Find an expression for $V_T$ in the form $A \cos (Bt+C)$, where $A, B$ and $C$ are real constants.

Hence write down the maximum voltage in the circuit.

$\theta =\frac{\mathrm{\pi }}{2}$.

$\theta =\mathrm{\pi }$.

$\theta =\frac{3\mathrm{\pi }}{2}$.

Find this value of $\theta$.

For this value of $\theta$, plot the approximate position of $z_{\theta }$ on the Argand diagram.

Write down the value of $u_1$.

Find the value of $u_6$.

Prove that $\{u_n\}$ is an arithmetic sequence, stating clearly its common difference.

Find Q.

Find CD.

Find D^–1.

find the values of $w^2$, $w^3$, and $w^4$.

draw $w$, $w^2$, $w^3$, and $w^4$ on the following Argand diagram.

Let $z=\frac{w}{2-\text{i}}$.

Find the value of $a$ for which successive powers of $z$ lie on a circle.

Show that A is an arithmetic sequence, stating its common difference d in terms of r.

A particular geometric sequence has u₁ = 3 and a sum to infinity of 4.

Find the value of d.

$z=2\text{i}$.

$z=1+\text{i}$.

Describe these two transformations and give the order in which they are applied.

Hence, or otherwise, find the value of $z$ when $w=2-\text{i}$.

Write down the value of $a$.

Find the value of $b$.

Let .

Find the value of $q$.

Use Sam’s measurements to calculate the area of the sector. Give your answer to four significant figures.

Find the upper bound and lower bound of the area of the sector.

Find, with justification, the largest possible percentage error if the answer to part (a) is recorded as the area of the sector.

In an arithmetic sequence, the sum of the 3rd and 8th terms is 1.

Given that the sum of the first seven terms is 35, determine the first term and the common difference.

(A + B)² = A² + 2AB + B²

 (A – kI)³ = A³ – 3kA² + 3k²A – k³

CA = B  C = $\frac{B}{A}$

Write down $2+5\text{i}$ in exponential form.

An equilateral triangle is to be drawn on the Argand plane with one of the vertices at the point corresponding to $2+5\text{i}$ and all the vertices equidistant from $0$.

Find the points that correspond to the other two vertices. Give your answers in Cartesian form.

Find the common ratio, $r$, for the sequence.

Find the least value of $n$ such that $u_n<\frac{1}{2}$.

the value of $d$;

the value of $r$;

Express w² and w³ in modulus-argument form.

Sketch on an Argand diagram the points represented by w⁰ , w¹ , w² and w³.

Show that the area of the quadrilateral Q is $\frac{21\sqrt{3}}{2}$.

Let $z=2\left(\text{cos}\frac{\pi }{n}+\text{i}\text{sin}\frac{\pi }{n}\right),n\in {\mathbb{Z}}^{+}$. The points represented on an Argand diagram by $z^0,z^1,z^2,\dots ,z^n$ form the vertices of a polygon $P_n$.

Show that the area of the polygon $P_n$ can be expressed in the form $a\left(b^n-1\right)\text{sin}\frac{\pi }{n}$, where $a,b\in \mathbb{R}$.

Find BA.

Calculate det (BA).

 Find A(A^–1B + 2A^–1)A.

Find the value of $a$.

Find the value of $b$.

Find the average number of earthquakes in a year with a magnitude of at least $7.2$.

Find $\text{P}(Y>100)$.

Show that A⁴ = 12A + 5I.

Let B = .

Given that B² – B – 4I = , find the value of $k$.

Write down the value of $a$.

Write down the value of $b$.

Write these equations as a matrix equation.

Solve the matrix equation.

Given matrices A, B, C for which AB = C and det A ≠ 0, express B in terms of A and C.

Find the matrix DA.

Find B if AB = C.

Find the coordinates of the point of intersection of the planes $x+2y+3z=5$, $2x-y+2z=7$, $3x-3y+2z=10$.

The square matrix X is such that X³ = 0. Show that the inverse of the matrix (I – X) is I + X + X².

Find A + B.

Find −3A. 

Find AB.

Find a relationship between $a$ and $b$ if the matrices  and  commute under matrix multiplication.

Find the value of $a$ if the determinant of matrix $M$ is −1.

Write down $M^{-1}$ for this value of $a$.

If A =  and B = , find 2 values of $x$ and $y$, given that AB = BA.

Find the vectors $\mathit{a}$ and $\mathit{b}$ in terms of $m$ and/or $c$.

Find the value of $\text{det\hspace{0.05em}}\mathit{M}$.

Show that the equation of the resulting line does not depend on $m$ or $c$.

$A$ and $B$ are 2 × 2 matrices, where  and . Find $B$

Write down the inverse matrix A⁻¹.

Express X in terms of A⁻¹ and B.

Hence, solve for X.

The sum of an infinite geometric sequence is $9$.

The first term is $4$ more than the second term.

Find the third term. Justify your answer.

Find the value of $a$ for which the system of equations does not have a unique solution.

Find the solution of the system of equations when $a=2$.

find the value of $d$.

determine the value of $\sum _{r=1}^N{u}_r$.

Find the determinant of A, where A = .

Show that $\text{lo}{\text{g}}_{r^2}x=\frac{1}{2}\text{lo}{\text{g}}_{r}x$ where $r,x\in {\mathbb{R}}^{+}$.

Express $y$ in terms of $x$. Give your answer in the form $y=p{x}^q$, where p , q are constants.

The region R, is bounded by the graph of the function found in part (b), the x-axis, and the lines $x=1$ and $x=\alpha$ where $\alpha >1$. The area of R is $\sqrt{2}$.

Find the value of $\alpha$.

Find det A.

Find the set of values of $k$ for which the system has a unique solution.

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.

Solve .

Given that AB = , find $a$.

Hence, or otherwise, find A^–1.

Find the matrix X, such that AX = C.

Show that M is non-singular.

 Calculate M².

 Show that det(M²) is positive.

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

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 $a, b\in \mathrm{\mathbb{Z}}$.

Using your answer to (a), or otherwise, find the long-term probability of the switch being in state $\text{A}$. Give your answer in the form $\frac{c}{d}$, where $c, d\in {\mathrm{\mathbb{Z}}}^{+}$.

By expressing $z_1$ and $z_2$ in modulus-argument form write down the modulus of $w$;

By expressing $z_1$ and $z_2$ in modulus-argument form write down the argument of $w$.

Find the smallest positive integer value of $n$, such that $w^n$ is a real number.

Write down the inverse, A^–1.

Given that XA + B = C, express X in terms of A^–1, B and C.

Given that B , and C , find X.

Find the values of $a$ and $b$ given that the matrix  is the inverse of the matrix .

For the values of $a$ and $b$ found in part (a), solve the system of linear equations

$$\begin{array}{c}x+2y-2z=5\\ 3x+by+z=0\\ -x+y-3z=a-1.\end{array}$$

Given that A is non-singular, prove that f is a bijection.

It is now given that A is singular.

By considering appropriate determinants, prove that f is not a bijection.

Solve $2\sin (x+{60}^{\circ })=\cos (x+{30}^{\circ }),0^{\circ }⩽x⩽{180}^{\circ }$.

Show that $\sin {105}^{\circ }+\cos {105}^{\circ }=\frac{1}{\sqrt{2}}$.

Find the modulus and argument of $z$ in terms of $\theta$. Express each answer in its simplest form.

Hence find the cube roots of $z$ in modulus-argument form.

Find the solution of ${\log }_{2}x-{\log }_{2}5=2+{\log }_{2}3$.

Write down the inverse of the matrix

A = 

Hence, find the point of intersection of the three planes.

$$\begin{array}{c}x-3y+z=1\\ 2x+2y-z=2\\ x-5y+3z=3\end{array}$$

A fourth plane with equation $x+y+z=d$ passes through the point of intersection. Find the value of $d$.

2A − B.

det (2A − B).

The square matrix X is such that X³ = 0. Show that the inverse of the matrix (I – X) is I + X + X².

Find the value of ${z_1}^3$.

Find the value of ${\left(\frac{z_1}{z_2}\right)}^4$ for $n=2$.

Find the least value of $n$ such that $z_1{z}_2\in {\mathrm{ℝ}}^{+}$.

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

Find the inverse of the matrix .

Hence solve the system of equations

$$x+2y+z=0$$

$$x+y+2z=7$$

$$2x+y+z=17$$

Given that A =  and B = , find X if BX = A – AB.

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

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

Estimate the value of Roger’s laptop after $5$ years.

Find the value of $k$.

Comment on the validity of your answer to part (b).

Find AB.

The matrix C = $\left(\begin{array}{c}20\\ 28\end{array}\right)$ and 2AB = C. Find the value of $x$.

Find how high the balloon will travel in the first $10$ minutes after it is launched.

The balloon is required to reach a height of at least $2520$ metres.

Determine whether it will reach this height.

Suggest a limitation of the given model.

Find the matrix A².

If det A² = 16, determine the possible values of $a$.

Consider the matrix A , where $x\in \mathbb{R}$.

Find the value of $x$ for which A is singular.

The matrices A, B, C and X are all non-singular 3 × 3 matrices.

Given that A^–1XB = C, express X in terms of the other matrices.

If A = and A² is a matrix whose entries are all 0, find $k$.

Given that M =  and that M² – 6M + kI = 0 find k.

In the following Argand diagram the point A represents the complex number $-1+4\text{i}$ and the point B represents the complex number $-3+0\text{i}$. The shape of ABCD is a square. Determine the complex numbers represented by the points C and D.

Find the values of the real number $k$ for which the determinant of the matrix  is equal to zero.

If  and det $A=14$, find the possible values of $p$.

Solve the equation $4^x+2^{x+2}=3$.

Solve the equation ${\log }_2(x+3)+{\log }_2(x-3)=4$.

The rate, $A$, of a chemical reaction at a fixed temperature is related to the concentration of two compounds, $B$ and $C$, by the equation

$A=k{B}^x{C}^y$, where $x$, $y$, $k\in \mathbb{R}$.

A scientist measures the three variables three times during the reaction and obtains the following values.

Find $x$, $y$ and $k$.