DP Further Mathematics HL Questionbank

• Syllabus

Topic 1 - Linear Algebra

Sub sections and their related questions

1.1

• SPNone.1.hl.TZ0.12a: (i)     Find the matrices ${\boldsymbol{A}^2}$ and ${\boldsymbol{A}^3}$ , and verify that...
• SPNone.1.hl.TZ0.12b: (i)     Suggest a similar expression for ${\boldsymbol{A}^n}$ in terms of $\boldsymbol{A}$...
• 14M.1.hl.TZ0.10: The matrix A is given by A =...
• 15M.2.hl.TZ0.7a: (i)     Show that ${({A^T})^{ - 1}} = {({A^{ - 1}})^T}$. (ii)     Show that...
• 15M.2.hl.TZ0.7b: A relation $R$ is defined on $S$ such that $A$ is related to $B$ if and only if there...
• 16M.1.hl.TZ0.14a: (i)     Explain why M is a square matrix. (ii)     Find the set of possible values of det(M).
• 259338: This is an example question for the example test. You can delete this question.
• 18M.1.hl.TZ0.2a: Show that A4 = 12A + 5I.
• 18M.1.hl.TZ0.2b: Let B = $\left[ {\begin{array}{*{20}{c}} 4&2 \\ 1&{ - 3} \end{array}} \right]$. Given...
• 18M.1.hl.TZ0.10c: A matrix M is said to be orthogonal if M TM = I where I is the identity. Show that Q is orthogonal.

1.2

• 10M.1.hl.TZ0.2a: Show that $R$ is an equivalence relation.
• 10M.1.hl.TZ0.2b: The relationship between $a$ , $b$ , $c$ and $d$ is changed to $ad - bc = n$ . State,...
• 12M.2.hl.TZ0.4A.a: Show that $f$ is a bijection if $\boldsymbol{A}$ is non-singular.
• 12M.2.hl.TZ0.4A.b: Suppose now that $\boldsymbol{A}$ is singular.   (i)     Write down the relationship between...
• SPNone.1.hl.TZ0.14a: Show that any matrix of this form is its own inverse.
• 14M.1.hl.TZ0.10: The matrix A is given by A =...
• 15M.2.hl.TZ0.7a: (i)     Show that ${({A^T})^{ - 1}} = {({A^{ - 1}})^T}$. (ii)     Show that...
• 16M.1.hl.TZ0.14a: (i)     Explain why M is a square matrix. (ii)     Find the set of possible values of det(M).
• 17M.2.hl.TZ0.4a.i: Find an expression for det(A) in terms of $\lambda$, simplifying your answer.
• 17M.2.hl.TZ0.4a.ii: Hence show that A is singular when $\lambda  = 1$ and find the other value of $\lambda$ for...

1.3

• SPNone.1.hl.TZ0.9a: By reducing the augmented matrix to row echelon form,   (i)     find the rank of the coefficient...
• SPNone.1.hl.TZ0.9b: For this value of $k$ , determine the solution.
• 14M.2.hl.TZ0.4: The matrix A is given by A =...
• 15M.1.hl.TZ0.12b: (i)     State the column rank of $M$. (ii)     Find the basis for the range of this...
• 16M.2.hl.TZ0.8b: (i)     Write down the matrices M$_1$, M$_2$ representing the permutations...
• 17M.2.hl.TZ0.4b.i: Explain how it can be seen immediately that B is singular without calculating its determinant.
• 17M.2.hl.TZ0.4b.ii: Determine the null space of B.
• 17M.2.hl.TZ0.4b.iii: Explain briefly how your results verify the rank-nullity theorem.
• 17M.2.hl.TZ0.4c: Prove, using mathematical induction, that B$^n = {8^{n - 2}}$B$^2$ for...

1.4

• SPNone.1.hl.TZ0.9a: By reducing the augmented matrix to row echelon form,   (i)     find the rank of the coefficient...
• SPNone.1.hl.TZ0.9b: For this value of $k$ , determine the solution.
• 17M.1.hl.TZ0.3a: Determine the value of $\lambda$ and the value of $\mu$ for which the equations are...
• 17M.1.hl.TZ0.3b: For these values of $\lambda$ and $\mu$, solve the equations.
• 17M.1.hl.TZ0.3c: State the rank of the matrix of coefficients, justifying your answer.
• 18M.1.hl.TZ0.8b: Explain why the equations have no unique solution when $a$ = 3.
• 18M.1.hl.TZ0.8c: Find all the solutions to the equations when $a$ = 3, $b$ = 10 in the form r = s +...

1.5

• SPNone.1.hl.TZ0.2a: Show that the following vectors form a basis for the vector space ${\mathbb{R}^3}$...
• SPNone.1.hl.TZ0.2b: Express the following vector as a linear combination of the above...
• 14M.2.hl.TZ0.4: The matrix A is given by A =...
• 15M.1.hl.TZ0.12b: (i)     State the column rank of $M$. (ii)     Find the basis for the range of this...
• 16M.1.hl.TZ0.12a: Explain why the set of position vectors of points whose coordinates satisfy \(x - y - z =...
• 16M.1.hl.TZ0.12b: (i)     Show that the set of position vectors of points whose coordinates satisfy...
• 17M.1.hl.TZ0.15a.i: By considering ${\alpha _1}$v$_1 + {\alpha _2}$v$_2 + {\alpha _3}$v$_3 = 0$, show that...
• 17M.1.hl.TZ0.15a.ii: Explain briefly why v$_1$, v$_2$, v$_3$ form a basis for vectors in ${\mathbb{R}^3}$.
• 17M.1.hl.TZ0.15b.i: Show that the...
• 17M.1.hl.TZ0.15b.ii: Express the vector $$\left[ {\begin{array}{*{20}{c}} 2 \\ 8 \\ 0 \end{array}} \right]$$ as a...

1.6

• 15M.1.hl.TZ0.12b: (i)     State the column rank of $M$. (ii)     Find the basis for the range of this...
• 18M.1.hl.TZ0.10a.i: determine the 2 × 2 matrix P which represents a reflection in the...
• 18M.1.hl.TZ0.10a.ii: determine the 2 × 2 matrix Q which represents an anticlockwise rotation of θ about the origin.
• 18M.1.hl.TZ0.10b: Describe the transformation represented by the matrix PQ.

1.7

• 18M.1.hl.TZ0.13a: Show that the linear transformation represented by M transforms any point on the line $y = x$...
• 18M.1.hl.TZ0.13b: Explain what happens to points on the line $4y + x = 0$ when they are transformed by M.

1.8

• 15M.2.hl.TZ0.5a: By considering the points $(1,{\text{ }}0)$ and $(0,{\text{ }}1)$ determine the...
• 15M.2.hl.TZ0.5b: Determine the matrix $A$ which represents a rotation from the direction...
• 15M.2.hl.TZ0.5c: A triangle whose vertices have coordinates $(0,{\text{ }}0)$, $(3,{\text{ }}1)$ and...
• 15M.2.hl.TZ0.5d: The matrix $B = {A^{ - 1}}XA$ represents a reflection in the line $y = mx$. Find the value of...
• 17M.2.hl.TZ0.9a: The point $(x,{\text{ }}y)$ is rotated through an anticlockwise angle $\alpha$ about the...
• 18M.1.hl.TZ0.4c: Write down the 2 × 2 matrices representing T3, T4 and T5.
• 18M.1.hl.TZ0.4d.i: Find the 2 × 2 matrix representing T.
• 18M.1.hl.TZ0.4d.ii: Give a geometric description of the transformation T.

1.9

• SPNone.2.hl.TZ0.8a: Given that the elements of a $2 \times 2$ symmetric matrix are real, show that   (i)     the...
• SPNone.2.hl.TZ0.8b: The matrix $\boldsymbol{A}$ is given...
• 14M.1.hl.TZ0.4: The matrix M is defined by M =...
• 16M.1.hl.TZ0.14b: (i)     Find the value of $a$. (ii)     Find the eigenvalues of N. (iii)     Find...
• 18M.1.hl.TZ0.13c: State the two eigenvalues of M.
• 18M.1.hl.TZ0.13d: State two eigenvectors of M which correspond to the two eigenvalues.