DP Mathematics: Applications and Interpretation Questionbank

Find a matrix $\mathit{P}$, with integer elements, such that $\mathit{T}=\mathit{P}\mathit{D}\mathbf{ }{\mathit{P}}^{-1}$, where $\mathit{D}$ is a diagonal matrix.

Hence, show that the long-term transition matrix ${\mathit{T}}^{\infty }$ is given by ${\mathit{T}}^{\infty }=\left(\begin{array}{cc}\frac{10}{17}& \frac{10}{17}\\ \frac{7}{17}& \frac{7}{17}\end{array}\right)$.

Determine, with justification, whether the equilibrium point $(0, 0)$ is stable or unstable.

Hence, or otherwise, determine the expected ratio of the number of patients Doctor Black would have compared to Doctor Green in the long term.

Hence, write down the general solution of the system.

Find the eigenvalues and corresponding eigenvectors of the matrix $\left(\begin{array}{cc}-4& 0\\ 3& -2\end{array}\right)$.

(i)   at $(4, 0)$.

(ii)  at $(-4, 0)$.

Sketch a phase portrait for the general solution to the system of coupled differential equations for $-6\le x\le 6$, $-6\le y\le 6$.

Write down a transition matrix $\mathit{T}$ indicating the annual population movement between clinics.

Find a prediction for the ratio of the number of patients Doctor Black will have, compared to Doctor Green, after two years.

By considering the determinant of a relevant matrix, show that the eigenvalues, $\lambda$, of $\mathit{A}$ satisfy the equation

${\lambda }^2-\alpha \lambda +\beta =0$,

where $\alpha$ and $\beta$ are functions of $a, b, c, d$ to be determined.

Verify that

${\mathit{A}}^2-\alpha \mathit{A}+\beta \mathit{I}=$ 0.

Assuming that $\mathit{A}$ is non-singular, use the result in part (b)(i) to show that

${\mathit{A}}^{-1}=\frac{1}{\beta }\left(\alpha \mathit{I}-\mathit{A}\right)$.

the eigenvectors.

Find the population of $\text{Y}$ at this value of $t$. Give your answer to the nearest $10$ herbivores.

Sketch the phase portrait for this system, for $x, y>0$.

On your sketch show

• the equation of the line defined by the eigenvector in the first quadrant
• at least two trajectories either side of this line using arrows on those trajectories to represent the change in populations as t increases

Find the value of $t$ at which $x=0$.

the eigenvalues.

Write down a condition on the size of the initial population of $\text{Y}$ if it is to avoid its population reducing to zero.

Hence write down the general solution of the system of equations.

It is sunny today. Find the probability that it will be sunny in three days’ time.

Find the eigenvalues and eigenvectors of $\mathit{T}$.

Hence find the long-term percentage of sunny days in Vokram.

Write down the matrix $\mathit{P}$.

Write down the matrix $\mathit{D}$.

Given that $y=\dot{x}$, show that $\dot{y}=-1.25x-3y$.

Find the eigenvalues of matrix $\mathit{A}$.

Write down the matrix $\mathit{A}$.

Find the eigenvectors of matrix $\mathit{A}$.

Given that when $t=0$ the shock absorber is displaced $8 \text{cm}$ and its velocity is zero, find an expression for $x$ in terms of $t$.

Find an eigenvector corresponding to the eigenvalue of $1$. Give your answer in the form $\left(\begin{array}{c}a\\ b\end{array}\right)$, where $a, b\in \mathrm{\mathbb{Z}}$.

Find the eigenvalues for the matrix $\left(\begin{array}{cc}0 & 1\\ -6 & -5\end{array}\right)$.

Hence state the long-term velocity of the particle.

Use the substitution $y=\frac{dx}{dt}$ to show that this equation can be written as

$\left(\begin{array}{c}\frac{dx}{dt}\\ \frac{dy}{dt}\end{array}\right)=\left(\begin{array}{cc}0 & 1\\ -6 & -5\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$.

Hence, find the Cartesian equation satisfied by $X$ and $Y$.

the curve in $X$ and $Y$ in part (e) (ii), giving a reason.

Show that .

Hence, show that .

State geometrically what transformation the matrix $\mathbf{R}$ represents.

Show that the matrix equation $\left(xy\right)\mathbf{M}\left(\begin{array}{c}x\\ y\end{array}\right)=\left(6\right)$ is equivalent to the Cartesian equation $\frac{5}{2}{x}^2+xy+\frac{5}{2}{y}^2=6$.

Find the Cartesian equation satisfied by $u$ and $v$ and state the geometric shape that this curve represents.

Show that $\left(\begin{array}{c}\frac{1}{\sqrt{2}}\\ \frac{-1}{\sqrt{2}}\end{array}\right)$ and $\left(\begin{array}{c}\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right)$ are unit eigenvectors and that they correspond to different eigenvalues.

the curve in $x$ and $y$ in part (b).

Write down the equations of two lines of symmetry for the curve in $x$ and $y$ in part (b).

Find the eigenvalues for $M$. For each eigenvalue find the set of associated eigenvectors.

Find matrix R.

Verify that .

Find the eigenvalues of $\mathit{M}$.

Find the eigenvectors of $\mathit{M}$.

Find the eigenvalues of the system of coupled first order equations given in part (a).

Given the matrix A =  find the values of the real number $k$ for which $\text{det}\left(A-kI\right)=0$ where 

Given that A =  and I = , find the values of $\lambda$ for which (A – $\lambda$I) is a singular matrix.

Find the value of $m$ and of $n$.

Use the result from part (b) (ii) to explain why A is non-singular.

Use the values from part (b) (i) to express A⁴ in the form $p$A+ $q$I where $p$, $q\in \mathbb{Z}$.

Find the values of $\lambda$ for which the matrix (A − $\lambda$I) is singular.

Hence show that I = $\frac{1}{5}$A (6I – A).

Show that the eigenvalues of A are real if ${\left(a-d\right)}^2+4bc⩾0$.

Deduce that the eigenvalues are real if A is symmetric.

Determine the eigenvalues of B.

Determine the corresponding eigenvectors.