MATH40006 Practice Paper 5

English review edition prepared on 4 October 2026 from a preserved source copy. It is a later presentation, not the historical study interface. Source checks and difficulty judgements describe the original material author's own process; they do not indicate Imperial College London endorsement.

Source SHA-256: 7435e6797e76ddc8bee35fac2443dd352e2485bd9610c8470a0a75c31caabfea
Source date: 2026-08-06

Notes — cell 1

# MATH40006 Revised Historical-Coverage Mock 5

**Time allowed:** 60 minutes 
**Total marks:** 50 
**Questions:** 3 (20 + 20 + 10) 
**Role:** 2021-22 gap repair: escape-time arrays, symbolic dynamics and ordered lookup

Answer all questions in this Jupyter notebook. Show testing code and outputs. Unless the question explicitly requests an in-place operation, functions should return a value without changing their inputs.

Code — cell 2

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt
import pandas as pd
from time import perf_counter
from copy import copy
import pickle
import ast
import math
sp.init_printing()

records=[('Ant','small insect'),('Bear','large mammal'),('Cat','domestic feline'),('Dog','domestic canine'),('Eel','long fish'),('Fox','wild canine'),('Goat','hoofed mammal'),('Hare','fast mammal')]

Notes — cell 3

## Question 1 (20 marks)

For each complex parameter c and initial value z0, iterate z <- z**2+c. Escape occurs when abs(z)>=2.

(a) Build a 220 by 220 complex grid on -1.6<=x,y<=1.6. **[3]**

(b) Create an escape-time array initially -1 and use a Boolean mask to record points already escaped at time 0. **[3]**

(c) Perform one iteration only on unresolved entries and record new escapes at time 1. **[3]**

(d) Write `escape_time(c,z0,max_iterations)` using masks, returning `max_iterations` for unresolved points and not changing the caller's z0. **[7]**

(e) Test one parameter-varying and one fixed-parameter case; display `log(times+1)` with correct origin and extent. **[4]**

Code — cell 4

# YOUR CODE HERE
raise NotImplementedError()

Notes — cell 5

## Question 2 (20 marks)

Let f(z)=z**2+a+i*b, with real parameters a,b.

(a) Use SymPy to find the fixed points and derivative. **[2]**

(b) For each fixed point form the squared derivative modulus using `re` and `im`. **[3]**

(c) Lambdify the two expressions and plot their level-1 contours on an a,b grid with equal axes. **[4]**

(d) Form f(f(z)), divide f(f(z))-z by f(z)-z, simplify, and solve the quotient for the genuine period-2 points. **[5]**

(e) Repeat the derivative-modulus and level-1 contour calculation for the period-2 points. **[4]**

(f) State what the level-1 curves represent dynamically. **[2]**

Code — cell 6

# YOUR CODE HERE
raise NotImplementedError()

Notes — cell 7

## Question 3 (10 marks)

A sorted list `records` of `(key,value)` pairs is supplied.

(a) Write a sequential lookup returning `'failed'`. **[2]**

(b) Give best- and worst-case bounds. **[1]**

(c) Write a robust iterative binary lookup with the same return convention. **[4]**

(d) Test a hit and two misses. **[1]**

(e) Convert to a dictionary and Series, perform the hit, and time repeated successful lookups in all four structures with a restrained comment. **[2]**

Code — cell 8

# YOUR CODE HERE
raise NotImplementedError()