DP Mathematics: Applications and Interpretation Questionbank

Complete the second column of the table by giving one example of a number from each set.

Use Giovanni's diagram to calculate the length of AX.

Use your graphic display calculator to write down r , Pearson’s product–moment correlation coefficient.

Use the regression line y on x to estimate Jerome’s examination score.

Hence state the long-term velocity of the particle.

By considering the image of $(0, 0)$, find $\mathit{q}$.

Hence find the angle and direction of the rotation represented by $\mathit{R}$.

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

Calculate the value of $\underset{n=1}{\overset{10}{\text{Σ}}}\left(625\times 1.{064}^{\left(n-1\right)}\right)$.

Find the first term and the common difference of the sequence.

Suppose that $u_1$ is the first term of a geometric series with common ratio $r$.

Prove, by mathematical induction, that the sum of the first $n$ terms, $s_n$ is given by

$s_n=\frac{u_1\left(1-r^n\right)}{1-r}$, where $n\in {\mathbb{Z}}^{+}$.

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

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

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

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

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

Find this value of $\theta$.

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 value of $b$.

Hence write down the maximum voltage in the circuit.

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

the amount Paul pays each month.

Find the value of $w_n$ for this path.

$l_n$.

Show that Eddie needs $144$ tiles.

Calculate the monthly repayment using option 1.

option 2.

Find the repayment made each quarter.

Calculate det (BA).

Calculate the number of positive terms in the sequence.

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

Sketch the graph of y = f (x), for −4 ≤ x ≤ 3 and −50 ≤ y ≤ 100.

Write down the distance Rosa runs in the third training session;

For that day find the weight of Sergei’s first lift.

Calculate the total amount Sophie would pay, using option 1.

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

Calculate the number of months it will take to repay the mortgage using option 2.

For the data from these seven species describe the correlation between the average body weight and the average weight of the brain.

Write down the value of $u_1$.

Calculate 14 000 USD in INR.

Calculate the price of a ticket in the 16th row.

Find Q.

Calculate the tea-shop’s total profit for the first 12 weeks.

Write down the median length of these leaves.

$b$.

$w_n$.

remaining in the bag at the end of the first day.

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

Show that $S_{12}=12{\log }_{2}x-66$.

Find the exact value of m and of c for these data.

Find the value of ${z_1}^3$.

Let .

Find the value of $q$.

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

state the coordinates, $\left(x_1, y_1\right)$, of its image in $U_1$.

Find the total vertical distance travelled by the ball from the point at which it is dropped until the fourth bounce.

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

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

Find the total amount paid for the car.

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.

Calculate the percentage error in the maximum number of daylight hours Boris recorded in the diagram.

Find the total area of the tiles in Eddie’s path. Give your answer in the form $a\times {10}^k$ where $1\le a<10$ and $k$ is an integer.

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

By considering the image of $(1, 0)$ and $(0, 1)$, show that

$\mathit{P}=\left(\begin{array}{cc}\frac{\sqrt{3}}{4} & \frac{1}{4}\\ -\frac{1}{4} & \frac{\sqrt{3}}{4}\end{array}\right)$.

Find the value of $a$ and the value of $b$.

Calculate the number of days that Scott can feed his dog with one bag of food.

Determine the amount that Scott expects to spend on dog food in $2025$. Round your answer to the nearest dollar.

Describe what the value in part (d)(i) represents in this context.

(i)   the gradient of this line in terms of $c$;

(ii)  the $y$-intercept of this line in terms of $A$.

Find the equation of the regression line for $\ln k$ on $\frac{1}{T}$.

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

Find the value of a and of b.

Write an expression for $C$ in terms of $pH$.

Use your graphic display calculator to find the equation of the tangent to the graph of y = f (x) at the point (–2, 38.75).

Give your answer in the form y = mx + c.

The matrices A, B, X are given by

A = , B = , X = , $\text{where}$ $a$, $b$, $c$, $d\in \mathbb{Q}$.

Given that AX + X = Β, find the exact values of $a$, $b$, $c$ and $d$.

The apple pie has a volume of $61 425 {\text{cm}}^3$.

Find the total number of slices Mia can cut from this pie.

Find the tenth term.

Use your regression line to estimate the average weight of the brain of grey wolves.

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

Write down these three equations as a matrix equation.

the eigenvalues.

Three drones are initially positioned at the points $\text{A}$, $\text{B}$ and $\text{C}$. After performing the movements listed above, the drones are positioned at points $\text{A}\prime$, $\text{B}\prime$ and $\text{C}\prime$ respectively.

Show that the area of triangle $\text{ABC}$ is equal to the area of triangle $\text{A}\prime \text{B}\prime \text{C}\prime$ .

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

Show that the maximum height reached by the ball after it has bounced for the sixth time is $68 \text{cm}$, to the nearest $\text{cm}$.

Write down three equations that express the information given above.

State whether, for all $n>k$, the university will have places available for all applicants. Justify your answer.

Find ${\mathit{P}}^2$.

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

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

Write down $E_{\theta }$.

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.

fed to the dog per day.

For option 2, find the minimum value of $x$ that Juliana would need to invest each year. Give your answer to the nearest dollar.

For option 1, determine the minimum amount Juliana would need to invest. Give your answer to the nearest dollar.

Calculate the pH of the coffee.

Determine whether the unknown liquid is more or less acidic than the coffee. Justify your answer mathematically.

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

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 amount Raul and Rosy will still owe the bank at the end of the first $10$ years.

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

Find a single transformation that is equivalent to the three transformations represented by matrix $\mathit{P}$.

Find the maximum value of $I_{\text{T}}$.

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

Find the value of $a$.

Find the value of $b$.

Given $0<M<8$, find the range for $N$.

Find the number of times, after the first bounce, that the maximum height reached is greater than $10 \text{cm}$.

Find the number of each type of ticket sold.

$a$.

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

Write down an equation satisfied by $\left(\begin{array}{c}a\\ b\end{array}\right)$.

Find the value of $a$.

Use $\mathit{P}=\mathit{R}\mathit{S}$ to find the matrix $\mathit{R}$.

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

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

Write down the common ratio of the sequence.

Write down each of the transformations in matrix form, clearly stating which matrix represents each transformation.

Find a single matrix $\mathit{P}$ that defines a transformation that represents the overall change in position.

Hence state what the value of ${\mathit{P}}^2$ indicates for the possible movement of the drone.

Hence find $\text{Re}\left(w_1+w_2\right)$ in the form $A \cos \left(x-a\right)$, where $A>0$ and $0<a\le \frac{\mathrm{\pi }}{2}$.

Find $w_1+w_2$ in the form ${\text{e}}^{\text{i}x}\left(c+\text{i}d\right)$.

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

Find the matrix X, such that AX = C.

Use your graphic display calculator to write down $\overline{y}$, the mean examination score.

the value of $r$;

Show the points represented by $z$ and $z-2a$ on the following Argand diagram.

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

Given that Phil’s aim is to own the house after 20 years, find the value for $P$ to the nearest dollar.

Express this volume in $\text{c}{\text{m}}^3$.

Show that the sum of the infinite sequence is $4{\log }_{2}x$.

Write down the value of $q$.

(A + B)² = A² + 2AB + B²

Find the area of ABCD.

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

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.

Hence find A⁻¹B.

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

CA = B  C = $\frac{B}{A}$

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

option 1.

Find the tea-shop’s profit during the 11th week.

In the mth week the tea-shop’s total profit exceeds the café’s total profit, for the first time since they both opened.

Find the value of m.

Write down the number of leaves with a length less than or equal to 8 cm.

Use your answer to part (a)(i) to calculate the amount remaining on her mortgage after the first 10 years.

Use your graphic display calculator to find the zero of f (x).

the total amount that Paul has to pay.

Show that A⁴ = 12A + 5I.

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

Find the total number of sticks used by Tomás for all 24 diagrams.

Calculate the café’s total profit for the first 12 weeks.

Find the common ratio.

Josh states: “Every integer is a natural number”.

Write down whether Josh’s statement is correct. Justify your answer.

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

Find an expression for $A_1$ and show that $A_2={1.004}^{2}x+1.004x$.

Show that $abc=8$.

For that day find how much weight was added after each lift.

Find matrix R.

Find the value of $b$.

Hence find the total amount of antibiotic, in mg, that Ted receives during the first $k$ days.

the total amount that Paul has to pay.

Find the range of the average body weights for these seven species of mammal.

Write down the value of $d$ and the value of $u_1$.

Write down the common ratio of the sequence.

Given that M³ can be written as a quadratic expression in M in the form aM² + bM + cI , determine the values of the constants a, b and c.

Find a quadratic expression in M for the inverse matrix M^–1.

Calculate the value of Siân’s investment after four years. Give your answer correct to two decimal places.

Write down A⁻¹.

Find D.

Find T₄.

Write down the distance Rosa runs in the $n$th training session.

Find the matrix DA.

Write down the inverse matrix A⁻¹.

Using your results from part (a) (ii), find T₁₀₀.

Find the interest paid on the loan.

For the data from these seven species calculate $r$, the Pearson’s product–moment correlation coefficient;

Find $r$.

Write down the value of $b$.

find AB.

Find $S$₄.

Write down the value of $a$.

write down A + B.

Write these equations as a matrix equation.

Express X in terms of A⁻¹ and B.

 Write down M⁻¹.

Write down the determinant of M.

Find the amount to be borrowed for this option.

Find the total amount paid for the car.

Find the interest paid on the loan.

Use the regression equation to estimate the value of y when x = 3.57.

A^–1 = I – A.

Find det A.

Find the common ratio.

 (A – kI)³ = A³ – 3kA² + 3k²A – k³

Show that $27m+9n+3p=18$.

Using this value of $a$, find $M^{-1}$ and hence solve the system of equations:

$-x+2y=-3$

$2x-y=3$

A³ = –I.

Find X.

Find the repayment made each quarter.

Show that ${\gamma }^2=\gamma -1$.

Find the sum to infinity of this sequence.

Find A + B.

Calculate the value of her savings after 10 years.

Find M².

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

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

Given that x_{k + 1} = x_k + a, find a.

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 $k$.

Find the annual interest rate, $r$.

Find the percentage error on Giovanni’s diagram.

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

Hence find the value of ${\left(1-\gamma \right)}^6$.

Calculate, in years, when the bicycle value will be less than 50 USD.

Find the common ratio of this sequence.

Show that one of the real roots is equal to 1.

Find $M^2$ in terms of $a$.

Find the value of $k$.

Find B if AB = C.

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

If all of the elements of M are positive, find the range of possible values for $a$.

Express w² and w³ in modulus-argument form.

Find the minimum value of $m$, where $m\in \mathrm{\mathbb{N}}$.

option 1.

Harinder chose option B. At the end of five years, Harinder converted this investment back to USD. The exchange rate, at that time, was 1 USD = 67.16 INR.

Calculate how much more money, in USD, Harinder earned by choosing option B instead of option A.

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

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

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

Use your graphic display calculator to find the coordinates of the local minimum point.

Show that M⁴ = 19M² + 40M + 30I.

Write down the value of the common difference, $d$

Find the common ratio.

Solve the matrix equation.

Calculate, in CAD, the total amount John pays for the bicycle.

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

Find the sum of the first ten terms.

It is now given that A is singular.

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

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

the curve in $x$ and $y$ in part (b).

 Calculate M².

Find the value of $u_6$.

Write down the value of $a$.

Hence or otherwise, find the minimum value of $Q$ that would permit David to withdraw annual amounts of $5000 indefinitely. Give your answer to the nearest dollar.

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

does X^–1 + Y^–1 have an inverse? Justify your conclusion.

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

Find the percentage error in Abdallah’s estimate.

Calculate the value of Imon’s investment after $5$ years.

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

Find the distance Carlos runs in the fifth month of training.

Hence or otherwise solve

$$x-3y=1$$

$$2x+z=2$$

$$4x+y+3z=-1$$

The function $f$ can also be written $f(x)=x\left(x-1\right)\left(rx-s\right)$ where $r$ and $s$ are integers. Find $r$ and $s$.

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

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 X and Y.

find the value of $d$.

Show that M³ = .

Find the determinant of A, where A = .

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

Determine the eigenvalues of B.

The company also makes a ladder that is 1050 cm long.

Find the maximum number of rungs in this 1050 cm long ladder.

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

Write down M⁴.

The sum of an infinite geometric sequence is 33.25. The second term of the sequence is 7.98. Find the possible values of $r$.

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

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

and hence find an expression for $A^n+{\left(A^n\right)}^{-1}$.

Boxes of mixed fruit are on sale at a local supermarket.

Box A contains 2 bananas, 3 kiwifruit and 4 melons, and costs $6.58.

Box B contains 5 bananas, 2 kiwifruit and 8 melons and costs $12.32.

Box C contains 5 bananas and 4 kiwifruit and costs $3.00.

Find the cost of each type of fruit.

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

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

Show that the total number of triangular panes, $S_n$, in the first $n$ levels is given by:

$S_n=n^2+4n$.

Find the common difference.

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

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

Hence solve M$\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}4\\ 8\end{array}\right)$.

Calculate the total amount Sophie would pay, using option 2.

Write down the inverse of the matrix A = .

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

Hence, solve for X.

Solve .

Hence calculate her monthly repayment for the final 10 years.

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 $\frac{5}{2}{x}^2+xy+\frac{5}{2}{y}^2=6$.

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.

Write down an equation, in terms of $u_1$ and $d$, for the amount of the drug that she receives on the seventh day.

Show that M is non-singular.

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

Find u₈.

Given that 2A + B = , find the value of $p$ and of $q$.

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 amount to be borrowed for this option.

Calculate the amount Yejin needs to have saved into her annuity fund, in order to meet her retirement goal.

Find the value of $\theta$ at which the enlargement scale factor equals zero.

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

Using your answer to part (e), find the value of $r$ which minimizes $A$.

Given that $S_{12}$ is equal to half the sum of the infinite geometric sequence, find $x$, giving your answer in the form $2^p$, where $p\in \mathbb{Q}$.

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

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

As soon as Mary was 18 she decided to invest $\mathrm{\$}15000$ of this money in an account of the same type earning 0.4% interest per month. She withdraws $\mathrm{\$}1000$ every year on her birthday to buy herself a present. Determine how long it will take until there is no money in the account.

On that day, Sergei made 12 successive lifts. Find the total combined weight of these lifts.

Determine the corresponding eigenvectors.

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

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

Mary’s grandparents wished for the amount in her account to be at least $\mathrm{\$}20000$ the day before she was 18. Determine the minimum value of the monthly deposit $\mathrm{\$}x$ required to achieve this. Give your answer correct to the nearest dollar.

Justify whether it is valid to use the regression line y on x to estimate Jerome’s examination score.

Write down the common difference, $d$.

Find the total amount of fuel pumped into the tank in the first $8$ hours.

Show that the tank will never be completely filled using this pump.

The daily amount of antibiotic Ted receives will first be less than 0.06 mg on the $k$ th day. Find the value of $k$.

Use Giovanni's diagram to find the length of BX, the horizontal displacement of the Tower.

Find the annual interest rate, $r$.

Write down the other two linear equations in $m$, $n$ and $p$.

Each triangular pane has an area of $1.84 {\text{m}}^2$.

Find the total area of the decorative glass face, if the maximum number of complete levels were built. Express your area to the nearest ${\text{m}}^2$.

Hence, or otherwise, find A^–1.

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

Using mathematical induction, prove that Mⁿ can be written as a quadratic expression in M for all positive integers n ≥ 3.

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

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

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

Find the common ratio in terms of $a$.

Write down the number of hours that the pump was pumping fuel into the tank.

Find the value of $m$ and of $n$.

Given that B , and C , find X.

Find A⁻¹.

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

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

Find $\frac{\text{d}A}{\text{d}r}$.

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

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.

The relationship between x and y can be modelled using the formula y = kxⁿ, where k ≠ 0 , n ≠ 0 , n ≠ 1.

By expressing ln y in terms of ln x, find the value of n and of k.

Giovanni adds a point D to his diagram, such that BD = 45 m, and another triangle is formed.

Find the angle of elevation of A from D.

After the four-year period, Siân withdraws $40000$ AUD from her savings account and uses this money to buy a car. It is known that the car will depreciate at a rate of $18$ % per year.

The value of the car will be $2500$ AUD after $t$ years.

Find the value of $t$.

Show that .

Find an approximation for the real interest rate for the money invested in the account.

Write down the value of $\underset{i=1}{\overset{9}{\text{Σ}}}{u}_i$.

Find the value of $u_1$.

Use Giovanni’s diagram to show that angle ABC, the angle at which the Tower is leaning relative to the
horizontal, is 85° to the nearest degree.

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

State which option Bryan should choose. Justify your answer.

State which option Bryan should choose. Justify your answer.

Find the sum of the first 8 terms.

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

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

Find D^–1.

Find the amount Phil would owe the bank after 20 years. Give your answer to the nearest dollar.

the annual interest rate of the loan.

Find $S$₁₀₀.

Write down the inverse of the matrix

A = 

det (2A − B).

Find CD.

Find the amount of antibiotic, in mg, that Ted receives on the fifth day.

option 2.

Yejin has just turned 28 years old. She currently has no retirement savings. She wants to save part of her salary each month into her annuity fund.

Calculate the amount Yejin needs to save each month, to meet her retirement goal.

The sum of the first $k$ terms is greater than $2500$.

Find the smallest possible value of $k$.

the number of months it will take for Paul to repay the loan.

Find the café’s profit during the 11th week.

Find the value of $a$ when $S_{\infty }=76$.

Find the exact value of $S_k$.

Show that the total value of Phil’s savings after 20 years is $\frac{({1.02}^{20}-1)P}{(1.02-1)}$.

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

The actual volume of the asteroid is found to be $6.074\times {10}^6 {\text{km}}^3$.

Find the percentage error in James’s estimate of the volume.

Calculate the total distance Carlos runs in the first year.

Find the least value of n for which S_n > 163.

Deduce that the eigenvalues are real if A is symmetric.

Find AB.

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

Show that there will be approximately 2645 fish in the lake at the start of 2020.

Find $d$, giving your answer as an integer.

find $x$ and $y$ in terms of $n$.

Calculate the amount of this investment, in INR, in this account after five years.

2A − B.

Hence, find the total number of panes in a glass face with $18$ levels.

Calculate James’s estimate of its volume, in ${\text{km}}^3$.

(i)     Write down a similar expression for $A_3$ and $A_4$.

(ii)     Hence show that the amount in Mary’s account the day before she turned 10 years old is given by $251({1.004}^{120}-1)x$.

The following diagram shows [AB], with length 2 cm. The line is divided into an infinite number of line segments. The diagram shows the first three segments.

The length of the line segments are $p\text{ cm},p^2\text{ cm},p^3\text{ cm},\dots$, where .

Show that $p=\frac{2}{3}$.

Write down the inverse, A^–1.

Hence find the value of n such that $\sum _{k=1}^n{x}_k=861$.

Calculate the total amount of the drug, in mg, that she receives.

Verify that

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

Find the volume of the smallest slice of pie.

Show that the volume of the tank is $624 000 {\text{m}}^3$, correct to three significant figures.

Calculate the time, in minutes, it takes for light from the Sun to reach the Earth.

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

Hence solve the system of equations

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

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

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

Hence show that M is a singular matrix.

Solve this matrix equation.

Tommaso wants to invest his money in an account such that his investment will increase to $1.5$ times the initial amount in $5$ years. Assume the account pays a nominal annual interest of $r\%$ compounded quarterly.

Determine the value of $r$.

An infinite geometric series is given as $S_{\mathrm{\infty }}=a+\frac{a}{\sqrt{2}}+\frac{a}{2}+\dots$, $a\in {\mathbb{Z}}^{+}$.

Find the largest value of $a$ such that .

Find the percentage error in your estimate in part (d).

Find the approximate number of fish in the lake at the start of 2042.

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

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

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

An orchestra has a sound intensity of 6.4 × 10⁻³W m⁻² . Calculate the intensity level, $L$ of the orchestra.

Find AB.

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

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

Find the inverse of the matrix .

Calculate angle $\mathrm{B}\hat{\mathrm{C}}\mathrm{D}$.

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

Write down the first three non-zero terms of $w_n$.

Find $h$, the height of the tank.

Given that AB = , find $a$.

Find the common ratio of the sequence.

Write down, in terms of $r$ and $h$, an equation for the volume of this water container.

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

Verify that .

A rock concert has an intensity level of 112 dB. Find the sound intensity, $S$.

Bryan chooses option B. The car dealership invests the money Bryan pays as soon as they receive it.

If they invest it in an account paying 0.4 % interest per month and inflation is 0.1 % per month, calculate the real amount of money the car dealership has received by the end of the 6 year period.

Find the matrix A².

Write down a formula for $A$, the surface area to be coated.

Find the least number of cans of water-resistant material that will coat the area in part (g).

If $M^2$ is equal to , find the value of $a$.

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

$z=2\text{i}$.

Write down the equation of the regression line $y$ on $x$, in the form $y=mx+c$.

Find, in hours, the shortest time from sunrise to sunset at point $\text{A}$ that is predicted by this model.

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

Bryan’s car depreciates at an annual rate of 25 % per year.

Find the value of Bryan’s car six years after it is purchased.

Show that (I − M)² = I − M where I is the identity matrix.

Diagram $n$ is formed with 52 sticks. Find the value of $n$.

Charlie ran on day $20$ of his fitness programme.

Find the values of $a$ for which the sum to infinity of the series exists.

Find the amount of fuel pumped into the tank in the $\text{13th}$ hour.

Find the number of triangular panes in the $12\text{th}$ level.

Write down the amount of the loan.

Find the amount of Daisy’s final payment.

the amount of interest paid by Daisy.

Find how much money Daisy saved by making one final payment after $5$ years.

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

Find the total cost of buying 2 tickets in each of the first 16 rows.

Calculate the value of Daisy’s investment after $2$ years.

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

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

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 the distance from the base of this ladder to the top rung.

Show that $a+d=1$.

Calculate Abdallah’s estimate for the area.

Show that $A=\pi r^2+\frac{1000000}{r}$.

Write down the second term of this sequence.

Hence, show that .

Find an expression for $bc$ in terms of $a$.

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.

Write down the value of the iron in the form $a\times {10}^k$ where $1\le a<10 , k\in \mathrm{\mathbb{Z}}$.

Find the maximum number of complete levels that Maegan can build.

Let $p=c^2$ and $q=c^3$. Find the value of $\sum _{n=1}^{20}{u}_n$.

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

Calculate the amount Pietro will have in his account after $5$ years. Give your answer correct to $2$ decimal places.

Find −3A. 

Find A².

Hence or otherwise find an equation for $L$ in the form $L\left(t\right)=p \sin (qt+r)+d$, where $p, q, r, d\in \mathrm{ℝ}$.

Let X be a 2 × 2 matrix such that AX = B. Find X.

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

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.

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

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

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

Suggest a limitation of the given model.

Let B = .

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

the total number of seats in the concert hall.

the number of seats in the last row.

Find the average number of visitors per concert in $2020$.

It is assumed that the concert hall will host $50$ concerts each year.

Use the average number of visitors per concert per year to predict the total number of people expected to attend the concert hall from when it opens until the end of $2025$.

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

the value of $d$;

Write down an equation, in terms of $u_1$ and $d$, for the amount of the drug that she receives on the eleventh day.

Find the value of this minimum area.

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

Find the value of d.

Find the value of $m$.

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

Show that $\text{BD}=93\text{ m}$ correct to the nearest metre.

Calculate the value of r.

David wishes to withdraw $5000 at the end of each year for a period of $n$ years. Show that an expression for the minimum value of $Q$ is

$\frac{5000}{1.028}+\frac{5000}{{1.028}^2}+\dots +\frac{5000}{{1.028}^n}$.

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

Hence, write down the general solution of the system.

Show that $F=3240$.

Find the total amount John has paid to insure his bicycle for the first 5 years.

 Show that det(M²) is positive.

Find the value of $n$ such that $u_n=0$.

Consider a geometric sequence where the first term is 768 and the second term is 576.

Find the least value of $n$ such that the $n$th term of the sequence is less than 7.

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

The rate of inflation during this 10 year period is 1.5% per year.

Calculate the real value of her savings after 10 years.

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

 Find A(A^–1B + 2A^–1)A.

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

Write down an expression for $A_n$ in terms of $x$ on the day before Mary turned 18 years old showing clearly the value of $n$.

Calculate the total distance, in kilometres, Rosa runs in the first 50 training sessions.

Find BA.

Find the value of her investment after a period of $5$ years.

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

Find an expression for the width of $U_n$ in centimetres.

hence find ${\mathit{b}}_1$.

Find the $pH$ value for a solution in which the hydrogen ion concentration is $5.2\times {10}^{-8}$.

Find the value of the bicycle during the 5th year. Give your answer to two decimal places.

Hence find the real value of Sophia’s investment at the end of $5$ years.

A game is played in which the arrow attached to the centre of the disc is spun and the sector in which the arrow stops is noted. If the arrow stops in sector $1$ the player wins $10$ points, otherwise they lose $2$ points.

Let $X$ be the number of points won

Find $\text{E}\left(X\right)$.

Find the value of $k$.

At the end of the $5$ years, Imon withdrew $x \text{SGD}$ from the fixed deposit account and reinvested this into a super-savings account with a nominal annual interest rate of $5.7\%$, compounded half-yearly.

The value of the super-savings account increased to $20 000 \text{SGD}$ after $18$ months.

Find the value of $x$.

Find the value of $r$.

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

Find the hydrogen ion concentration in a solution with $pH 4.2$. Give your answer in the form $a\times 10k$ where $1\le a<10$ and $k$ is an integer.

The following diagram shows [CD], with length $b\text{ cm}$, where $b>1$. Squares with side lengths $k\text{ cm},k^2\text{ cm},k^3\text{ cm},\dots$, where , are drawn along [CD]. This process is carried on indefinitely. The diagram shows the first three squares.

The total sum of the areas of all the squares is $\frac{9}{16}$. Find the value of $b$.

Write down $z_1$ and $z_2$ in exponential form, with a constant modulus.

Write your answer to part (b)(ii) in the form $a\times {10}^k$, where .

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

Given that $q=8d^2$, find the other two real roots.

Daniella ran on day $20$ of her fitness programme.

A triangular field $\text{ABC}$ is such that $\text{AB}=56 \text{m}$ and $\text{BC}=82 \text{m}$, each measured correct to the nearest metre, and the angle at $\text{B}$ is equal to $105°$, measured correct to the nearest $5°$.

Calculate the maximum possible area of the field.

Find the value of $k$.

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

Determine whether it will reach this height.

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

Calculate Katya’s approximation of $\mathrm{\pi }$, correct to four decimal places.

Calculate the percentage error in using Katya’s four decimal place approximation of $\mathrm{\pi }$, compared to the exact value of $\mathrm{\pi }$ in your calculator.

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

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

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

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 ${\mathit{T}}^{\infty }=\left(\begin{array}{cc}\frac{10}{17}& \frac{10}{17}\\ \frac{7}{17}& \frac{7}{17}\end{array}\right)$.

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

(i)   at $(4, 0)$.

(ii)  at $(-4, 0)$.

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

Plot the position of $z$ on an Argand Diagram.

After the enlargement scale factor equals zero, Ben continues to rotate the image for another two revolutions.

Describe the animation for these two revolutions, stating the final position of the sequence of squares.

Express $z$ in the form $z=a{\text{e}}^{\text{i}b}$, where $a, b\in \mathrm{ℝ}$, giving the exact value of $a$ and the exact value of $b$.

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

the eigenvectors.

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

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

Hence find the image of $(1, 1)$ after it is rotated $135°$ and enlarged.

Given the width of a pixel is approximately $0.025 \text{cm}$, find the number of squares in the final image.

show that ${\mathit{b}}_n=\left(\begin{array}{c}8\left(1-2^{-n}\right)\\ 8\left(1-2^{-n}\right)\end{array}\right)$.

Hence or otherwise, find the coordinates of the top left-hand corner in $U_7$.

Find, $s$, in the form $s\left(\theta \right)=m\theta +c$.

Find the phase shift of $I_{\text{T}}$.

Use the trapezoidal rule with an interval width of $0.4$ to estimate the total amount of carbon dioxide emitted during these two hours.

The real amount of carbon dioxide emitted during these two hours was $72$ tonnes.

Find the percentage error of the estimate found in part (a).

Find the amount they will pay the bank each month.

Using your answers to parts (a) and (b)(i), calculate how much interest they will have paid in total during the first $10$ years.

Calculate the percentage increase in applications from the first year to the second year.

Find an expression for $u_n$.

Find the number of student applications the university expects to receive when $n=11$. Express your answer to the nearest integer.

Write down an expression for $v_n$ .

Calculate the total amount of acceptance fees paid to the university in the first $10$ years.

Find $k$.

Write down the 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 $t$.