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EDEXCEL A LEVEL DECISION MATHS 1
2025 SPECIFICATION EXAM QUESTION BANK
Total Questions: 260 (Multiple Choice with Full
Explanations)
ALGORITHMS
- Which of the following best defines an algorithm? A) A mathematical proof B) A set of step-by-step instructions to solve a problem C) A computer programming language D) A random process for decision making
Answer: B Explanation: An algorithm is a finite sequence of well-defined, computer-implementable instructions to solve a class of problems or perform a computation.
- What is the main purpose of using flowcharts in algorithm design? A) To write computer code directly B) To visually represent the logical flow of an algorithm C) To perform mathematical calculations D) To store data efficiently
Answer: B Explanation: Flowcharts use standardized symbols to represent different operations and decisions, making the algorithm's logic easy to understand visually.
- Which symbol is used to represent a decision in a flowchart? A) Rectangle B) Diamond C) Parallelogram D) Oval
Answer: B Explanation: Diamonds represent decision points where the flow branches based on a condition.
- In the context of algorithms, what does "pseudocode" refer to? A) A high-level description of an algorithm using structured English B) A specific programming language
C) A type of flowchart D) Machine code instructions
Answer: A Explanation: Pseudocode uses natural language mixed with programming constructs to describe algorithms without strict syntax rules.
- Which of the following is NOT a characteristic of a good algorithm? A) Finiteness B) Unambiguity C) Input/output specification D) Subjectivity
Answer: D Explanation: Good algorithms are objective, not subjective. They should be finite, unambiguous, have defined inputs/outputs, and be effective.
- What does the term "bubble sort" refer to? A) A searching algorithm B) A sorting algorithm that repeatedly steps through a list C) A graph traversal algorithm D) A mathematical optimization technique
Answer: B Explanation: Bubble sort compares adjacent elements and swaps them if they are in the wrong order, "bubbling" larger elements to the end.
- In the bubble sort algorithm, after the first pass through an n-element list: A) The largest element is in its correct position B) All elements are sorted C) The smallest element is in its correct position D) No elements have moved
Answer: A Explanation: After the first pass, the largest element has "bubbled" to the end of the list and is in its final position.
- The time complexity of bubble sort in the worst case is: A) O(1) B) O(log n) C) O(n) D) O(n²)
Answer: D
B) C
C) P
D) W
Answer: A Explanation: K represents a complete graph with n vertices where every pair of distinct vertices is connected by an edge.
- Which of the following describes a bipartite graph? A) Vertices can be divided into two sets with edges only between sets B) All vertices have even degree C) The graph is completely connected D) It contains no cycles
Answer: A Explanation: In a bipartite graph, vertices are partitioned into two sets with edges only connecting vertices from different sets.
- What is a planar graph? A) A graph that can be drawn in a plane without edges crossing B) A graph with only straight edges C) A graph with vertices arranged in a plane D) A graph with equal numbers of vertices and edges
Answer: A Explanation: A planar graph can be embedded in the plane without edge crossings.
- Euler's formula for connected planar graphs states: A) v - e + f = 2 B) v + e - f = 2 C) v + e + f = 2 D) v - e - f = 2
Answer: A Explanation: For connected planar graphs: v - e + f = 2, where v = vertices, e = edges, f = faces.
- A graph is Eulerian if: A) It has an Eulerian trail that uses every edge exactly once B) It has a Hamiltonian cycle C) It is planar D) All vertices have the same degree
Answer: A
Explanation: An Eulerian graph contains an Eulerian circuit that traverses each edge exactly once and returns to the starting vertex.
- A connected graph is Eulerian if and only if: A) Every vertex has even degree B) Every vertex has odd degree C) The sum of degrees is even D) It has an even number of vertices
Answer: A Explanation: A connected graph is Eulerian if and only if every vertex has even degree.
- What is a Hamiltonian cycle? A) A cycle that visits every vertex exactly once B) A path that uses every edge exactly once C) The shortest path between two vertices D) A cycle with minimum weight
Answer: A Explanation: A Hamiltonian cycle visits each vertex exactly once and returns to the starting vertex.
- The travelling salesman problem seeks: A) The shortest Hamiltonian cycle in a weighted graph B) The longest path in a graph C) The minimum spanning tree D) An Eulerian circuit
Answer: A Explanation: TSP finds the minimum weight Hamiltonian cycle visiting each vertex exactly once.
ALGORITHMS ON GRAPHS
- Dijkstra's algorithm is used to find: A) The shortest path from a source to all other vertices B) The minimum spanning tree C) Eulerian circuits D) Hamiltonian cycles
Answer: A Explanation: Dijkstra's algorithm finds shortest paths from a single source vertex to all other vertices in a weighted graph.
- What is the key requirement for Dijkstra's algorithm to work correctly?
Explanation: Prim's algorithm maintains a tree that grows by adding the cheapest edge connecting the tree to a vertex not yet in the tree.
- Kruskal's algorithm finds a minimum spanning tree by: A) Sorting edges by weight and adding them if they don't create a cycle B) Starting from a vertex and adding cheapest connecting edges C) Removing heaviest edges first D) Using Dijkstra's approach
Answer: A Explanation: Kruskal's algorithm sorts all edges by weight and adds them in order, skipping those that would create cycles.
- What data structure is useful for cycle detection in Kruskal's algorithm? A) Disjoint set union (Union-Find) B) Stack C) Queue D) Priority queue
Answer: A Explanation: Union-Find data structure efficiently checks if adding an edge creates a cycle.
- The time complexity of Kruskal's algorithm is: A) O(E log E) B) O(V²) C) O(V log V) D) O(V + E)
Answer: A Explanation: Sorting edges takes O(E log E), and Union-Find operations are nearly O(1).
- Which algorithm is better for dense graphs? A) Prim's algorithm B) Kruskal's algorithm C) Dijkstra's algorithm D) Depth-first search
Answer: A Explanation: Prim's with adjacency matrix is O(V²) and works well for dense graphs, while Kruskal's O(E log E) is better for sparse graphs.
LINEAR PROGRAMMING
- Linear programming involves:
A) Optimizing a linear objective function subject to linear constraints B) Solving linear equations C) Programming linear algorithms D) Linear regression analysis
Answer: A Explanation: LP maximizes or minimizes a linear objective function subject to linear inequality constraints.
- The feasible region in linear programming is: A) The set of points satisfying all constraints B) The area where the objective function is maximum C) The boundary of the constraints D) The solution to the equations
Answer: A Explanation: The feasible region contains all points that satisfy all constraints simultaneously.
- For a linear programming problem, the optimal solution always occurs: A) At a vertex (corner point) of the feasible region B) At the center of the feasible region C) Anywhere on the boundary D) Outside the feasible region
Answer: A Explanation: For linear programming, if an optimal solution exists, at least one occurs at a vertex of the feasible region.
- The simplex method: A) Moves from vertex to vertex improving the objective B) Uses graphical analysis C) Solves only 2-variable problems D) Is a geometric method
Answer: A Explanation: The simplex method algebraically moves from one basic feasible solution (vertex) to an adjacent one, improving the objective.
- In the simplex method, what indicates optimality? A) All coefficients in the objective row are non-negative (maximization) B) All coefficients are negative C) The RHS values are all positive D) The tableau is square
B) Positive C) Negative D) Undefined
Answer: A Explanation: Non-binding constraints (with slack) have zero shadow price since additional units wouldn't improve the objective.
CRITICAL PATH ANALYSIS
- Critical path analysis is used for: A) Project scheduling and management B) Finding shortest paths in graphs C) Linear programming D) Sorting algorithms
Answer: A Explanation: CPA determines the minimum project duration and critical activities that cannot be delayed.
- In a project network, an activity's earliest start time is: A) Maximum of earliest finish times of its predecessors B) Minimum of earliest finish times C) Its duration D) Always zero
Answer: A Explanation: ES = max{EF of all immediate predecessors}.
- The latest finish time for an activity is: A) Minimum of latest start times of its successors B) Maximum of latest start times C) Earliest finish plus duration D) Project duration
Answer: A Explanation: LF = min{LS of all immediate successors}.
- Total float for an activity is calculated as: A) LS - ES or LF - EF B) ES - LS C) Duration × 2 D) Project duration minus activity duration
Answer: A Explanation: Total float = LS - ES = LF - EF, representing how long an activity can be delayed without affecting project completion.
- Activities on the critical path have: A) Zero total float B) Maximum float C) Negative float D) Float equal to duration
Answer: A Explanation: Critical activities have zero float; any delay delays the entire project.
- The project duration is determined by: A) The length of the critical path B) Sum of all activity durations C) The shortest path D) Average of all paths
Answer: A Explanation: The longest path through the network (critical path) determines minimum project duration.
- What does "dummy activity" represent in CPA? A) A dependency with no time or resource consumption B) An activity that can be skipped C) A zero-duration real activity D) A waiting period
Answer: A Explanation: Dummy activities show logical dependencies without consuming time or resources.
- In CPA, resource leveling aims to: A) Smooth resource usage over time B) Minimize project duration C) Maximize resource utilization D) Eliminate all floats
Answer: A Explanation: Resource leveling reduces fluctuations in resource requirements by shifting non-critical activities.
- The purpose of Gantt charts in project management is to: A) Visually represent the project schedule
Answer: A Explanation: A perfect matching covers every vertex exactly once.
- In bipartite matching, the Hungarian algorithm solves: A) The assignment problem B) The travelling salesman problem C) Minimum spanning tree D) Shortest path problem
Answer: A Explanation: The Hungarian algorithm solves the assignment problem (weighted bipartite matching).
- The augmenting path method for finding maximum matchings: A) Increases matching size by alternating paths B) Starts with a perfect matching C) Uses linear programming D) Only works for complete graphs
Answer: A Explanation: Augmenting paths have alternating matched and unmatched edges, allowing the matching to be increased.
TRANSPORTATION PROBLEMS
- The transportation problem aims to: A) Minimize total shipping cost from sources to destinations B) Maximize profit from transportation C) Balance supply and demand exactly D) Find shortest routes
Answer: A Explanation: The classic transportation problem minimizes total cost of shipping from multiple sources to multiple destinations.
- A balanced transportation problem requires: A) Total supply = total demand B) Supply > demand C) Demand > supply D) Equal number of sources and destinations
Answer: A Explanation: In balanced problems, Σ supply = Σ demand; unbalanced problems use dummy sources/destinations.
- The northwest corner method is used to find: A) An initial basic feasible solution B) The optimal solution C) Shadow prices D) Opportunity costs
Answer: A Explanation: The northwest corner rule provides a starting solution for transportation simplex.
- The stepping stone method evaluates: A) Empty cells for potential improvement B) Only basic cells C) Supply and demand values D) Transportation costs
Answer: A Explanation: The stepping stone method checks non-basic cells for potential cost reduction.
- MODI method stands for: A) Modified Distribution method B) Modified Dual Iteration C) Maximum Optimal Distribution D) Minimum Opportunity Distribution
Answer: A Explanation: MODI method calculates improvement indices more efficiently than stepping stone.
DYNAMIC PROGRAMMING
- Dynamic programming is based on: A) Breaking problems into overlapping subproblems B) Linear programming principles C) Graph algorithms only D) Random optimization
Answer: A Explanation: Dynamic programming solves complex problems by combining solutions to overlapping subproblems.
- The principle of optimality states: A) An optimal policy has optimal subpolicies B) All solutions are equally good C) Greedy choices are always optimal
- A saddle point in a payoff matrix occurs when: A) Maximin = Minimax B) All payoffs are equal C) The game is unfair D) Players cooperate
Answer: A Explanation: Saddle point represents equilibrium in pure strategies for zero-sum games.
- Mixed strategies involve: A) Randomizing among pure strategies B) Combining multiple games C) Changing strategies mid-game D) Cooperative play
Answer: A Explanation: Mixed strategies assign probabilities to pure strategies.
- The value of a game is: A) The expected payoff when both play optimally B) The maximum possible payoff C) The sum of all payoffs D) The number of strategies
Answer: A Explanation: Game value is the expected outcome under optimal mixed strategies.
- Dominant strategy is one that: A) Is best regardless of opponent's choices B) Is best only with cooperation C) Always gives minimum payoff D) Requires randomization
Answer: A Explanation: A dominant strategy yields the highest payoff for all opponent strategies.
FLOWS IN NETWORKS
- The maximum flow problem finds: A) The greatest possible flow from source to sink B) The shortest path C) The minimum cost flow D) An Eulerian circuit
Answer: A Explanation: Maximum flow determines the maximum amount that can flow through a network.
- The max-flow min-cut theorem states: A) Maximum flow = minimum cut capacity B) Maximum flow = total edge capacity C) Minimum flow = maximum cut D) Flow equals cut number
Answer: A Explanation: The value of maximum flow equals the capacity of the minimum cut.
- The Ford-Fulkerson method solves maximum flow using: A) Augmenting paths B) Shortest paths C) Minimum spanning trees D) Linear programming
Answer: A Explanation: Ford-Fulkerson repeatedly finds augmenting paths to increase flow.
- An augmenting path in flow networks: A) Has residual capacity > 0 B) Has zero capacity C) Is the shortest path D) Uses only forward edges
Answer: A Explanation: Augmenting paths have positive residual capacity allowing flow increase.
- The Edmonds-Karp algorithm implements Ford-Fulkerson by: A) Using BFS to find shortest augmenting paths B) Using DFS randomly C) Using Dijkstra's algorithm D) Using greedy selection
Answer: A Explanation: Edmonds-Karp uses BFS to find shortest augmenting paths, ensuring polynomial time.
RECURRENCE RELATIONS
- A recurrence relation defines: A) A sequence recursively in terms of previous terms
- The adjacency matrix of a graph with n vertices is: A) An n×n matrix where aᵢ = 1 if edge (i,j) exists B) An n×n matrix of edge weights C) An n×2 matrix of edges D) A diagonal matrix
Answer: A Explanation: Adjacency matrix A has A[i][j] = 1 if vertices i and j are adjacent.
- The number of walks of length k from i to j is given by: A) The (i,j) entry of Aᵏ B) The sum of row i in A C) The determinant of A D) The trace of A
Answer: A Explanation: Aᵏ[i][j] counts walks of length k from i to j.
- The incidence matrix has rows for: A) Vertices and columns for edges B) Edges and columns for vertices C) Only vertices D) Only edges
Answer: A Explanation: Incidence matrix: rows = vertices, columns = edges, entries indicate incidence.
- A graph is connected if and only if: A) Its adjacency matrix is irreducible B) Its determinant is zero C) It has an Eulerian circuit D) All vertices have even degree
Answer: A Explanation: For undirected graphs, connectedness means the adjacency matrix is irreducible.
- The Laplacian matrix L is defined as: A) L = D - A where D=degree matrix, A=adjacency B) L = A + D C) L = A × D D) L = A⁻¹
Answer: A
Explanation: Laplacian L = D - A is important for graph properties.
BINOMIAL DISTRIBUTION (DECISION CONTEXT)
- In decision mathematics, binomial distribution models: A) Number of successes in n independent trials B) Continuous random variables C) Time between events D) Normal distributions
Answer: A Explanation: Binomial: X ~ B(n,p) counts successes in n independent Bernoulli trials.
- The mean of binomial distribution B(n,p) is: A) np B) n(1-p) C) √(np(1-p)) D) p
Answer: A Explanation: E[X] = np for binomial distribution.
- The variance of B(n,p) is: A) np(1-p) B) np C) n²p D) p(1-p)
Answer: A Explanation: Var(X) = np(1-p) for binomial.
- Conditions for binomial distribution include: A) Fixed number of trials, constant probability, independence B) Variable trials, changing probability C) Continuous outcomes D) Time-dependent probabilities
Answer: A Explanation: Binomial requires fixed n, constant p, independent trials, two outcomes.
- Normal approximation to binomial is reasonable when: A) np > 5 and n(1-p) > 5 B) n > 10 C) p = 0.5 exactly
