# Recent questions and answers in Engineering Mathematics

1
We are given a set $X = \{X_1,\ldots,X_n\}$ where $X_i=2^i$. A sample $S\subseteq X$ is drawn by selecting each $X_i$ independently with probability $P_i = \frac{1}{2}$ . The expected value of the smallest number in sample $S$ is: $\left(\frac{1}{n}\right)$ $2$ $\sqrt n$ $n$
2
Let $P =\sum \limits_ {i\;\text{odd}}^{1\le i \le 2k} i$ and $Q = \sum\limits_{i\;\text{even}}^{1 \le i \le 2k} i$, where $k$ is a positive integer. Then $P = Q - k$ $P = Q + k$ $P = Q$ $P = Q + 2k$
3
$\int^{\pi/4}_0 (1-\tan x)/(1+\tan x)\,dx$ $0$ $1$ $\ln 2$ $1/2 \ln 2$
1 vote
4
Isn’t the 3) statement is wrong it should be [(p->r)^(q->r)]->[(p∨q)->r] \
5
Given a set of elements $N = {1, 2, ..., n}$ and two arbitrary subsets $A⊆N$ and $B⊆N$, how many of the n! permutations $\pi$ from $N$ to $N$ satisfy $\min(\pi(A)) = \min(\pi(B))$, where $\min(S)$ is the smallest integer in the set of integers $S$, and $\pi$(S) is the set of integers obtained by ... $n! \frac{|A ∩ B|}{|A ∪ B|}$ $\dfrac{|A ∩ B|^2}{^n \mathrm{C}_{|A ∪ B|}}$
6
Let $A$ be a set with $n$ elements. Let $C$ be a collection of distinct subsets of $A$ such that for any two subsets $S_1$ and $S_2$ in $C$, either $S_1 \subset S_2$ or $S_2\subset S_1$. What is the maximum cardinality of $C?$ $n$ $n+1$ $2^{n-1} + 1$ $n!$
7
Consider a set $U$ of $23$ different compounds in a chemistry lab. There is a subset $S$ of $U$ of $9$ compounds, each of which reacts with exactly $3$ compounds of $U$. Consider the following statements: Each compound in U \ S reacts with an odd number of ... in U \ S reacts with an even number of compounds. Which one of the above statements is ALWAYS TRUE? Only I Only II Only III None.
8
Let $S$ be a set of $n$ elements $\left\{1, 2,\ldots, n\right\}$ and $G$ a graph with $2^{n}$ vertices, each vertex corresponding to a distinct subset of $S$. Two vertices are adjacent iff the symmetric difference of the corresponding sets has exactly $2$ elements. ... $G$ has the same degree. What is the degree of a vertex in $G$? How many connected components does $G$ have?
9
Out of a group of $21$ persons, $9$ eat vegetables, $10$ eat fish and $7$ eat eggs. $5$ persons eat all three. How many persons eat at least two out of the three dishes?
10
Consider the following set of propositions 1. I carry an umbrella only if it rains. 2. If I carry an umbrella I use it. 3. I get wet in rain only if I don't use umbrella. 4. It rains today. Possible Conclusion is: A. I will not get wet today. B. I will get wet today. C. I might get wet today. D. None of the above.
11
Which of the following predicate calculus statements is/are valid? $(\forall (x)) P(x) \vee (\forall(x))Q(x) \implies (\forall (x)) (P(x) \vee Q(x))$ $(\exists (x)) P(x) \wedge (\exists (x))Q(x) \implies (\exists (x)) (P(x) \wedge Q(x))$ ... $(\exists (x)) (P(x) \vee Q(x)) \implies \sim (\forall (x)) P(x) \vee (\exists (x)) Q(x)$
12
Which one of the following predicate formulae is NOT logically valid? Note that $W$ is a predicate formula without any free occurrence of $x$. $\forall x (p(x) \vee W) \equiv \forall x \: ( px) \vee W$ $\exists x(p(x) \wedge W) \equiv \exists x \: p(x) \wedge W$ ... $\exists x(p(x) \rightarrow W) \equiv \forall x \: p(x) \rightarrow W$
13
A relation $R$ is said to be circular if $a\text{R}b$ and $b\text{R}c$ together imply $c\text{R}a$. Which of the following options is/are correct? If a relation $S$ is reflexive and symmetric, then $S$ is an equivalence relation. If a relation $S$ ... $S$ is an equivalence relation. If a relation $S$ is transitive and circular, then $S$ is an equivalence relation.
14
If $f(x)$ is defined as follows, what is the minimum value of $f(x)$ for $x \in (0, 2]$ ? $f(x) = \begin{cases} \frac{25}{8x} &\text{ when } x \leq \frac{3}{2} \\ x+ \frac{1}{x} &\text { otherwise}\end{cases}$ $2$ $2 \frac{1}{12}$ $2\frac{1}{6}$ $2\frac{1}{2}$
15
Suppose that there are 9 faculty members in the mathematics department and 11 in the computer science department. How many ways are there to select a committee to develop a discrete mathematics course at a school if the committee is to consist of three faculty members from the ... where to use Sum rule. How to decide on Product rule or Sum rule? Please use some basic example for both the cases.
16
Let $R$ be a binary relation on $A = \{a, b, c, d, e, f, g, h\}$ represented by the following two component digraph. Find the smallest integers $m$ and $n$ such that $m < n$ and $R^m = R^n$.
17
Let $R$ be a reflexive and transitive relation on a set $A$. Define a new relation $E$ on $A$ as $E=\{(a, b) \mid (a, b) \in R \text{ and } (b, a) \in R \}$ Prove that $E$ is an equivalence relation on $A$. Define a relation $\leq$ on the equivalence classes of $E$ as $E_1 \leq E_2$ if $\exists a, b$ such that $a \in E_1, b \in E_2 \text{ and } (a, b) \in R$. Prove that $\leq$ is a partial order.
18
Let $p, q, r$ ... $(\neg p \wedge r) \vee (r \rightarrow (p \wedge q))$
19
Consider the set of integers $\{1,2,3,4,6,8,12,24\}$ together with the two binary operations LCM (lowest common multiple) and GCD (greatest common divisor). Which of the following algebraic structures does this represent? group ring field lattice
20
There are two elements $x,\:y$ in a group $(G,*)$ such that every element in the group can be written as a product of some number of $x$'s and $y$'s in some order. It is known that $x*x=y*y=x*y*x*y=y*x*y*x=e$ where $e$ is the identity element. The maximum number of elements in such a group is ____.
21
Let $(A, *)$ be a semigroup, Furthermore, for every $a$ and $b$ in $A$, if $a \neq b$, then $a*b \neq b*a$. Show that for every $a$ in $A$, $a*a=a$ Show that for every $a$, $b$ in $A$, $a*b*a=a$ Show that for every $a,b,c$ in $A$, $a*b*c=a*c$
22
Given a boolean function $f (x_1, x_2, \ldots, x_n),$ which of the following equations is NOT true? $f (x_1, x_2, \ldots, x_n) = x_1'f(x_1, x_2, \ldots, x_n) + x_1f(x_1, x_2, \ldots, x_n)$ $f (x_1, x_2, \ldots, x_n) = x_2f(x_1, x_2, \ldots , x_n) + x_2'f(x_1, x_2, \ldots ,x_n)$ ... $f (x_1, x_2, \ldots , x_n) = f(0, x_2, , x_n) + f(1, x_2, \ldots, x_n)$
23
Let $R$ denote the set of real numbers. Let $f:R\times R \rightarrow R \times R$ be a bijective function defined by $f(x,y) = (x+y, x-y)$. The inverse function of $f$ is given by $f^{-1} (x,y) = \left( \frac {1}{x+y}, \frac{1}{x-y}\right)$ $f^{ -1} (x,y) = (x-y , x+y)$ $f^{-1} (x,y) = \left( \frac {x+y}{2}, \frac{x-y}{2}\right)$ $f^{-1}(x,y)=\left [ 2\left(x-y\right),2\left(x+y\right) \right ]$
24
Let $N$ be the set of natural numbers. Consider the following sets, $P:$ Set of Rational numbers (positive and negative) $Q:$ Set of functions from $\{0,1\}$ to $N$ $R:$ Set of functions from $N$ to $\{0, 1\}$ $S:$ Set of finite subsets of $N$ Which of the above sets are countable? $Q$ and $S$ only $P$ and $S$ only $P$ and $R$ only $P, Q$ and $S$ only
25
Consider the set $\{a, b, c\}$ with binary operators $+$ and $*$ defined as follows: ... $(b * x) + (c * y) = c$ The number of solution(s) (i.e., pair(s) $(x, y)$ that satisfy the equations) is $0$ $1$ $2$ $3$
26
Suppose a fair six-sided die is rolled once. If the value on the die is $1, 2,$ or $3,$ the die is rolled a second time. What is the probability that the sum total of values that turn up is at least $6$ ? $\dfrac{10}{21}$ $\dfrac{5}{12}$ $\dfrac{2}{3}$ $\dfrac{1}{6}$
27
Two people, $P$ and $Q$, decide to independently roll two identical dice, each with $6$ faces, numbered $1$ to $6$. The person with the lower number wins. In case of a tie, they roll the dice repeatedly until there is no tie. Define a trial as a ... equi-probable and that all trials are independent. The probability (rounded to $3$ decimal places) that one of them wins on the third trial is ____
28
Seven (distinct) car accidents occurred in a week. What is the probability that they all occurred on the same day? $\dfrac{1}{7^7}\\$ $\dfrac{1}{7^6}\\$ $\dfrac{1}{2^7}\\$ $\dfrac{7}{2^7}\\$
29
Let $G$ be a group of order $6$, and $H$ be a subgroup of $G$ such that $1<|H|<6$. Which one of the following options is correct? Both $G$ and $H$ are always cyclic $G$ may not be cyclic, but $H$ is always cyclic $G$ is always cyclic, but $H$ may not be cyclic Both $G$ and $H$ may not be cyclic
30
Let $G =\{a_1,a_2, \dots ,a_{12}\}$ be an Abelian group of order $12$ . Then the order of the element $( \prod_{i=1}^{12} a_i)$ is $1$ $2$ $6$ $12$
31
Suppose that the number plate of a vehicle contains two vowels followed by four digits. However, to avoid confusion, the letter $‘O’$ and the digit $‘0’$ are not used in the same number plate. How many such number plates can be formed? $164025$ $190951$ $194976$ $219049$
32
I think only d) is correct
33
Which of the following statements is true for every planar graph on $n$ vertices? The graph is connected The graph is Eulerian The graph has a vertex-cover of size at most $\frac{3n}{4}$ The graph has an independent set of size at least $\frac{n}{3}$
34
Let $G=(V, E)$ be an undirected unweighted connected graph. The diameter of $G$ is defined as: $\text{diam}(G)=\displaystyle \max_{u,v\in V} \{\text{the length of shortest path between$u$and$v$}\}$ Let $M$ be the adjacency matrix of $G$. Define graph $G_2$ on the same set of ... $\text{diam}(G_2) = \text{diam}(G)$ $\text{diam}(G)< \text{diam}(G_2)\leq 2\; \text{diam}(G)$
35
Let $G$ be any connected, weighted, undirected graph. $G$ has a unique minimum spanning tree, if no two edges of $G$ have the same weight. $G$ has a unique minimum spanning tree, if, for every cut of $G$, there is a unique minimum-weight edge crossing the cut. Which of the following statements is/are TRUE? I only II only Both I and II Neither I nor II
36
The line graph $L(G)$ of a simple graph $G$ is defined as follows: There is exactly one vertex $v(e)$ in $L(G)$ for each edge $e$ in $G$. For any two edges $e$ and $e'$ in $G$, $L(G)$ has an edge between $v(e)$ and $v(e')$, if and only if $e$ and $e'$ are ... line graph of a planar graph is planar. (S) The line graph of a tree is a tree. $P$ only $P$ and $R$ only $R$ only $P, Q$ and $S$ only
1 vote
37
Let $\theta=2\pi/67$. Now consider the matrix $A = \begin{pmatrix} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta \end{pmatrix}$. Then the matrix $A^{2010}$ is $\begin{pmatrix} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta \end{pmatrix}$ ... $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$
Let $\{a_n\}$ be a sequence of real numbers. Then $\underset{n \to \infty}{\lim} a_n$ exists if and only if $\underset{n \to \infty}{\lim} a_{2n}$ and $\underset{n \to \infty}{\lim} a_{2n+2}$ exists $\underset{n \to \infty}{\lim} a_{2n}$ ... $\underset{n \to \infty}{\lim} a_{3n}$ exist none of the above
In an undirected connected planar graph $G$, there are eight vertices and five faces. The number of edges in $G$ is _________.