DP Mathematics: Applications and Interpretation Questionbank

The following diagram shows a corner of a field bounded by two walls defined by lines $L_1$ and $L_2$. The walls meet at a point $\text{A}$, making an angle of $40°$.

Farmer Nate has $7 \text{m}$ of fencing to make a triangular enclosure for his sheep. One end of the fence is positioned at a point $\text{B}$ on $L_2$, $10 \text{m}$ from $\text{A}$. The other end of the fence will be positioned at some point $\text{C}$ on $L_1$, as shown on the diagram.

He wants the enclosure to take up as little of the current field as possible.

Find the minimum possible area of the triangular enclosure $\text{ABC}$.

Solve the equation ${\sec }^{2}x+2\tan x=0,0⩽x⩽2\pi$.

Find the value of $\sin \frac{\pi }{4}+\sin \frac{3\pi }{4}+\sin \frac{5\pi }{4}+\sin \frac{7\pi }{4}+\sin \frac{9\pi }{4}$.

Show that $\frac{1-\cos 2x}{2\sin x}\equiv \sin x,x\ne k\pi$ where $k\in \mathbb{Z}$.

Use the principle of mathematical induction to prove that

$\sin x+\sin 3x+\dots +\sin (2n-1)x=\frac{1-\cos 2nx}{2\sin x},n\in {\mathbb{Z}}^{+},x\ne k\pi$ where $k\in \mathbb{Z}$.

Hence or otherwise solve the equation $\sin x+\sin 3x=\cos x$ in the interval .

Use the cosine rule to show that $r^2-12\sqrt{3}r+144-p^2=0$.

Calculate the two corresponding values of PQ.

Hence, find the area of the smaller triangle.

Determine the range of values of $p$ for which it is possible to form two triangles.

Use the cosine rule to find the two possible values for AC.

Find the difference between the areas of the two possible triangles ABC.

Show that $\text{sin}\theta =\frac{\sqrt{15}}{4}$.

Find the two possible values for the length of the third side.

Find the set of values of $k$ that satisfy the inequality .

The triangle ABC is shown in the following diagram. Given that , find the range of possible values for AB.

Find the value of $\text{tan}\theta$.

Line $L$ passes through the origin and has a gradient of $\text{tan}\theta$. Find the equation of $L$.

Find the derivative of $f$.

The following diagram shows the graph of $f$  for 0 ≤ $x$ ≤ 3. Line $M$ is a tangent to the graph of $f$ at point P.

Given that $M$ is parallel to $L$, find the $x$-coordinate of P.