DP Mathematics: Analysis and Approaches Questionbank

Write down the value of the iron in the form $a\times {10}^k$ where $1\le a<10 , k\in \mathrm{\mathbb{Z}}$.

Calculate James’s estimate of its volume, in ${\text{km}}^3$.

The actual volume of the asteroid is found to be $6.074\times {10}^6 {\text{km}}^3$.

Find the percentage error in James’s estimate of the volume.

Write down the radius of the planet.

The volume of the planet can be expressed in the form $\mathrm{\pi }\left(a\times {10}^k\right) {\text{km}}^3$ where $1\le a<10$ and $k\in \mathrm{\mathbb{Z}}$.

Find the value of $a$ and the value of $k$.

Find the value of $l$.

Hence, find the volume of the cone.

Find $\text{AV}$.

A second ornament is in the shape of a cuboid with a rectangular base of length $2x \text{cm}$, width $x \text{cm}$ and height $y \text{cm}$. The cuboid has the same volume as the pyramid.

The cuboid has a minimum surface area of $S {\text{cm}}^2$. Find the value of $S$.

Find the distance between $L$ and $\prod _3$.

Show that the three planes do not intersect.

Find a vector equation of $L$, the line of intersection of $\prod _1$ and $\prod _2$.

Verify that the point $\text{P}(1, -2, 0)$ lies on both $\prod _1$ and $\prod _2$.

Find the height of the pyramid, $\text{VX}$.

Find the volume of the sphere expressing your answer in the form $a\times {10}^k$, and $k\in \mathbb{Z}$.

The sphere is to be melted down and remoulded into the shape of a cone with a height of 14.8 cm.

Find the radius of the base of the cone, correct to 2 significant figures.

Calculate the radius of the base of the cone which has been removed.

Calculate the curved surface area of the cone which has been removed.

Calculate the curved surface area of the remaining solid.

Find the length $\text{AB}$, giving your answer to four significant figures.

Find the total area of the two rectangles that make up the roof.

Find the maximum number of complete panels that can be fitted to the whole roof.

Find the percentage error in her estimate.

Olivia investigates arranging the panels, such that the longer edge of each panel is parallel to $\text{AB}$ or $\text{AC}$.

State whether this new arrangement will allow Olivia to fit more solar panels to the roof. Justify your answer.

Find the volume of one cone-shaped container.

Find the slant height of the cone-shaped container.

Show that the total surface area of the cone-shaped container is 314 cm², correct to three significant figures.

Find the height, $h$, of this cylinder-shaped container.

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

State whether or not they should replace the cone-shaped containers with cylinder‑shaped containers. Justify your conclusion.

Show that the volume of a cone shaped glass is $160\text{ c}{\text{m}}^3$, correct to 3 significant figures.

Calculate the radius, $r$, of a hemisphere shaped glass.

Find the cost of $100\text{ c}{\text{m}}^3$ of chocolate mousse.

Show that there is $20\text{ c}{\text{m}}^3$ of orange paste in each special dessert.

Find the total cost of the ingredients of one special dessert.

Find the value of $x$.

Calculate the volume of this pan.

Find the radius of the sphere in cm, correct to one decimal place.

Find the value of $a$.

Find the temperature that the pizza will be 5 minutes after it is taken out of the oven.

Calculate, to the nearest second, the time since the pizza was taken out of the oven until it can be eaten.

In the context of this model, state what the value of 19 represents.

Calculate the area of triangle EAD.

Calculate the total volume of the barn.

Calculate the length of MN.

Calculate the length of AE.

Show that Farmer Brown is incorrect.

Calculate the total length of metal required for one support.

Find an expression for the slant height of the cone in terms of r.

The total external surface area of the pencil case rounded to 3 significant figures is 570 cm².

Using your graphic display calculator, calculate the value of r.

Write down a formula for $A$, the surface area to be coated.

Express this volume in $\text{c}{\text{m}}^3$.

Write down, in terms of $r$ and $h$, an equation for the volume of this water container.

Show that $A=\pi r^2+\frac{1000000}{r}$.

Find $\frac{\text{d}A}{\text{d}r}$.

Using your answer to part (e), find the value of $r$ which minimizes $A$.

Find the value of this minimum area.

Find the least number of cans of water-resistant material that will coat the area in part (g).

Find the radius, $r$, of the circular base of the cone.

Find the slant height, $l$, of the cone.

Find the curved surface area of the cone.

Find the volume of water in the container.

Find the value of $h$.

Calculate the volume of this hemisphere in cm³.

Give your answer correct to one decimal place.

Write down your answer to part (a) correct to the nearest integer.

Write down your answer to part (b) in the form a × 10^k , where 1 ≤ a < 10 and k ∈ $\mathbb{Z}$.

Write down the value of x.

Calculate the volume of the paperweight.

1 cm³ of glass has a mass of 2.56 grams.

Calculate the mass, in grams, of the paperweight.

Write down an equation for the area, $A$, of the curved surface in terms of $r$.

Find $\frac{\text{d}A}{\text{d}r}$.

Find the value of $r$ when the area of the curved surface is maximized.

Calculate the volume of the balloon.

Calculate the radius of the balloon following this increase.

Find the volume, in cm³, of this calculator case.