Find $S_{\infty}$ in terms of $r$.

Given that $S_5=160$, find $r$.

Hence find $S_{\infty}$.

Write $C_2$ as an inequality in $x$ and $y$, where $z=x+iy$.

From the diagram, state the maximum and minimum values of $\mathrm{Re}(z)$ on the boundary.

Find the range of $\arg(z-3)$ for points on the boundary above the real axis.

A point $w$ lies in $C_2$ with $\arg(w-3)=\pi/6$ and $\mathrm{Im}(w)>0$. Find the largest possible value of $|w|$.

Write the system in the form $A x=b$ and find $A^{-1}$.

Hence solve for $(x,y)$.

Verify your solution using technology (GDC), showing appropriate working.

Find $a$ and $c$ so that $f$ is continuous and differentiable at $x=2$.

With those values, find $f^{-1}(x)$ for the branch $x\ge 2$, stating its domain.

Use your GDC to find Pearson’s correlation coefficient $r$ and the equation of the least-squares regression line of $y$ on $x$.

Interpret the value of $r$ in context.

Use the regression line to predict $y$ when $x=9$, commenting on the reasonableness of your prediction.

Show that the gradient of $M$ at $x=0$ is $1$. Hence write an equation for the tangent at $O$.

Using your GDC, find the $x$-coordinate $x_P>0$ of the intersection of $M$ with the line $y=(1/2)x$.

Find the exact area under $M$ between $x=0$ and $x=\ln 2$.

Hence evaluate $\int_{0}^{\ln 2}\left[xe^{-2x}-(1/2)x\right],dx$.

Find the probability that, in a sample of $12$ components, at least $11$ are non-defective.

Find the expected number of non-defective components in $50$ trials.

Comment on whether a binomial model is suitable in this context.

Find $f'(x)$ and solve $f'(x)=0$.

Show that the stationary point is a maximum.

Find the maximum value of $f$.

Find the angle between $\mathbf{a}$ and $\mathbf{b}$.

Find the component of $\mathbf{a}$ in the direction of $\mathbf{b}$.

Find a unit vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$.

Find the probability that in a $10$-minute interval there are exactly $3$ calls.

Two independent lines have rates $12$ per hour and $7$ per hour. Find the mean and variance of the total number of calls per hour.

Briefly justify why a Poisson model is appropriate here.

Using the matrix, list the neighbours of vertex $C$ and their edge weights.

Verify the statement in the image: “The shortest path from $D$ to $F$ has total weight $6$” by giving one such path and showing its weight.

Use Dijkstra’s algorithm (or a clear GDC table) to find the shortest distances from $A$ to all vertices. State the distance to $E$.

From the graph, state the equation of the vertical asymptote and the $x$-intercept.

Describe the transformation that maps $y=\ln x$ to $y=\ln(x-4)$.

Solve $\ln(x-4)=2$.

Find $\frac{d}{dx}\left[\ln(x-4)\right]$.

Use your GDC to find the least-squares exponential regression and state $k$ and $r$, correct to three significant figures.

Predict $M$ at $t=5$. State the units.

Comment on extrapolation in this context.

Find $k$.

Write the model for $y(t)$. You may give $k$ exactly or as a decimal.

Find $P(X>54.5)$. Use your GDC appropriately and show the $z$-calculation.

A sample of $36$ items is taken. Find a $95$ confidence interval for the mean weight.

Interpret your interval in the context of item weights.

Find $\mathbf{s}_1$ and $\mathbf{s}_2$.

Find the steady-state vector $\mathbf{s}$ and state its meaning in context.