AQA A-Level Computer Science: Fundamentals of Functional Programming — Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs — examination boards hold copyright in their own papers. Use these alongside the official past papers available free from your board.

Related: Functional programming study guide | Functional programming revision notes

These questions cover section 4.12 of the AQA A-Level Computer Science (7517) specification: the functional programming paradigm (4.12.1), writing functional programs (4.12.2) and list processing (4.12.3). Code is written in Haskell notation: foldl folds from the left, foldr from the right, : prepends and ++ joins lists.

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Section A: the paradigm

1. A function is defined as f: {1, 2, 3} -> {0, 1, 2, …, 9} with the rule f(x) = 2x. (a) State the domain and the co-domain of f. [2] (b) List the members of the co-domain that are outputs of f, and explain why this does not stop {0, …, 9} being a valid co-domain. [2]

2. Explain what is meant by saying that functions are first-class objects. [3]

3. The function multiply takes two integers and returns their product. (a) Give its function type using the Cartesian product. [1] (b) Explain why it can be said to take only one argument. [1]

4. A function is defined:

volume :: Int -> Int -> Int -> Int
volume l w h = l * w * h

(a) State the type of volume 2 and of volume 2 3. [2] (b) State the value of volume 2 3 4. [1] (c) Explain what is meant by partial function application, using volume 2 as an example. [2]

5. f x = 3 * x and g x = x - 1, both with domain and co-domain the integers. (a) Evaluate (f . g) 5 and (g . f) 5. [2] (b) Explain what is meant by the composition g ∘ f, stating which function is applied first. [2]

Section B: higher-order functions

6. Define a higher-order function. [1]

7. Evaluate: (a) map (+3) [2, 4, 6] [1] (b) filter (> 10) [5, 12, 8, 20] [1] (c) foldl (*) 1 [1, 2, 3, 4] [1] (d) foldl (-) 0 [5, 3, 1] [1] (e) foldr (-) 0 [5, 3, 1] [2]

8. words = ["sun", "cloud", "rain", "sky"]. Write a single expression using filter, map and a fold that returns the total number of letters in the words longer than three letters, and state its value. [4]

9. For each, state whether map, filter or fold is most suitable: (a) converting a list of temperatures from Celsius to Fahrenheit (b) finding the largest value in a list (c) keeping only the passwords at least 12 characters long. [3]

Section C: lists

10. xs = [7, 2, 9]. State the result of: (a) head xs (b) tail xs (c) head (tail xs) (d) tail (tail (tail xs)) (e) null [] (f) length xs (g) 1 : xs (h) xs ++ [5] [8]

11. Explain the difference between the head and the tail of a list. [2]

12. (a) Write a recursive function len that returns the length of a list, using head and tail pattern matching. [3] (b) Show how len [7, 2, 9] is evaluated. [2]

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Answers

1. (a) Domain: {1, 2, 3} [1]. Co-domain: {0, 1, 2, …, 9} [1]. (b) {2, 4, 6} [1]. The co-domain is the set from which outputs are chosen; not every member has to be an output [1].

2. A first-class object can appear in expressions [1], be assigned to a variable [1], and be passed as an argument to a function and returned as the result of a function call [1].

3. (a) multiply: integer × integer -> integer [1]. (b) Its single argument is a pair of integers, e.g. (3, 4), from the Cartesian product integer × integer [1].

4. (a) volume 2 :: Int -> Int -> Int [1]; volume 2 3 :: Int -> Int [1]. (b) 24 [1]. (c) Supplying fewer arguments than the function takes returns a new function that expects the remaining arguments [1]: volume 2 is a function that takes a width and a height and returns 2 × width × height [1].

5. (a) (f . g) 5 = f(g(5)) = f(4) = 12 [1]; (g . f) 5 = g(15) = 14 [1]. (b) Composition combines two functions to make a new function [1]; in g ∘ f, f is applied first and g is applied to the result returned by f [1].

6. A function that takes a function as an argument, returns a function as its result, or both [1].

7. (a) [5, 7, 9] [1] (b) [12, 20] [1] (c) 24 [1] (d) ((0 - 5) - 3) - 1 = -9 [1] (e) 5 - (3 - (1 - 0)) = 5 - 2 = 3 [1 working, 1 answer].

8.

foldl (+) 0 (map length (filter (\w -> length w > 3) words))

filter with a correct condition [1] -> [“cloud”, “rain”]; map length [1] -> [5, 4]; fold with + and 0 [1]; value 9 [1].

9. (a) map [1] (b) fold – e.g. foldl max with a suitable starting value [1] (c) filter [1].

10. (a) 7 (b) [2, 9] (c) 2 (d) [] (e) True (f) 3 (g) [1, 7, 2, 9] (h) [7, 2, 9, 5] [1 each].

11. The head is the first element of the list [1]; the tail is the list of all the remaining elements, which may be the empty list [1].

12. (a)

len :: [a] -> Int
len []     = 0
len (_:xs) = 1 + len xs

Base case for the empty list returning 0 [1]; pattern splitting head and tail [1]; recursive call on the tail adding 1 [1]. (b) len [7, 2, 9] = 1 + len [2, 9] = 1 + (1 + len [9]) = 1 + (1 + (1 + len [])) [1] = 1 + 1 + 1 + 0 = 3 [1].

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Where marks are usually lost

• Composing in the wrong order: (f . g) x means g first.
• Giving the type of a partially applied function as a value, or dropping one of the remaining arrows.
• Folding subtraction from the wrong end.
• Writing tail [9] as 9 or “nothing” instead of [].
• Writing the condition of filter correctly but then applying map to the original list rather than the filtered one.

Frequently asked questions

Will I have to write full Haskell programs in the exam? The written paper expects you to read and evaluate short functional expressions and to understand the terms. Writing simple functional programs is part of the course experience the specification requires, so practise in a real language.

Is reduce the same as fold? Yes. The 7517 specification names the function “reduce or fold”: it reduces a list to a single value by repeatedly applying a combining function.