# Recent questions and answers in Engineering Mathematics 1
Let $S = \{1, 2, 3,\ldots, m\}, m >3.$ Let $X_1,\ldots,X_n$ be subsets of $S$ each of size $3.$ Define a function $f$ from $S$ to the set of natural numbers as, $f(i)$ is the number of sets $X_j$ that contain the element $i.$ That is $f(i)=\left | \left\{j \mid i\in X_j \right\} \right|$ then $\sum_{i=1}^{m} f(i)$ is: $3m$ $3n$ $2m+1$ $2n+1$
2
Which of the following pairs of propositions are not logically equivalent? $((p \rightarrow r) \wedge (q \rightarrow r))$ and $((p \vee q) \rightarrow r)$ $p \leftrightarrow q$ and $(\neg p \leftrightarrow \neg q)$ $((p \wedge q) \vee (\neg p \wedge \neg q))$ and $p \leftrightarrow q$ $((p \wedge q) \rightarrow r)$ and $((p \rightarrow r) \wedge (q \rightarrow r))$
3
How many ways are there to pack six copies of the same book into four identical boxes, where a box can contain as many as six books? $4$ $6$ $7$ $9$
4
The number of positive integers not exceeding $100$ that are either odd or the square of an integer is _______ $63$ $59$ $55$ $50$
5
A binary relation $R$ on $\mathbb{N} \times \mathbb{N}$ is defined as follows: $(a, b) R(c, d)$ if $a \leq c$ or $b \leq d$. Consider the following propositions: $P:$ $R$ is reflexive. $Q:$ $R$ is transitive. Which one of the following statements is TRUE? Both $P$ and $Q$ are true. $P$ is true and $Q$ is false. $P$ is false and $Q$ is true. Both $P$ and $Q$ are false.
6
Suppose we uniformly and randomly select a permutation from the $20 !$ permutations of $1, 2, 3\ldots ,20.$ What is the probability that $2$ appears at an earlier position than any other even number in the selected permutation? $\left(\dfrac{1}{2} \right)$ $\left(\dfrac{1}{10}\right)$ $\left(\dfrac{9!}{20!}\right)$ None of these
7
An unbalanced dice (with $6$ faces, numbered from $1$ to $6$) is thrown. The probability that the face value is odd is $90\%$ of the probability that the face value is even. The probability of getting any even numbered face is the same. If the probability that the face is ... one of the following options is closest to the probability that the face value exceeds $3$? $0.453$ $0.468$ $0.485$ $0.492$
8
The value of the integral $\displaystyle{}\int_{-1}^1 \dfrac{x^2}{1+x^2} \sin x \sin 3x \sin 5x dx$ is $0$ $\frac{1}{2}$ $– \frac{1}{2}$ $1$
1 vote
9
Let $A,B,C,D$ be $n\times n$ matrices, each with non-zero determinant. If $ABCD=1$, then $B^{-1}$ is: $D^{-1}C^{-1}A^{-1}$ $CDA$ $ADC$ Does not necessarily exist.
10
Let $f(x)$ be the continuous probability density function of a random variable $x$, the probability that $a < x \leq b$, is : $f(b-a)$ $f(b) - f(a)$ $\int\limits_a^b f(x) dx$ $\int\limits_a^b xf (x)dx$
11
How many $4$-digit even numbers have all $4$ digits distinct $2240$ $2296$ $2620$ $4536$
12
There are four bus lines between $A$ and $B$; and three bus lines between $B$ and $C$. The number of way a person roundtrip by bus from $A$ to $C$ by way of $B$ will be $12$ $7$ $144$ $264$
13
Two $n$ bit binary strings, $S_1$ and $S_2$ are chosen randomly with uniform probability. The probability that the Hamming distance between these strings (the number of bit positions where the two strings differ) is equal to $d$ is $\dfrac{^{n}C_{d}}{2^{n}}$ $\dfrac{^{n}C_{d}}{2^{d}}$ $\dfrac{d}{2^{n}}$ $\dfrac{1}{2^{d}}$
14
Consider the following graph $L$ and find the bridges,if any. No bridge $\{d,e\}$ $\{c,d\}$ $\{c,d\}$ and $\{c,f\}$
15
Mala has the colouring book in which each English letter is drawn two times. She wants to paint each of these $52$ prints with one of $k$ colours, such that the colour pairs used to colour any two letters are different. Both prints of a letter can also be coloured with the same colour. What is the minimum value of $k$ that satisfies this requirement? $9$ $8$ $7$ $6$
1 vote
16
The value of $p$ such that the vector $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ is an eigen vector of the matrix $\begin{bmatrix} 4 & 1 & 2 \\ p & 2 & 1 \\ 14 & -4 & 10 \end{bmatrix}$ is $15$ $16$ $17$ $18$
17
The number of distinct simple graphs with up to three nodes is $15$ $10$ $7$ $9$
18
Let $A$ be an invertible $10 \times 10$ matrix with real entries such that the sum of each row is $1$. Then The sum of the entries of each row of the inverse of $A$ is $1$ The sum of the entries of each column of the inverse of $A$ is $1$ The trace of the inverse of $A$ is non-zero None of the above
19
Let G be a connected planar graph with 10 vertices. If the number of edges on each face is three, then the number of edges in G is_______________.
1 vote
20
If $f(x)=k$ exp, $\{ -(9x^2-12x+13)\}$, is a $p, d, f$ of a normal distribution ($k$, being a constant), the mean and standard deviation of the distribution: $\mu = \frac{2}{3}, \sigma = \frac{1}{3 \sqrt{2}}$ $\mu = 2, \sigma = \frac{1}{\sqrt{2}}$ $\mu = \frac{1}{3}, \sigma = \frac{1}{3 \sqrt{2}}$ $\mu = \frac{2}{3}, \sigma = \frac{1}{ \sqrt{3}}$
21
The limit $\underset{n\rightarrow \infty }{\lim}\:n^{2}\int_{0}^{1}\:\frac{1}{\left ( 1+x^{2} \right )^{n}}\:dx$ is equal to $1$ $0$ $+\infty$ $1/2$
22
A solution for the differential equation $x’(t) + 2x(t) = \delta(t)$ with initial condition $x(\overline{0}) = 0$ $e^{-2t}u(t)$ $e^{2t}u(t)$ $e^{-t}u(t)$ $e^{t}u(t)$
23
15. a) How many cards must be chosen from a standard deck of 52 cards to guarantee that at least two of the four aces are chosen? b) How many cards must be chosen from a standard deck of 52 cards to guarantee that at least two of the four aces and at least ... many cards must be chosen from a standard deck of 52 cards to guarantee that there are at least two cards of each of two different kinds?
24
How many pairs $(x,y)$ such that $x+y <= k$, where x y and k are integers and $x,y>=0, k > 0$. Solve by summation rules. Solve by combinatorial argument.
25
Solve the recurrence relation for the number of rounds in the tournament described in question $14.$
26
How many rounds are in the elimination tournament described in question $14$ when there are $32$ teams?
27
Suppose that there are $n = 2^{k}$ teams in an elimination tournament, where there are $\frac{n}{2}$ games in the first round, with the $\frac{n}{2} = 2^{k-1}$ winners playing in the second round, and so on. Develop a recurrence relation for the number of rounds in the tournament.
28
Give a big-O estimate for the function $f$ given below if $f$ is an increasing function. $f (n) = 2f (n/3) + 4 \:\text{with}\: f (1) = 1.$
1 vote
29
Find $f (n)$ when $n = 3k,$ where $f$ satisfies the recurrence relation $f (n) = 2f (n/3) + 4 \:\text{with}\: f (1) = 1.$
30
Give a big-O estimate for the function $f$ in question $10$ if $f$ is an increasing function.
31
Find $f (n)$ when $n = 2^{k},$ where $f$ satisfies the recurrence relation $f (n) = f (n/2) + 1 \:\text{with}\: f (1) = 1.$
32
Suppose that $f (n) = f (n/5) + 3n^{2}$ when $n$ is a positive integer divisible by $5, \:\text{and}\: f (1) = 4.$ Find $f (5)$ $f (125)$ $f (3125)$
33
Suppose that $f (n) = 2f (n/2) + 3$ when $n$ is an even positive integer, and $f (1) = 5.$ Find $f (2)$ $f (8)$ $f (64)$ $(1024)$
34
Suppose that $f (n) = f (n/3) + 1$ when $n$ is a positive integer divisible by $3,$ and $f (1) = 1.$ Find $f (3)$ $f (27)$ $f (729)$
35
How many operations are needed to multiply two $32 \times 32$ matrices using the algorithm referred to in Example $5?$
36
Determine a value for the constant C in Example $4$ and use it to estimate the number of bit operations needed to multiply two $64$-bit integers using the fast multiplication algorithm.
37
Express the fast multiplication algorithm in pseudocode.
Multiply $(1110)_{2} \:\text{and}\: (1010)_{2}$ using the fast multiplication algorithm.
How many comparisons are needed to locate the maximum and minimum elements in a sequence with $128$ elements using the algorithm in Example $2$?
How many comparisons are needed for a binary search in a set of $64$ elements?