardCh1 · Matrices and transformations· Matrices
Matrices are fundamental tools in mathematics, used to represent linear transformations and solve systems of equations. Unlike scalar multiplication, the order of matrix multiplication is crucial.
(a) For the matrices P=(23−14) and Q=(1−205), show that PQ=QP. [6]
(b) Deduce what this implies about matrix multiplication in general. [4]
Q33[8 marks]mediumCh1 · Matrices and transformations· Proving results in trigonometry
The rotation matrix provides a powerful tool for linking geometric transformations with trigonometric identities.
(a) Derive the matrix for a rotation of angle A anticlockwise about the origin. [4]
(b) By considering the product of two rotation matrices, prove the trigonometric identity for sin(A+B). [4]
Q34[5 marks]easyCh1 · Matrices and transformations· Transformations
Transformations are a fundamental concept in linear algebra, mapping points or shapes from one position to another.
(a) Define what is meant by a 'linear transformation'. [2]
(b) Give two properties that are preserved under a linear transformation. [3]
Q35[8 marks]mediumCh1 · Matrices and transformations· Matrices
Matrix multiplication is a key operation in linear algebra, but it has specific rules regarding when it can be performed.
(a) Calculate the product AB for A=(4213) and B=(5−2). [4]
(b) Determine if BA is conformable for multiplication, explaining your reasoning. [4]
Q36[4 marks]easyCh1 · Matrices and transformations· Associativity and commutativity
Matrix multiplication has specific rules regarding the order and dimensions of matrices. Consider two matrices, P and Q.
(a) Give an example of two 2x2 matrices, P and Q, such that PQ is defined. [2]
(b) Identify the order of the product PQ. [2]
Q37[10 marks]hardCh1 · Matrices and transformations· Shears
A transformation is represented by the shear matrix M = \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}.
(a) Show that the line y=mx is an invariant line for this shear if m=0. [3]
(b) Determine the invariant points for the shear represented by M = \begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}. [4]
(c) Find the equation of the invariant line for the shear given in part (b). [3]
Q38[6 marks]easyCh1 · Matrices and transformations· Working with matrices
Given the matrices A=(3−125) and B=(06−41).
(a) Calculate A+B. [3]
(b) Calculate 2A. [3]
Q39[5 marks]easyCh1 · Matrices and transformations· Invariant lines
In the study of matrix transformations, certain geometric features remain unchanged. These are often referred to as invariant properties.
(a) Define the term 'invariant line' in the context of matrix transformations. [2]
(b) State the general approach to finding invariant lines for a given 2x2 matrix transformation. [3]
Q40[5 marks]easyCh1 · Matrices and transformations· Invariant points
Transformations can map points from one position to another. Sometimes, a point remains in its original position after a transformation.
(a) Define what is meant by an invariant point under a transformation. [2]
(b) Find the invariant point for the transformation represented by the matrix (100−1). [3]
Q41[12 marks]hardCh1 · Matrices and transformations· Successive transformations
A geometric shape undergoes two successive transformations in the coordinate plane.
(a) Find the single matrix P representing a rotation of 45 degrees anticlockwise about the origin followed by an enlargement with scale factor 2 about the origin. [4]
(b) Describe the single transformation represented by P. [4]
(c) Compare the effect of applying the transformations in the reverse order. [4]
Q42[10 marks]hardCh1 · Matrices and transformations· Associativity and commutativity
Matrix multiplication is a fundamental operation in linear algebra, but it differs from scalar multiplication in several key aspects. Its definition and properties are strictly governed by the dimensions of the matrices involved.
(a) Discuss the conditions under which matrix multiplication is defined, and explain why these conditions are crucial for associativity. [6]
(b) Evaluate whether there are specific types of matrices for which multiplication is commutative, providing an example. [4]
Q43[7 marks]mediumCh1 · Matrices and transformations· Multiplication of matrices
Consider the matrices A and B given as:
A=(23−14)B=(1−2035−1)
(a) Calculate the product AB. [4]
(b) Determine if BA can be calculated, and if so, state its order. [3]
Q44[11 marks]hardCh1 · Matrices and transformations· Properties of matrix multiplication
Transformations can be represented by matrices, and successive transformations can be combined using matrix multiplication. Consider a rotation about the origin.
(a) Derive the general matrix for a rotation of θ degrees anticlockwise about the origin, and then apply it to reflect the unit square shown in Fig 1.1 in the x-axis. [7]
(b) Explain how the resulting transformation matrix demonstrates the property of matrix multiplication when combining transformations. [4]
Q45[9 marks]mediumCh1 · Matrices and transformations· Invariant points
A linear transformation in a 2D plane is given by the matrix M=(2011).
(a) Determine the invariant points for this transformation. [6]
(b) Show that any point on the line y=0 is an invariant point for this transformation. [3]
Q46[4 marks]easyCh1 · Matrices and transformations· Special matrices
In matrix algebra, certain types of matrices have unique properties and specific names due to their structure and function.
(a) Identify the special name for a square matrix with 1s on the leading diagonal and zeros elsewhere. [2]
(b) Write down a 3x3 example of such a matrix. [2]
Q47[6 marks]easyCh1 · Matrices and transformations· Summary of transformations in two dimensions
Matrices can be used to represent various transformations in two dimensions.
(a) Identify the transformation represented by \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}. [2]
(b) Describe the transformation represented by \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}. [4]
Q48[8 marks]mediumCh1 · Matrices and transformations· Properties of matrix multiplication
The identity matrix, denoted by I, plays a crucial role in matrix algebra, similar to the number 1 in scalar multiplication.
(a) Show that the identity matrix I acts as a multiplicative identity for any general 2x2 matrix A, i.e., IA = AI = A. [6]
(b) State the order of the identity matrix used in part (a). [2]
Q49[9 marks]mediumCh1 · Matrices and transformations· Working with matrices
Fig 1.1 provides information about ingredient costs and quantities for two recipes.
(a) Calculate the total cost of ingredients for each recipe. [5]
(b) Compare the cost per serving of Recipe 1 and Recipe 2. [4]
Q50[8 marks]mediumCh1 · Matrices and transformations· Invariant lines
A transformation is represented by the matrix M=(2−114).
(a) Find the equations of the invariant lines for this transformation. [6]
(b) Determine if any of these invariant lines are also lines of invariant points. [2]
Q51[9 marks]mediumCh1 · Matrices and transformations· Invariant lines
Fig 1.1 shows an object triangle ABC and its image A'B'C' after a linear transformation. The vertices of the object triangle are A(1,1), B(3,1), and C(1,3). The corresponding vertices of the image triangle are A'(1,1), B'(5,1), and C'(1,5).
(a) Determine the equation of the invariant line for the transformation shown in Fig 1.1. [6]
(b) Draw the invariant line on the provided grid in Fig 1.1 and label it. [3]
Q52[8 marks]mediumCh1 · Matrices and transformations· Invariant lines
Fig 1.1 shows the unit square (with vertices at (0,0), (1,0), (1,1), (0,1)) and its image after transformation by the matrix M = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}.
(a) Identify the invariant lines for the transformation represented by M from Fig 1.1. Write down their equations. [5]
(b) Illustrate how a point on one of these invariant lines moves under the transformation M, by choosing a specific point and its image from the diagram. [3]
Q53[9 marks]mediumCh1 · Matrices and transformations· Invariant lines
A linear transformation is defined by the matrix M=(42−20).
(a) Show that the transformation represented by M has exactly two distinct invariant lines, and find their equations. [7]
(b) Deduce the intersection point of these two invariant lines. [2]
Q54[11 marks]hardCh1 · Matrices and transformations· Invariant lines
The general 2x2 matrix for a linear transformation is M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}.
(a) Derive the general conditions for this matrix M to have:
(i) no invariant lines,
(ii) exactly one invariant line,
(iii) infinitely many invariant lines. [7]
(b) Analyse the special case of the identity matrix I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} in terms of invariant lines based on your derivation in part (a). [4]
Q55[10 marks]hardCh1 · Matrices and transformations· Invariant lines
A transformation is represented by the matrix T = \begin{pmatrix} 1 & a \\ 2 & 3 \end{pmatrix}, where 'a' is a constant.
(a) Determine the values of the constant 'a' for which this transformation has only one invariant line. Find the equation of this invariant line for these values of 'a'. [8]
(b) Explain the geometrical significance of having only one invariant line. [2]
Q56[7 marks]mediumCh1 · Matrices and transformations· Invariant lines
A reflection in the line y=x is represented by the matrix T = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.
(a) Find the equations of the invariant lines for this transformation. [5]
(b) Sketch the coordinate axes and the invariant lines found in part (a). [2]
Q57[12 marks]hardCh1 · Matrices and transformations· Invariant lines
The transformation T is represented by the matrix M=(56−3−4).
(a) Find the equations of all invariant lines under T. [8]
(b) Discuss the relationship between the invariant lines found in part (a) and the eigenvectors of the matrix representing T. [4]
Q58[8 marks]mediumCh1 · Matrices and transformations· Shears
Fig 1.15 shows the unit square and its image under a shear transformation.
(a) State the coordinates of the original unit vectors i and j from Figure 1.15.
(b) Find the images of the unit vectors i and j after the shear transformation shown in Figure 1.15.
(c) Determine the matrix that represents this shear transformation, using the images of the unit vectors.
Q59[10 marks]hardCh1 · Matrices and transformations· Shears
Fig 1.15 shows the unit square and its image under a shear transformation.
(a) Identify the fixed line for the shear transformation shown in Figure 1.15.
(b) Determine the shear factor k for this transformation, by observing the coordinates of the image of the point (0,1).
(c) Calculate the matrix that represents this shear transformation.
(d) Explain why the area of the transformed unit square is equal to the area of the original unit square.
Q60[8 marks]mediumCh1 · Matrices and transformations· Transformations
Fig 1.7 shows the unit vectors i and j, and their images i' and j' after a transformation. The original unit square has vertices (0,0), (1,0), (1,1), (0,1). The transformed unit square has vertices (0,0), (0,1), (1,1), (1,0).
(a) Identify the coordinates of the origin O and the unit vectors i and j from Fig 1.7.
(b) Calculate the image of the unit vectors i and j after the transformation shown in Fig 1.7.
(c) Determine the matrix that represents this transformation.
Q61[11 marks]hardCh1 · Matrices and transformations· Invariant lines
Fig 1.20 shows a reflection in the line l. Point P(2,1) is reflected to P'(1,2). The line l passes through Q(3,3) and R(0,0).
(a) Determine the matrix M for the reflection shown in Fig 1.20.
(b) Find the equation of the line perpendicular to l that passes through P(2,1).
(c) Show that the line found in part (b) is an invariant line under the transformation M, by taking a general point on this line and showing its image also lies on the same line.
Q62[5 marks]easyCh1 · Matrices and transformations· Matrices
Fig 1.1 shows direct flights between countries by one airline, represented by arrows indicating flight paths.
(a) Identify the number of nodes in the flight network.
(b) Write down the adjacency matrix M for the flight network, where M_ij = 1 if there is a direct flight from country i to country j, and 0 otherwise. Assume the order of countries is Philippines, Singapore, Australia, New Zealand, UK.
(c) Calculate the number of direct flights from Singapore according to the matrix M.
Q63[8 marks]mediumCh1 · Matrices and transformations· Finding the transformation represented by a given matrix
Fig 1.4 illustrates a transformation of an object triangle ABC to its image A'B'C'.
(a) Describe fully the transformation shown in Figure 1.4, mapping triangle ABC to A'B'C'.
(b) Calculate the matrix that represents this transformation, using the coordinates of the vertices of triangle ABC and its image A'B'C'.
(c) Sketch the image of the point P(2,0) under this transformation and state its coordinates.
Q64[12 marks]hardCh1 · Matrices and transformations· Invariant points
Fig 1.20 shows a reflection in line l, which is the line y=x.
(a) Determine the matrix M for the reflection in line l (y=x) as seen in Fig 1.20.
(b) Find the general equation for an invariant point (x,y) under this transformation M.
(c) Show that any point on the line y=x is an invariant point for this transformation. Give an example of such a point from the diagram.
Q65[9 marks]mediumCh1 · Matrices and transformations· Successive transformations
Fig 1.18 illustrates successive transformations A and B on a point P. Point P is mapped to A(P) by transformation A. Then, A(P) is mapped to B(A(P)) by transformation B.
(a) Determine the matrix for transformation A, which maps P to A(P).
(b) Find the matrix for transformation B, which maps A(P) to B(A(P)).
(c) Calculate the determinant of the composite transformation matrix BA. What does this imply about the area scaling?
Q66[9 marks]mediumCh1 · Matrices and transformations· Transformations
Fig 1.4 illustrates a transformation of an object triangle ABC to its image A'B'C'.
(a) Identify the coordinates of the vertices of the image triangle A'B'C' in Figure 1.4.
(b) Calculate the area of the image triangle A'B'C' and compare it to the area of the object triangle ABC.
(c) Describe the relationship between the determinant of the transformation matrix and the change in area observed.
Q67[8 marks]mediumCh1 · Matrices and transformations· Finding the matrix that represents a given transformation
Fig 1.4 illustrates a transformation of an object triangle ABC to its image A'B'C'.
(a) Determine the matrix for the transformation shown in Figure 1.4.
[3]
(b) Find the image of the point (-2, -3) under this transformation.
[3]
(c) Calculate the area of the object triangle ABC from Figure 1.4.
[2]
Q68[9 marks]mediumCh1 · Matrices and transformations· Invariant lines
Fig 1.7 shows the unit vectors i and j and their images after a transformation.
(a) Determine the matrix M for the transformation shown in Figure 1.7.
[3]
(b) Find the equation of the line of invariant points for this transformation.
[3]
(c) Verify that the line y = -x is an invariant line for this transformation by finding the image of two points on it.
[3]
Q69[10 marks]hardCh1 · Matrices and transformations· Successive transformations
Fig 1.18 illustrates successive transformations A and B on a point P(1,2). P is mapped to A(P), and A(P) is then mapped to B(A(P)).
(a) Determine the matrix A that maps P to A(P).
(b) Calculate the matrix B that maps A(P) to B(A(P)).
(c) Find the single matrix C that represents the composite transformation B followed by A, i.e., C = BA. Show that applying C to P results in B(A(P)).
Q70[8 marks]mediumCh1 · Matrices and transformations· Invariant lines
Fig 1.20 shows a reflection in line l. Point P(2,1) is reflected to P'(1,2). Points Q and R are also shown.
(a) Identify the equation of the invariant line l in Figure 1.20.
(b) Describe another set of lines that are invariant under this reflection, by observing the relationship between the object, image, and line l.
(c) Verify that the line y = -x + 3 is an invariant line for this transformation by finding the image of two points on it, using the matrix for reflection in y=x.
Q71[9 marks]mediumCh1 · Matrices and transformations· Successive transformations
Fig 1.18 illustrates successive transformations A and B on a point P(1,2). P is mapped to A(P), and A(P) is then mapped to B(A(P)).
(a) Describe the transformation A that maps P to A(P).
(b) Describe the transformation B that maps A(P) to B(A(P)).
(c) Determine the matrix BA and calculate the image of the point (3,4) under this composite transformation.
Q72[8 marks]mediumCh1 · Matrices and transformations· Finding the transformation represented by a given matrix
Fig 1.6 shows the unit vectors i and j and their images, i′ and j′, respectively, after a transformation. The unit square is transformed into the shaded region.
(a) Describe fully the transformation represented by the matrix derived from Figure 1.6.
(b) Calculate the determinant of the transformation matrix.
(c) Find the area of the transformed unit square shown in Figure 1.6.
Q73[11 marks]hardCh1 · Matrices and transformations· Finding the transformation represented by a given matrix
Fig 1.7 shows the unit vectors i and j, and their images i' and j' after a transformation. The original unit square has vertices (0,0), (1,0), (1,1), (0,1). The transformed unit square has vertices (0,0), (0,1), (1,1), (1,0).
(a) Describe fully the transformation represented by the matrix derived from Fig 1.7.
(b) Calculate the determinant of the transformation matrix.
(c) Find the equation of the invariant line for this transformation. Explain why this line is invariant.
Q74[6 marks]mediumCh1 · Matrices and transformations· Transformations
Fig 1.3 shows a transformation of an object triangle ABC to its image A'B'C'.
(a) Identify the coordinates of the vertices of the object triangle ABC from Fig 1.3.
(b) Calculate the midpoint of side AB for the object triangle ABC.
(c) Sketch the image of this midpoint under the transformation and state its coordinates.
Q75[8 marks]mediumCh1 · Matrices and transformations· Reflection in x-axis
Fig 1.12 shows the unit square and its image after a transformation.
(a) Describe fully the transformation shown in Figure 1.12.
(b) Calculate the determinant of the transformation matrix from Figure 1.12.
(c) Find the image of the point (-2,3) under this transformation.
Q76[8 marks]mediumCh1 · Matrices and transformations· Successive transformations
Fig 1.18 illustrates successive transformations A and B on a point P(1,2).
(a) Determine the matrix M_A that represents the transformation A from Fig 1.18.
(b) Find the inverse matrix M_A^(-1).
(c) Apply M_A^(-1) to the point A(P) and state the result.
Q77[10 marks]hardCh1 · Matrices and transformations· Matrices
Fig 1.1 shows direct flights between countries by one airline.
(a) Construct the adjacency matrix M for the flight network shown in Figure 1.1, where M_ij = 1 if there is a direct flight from country i to country j, and 0 otherwise. Use the order Philippines, Singapore, Australia, New Zealand, UK.
(b) Calculate the matrix M^2.
(c) Interpret the element in the third row, fifth column of M^2 in the context of flight paths.
Q78[12 marks]hardCh1 · Matrices and transformations· Successive transformations
Fig 1.18 illustrates successive transformations A and B on a point P(1,2).
(a) Determine the matrix M_A that represents the transformation A in Figure 1.18.
[3]
(b) Find the matrix M_B that represents the transformation B in Figure 1.18.
[3]
(c) Show that the transformation A is a rotation about the origin. Calculate the angle of rotation. Show that the transformation B is a reflection. State the line of reflection.
[6]
Q79[5 marks]mediumCh1 · Matrices and transformations· Reflection in x-axis
Fig 1.12 shows the unit square and its image after a transformation.
(a) State the coordinates of the unit vectors I and J from Figure 1.12.
(b) Write down the image coordinates I' and J' after the transformation.
(c) Calculate the transformation matrix M based on I' and J'.
