Geometry and trigonometry: 7 IB Maths AI SL practice questions

A solid right circular cone has a base radius of 21 cm and a slant height of 35 cm. A smaller right circular cone has a height of 12 cm and a slant height of 15 cm, and is removed from the top of the larger cone, as shown in the diagram.

(a) Calculate the radius of the base of the cone which has been removed.[2]

(b) Calculate the curved surface area of the cone which has been removed.[2]

(c) Calculate the curved surface area of the remaining solid.[2]

The coordinates of point A are $(6, -7)$ and the coordinates of point B are $(-6, 2)$. Point M is the midpoint of AB.

(a) Find the coordinates of M.[2]

$L_1$ is the line through A and B.

(b) Find the gradient of $L_1$.[2]

The line $L_2$ is perpendicular to $L_1$ and passes through M.

(c) (i) Write down the gradient of $L_2$.[2]

(ii) Write down, in the form $y = mx + c$, the equation of $L_2$.[2]

A balloon in the shape of a sphere is filled with helium until the radius is 6 cm.

(a) Calculate the volume of the balloon.[2]

The volume of the balloon is increased by 40%.

(b) Calculate the radius of the balloon following this increase.[4]

The point A has coordinates $(4, -8)$ and the point B has coordinates $(-2, 4)$.

(a) Write down the coordinates of C, the midpoint of line segment AB.[2]

The point D has coordinates $(-3, 1)$.

(b) Find the gradient of the line DC.[2]

(c) Find the equation of the line DC. Write your answer in the form $ax + by + d = 0$ where $a$, $b$ and $d$ are integers.[2]

The base of an electric iron can be modelled as a pentagon ABCDE, where:

BCDE is a rectangle with sides of length $(x + 3)$ cm and $(x + 5)$ cm;

ABE is an isosceles triangle, with $AB = AE$ and a height of $x$ cm;

the area of ABCDE is 222 cm$^2$.

(a)(i) Write down an equation for the area of ABCDE using the above information.[3]

(a)(ii) Show that the equation in part (a)(i) simplifies to $3x^2 + 19x - 414 = 0$.[4]

(b) Find the length of CD.[2]

(c) Show that angle $BAE = 67.4°$, correct to one decimal place.[3]

Insulation tape is wrapped around the perimeter of the base of the iron, ABCDE.

(d) Find the length of the perimeter of ABCDE.[3]

F is the point on AB such that $BF = 8$ cm. A heating element in the iron runs in a straight line, from C to F.

(e) Calculate the length of CF.[4]

The point A has coordinates $(1, 2, 1)$ and the point B has coordinates $(3, 5, 2)$.

(a) Find $AB$.[2]

Triangle ABC is right-angled with its right angle at B. The point C has coordinates $(2, 8, k)$.

(b) Find the value of $k$.[4]

(c) Calculate the size of $\angle BAC$.[3]

A factory packages coconut water in cone-shaped containers with a base radius of 5.2 cm and a height of 13 cm.

(a) Find the volume of one cone-shaped container.[2]

(b) Find the slant height of the cone-shaped container.[2]

(c) Show that the total surface area of the cone-shaped container is 314 cm², correct to three significant figures.[3]

The factory designers are currently investigating whether a cone-shaped container can be replaced with a cylinder-shaped container with the same radius and the same total surface area.

(d) Find the height, $h$, of this cylinder-shaped container.[4]

The factory director wants to increase the volume of coconut water sold per container.

(e) State whether or not they should replace the cone-shaped containers with cylinder-shaped containers. Justify your conclusion.[4]