Geometry and Trigonometry — Free Maths AI HL Practice Questions

Find the value of $c$ and hence write down the equation of the model. [2 marks]

A rival company charges no booking fee but an hourly rate of 6.50 USD. Determine the number of hours at which both companies charge the same amount. [1 mark]

Hence, find the minimum number of candles that must be sold each week for the business to break even. [2 marks]

The business owner claims the model predicts that selling $0$ candles results in a loss of $\text{USD }70$. Evaluate whether this interpretation is meaningful in the context of the model. [2 marks]

A second aircraft flies at a different altitude and follows the path $$L_2: \mathbf{r} = \begin{pmatrix}12\\20\\4\end{pmatrix} + s\begin{pmatrix}4\\-2\\0\end{pmatrix}$$ where $s$ is measured in minutes after 14:00. Determine the time, to the nearest second, at which the two aircraft are closest, and find the minimum distance between them. [3 marks]

Analyse whether a collision between the two aircraft is possible. Justify your answer with reference to your results from parts (a) and (b). [2 marks]

Find the gradient of the line. [2 marks]

A second line has equation $y = -\dfrac{1}{2}x + k$, where $k$ is a constant. Determine the value of $k$ such that the two lines intersect at $x = 2$. [2 marks]

Calculate the length of the ladder. Give your answer correct to 3 significant figures. [2 marks]

The base of the ladder is pushed $0.3\,\text{m}$ further from the wall. Determine the new angle the ladder makes with the ground, correct to 3 significant figures. [1 mark]

Show that $h = (x + 50)\tan 35^\circ$. [1 mark]

Find the value of $x$, giving your answer correct to 3 significant figures. [2 marks]

Hence find the height of the cliff, giving your answer correct to 3 significant figures. [1 mark]

Using triangle $ACD$ and your result from part (a), show that $$x = \frac{50\tan 35°}{\tan 52° - \tan 35°}$$ [3 marks]

Hence calculate the height of the cliff $h$, giving your answer correct to 3 significant figures. [2 marks]

The surveyor's measuring instrument is held at eye level, $1.6\,\text{m}$ above the ground. Determine the percentage error introduced by ignoring this height when calculating $h$. [2 marks]

Calculate the height of the flagpole. Give your answer correct to 3 significant figures. [2 marks]

A second surveyor stands at point $Q$, which is $8\,\text{m}$ further from the base of the flagpole than point $P$, on the same level ground. Determine the angle of elevation to the top of the flagpole from point $Q$, giving your answer correct to 1 decimal place. [2 marks]

Calculate the distance $AB$, giving your answer correct to 3 significant figures. [3 marks]

The surveyor claims that if the angle $A\hat{C}B$ were increased to $90°$ while $AC$ and $BC$ remained unchanged, the distance $AB$ would increase by more than $10\,\text{m}$. Determine whether this claim is correct, justifying your answer. *(: triangle with vertices labelled $A$, $B$, $C$; no side or angle values marked.)* [3 marks]

Hence write down the two possible values of angle $B$, correct to the nearest degree. [1 mark]

Determine which value(s) of $B$ give a valid triangle. For each valid triangle, find the corresponding length of side $c$, correct to one decimal place. [4 marks]

Point $D$ lies on $BC$ such that $AD$ bisects angle $\hat{A}$. Using the angle bisector theorem and the cosine rule, determine the length of $AD$, correct to 3 significant figures. [4 marks]

Calculate the length of side $QR$, correct to 3 significant figures. [3 marks]

Determine the equation of the cost function in the form $C = mx + c$, where $m$ and $c$ are constants. [2 marks]

Each panel is sold for USD 180. Determine the minimum number of panels the company must sell to make a profit, and justify your answer. [2 marks]

Write down the meaning of the gradient in this context. [1 mark]

Determine the equation of the line in the form $T = mt + c$. [2 marks]

The solution is considered safe to handle when its temperature falls below $30\,^\circ\text{C}$. A technician claims the solution will be safe to handle after exactly 25 minutes. Determine whether the technician's claim is correct, justifying your answer. [1 mark]

Interpret the value of $m$ and the value of $c$ in context. [2 marks]

The business owner claims that selling $180$ candles per month is sufficient to achieve a monthly profit of at least $\text{USD }1100$. Determine whether this claim is correct, justifying your answer. [3 marks]

Find the coordinates of the midpoint $M$ of side $QR$. [1 mark]

Calculate the area of triangle $PQR$ in square units. [2 marks]

A straight path is built from $P$ through $M$ and extended until it meets side $PR$ at point $N$. Find the coordinates of $N$ and hence determine the ratio $PN : NR$. [3 marks]

Using your answer to part (a) and the angle of elevation from $P$, form an equation in $x$ alone and hence show that $$x = \frac{50\tan 32^\circ}{\tan 48^\circ - \tan 32^\circ}$$ [3 marks]

Calculate the height of the cliff $h$, correct to 3 significant figures. [1 mark]

The surveyor estimates that each angle of elevation could have a measurement error of up to $\pm 1^\circ$. Determine the maximum possible value of $h$ under this error, and hence evaluate whether the measurement error has a significant effect on the calculated height. [4 marks]

Given that $h = 2d$, find the value of $\theta$. [3 marks]

A safety regulation states that for a ladder of this length, the foot must be placed at least $1.5\,\text{m}$ from the wall. Determine whether the angle found in part (b) satisfies this regulation. Justify your answer. [1 mark]

The wind changes and the kite moves so that the string now makes angle of $40^\circ$ with the horizontal. The length of the string remains the same. (i) Determine the new vertical height of the kite above the ground. Give your answer correct to 3 significant figures. [2 marks]
(ii) Determine how much further the kite is horizontally from its original position. Give your answer correct to 3 significant figures. [2 marks]

Safety regulations require that the angle of inclination does not exceed $8°$ for the same vertical rise of $0.8\,\text{m}$. (i) Determine the minimum length of the new ramp that satisfies this regulation, correct to 3 significant figures. [2 marks]
(ii) The new ramp must fit within a loading bay of horizontal length $5.70\,\text{m}$. Determine whether the new ramp fits within the loading bay. [2 marks]

Determine the distance $AD$. Give all answers correct to 3 significant figures. [3 marks]

A farmer wants to build a fence from one vertex perpendicular to the opposite side. Determine the length of the shortest such fence, justifying your answer. Give all answers correct to $3$ significant figures. [3 marks]

Determine the bearing of $R$ from $P$. Give all answers correct to 3 significant figures. [3 marks]

Determine the distance $AB$. Give all answers correct to $3$ significant figures. [4 marks]

The cable consists of two straight segments, from $A$ to $P$ and from $P$ to $B$, where $P$ has $x$-coordinate $x$. Show that the total cable length $L$, in kilometres, satisfies $$L = \sqrt{(x-2)^2 + \left(\tfrac{1}{2}x - 3\right)^2} + \sqrt{(x-10)^2 + \left(\tfrac{1}{2}x - 5\right)^2}.$$ [3 marks]

Using your GDC, find the minimum value of $L$ and the coordinates of $P$ that achieve this minimum. [2 marks]

The reflection of $A(2,5)$ across the line $y = \dfrac{1}{2}x + 2$ is the point $A'(6, 1)$. Explain, using a geometric argument, why the minimum total cable length equals the straight-line distance $A'B$, and verify this numerically. [2 marks]

Let the coordinates of $Q$ be $(x,\,-2x+3)$. Show that the sum of squared distances $$\mathcal{S} = QS^2 + QT^2$$ can be expressed as $\mathcal{S} = 10x^2 - 20x + 100$. [3 marks]

Find the coordinates of $Q$ that minimise $\mathcal{S}$, and show that $Q$ is the midpoint $M$ of $ST$. [2 marks]

The perpendicular bisector of $ST$ passes through $M$. Using this property, explain why $M$ minimises $\mathcal{S}$ for any point on line $ST$, and deduce what happens to $\mathcal{S}$ as $Q$ moves away from $M$ along the belt. [2 marks]

Show that $\cos(\theta(t))=\sqrt{1-t^2}$ for $0\leq t\leq 1$, and hence prove that $\omega(t)=\dfrac{1}{\sqrt{1-t^2}}$. [4 marks]

Using your result from (b), evaluate $\displaystyle\int_{0}^{1/2}\omega(t)\,dt$ and verify that this equals $\theta\!\left(\tfrac{1}{2}\right)-\theta(0)$. Hence interpret, in terms of arc length on the unit circle, what this integral represents. [3 marks]

(i) Show that the angular velocity is $\omega(t) = \dfrac{2\pi}{3}\cos(2t)$. [2]
(ii) Calculate the maximum angular speed of the arm, justifying your answer using properties of $\cos(2t)$. [2 marks]

The tip of the arm can only reach points on the unit circle corresponding to angles within the range of $\theta(t)$. - Determine the range of $\theta(t)$. - Hence deduce the arc of the unit circle that the tip can never reach, giving your answer as a set of angles $\theta$ that are inaccessible. - Verify that the point $(-1, 0)$ lies in this inaccessible arc. [3 marks]

A second surveyor, working from a different reference, models a triangle $PQR$ in which $PQ = 120\,\text{m}$, $QR = 150\,\text{m}$, angle $Q = 36.9°$, and the side opposite angle $P$ is $PR = d\,\text{m}$. Using the Law of Sines applied to triangle $ABC$ from part (a), with the known angle $B$ and its opposite side $CA = 90\,\text{m}$, find the two possible values of angle $A$, correct to one decimal place. [4 marks]

Justify why the Law of Sines produces two candidate values for angle $A$, and evaluate which value is geometrically valid for triangle $ABC$. Support your reasoning with reference to the angle sum of a triangle and the relative lengths of the sides. Clarification of labelling for (b): In triangle $ABC$, angle $B \approx 36.9°$ is opposite side $CA = 90\,\text{m}$. Side $BC = 150\,\text{m}$ is opposite angle $A$. Apply the Law of Sines to find angle $A$. [3 marks]

Calculate $h$, the vertical height of the top of the tower above the level ground, correct to the nearest metre. [3 marks]

Determine the angle of elevation of the top of the tower from point $B$. Hence justify whether the angle of elevation from $B$ is greater than, equal to, or less than that from $A$, explaining your reasoning in terms of the relative distances $AT$ and $BT$. [3 marks]