Geometry and Trigonometry — Free Maths AI SL Practice Questions

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

A competitor charges according to the model $C = 6t + 5$. Determine the number of hours for which both companies charge the same amount, and state which company is cheaper for a $4$-hour hire. [2 marks]

Find the monthly rental cost for an office with a floor area of $75\,\text{m}^2$. [1 mark]

The business has a monthly rental budget of USD 2000. Determine the maximum floor area the business can rent, and comment on whether the $y$-intercept of the model has a meaningful interpretation in this context. [3 marks]

Determine the coordinates of point $R$. Airport B is located at point $B(k,\,61)$, and the aircraft's flight path passes directly over airport B at some time after 14:24. [2 marks]

(i) Determine the value of $k$. [1]
(ii) Hence, determine the time at which the aircraft flies directly over airport B. [2 marks]

State the value of $m$ and explain its meaning in this context. [1 mark]

Determine the time at which the tank will be empty. State one assumption required for your answer to be valid. [2] > 5-mark version (if total must stay at 5): (a) Find the equation of the line in the form $W = mt + c$, showing all working. [3] (b) State the value of $m$ and explain its meaning in this context. [1] (c) Determine the time at which the tank will be empty. [1 mark]

Find the coordinates of the point where line $L$ intersects the $x$-axis. [1 mark]

A second line, $M$, is perpendicular to $L$ and passes through $Q(6,\,1)$. Determine the coordinates of the point where line $M$ intersects the $y$-axis. [3 marks]

Find the length of the ladder $l$. [2 marks]

The top of the ladder rests against the wall at height $h$. Find $h$, giving your answer correct to 3 significant figures. [2 marks]

Show that $h = (d - 50)\tan 52^\circ$. [1 mark]

Hence find the height of the cliff. Give your answer correct to 3 significant figures. [3 marks]

Hence, determine the value of $x$, correct to 3 significant figures. [3 marks]

Hence, determine the height $h$, correct to 3 significant figures. [1 mark]

The surveyor's clinometer has a measurement uncertainty of $\pm 1^\circ$. Using $34^\circ$ instead of $35^\circ$ for the angle at A gives $h = 171\,\text{m}$ (3 s.f.). Evaluate whether the surveyor's measurement of $h$ is reliable given this uncertainty. [2 marks]

Find the height of the tree. [2 marks]

The surveyor moves closer to the tree until the angle of elevation becomes $50°$. Find how much closer, in metres, the surveyor has moved. [2 marks]

Given that the foot of the ladder is $2.5$ metres from the wall, find the height $y$ of the top of the ladder above the ground. Give your answer correct to 3 significant figures. [3 marks]

The ladder slips so that the foot moves to $3.1$ metres from the wall. The length $L$ of the ladder does not change. Determine the new height of the top of the ladder above the ground, and find the decrease in height. Give both answers correct to 3 significant figures. [3 marks]

Find the distance $AC$. Give your answer correct to 3 significant figures. [4 marks]

The perpendicular distance from $C$ to the line $AB$ represents the width of the river. Find this width, correct to 3 significant figures. [2 marks]

Calculate the length of the perpendicular path $BP$, giving your answer correct to the nearest metre. [3 marks]

Determine the angle that the path $BP$ makes with side $AB$, giving your answer correct to 1 decimal place. [3 marks]

Find the possible values of angle $PQR$. Give your answers correct to the nearest degree. [4 marks]

Hence find the area of triangle $PQR$. Give your answer correct to 3 significant figures. [2 marks]

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

The fixed cost of production is the cost when $n = 0$. A manager claims that producing 40 units costs exactly twice the fixed cost. Determine whether the manager's claim is correct, showing all working. [2 marks]

Calculate the gradient of $L$, giving your answer correct to 3 significant figures. [2 marks]

Using your answers to parts (a) and (b), determine the altitude at which the model predicts a temperature of $-5\,^\circ\text{C}$. [2 marks]

Determine the equation of the line in the form $P = mn + c$, where $m$ and $c$ are constants. [2 marks]

The business owner claims that renting out more scooters always increases daily profit, so the business should aim to rent as many scooters as possible. Using your equation from part (b), identify the minimum number of scooters that must be rented for the business to make a positive daily profit, and evaluate whether the owner's claim is supported by this model. [2 marks]

Determine the equation of line $L$ in the form $y = mx + c$. [2 marks]

A second line $M$ is perpendicular to $L$ and passes through the point $P(0, 5)$. Find the coordinates of the point where $L$ and $M$ intersect, and hence determine the exact distance from $P$ to that intersection point. [2 marks]

Show that the length of side $AC = \sqrt{20}$ m. [2 marks]

Hence calculate the area of triangle $ABC$, giving your answer in exact form. [2 marks]

Calculate the angle of elevation from $Q$ to $T$, giving your answer to the nearest degree. [2 marks]

The surveyor claims that the angle $\angle TPQ$ (the angle at $P$ in triangle $TPQ$, where $T$ is the cliff top) is greater than $10°$. Determine whether this claim is correct, showing all working. [2 marks]

The ladder is repositioned so that it makes angle of $70°$ with the ground. The ladder still has length $6\,\text{m}$. (i) Calculate the new height the ladder reaches up the wall. Give your answer correct to 3 significant figures. [2 marks]
(ii) Determine how much higher up the wall the ladder reaches in this new position compared to its original position. Give your answer correct to 3 significant figures. [2 marks]

Calculate the vertical height of the kite above the ground. Give your answer correct to 3 significant figures. [2 marks]

The wind changes and the kite moves so that its new vertical height is $25\,\text{m}$, while the string length remains $30\,\text{m}$. The kite is now directly above a point Y on the ground. Calculate the horizontal distance XY between points X and Y. Give your answer correct to 3 significant figures. [2 marks]

Calculate the length of the ramp surface, correct to 3 significant figures. [2 marks]

Building regulations require the angle of inclination to be no more than $5^\circ$. The ramp surface length from part (b) is kept fixed. Determine the minimum horizontal length required for the ramp to comply with regulations, correct to 3 significant figures, and state whether the ramp now fits within the original $12\,\text{m}$ horizontal space. [2 marks]

Calculate the vertical distance from the attachment point on the wall down to the ship. Give your answer correct to 3 significant figures. [2 marks]

The tide rises and the ship moves vertically upward by $3.5\,\text{m}$. The ship also drifts horizontally $2.0\,\text{m}$ closer to the wall. The attachment point on the wall is unchanged. Calculate the new length of cable required to reach the ship, and the new angle the cable makes with the vertical wall. Give both answers correct to 3 significant figures. Total: 8 marks [4 marks]

The surveyor places a marker at point $D$ on the same bank as $A$, such that $A$, $C$, and $D$ are collinear (i.e. $D$ lies on the line $AC$ extended beyond $C$). Triangle $BCD$ isosceles with $BC = CD$. Calculate the angle $B\hat{D}C$. [3 marks]

Calculate the distance $PB$. [2 marks]

A rescue helicopter is stationed at the midpoint $M$ of $PQ$. Calculate the distance from $M$ to $B$. [3 marks]

A sprinkler is placed at the vertex $C$, where the $15.0\,\text{m}$ and $20.0\,\text{m}$ sides meet. Calculate the angle at vertex $C$. [2 marks]

The sprinkler at $C$ can water a circular region of radius $8.00\,\text{m}$. The sprinkler rotates through the angle found in part (b). Determine the area of the garden that the sprinkler can water, and hence determine what percentage of the total garden area is watered. Formulae provided: - Law of Cosines: $a^2 = b^2 + c^2 - 2bc\cos A$ - Area of a triangle: $\frac{1}{2}ab\sin C$ - Area of a sector: $\frac{1}{2}r^2\theta$ (where $\theta$ is in radians) - Area of a circle: $\pi r^2$ [2 marks]

Using the sine rule, calculate the distance $SL$ and the distance $TL$. [3 marks]

Calculate the area of the field. [2 marks]

A straight path is built from $B$ to a point $D$ on $AC$, where $BD \perp AC$. Using your answers to parts (a) and (b), calculate the length of $BD$. [1 mark]

Hence find the angle $\angle BAC$. [1 mark]