DP Mathematics: Applications and Interpretation Questionbank

Find an expression for the acceleration of the ball at time $t$.

Show that the ball is moving in a circle with its centre at $\text{O}$ and state the radius of the circle.

Hence show that the velocity of the ball is always perpendicular to the position vector of the ball.

Find an expression for the velocity of the ball at time $t$.

Find the value of $t$ at the instant the string breaks.

How many complete revolutions has the ball completed from $t=0$ to the instant at which the string breaks?

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.

Find $\text{FÂO}$, the angle the rope makes with the platform.

Find the velocity vector of $P$.

Show that the acceleration vector of $P$ is never parallel to the position vector of $P$.

Given that v is perpendicular to B, find the value of $d$.

The force, F, produced by P moving in the magnetic field is given by the vector equation F = $a$v × B, $a\in {\mathbb{R}}^{+}$.

Given that | F | = 14, find the value of $a$.

Find and simplify an expression for a • b in terms of $t$.

Hence or otherwise, find the values of $t$ for which the angle between a and b is obtuse .

Let a = $\left(\begin{array}{c}2\\ k\\ -1\end{array}\right)$ and b = $\left(\begin{array}{c}-3\\ k+2\\ k\end{array}\right)$, $k\in \mathbb{R}$.

Given that a and b are perpendicular, find the possible values of $k$.

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

Let $\overrightarrow{\text{AC}}=\left(\begin{array}{c}3\\ 0\\ 0\end{array}\right)$. Find $\mathrm{B}\hat{\mathrm{A}}\mathrm{C}$.

Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.
This is shown in the following diagram.

The vectors p , q and r are shown on the diagram.

Find p•(p + q + r).

Find $\overrightarrow{\text{PQ}}$.

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

Find the angle between PQ and PR.

Find the area of triangle PQR.

Hence or otherwise find the shortest distance from R to the line through P and Q.

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 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 $n$.

The magnitudes of two vectors, u and v, are 4 and $\sqrt{3}$ respectively. The angle between u and v is $\frac{\pi }{6}$.

Let w = u − v. Find the magnitude of w.

parallel.

perpendicular.

Hence or otherwise, find the area of the triangle ABC.

Find the vector $\overrightarrow{\text{AC}}$.

Find the vector $\overrightarrow{\text{AB}}$.