DP Mathematics: Applications and Interpretation Questionbank

Express $\overrightarrow{\text{AB}}$ in terms of $m$.

Find the value of $m$.

Consider a unit vector $\mathit{u}$, such that $\mathit{u}=p\mathit{i}-\frac{2}{3}\mathit{j}+\frac{1}{3}\mathit{k}$, where $p>0$.

Point $\text{C}$ is such that $\overrightarrow{\text{BC}}=9\mathit{u}$.

Find the coordinates of $\text{C}$.

In the case that $p<0$, determine whether the lines intersect.

Find the possible value(s) for $p$.

Find the value of $t$ when the ship will be closest to the lighthouse.

Find the position vector $\overrightarrow{\text{OS}}$ of the ship at time $t$ hours.

An alarm will sound if the ship travels within $20$ kilometres of the lighthouse.

State whether the alarm will sound. Give a reason for your answer.

Write down an expression for the position vector $\mathit{r}$ of the ship, $t$ hours after $1:00 \text{pm}$.

Find the time at which the bearing of the ship from the harbour is $045˚$.

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

Find the time at which the aircraft is 4 km above the ground.

verify that it would pass directly over the airport.

state the height of the aircraft at this point.

find the time at which it would fly directly over the airport.

Given that the velocity of the aircraft, after the adjustment of the angle of descent, is $\left(\begin{array}{c}-150\\ -50\\ a\end{array}\right)\text{km}{\text{h}}^{-1}$, find the value of $a$.

Write down a vector equation for the displacement, r of the aircraft in terms of $t$.

Find the direct distance of the aircraft from the airport at this point.

find the value of $a$;

determine the coordinates of the point of intersection P.

Show that $\overrightarrow{\text{AB}}=\left(\begin{array}{c}2\\ -1\\ 1\end{array}\right)$.

Find a vector equation for $L$.

Find the value of $p$.

The point D has coordinates $(q^2,0,q)$. Given that $\overrightarrow{\text{DC}}$ is perpendicular to $L$, find the possible values of $q$.

Show that $\overrightarrow{\text{AB}}=\left(\begin{array}{c}6\\ 8\\ -5\end{array}\right)$

Find a vector equation for L.

Point C (k , 12 , −k) is on L. Show that k = 14.

Find $\overrightarrow{\text{OB}}\bullet \overrightarrow{\text{AB}}$.

Write down the value of angle OBA.

Point D is also on L and has coordinates (8, 4, −9).

Find the area of triangle OCD.

Find a vector equation of the line that passes through P and Q.

The line through P and Q is perpendicular to the vector 2i $+$ nk. Find the value of $n$.

Find the gradient of $L$.

Find u.

Find the acute angle between $y=x$ and $L$.

Find $\left(f\circ f\right)\left(x\right)$.

Hence, write down $f^{-1}\left(x\right)$.

Hence or otherwise, find the obtuse angle formed by the tangent line to $f$ at $x=8$ and the tangent line to $f$ at $x=2$.

The vector equation of line $L$ is given by r $=\left(\begin{array}{c}-1\\ 3\\ 8\end{array}\right)+t\left(\begin{array}{c}4\\ 5\\ -1\end{array}\right)$.

Point P is the point on $L$ that is closest to the origin. Find the coordinates of P.

Find a vector equation for L₁.

The vector $\left(\begin{array}{c}2\\ p\\ 0\end{array}\right)$ is perpendicular to $\overrightarrow{\text{AB}}$. Find the value of p.

Find $\left|\overrightarrow{\text{AB}}\right|$.

Find the value of $y$.

Show that $\overrightarrow{\text{AC}}=\left(\begin{array}{c}8\\ -10\\ -1\end{array}\right)$.

Find the angle between $\overrightarrow{\text{AB}}$ and $\overrightarrow{\text{AC}}$.

Find the area of triangle ABC.

Find $c$.

Show that the two submarines would collide at a point P and write down the coordinates of P.

Find the distance between the two submarines at this time.

A second line, $L_2$, is parallel to $L_1$ and passes through (1, 2, 3).

Write down a vector equation for $L_2$.

Find the value of t when the two submarines are closest together.

Find the value of t when submarine B passes through P.