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SL 3.1—3d space, volume, angles, distance, midpoints - IB Questionbank | RevisionDojo

Practice SL 3.1—3d space, volume, angles, distance, midpoints with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Consider a triangle $XYZ$ such that $X$ has coordinates $(0, 0, 0), Y$ has coordinates $(0, 1, 2)$ and $Z$ has coordinates $(2b, 0, b - 1)$ where $b < 0$

Let $N$ be the midpoint of the line segment $XZ$ .

Find, in terms of $b$ , a Cartesian equation of the plane $\Pi$ containing this triangle.

[5]

Find, in terms of $b$ , the equation of the line $L$ which passes through N and is perpendicular to the plane $П$ .

[2]

Question 2

HLPaper 1

Consider a water tank in the shape of an inverted cone with a height of 10 meters and a base radius of 5 meters. Water is being pumped into the tank at a rate of 3 cubic meters per minute.

Find the rate at which the water level is rising when the water is 4 meters deep.

[5]

Question 3

SLPaper 2

A research drone is tethered by two cables, $AP$ and $BP$ , attached to points $A$ and $B$ on level ground.
The points $A$ and $B$ are 80 m apart. The drone $P$ is directly above the point $P_0$ on the ground.
In the horizontal triangle $ABP_0$ , the angle at $P_0$ between the lines $P_0A$ and $P_0B$ is $60^\circ$ .

At a certain instant, the angle of elevation of the drone from $A$ is $28^\circ$ and from $B$ is $35^\circ$ .

Draw a fully-labelled 3-D diagram showing $A,B,P_0,P$ and all known angles.

[2]

Let the horizontal distances $AP_0 = x$ m and $BP_0 = y$ m.
Show that
$x\tan28^\circ = y\tan35^\circ.$

[2]

Using the cosine rule in triangle $ABP_0$ , show that
$80^2 = x^2 + y^2 - 2xy\cos60^\circ.$

[2]

Use your results from part 2 and 3 to find the height $h$ of the drone above the ground.
Give your answer to the nearest metre.

[3]

The drone then moves horizontally so that its projection $P_0$ traces out a circular path of radius 80 m centred at a fixed point on the ground.
Calculate, in metres and in radians, the length of the arc of this path when the bearing of $P_0$ from $A$ changes from $050^\circ$ to $130^\circ$ .

[3]

The tension in each cable is proportional to its length.
If the tension in $AP$ is $T_A = k|AP|$ and in $BP$ is $T_B = k|BP|$ , find the ratio $\dfrac{T_A}{T_B}$ at this instant.

[3]

Question 4

SLPaper 2

A rescue helicopter $P$ hovers above a mountain valley. It is connected by two winch cables to huts $A$ and $B$ on level ground.
The distance between the huts is 90 m. The helicopter is vertically above point $P_0$ , and in the horizontal triangle $ABP_0$ , $\angle AP_0B = 70^\circ$ .

The angle of elevation of the helicopter is $32^\circ$ from $A$ and $45^\circ$ from $B$ .

Sketch a fully-labelled 3-D diagram showing $A,B,P_0,P$ and all given angles.

[2]

Let $AP_0=x$ m and $BP_0=y$ m.
Show that
$x\tan32^\circ = y\tan45^\circ.$

[2]

Using the cosine rule in triangle $ABP_0$ , write down an equation relating $x$ and $y$ .

[2]

Use your results to find the height $h$ of the helicopter above the ground, correct to the nearest metre.

[2]

The point $P_0$ later moves horizontally around a circular path of radius 90 m centred at a base camp.
Find the length of the arc, in metres and radians, when its bearing from $A$ changes from $010^\circ$ to $100^\circ$ .

[3]

If tensions are $T_A = k|AP|$ and $T_B = k|BP|$ , find $\dfrac{T_A}{T_B}$ .

[3]

Question 5

SLPaper 2

A hot-air balloon $P$ is anchored by two ropes to points $A$ and $B$ on level ground.
The distance $AB$ is 75 m. The balloon is vertically above $P_0$ , and $\angle AP_0B = 60^\circ$ in the horizontal triangle $ABP_0$ .

The angles of elevation of the balloon are $28^\circ$ from $A$ and $37^\circ$ from $B$ .

Draw a fully-labelled 3-D diagram showing $A,B,P_0,P$ and all the given angles.

[2]

Let $AP_0=x$ m and $BP_0=y$ m.
Show that
$x\tan28^\circ = y\tan37^\circ.$

[2]

Using the cosine rule in triangle $ABP_0$ , write an equation relating $x$ and $y$ .

[2]

Find the height $h$ of the balloon, to the nearest metre.

[2]

The projection $P_0$ moves horizontally on a circle of radius 75 m.
Find, in metres and radians, the length of the arc when the bearing of $P_0$ from $A$ changes from $045^\circ$ to $115^\circ$ .

[3]

If $T_A = k|AP|$ and $T_B = k|BP|$ , find $\dfrac{T_A}{T_B}$ .

[3]

Question 6

SLPaper 2

Given a cylinder having diameter $x$ and height $h$ . It's volume is 25 and total area is expressed as an expression in A as $\frac{\pi}{2} x^{2}+\frac{2 n}{x}$ .

Find the value of $n$ .

[6]

Value of $x$ which makes the total surface area minimum.

[3]

Question 7

SLPaper 2

A mall is to be constructed with a concrete slab foundation. In order to fit this foundation, a rectangular section of earth measuring 10 m by 20 m is removed to a depth of 1.5 m. Now the removed earth is used to create a hemispherical structure in order to reduce the waste.

Find the diameter of the hemispherical structure.

[4]

Now the architect decided that the cylindrical structure of height 2 m would be much better than the hemispherical one. Given that the maximum straight line distance that is available on the site is 12 m. Find the radius of cylindrical structure and show cylindrical structure is not suitable for the site.

[4]

Question 8

SLPaper 1

Find the volume and surface area of the cylinder with dimensions $r=4$ and $h=10$

[4]

Find the volume and surface area of the sphere with dimensions $r=6$

[4]

Question 9

SLPaper 2

A cylindrical tank that stores water has a height of $12.3 m$ and a volume of $503.7 m^3$ .

Find the radius of the base of the cylinder.

[2]

They would like to store the water in a cone instead. Find the height of a cone which has the same radius and the same volume as the cylinder.

[4]

Question 10

HLPaper 1

The points C and D are given by ${\text{C}}(0, 3, -6)$ and ${\text{D}}(6, -5, 11)$ .

The plane Ω is defined by the equation $4x - 3y + 2z = 20$ .

Find a vector equation of the line $M$ passing through the points C and D.

[3]

Find the coordinates of the point of intersection of the line $M$ with the plane $\Omega$ .

[3]

SL 3.1—3d space, volume, angles, distance, midpoints Questionbank