DP Further Mathematics HL Questionbank

Description

The aim of this topic is to provide the opportunity for students to engage in logical reasoning, algorithmic thinking and applications.

Directly related questions

• 18M.2.hl.TZ0.3c.ii: Explain the consequences of having a Hamiltonian cycle for Pauline’s visit to the ground floor of...
• 18M.2.hl.TZ0.3c.i: Write down a Hamiltonian cycle for the graph G.
• 18M.2.hl.TZ0.3b.ii: Explain the consequences of having an Eulerian trail but not an Eulerian circuit, for Pauline’s...
• 18M.2.hl.TZ0.3b.i: Explain why the graph G has an Eulerian trail but not an Eulerian circuit.
• 18M.2.hl.TZ0.3a: Draw a graph G to represent this floorplan where the rooms are represented by the vertices and an...
• 18M.2.hl.TZ0.3d: Use the nearest-neighbour algorithm to determine a possible route and an upper bound for the...
• 18M.2.hl.TZ0.3e: By removing Z, use the deleted vertex algorithm to determine a lower bound for the length of her...
• 18M.1.hl.TZ0.3b: 1000 is a number written in base 10. Find this as a number written in base 7.
• 18M.1.hl.TZ0.3a: A number written in base 5 is 4303. Find this as a number written in base 10.
• 18M.1.hl.TZ0.12b: Consider \({v_n}\) which satisfies the recurrence...
• 18M.1.hl.TZ0.12a: Solve the recurrence relation \({u_n} = 4{u_{n - 1}} - 4{u_{n - 2}}\) given that...
• 18M.1.hl.TZ0.1b: Hence find integers s and t such that 74s + 383t = 1.
• 18M.1.hl.TZ0.1a: Use the Euclidean algorithm to find the greatest common divisor of 74 and 383.
• 16M.2.hl.TZ0.5c: (i)     Find an expression for \({w_n}\) in terms of \(n\). (ii)     Show that \({w_{3n}} = 0\)...
• 16M.2.hl.TZ0.5b: The sequence \(\{ {v_n}:n \in {\mathbb{Z}^ + }\} \) satisfies the recurrence relation...
• 16M.2.hl.TZ0.5a: (i)     Find an expression for \({u_n}\) in terms of \(n\). (ii)     Show that the sequence...
• 16M.2.hl.TZ0.1e: Explain how the result in part (b) can be used to find a different upper bound and state its value.
• 16M.2.hl.TZ0.1d: Deduce an upper bound for the total weight of a closed walk of minimum weight which visits every...
• 16M.2.hl.TZ0.1c: Use Kruskal’s algorithm to find a minimum weight spanning tree and state its weight.
• 16M.2.hl.TZ0.1b: Determine whether or not the graph is Hamiltonian.
• 16M.2.hl.TZ0.1a: Determine whether or not the graph is Eulerian.
• 16M.1.hl.TZ0.6c: Prove by mathematical induction that your formula is valid for all \(n \in {\mathbb{Z}^ + }\).
• 16M.1.hl.TZ0.6b: Conjecture a formula for \({H_n}\) in terms of \(n\), for \(n \in {\mathbb{Z}^ + }\).
• 16M.1.hl.TZ0.6a: Find \({H_2}\), \({H_3}\) and \({H_4}\).
• 16M.1.hl.TZ0.10c: Express the hexadecimal (base 16) number \({\text{ABB}}{{\text{A}}_{{\text{16}}}}\) in binary.
• 16M.1.hl.TZ0.10b: Hence show that an integer is divisible by 3 if and only if the difference between the sum of its...
• 16M.1.hl.TZ0.10a: Show that \({2^n} \equiv {( - 1)^n}(\bmod 3)\), where \(n \in \mathbb{N}\).
• 16M.1.hl.TZ0.3c: Find the solution satisfying \(xy = 2014\).
• 16M.1.hl.TZ0.3b: Hence find the solution with minimum positive value of \(xy\).
• 16M.1.hl.TZ0.3a: Find the general solution to this equation.
• 16M.1.hl.TZ0.9a: Use the Euclidean algorithm to find \(\gcd (162,{\text{ }}5982)\).
• 17M.2.hl.TZ0.3b: Determine the second-degree recurrence relation satisfied by \(\{ {v_n}\} \).
• 17M.2.hl.TZ0.3a.iii: Determine the value...
• 17M.2.hl.TZ0.3a.ii: Given that \({u_1} = 12,{\text{ }}{u_2} = 6\), show...
• 17M.2.hl.TZ0.3a.i: Write down the auxiliary equation.
• 17M.2.hl.TZ0.1b: Use Dijkstra’s algorithm to find the path of minimum total weight joining A to D, stating this...
• 17M.2.hl.TZ0.1a.ii: Write down an Eulerian trail.
• 17M.2.hl.TZ0.1a.i: State what features of the graph enable you to state that \(G\) contains an Eulerian trail but no...
• 17M.1.hl.TZ0.11c.i: Explain why a graph containing a cycle of length three cannot be bipartite. 
• 17M.1.hl.TZ0.11a.i: Without attempting to draw \(J\), verify that J satisfies the handshaking lemma;
• 17M.1.hl.TZ0.2b: Hence find the smallest value of \(x\) greater than 100 satisfying the linear congruence...
• 17M.1.hl.TZ0.2a: Consider the linear congruence \(ax \equiv b(\bmod p)\) where...
• 15M.2.hl.TZ0.3c: Show using proof by strong induction that the solution is correct.
• 15M.2.hl.TZ0.3b: Solve the recurrence relation.
• 15M.2.hl.TZ0.3a: Show that for \(n \geqslant 2,{\text{ }}{x_n} = 4{x_{n - 1}} - 3{x_{n - 2}}\).
• 15M.1.hl.TZ0.9d: Show that \(121\) is always a square number in any base greater than \(2\).
• 15M.1.hl.TZ0.9c: (i)     Given that \(N = 899,{\text{ }}r = 10\) and \(s = 12\) find the values of...
• 15M.1.hl.TZ0.9b: Given that \(N = 365,{\text{ }}r = 10\) and \(s = 7\) find the values of...
• 15M.1.hl.TZ0.9a: In base \(5\) an integer is written \(1031\). Express this integer in base \(10\).
• 15M.1.hl.TZ0.4f: Draw the complement \(G'\) of \(G\).
• 15M.1.hl.TZ0.4e: State whether or not the simple graph \(G\) is bipartite, giving a reason for your answer.
• 15M.1.hl.TZ0.4d: State whether or not \(G\) is planar, giving a reason for your answer.
• 15M.1.hl.TZ0.4c: Show that \(G\) has a Hamiltonian cycle.
• 15M.1.hl.TZ0.4b: Explain why \(G\) does not contain an Eulerian circuit.
• 15M.1.hl.TZ0.4a: Draw the simple graph \(G\).
• 15M.1.hl.TZ0.2b: Find the values of \(x\) and \(y\) satisfying the equation for which \(x\) has the smallest...
• 15M.1.hl.TZ0.2a: Find the general solution to the Diophantine equation \(3x + 5y = 7\).
• 11M.1.hl.TZ0.3a: Prove that the number \(14 641\) is the fourth power of an integer in any base greater than \(6\).
• 11M.1.hl.TZ0.4a: Prove that if \({\rm{gcd}}(a,b) = 1\) and \({\rm{gcd}}(a,c) = 1\) , then \({\rm{gcd}}(a,bc) = 1\) .
• 11M.1.hl.TZ0.4b: (i)     A simple graph has e edges and v vertices, where \(v > 2\) . Prove that if all the...
• 11M.2.hl.TZ0.1a: Draw a planar graph to represent this map.
• 11M.2.hl.TZ0.1b: Write down the adjacency matrix of the graph.
• 11M.2.hl.TZ0.1c: List the degrees of each of the vertices.
• 11M.2.hl.TZ0.1d: State with reasons whether or not this graph has   (i)     an Eulerian circuit;   (ii)     an...
• 11M.2.hl.TZ0.1e: Find the number of walks of length \(4\) from E to F.
• 11M.2.hl.TZ0.4a: Given the linear congruence \(ax \equiv b({\rm{mod}}p)\) , where \(a\) , \(b \in \mathbb{Z} \) ,...
• 11M.2.hl.TZ0.4b: (i)     Solve \(17x \equiv 14(\bmod 21)\) . (ii)     Use the solution found in part (i) to find...
• 10M.1.hl.TZ0.3b: (i)     Draw \(G'\), the complement of \(G\) . (ii)     Write down the degrees of all the...
• 10M.1.hl.TZ0.4: Given that \({n^2} + 2n + 3 \equiv N(\bmod 8)\) , where \(n \in {\mathbb{Z}^ + }\) and...
• 10M.1.hl.TZ0.3a: (i)     Write down the adjacency matrix for \(G\) . (ii)     Find the number of walks of length...
• 10M.2.hl.TZ0.3a: (i)     Explain briefly what features of the graph enable you to state that \(G\) has an Eulerian...
• 10M.2.hl.TZ0.3b: (i)     Use Kruskal’s algorithm to find and draw the minimum spanning tree for \(G\) . Your...
• 10M.2.hl.TZ0.3c: Use Dijkstra’s algorithm to find the path of minimum total weight joining A to D, and state its...
• 09M.1.hl.TZ0.4: Prove that \(3k + 2\) and \(5k + 3\) , \(k \in \mathbb{Z}\) are relatively prime.
• 09M.1.hl.TZ0.6c: Show that \(\sum\limits_{n = 1}^{100} {n! \equiv 3(\bmod 15)} \) .
• 09M.2.hl.TZ0.3A.a: Use Kruskal’s algorithm to find a minimum spanning tree for the weighted graph shown below. State...
• 09M.2.hl.TZ0.3A.b: For the travelling salesman problem defined by this graph, find   (i)     an upper bound;  ...
• 09M.2.hl.TZ0.3B.a: Given that the integers \(m\) and \(n\) are such that \(3|({m^2} + {n^2})\) , prove that \(3|m\)...
• 09M.2.hl.TZ0.3B.b: Hence show that \(\sqrt 2 \) is irrational.
• 13M.1.hl.TZ0.1a: (i)     Use the Euclidean algorithm to find gcd(\(6750\), \(144\)) . (ii)     Express your...
• 13M.1.hl.TZ0.1b: Consider the base \(15\) number CBA, where A, B, C represent respectively the digits ten, eleven,...
• 13M.2.hl.TZ0.4a: (i)     Draw the planar graph \(H\) that represents these mutual friendships. (ii)     State how...
• 13M.2.hl.TZ0.4b: Determine, giving reasons, whether \(H\) has   (i)     a Hamiltonian path;   (ii)     a...
• 13M.2.hl.TZ0.4c: Verify Euler’s formula for \(H\) .
• 13M.2.hl.TZ0.4d: State, giving a reason, whether or not \(H\) is bipartite.
• 13M.2.hl.TZ0.4e: Write down the adjacency matrix for \(H\) .
• 13M.2.hl.TZ0.4f: David wishes to send a message to Grace, in a sealed envelope, through mutual friends. In how...
• 08M.1.hl.TZ0.1a: Determine whether or not \(G\) is bipartite.
• 08M.1.hl.TZ0.1b: (i)     Write down the adjacency matrix for \(G\). (ii)     Find the number of distinct walks of...
• 08M.2.hl.TZ0.1A.a: The graph \(G\) has the following cost adjacency matrix. Draw \(G\) in planar form.
• 07M.1.hl.TZ0.4b: Find the general solution to the diophantine equation \(275x + 378y = 1\) .
• 08M.2.hl.TZ0.1B.a: Given that \(ax \equiv b(\bmod p)\) where \(a,b,p,x \in {\mathbb{Z}^ + }\) , \(p\) is prime and...
• 08M.2.hl.TZ0.1B.b: Hence solve the simultaneous linear congruences\[3x \equiv 4(\bmod 5)\]\[5x \equiv 6(\bmod...
• 07M.1.hl.TZ0.4a: Use the Euclidean Algorithm to show that \(275\) and \(378\) are relatively prime.
• 07M.2.hl.TZ0.1a: (i)     Explain briefly why \(G\) has no Eulerian circuit. (ii)     Determine whether or not...
• 07M.2.hl.TZ0.1b: The cost adjacency matrix of a graph with vertices P, Q, R, S, T, U is given by Use Dijkstra’s...
• 12M.1.hl.TZ0.2a: Express the number \(47502\) as a product of its prime factors.
• 12M.1.hl.TZ0.2b: The positive integers \(M\) , \(N\) are such that \(\gcd (M,N) = 63\) and \(lcm(M,N) = 47502\) ....
• 12M.1.hl.TZ0.6a: Using mathematical induction or otherwise, prove that the number \({(1020)_n}\) , that is the...
• 12M.1.hl.TZ0.6b: Explain briefly why the case \(n = 2\) has to be excluded.
• 12M.2.hl.TZ0.2A.a: (i)     Show that \(H\) is bipartite. (ii)     Draw \(H\) as a planar graph.
• 12M.2.hl.TZ0.2A.b: (i)     Explain what feature of \(H\) guarantees that it has an Eulerian circuit. (ii)     Write...
• 12M.2.hl.TZ0.2A.c: (i)     Find the number of different walks of length five joining A to B. (ii)     Determine how...
• 12M.2.hl.TZ0.2A.d: Find the maximum number of extra edges that can be added to \(H\) while keeping it simple, planar...
• 12M.2.hl.TZ0.2B.a: Find the smallest positive integer \(m\) such that \({3^m} \equiv 1(\bmod 22)\) .
• 12M.2.hl.TZ0.2B.b: Given that \({3^{49}} \equiv n(\bmod 22)\) where \(0 \le n \le 21\) , find the value of \(n\) .
• 12M.2.hl.TZ0.2B.c: Solve the equation \({3^x} \equiv 5(\bmod 22)\) .
• SPNone.1.hl.TZ0.3a: Determine the value of \(b\) .
• SPNone.1.hl.TZ0.3b: Find the representation of \(N\)   (i) in base \(10\);   (ii) in base \(12\).
• SPNone.1.hl.TZ0.7a: Given that \(a \equiv b(\bmod p)\) , show that \({a^n} \equiv {b^n}(\bmod p)\) for all...
• SPNone.1.hl.TZ0.7b: Show that \({29^{13}} + {13^{29}}\) is exactly divisible by \(7\).
• SPNone.1.hl.TZ0.13: A sequence \(\left\{ {{u_n}} \right\}\) satisfies the recurrence relation...
• SPNone.2.hl.TZ0.6a: A connected planar graph has \(e\) edges, \(f\) faces and \(v\) vertices. Prove Euler’s relation,...
• SPNone.2.hl.TZ0.6b: (i)     A simple connected planar graph with \(v\) vertices, where \(v \ge 3\) , has no circuit...
• SPNone.2.hl.TZ0.6c: The graph \(P\) has the following adjacency table, defined for vertices A to H, where each...
• 14M.1.hl.TZ0.1: Find the positive square root of the base 7 number \({(551662)_7}\), giving your answer as a base...
• 14M.1.hl.TZ0.15: (a)     Show that the solution to the linear congruence \(ax \equiv b(\bmod p)\), where...
• 14M.2.hl.TZ0.3: The vertices and weights of the graph \(G\) are given in the following table.   (a)     (i)  ...
• 14M.2.hl.TZ0.6: (a)     Consider the recurrence relation \(a{u_{n + 1}} + b{u_n} + c{u_{n - 1}} = 0\). Show that...