IB Maths AI HL · Paper 2

Both dog and cat is 14%. Dog only is $68-14$, cat only is $36-14$, and the rest have neither.

John picks 10 students from the survey without putting them back, so the choices are not independent and the probability of a dog changes from pick to pick.

C$(7,8)$ and D$(7,5)$ have a vertical segment between them, so the bisector is horizontal through the midpoint $(7,6.5)$.

Midpoint of B$(3,6)$ and C$(7,8)$ is $(5,7)$. The gradient of BC is $\frac{8-6}{7-3}=\frac{1}{2}$, so the bisector has gradient $-2$.

The area under the northern boundary from $x=0$ to $x=70$:

A cycle that visits every office once and returns to the start.

An Eulerian circuit needs every vertex to have an even degree. A has three edges (to B, E and F), which is odd, so there is no Eulerian circuit.

Find the shortest route in each case.

From E choose the nearest unvisited office each time, using the table.

Deleting a different vertex can give a different lower bound, so 110 might not be the best (largest) lower bound. Repeating the algorithm for each vertex and taking the biggest result gives a better one.

The surfaces meet at the top, where $f(p)=g(p)$ with $g(x)=(x-1{)}^4$.

The top of the glass is at $y=f(p)=13.33$. The liquid sits above the inner surface $y=\frac{1}{2}{x}^3+1$, so $x^2={\left(2(y-1)\right)}^{2/3}$.

The glass is the region between the two curves. With shells:

The total $T=W+P$ has mean $7+3=10$ and variance $0.09+0.0256=0.1156$.

For eight doors the total has mean 80 and variance $8\times 0.1156=0.9248$.

Use the statistics mode on the seven thicknesses.