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Discrete Mathematics Exam Prep
English 63 leveled MCQs Free
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Sample questions with model answers
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10 marks Long answer
Solve the following recurrence relations by the methods of characteristic roots: $a_n - 4a_{n-1} + 3a_{n-2} = 2^n, n \ge 2, a_0 = 1, a_1 = 11$.
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Apply the characteristic equation and find the particular solution.
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10 marks Long answer
Solve the recurrence relation $a_{n+2} + 4a_{n+1} + 4a_n = 7, n \ge 0, a_0 = 1, a_1 = 2$. By Generating function.
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Use generating functions to derive the closed form of the sequence.
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10 marks Long answer
Apply Prim's algorithm and Kruskal's Algorithm to find the minimum spanning tree (MST) of the graph with edges: (A-B, 4), (A-C, 2), (B-C,1), (B-D, 5), (C-D, 3). Also find the total weight of the minimal spanning tree.
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MST weight is 6.
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10 marks Long answer
Construct the expression tree for the expression $((3+5)/2) - ((7*2)/7)$. Traverse the tree in preorder, in-order and post-order traversals. Also find prefix, infix and postfix notations of the given expression.
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Prefix: - / + 3 5 2 / 7 14; Infix: ((3+5)/2) - ((7*2)/7); Postfix: 3 5 + 2 / 7 2 * 7 / -.
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8 marks Long answer
Solve $a_n - 3a_{n-1} + 2a_{n-2} = 0, a_0 = 1, a_1 = 2$.
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The solution is $a_n = 2^n + n$.
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6 marks Long answer
Prove the validity of the following arguments: $\neg r \rightarrow (s \rightarrow \neg t)$, $\neg v \lor w$, $p \rightarrow s$, $\neg w$, $\therefore t \rightarrow p$.
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Use truth tables or logical deduction to show the conclusion follows from the premises.
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6 marks Long answer
Define minimal spanning tree. Using Prim's algorithm, find a minimal spanning tree for the following weighted graph.
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A minimal spanning tree connects all vertices with minimum total edge weight; apply Prim's algorithm step by step.
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5 marks Long answer
Explain the following with example. (i) complete graph (ii) Bipartite graph (iii) complete Bipartite graph.
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Complete graph: all vertices connected; Bipartite: two sets of vertices; Complete Bipartite: every vertex of one set connected to all vertices of the other.
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5 marks Long answer
If G is a planar connected graph with n vertices, m edges and r regions then show that $r+n-m = 2$.
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Use Euler's formula for planar graphs to derive the relationship.
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4 marks Medium answer
Construct truth table if : $(P \rightarrow \neg R) \rightarrow (\neg Q \land R) \leftrightarrow (Q \rightarrow R)$.
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Construct a truth table for all combinations of truth values for P, Q, and R.
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4 marks Medium answer
Give an indirect proof for the following statement: If x and y are integers such that xy is odd, then x and y are both odd.
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Assume x or y is even, leading to a contradiction since xy cannot be odd.
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4 marks Medium answer
Obtain the Principal Disjunctive Normal Form for $(P \land Q) \lor (\neg P \land R) \lor (Q \land R)$.
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The PDNF is $PQR + PQ + \neg PQR$.
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4 marks Medium answer
In how many ways can the 26 letters of the English alphabets be permuted so that none of the patterns CAR, DOG, PUN, BYTE occur?
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Calculate using the principle of inclusion-exclusion.
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4 marks Medium answer
Let $X = \{1,2,5,10,20,40,60,120\}$ and the relation R defined by aRb if and only if a divides b. Check it is Lattice or not?
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Check the properties of meet and join to verify if R forms a lattice.
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4 marks Medium answer
Find the Identity and Inverse elements of the group with the operator * defined as (i) $(a, b) * (c, d) = \{ad-bc, ac+bd\}$ (ii) $a*b = (ab)/3$.
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Identity for (i) is (0,0), and for (ii) is 1.
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4 marks Medium answer
A drawer contains ten black and ten white socks. What is the least the least number of socks one must pull out to be sure to get a matched pair?
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At least 3 socks must be pulled out.
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4 marks Medium answer
Find the number of integral solutions to the equation $x_1 + x_2 + x_3 + x_4 + x_5 = 20$ where $x_1 \ge 3, x_2 \ge 2, x_3 \le 4, x_4 \ge 6, x_5 \ge 0$.
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Use generating functions or combinatorial methods to find the solutions.
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4 marks Medium answer
How many 2-digit or 3-digit numbers can be formed using the digits 1, 3, 4, 5, 6, 8 and 9 if no repetition is allowed?
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Calculate the combinations for 2-digit and 3-digit separately.
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4 marks Medium answer
In how many ways can a committee of five ladies and four gents be chosen from nine ladies and 15 gents, if a particular person A refuses to take part if lady B is on the committee?
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Use combinatorial counting considering the restriction.
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4 marks Medium answer
Find the 5-digit integers that contain the number 6 exactly once.
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Count the valid placements of 6 in the 5-digit number.
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4 marks Medium answer
State the Sum of degrees theorem and find N when a graph has N vertices of degree 1, 2 vertices of degree 2, 4 vertices of degree 3 and 3 vertices of degree 4.
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The sum of degrees equals twice the number of edges; calculate N based on given degrees.
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4 marks Medium answer
Show that $p \rightarrow q$ is equivalent to $\neg p \lor q$ using truth tables.
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Truth tables confirm the equivalence.
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4 marks Medium answer
Translate into predicate logic: (i) All students are intelligent. (ii) Some birds cannot fly.
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i) ∀x (Student(x) → Intelligent(x)), ii) ∃y (Bird(y) ∧ ¬Fly(y)).
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4 marks Medium answer
Out of 100 students, 60 like tea, 45 like coffee and 20 like both. How many like neither tea nor coffee?
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20 students like neither.
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4 marks Medium answer
Let $f(x)=x+2$ and $g(x)=2x$. Find $(f \circ g)(3)$ and $(g \circ f)(3)$.
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(f \circ g)(3) = 8, (g \circ f)(3) = 10.
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4 marks Medium answer
If $f(x)=3x+5$, find the inverse $f^{-1}(x)$.
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The inverse is $f^{-1}(x) = \frac{x-5}{3}$.
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4 marks Medium answer
Find the coefficient of $x^4$ in the expansion of $(1+x)^{12}$.
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The coefficient is 495.
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4 marks Medium answer
Using multinomial theorem, find the coefficient of $a^2 b^2 c^2$ in $(a+b+c)^6$.
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The coefficient is 90.
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4 marks Medium answer
Find the middle term(s) in the expansion of $(x+3)^{10}$.
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The middle term is 945.
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4 marks Medium answer
Show that $\sum_{k=0}^{n} \binom{n}{k} = 2^n$.
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It follows from the binomial theorem.
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4 marks Medium answer
Solve $a_n = a_{n-1} + n, a_0 = 0$.
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a_n = \frac{n(n+1)}{2}.
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4 marks Medium answer
Solve $a_n = a_{n-1} + 2n, a_0 = 1$.
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a_n = n^2 + n + 1.
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3 marks Medium answer
Distinguish between implication and equivalence with examples.
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Implication is $p \rightarrow q$, equivalence is $p \leftrightarrow q$.
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3 marks Medium answer
Test whether $(p \leftrightarrow q)$ is equivalent to $(p \rightarrow q) \land (q \rightarrow p)$.
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Yes, they are equivalent.
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3 marks Medium answer
Differentiate between universal quantifier ($\forall$) and existential quantifier ($\exists$) with examples.
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Universal: ∀x P(x), Existential: ∃y Q(y).
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3 marks Medium answer
Write the negation of the statement: $\forall x (P(x) \lor Q(x))$.
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The negation is $\exists x (\neg P(x) \land \neg Q(x))$.
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3 marks Medium answer
State and explain the principle of inclusion-exclusion.
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It calculates the size of unions by subtracting intersections.
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3 marks Medium answer
Find the number of integers between 1 and 200 divisible by 3 or 5.
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The count is 66.
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3 marks Medium answer
Define composition of functions with an example.
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Composition: $(f \circ g)(x) = f(g(x)).
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3 marks Medium answer
A committee of 6 is formed from a group of 8 men and 7 women. Find the number of ways of selecting the committee with 3 men, 3 women.
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The number of ways is 280.
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2 marks Short answer
Give Truth table for the propositional formula $(p \rightarrow q) \rightarrow q$.
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The truth table shows the output of the formula based on the truth values of p and q.
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2 marks Short answer
State the converse, opposite, and contrapositive to the following: If triangle ABC is a right-angled triangle, then $|AB|^2 + |BC|^2 = |AC|^2$.
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Converse: If $|AB|^2 + |BC|^2 = |AC|^2$, then triangle ABC is a right-angled triangle.
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2 marks Short answer
Define equivalence relation.
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An equivalence relation is a binary relation that is reflexive, symmetric, and transitive.
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2 marks Short answer
Find fog(x) if $f(x) = x^2 + 1$ and $g(x) = 2x – 1$.
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fog(x) = (2x - 1)^2 + 1.
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2 marks Short answer
Find the number of all possible five letter words from the word “DADDY”.
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There are 30 distinct five-letter words from 'DADDY'.
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2 marks Short answer
What is the coefficient of $x^3y$ in $(x+y)^{10}$?
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The coefficient is 120.
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2 marks Short answer
Find the generating function for the sequence $0^2, 1^2, 2^2, 3^2, \ldots$.
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The generating function is $\frac{x^2}{(1-x)^3}$.
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2 marks Short answer
A person deposits Rs: 1,000/- that yields interest compounded 9% yearly. Set up the recurrence relation for the amount in the account at the end of 'n' years.
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The recurrence relation is $A_n = 1.09 A_{n-1}$, with $A_0 = 1000$.
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2 marks Short answer
Explain is there a simple graph corresponding to degree sequence (1,1,2,3).
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Yes, a simple graph can be constructed with vertices of degrees 1, 1, 2, 3.
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2 marks Short answer
Define Hamiltonian graphs.
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A Hamiltonian graph is a graph that contains a Hamiltonian cycle, visiting each vertex exactly once.
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2 marks Short answer
Check whether the statement $(p \lor eg p)$ is a tautology.
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Yes, it is a tautology.
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2 marks Short answer
Translate the statement 'All humans are mortal' into predicate logic.
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∀x (Human(x) → Mortal(x))
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2 marks Short answer
Define the recursive function: $f(0)=1, f(n)=2f(n-1)$. Find $f(3)$.
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f(3) = 8.
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2 marks Short answer
Show that the set of natural numbers under addition is a monoid but not a group.
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It has an identity (0) but lacks inverses.
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2 marks Short answer
In how many ways can the letters of the word MISSISSIPPI be arranged?
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The arrangements = 34650.
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2 marks Short answer
Find the coefficient of $x^4$ in the expansion of $(1+x)^{10}$.
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The coefficient is 210.
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2 marks Short answer
Solve the recurrence relation $a_n = 3a_{n-1}, a_0 = 2$.
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a_n = 2 * 3^n.
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2 marks Short answer
If $a_n=2^n$, write its generating function.
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The generating function is $\frac{1}{1-2x}$.
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2 marks Short answer
How many edges does a tree with 8 vertices have?
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A tree with 8 vertices has 7 edges.
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2 marks Short answer
Construct a complete binary tree of height 3. How many leaf nodes does it have?
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It has 8 leaf nodes.
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