Paper 2 CS Discrete Structures and Optimization Part 4

With the help of dirac’s theorem, we can prove above three statements.

“0 and 100 → Counting sequentially:
0,6,12,18,24,30,36,42,48,54,60,66,72,78,84,90,96 Total=17
–6 and 34 → Counting sequentially: -6,0,6,12,18,24,30
Total=7 “

“Step-1: Given data,
f(x) = x​^3​ – 4x, g(x) = 1/(x​^ 2​ + 1) and h(x) = x​ ^4
hog(x)=h(1/(x​ ^2​ + 1))
=h(1/(x​ ^2​ )+1)^​ 4
= 1/(x​^ 2​ +1)​^ 4
= (x​ ^2​ +1)​ ^-4
hogof(x)= hog(x^​ 3​ -4x)
= h(1/(x^​ 3​ -4x)​^ 2​ +1)
= h(1/(x^​ 3​ -4x)^​ 2​ +1)​^ 4
= h((x​^ 3​ -4x)​^ 2​ +1)^​ -4
So, option D id is correct answer. “

“Given data,
Sequence F​ 00​ defined as
F​ 00​ (0) = 1,
F​ 00​ (1) = 1,
F​ 00​ (n) = ((10 ∗ F​ 00​ (n – 1) + 100)/ F​ 00​ (n – 2)) for n ≥ 2
Let n=2
F​ 00​ (2) = (10 * F​ 00​ (1) + 100) / F​ 00​ (2 – 2)
= (10 * 1 + 100) / 1
= (10 + 100) / 1
= 110
Let n=3
F​ 00​ (3) = (10 * F​ 00​ (2) + 100) / F​ 00​ (3 – 2)
= (10 * 110 + 100) / 1
= (1100 + 100) / 1
= 1200
Similarly, n=4
F​ 00​ (4) = (10 * F​ 00​ (3) + 100) / F​ 00​ (4 – 2)
= (12100) / 110
= 110
F​ 00​ (5) = (10 * F​ 00​ (4) + 100) / F​ 00​ (5 – 2)
= (10*110 + 100) / 1200
= 1
The sequence will be (1, 110, 1200,110, 1). “

Take negation inside.

“Total distinguishable permutations means no repetition.
BANANA= 6 letters
B is appearing 1 time
A is appearing 3 times
N is appearing 2 times
So, = 6! / (3! * 2!)
= 60 “

The sum of the series: (1/2) + (1/3) + (1/4) – (1/6) + (1/8) + (1/9) + (1/16) – (1/12) + … + α is 1 + log(√2)3.

“→ In a Lattice if Upper Bound and Lower exists then it is called Bounded Lattice.
→ A bounded lattice is an algebraic structure of the form (L, ∨, ∧, 0, 1) such that (L, ∨, ∧) is a lattice, 0 (the lattice’s bottom) is the identity element for the join operation ∨, and 1 (the lattice top) is the identity element for the meet operation ∧.
→ Let ‘L’ be a lattice w.r.t R if there exists an element I∈L such that (aRI)∀x∈L, then I is called Upper Bound of a Lattice L.
Similarly, if there exists an element O∈L such that (ORa)∀a∈L, then O is called Lower Bound of Lattice L. “

The relation R is {(1,3), (1,5),(2,3),(2,5),(3,5),(4,5) } where (a,b) ∈ R if a The R-1 is { (3,1),(5,1),(3,2),(5,2),(5,3),(5,4) } where (a,b) ∈ R-1 if a>b.

“11=1*1=1 which is a factor of 11
12=1*2=2 which is a factor of 12
15=1*5=5 which is a factor of 15
24=2*4=8 which is a factor of 24
36=3*6=12 which is a factor of 36 “

“Some properties are tautologies:
1. The negation of a contradiction is a tautology.
2. The disjunction of two contingencies can be a tautology.
3. The conjunction of two tautologies is a tautology. “

“A) numbers divisible by 2: 200/2 = 100
B) numbers divisible by 3: 200/3 = 66
C) numbers divisible by 5: 200/5 = 40
Counting twice:
AB) numbers divisible by 6: 200/6 = 33
AC) numbers divisible by 10: 200/10 = 20
BC) numbers divisible by 15: 200/15 = 13
Counting 3 times:
ABC) numbers divisible by 30: 200/30 = 6
Total of numbers = A + B + C – AB – AC – BC + ABC = 100 + 66 + 40 – 33 – 20 -13 + 6 = 146 “

“Sum upto n terms = 1/(1*2) + 1/(2*3) + 1/(3*4) + …….. + 1/(n*(n+1))
where
1st term = 1/(1*2)
2nd term = 1/(2*3)
3rd term = 1/(3*4)
.
.
.
.
n-th term = 1/(n*(n+1))
n-th term = 1/(n*(n+1))
i.e. the k-th term is of the form 1/(k*(k+1))
which can further be written as k-th term = 1/k – 1/(k+1)
So, sum upto n terms can be calculated as:
(1/1 – 1/1+1) + (1/2 – 1/2+1) + (1/3 – 1/3+1) + ……… + (1/n-1 – /1n) + (1/n – 1/n+1)
= (1 – 1/2) + (1/2 – 1/3) + (1/3 – 1/4) + ……… + (1/n-1 – 1/n) + (1/n – 1/n+1)
= 1 – 1/n+1
= ((n+1) – 1)/n+1
= n/n+1 “

“• Declarative sentences are propositions.
• Sentences that assert a fact that could either be true or false. “

“• A square matrix of order ’r’ is nonsingular if its determinant is non zero and therefore its rank is “r”. It’s all rows and columns are linearly independent and it is invertible.
• Rank of singular matrix is less than “r”. “

Given a connected graph G, a connected subgraph that is both a tree and contains all the vertices of G is called a spanning tree for G.Given a connected graph G, a connected subgraph that is both a tree and contains all the vertices of G is called a spanning tree for G.

“• If a determinant D vanishes for x = a, then (x – a) is a factor of D, in other words, if two rows (or two columns) become identical for x = a, then (x-a) is a factor of D.
• In general, if k rows (or k columns) become identical (x=a) when a is substituted for x, then (x-a)r-1 is a factor of D. “

“As the order of people does not matter, it is C4 12
C(n,r) = C(12,4)
= 12! / [(4!(12−4)!)]
= 495
Hence, a committee of 4 people be selected from a group of 12 people in 495 ways. “

“• Three digit odd numbers implies the numbers would only be made of digits 1 , 3 , 5 , 7 , 9 With repetition of digits we would have had 5 * 5 * 5 = 125
• But for every hundreds, maximum of 4 tens is possible (avoiding the duplicate of digit used in hundreds)
• And each of these 4 tens, maximum of 3 units is possible (avoiding the duplicate of digits used in tens)
• 5 * 4 * 3 = 60 “

The root mean square deviation when measured from the mean is least value.

The sum of the deviations from the mean is zero. This will always be the case as it is a property of the sample mean, i.e., the sum of the deviations below the mean will always equal the sum of the deviations above the mean. However, the goal is to capture the magnitude of these deviations in a summary measure.

The exponential distribution is not the same as the class of exponential families of distributions, which is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes the normal distribution, binomial distribution, gamma distribution, Poisson, and many others.

Symmetrical distribution occurs when the values of variables occur at regular frequencies and the mean, median and mode occur at the same point. In graph form, symmetrical distribution often appears as a bell curve. If a line were drawn dissecting the middle of the graph, it would show two sides that mirror each other.

Standard formula

“→ A discrete variable is a variable whose value is obtained by counting.
Examples:
number of students present
number of red marbles in a jar
number of heads when flipping three coins
students’ grade level
→ A discrete random variable X has a countable number of possible values.
Example:
Let X represent the sum of two dice.
→ A discrete random variable is one which may take on only a countable number of distinct values such as 0,1,2,3,4,…….. Discrete random variables are usually (but not necessarily) counts. If a random variable can take only a finite number of distinct values, then it must be discrete.
Examples of discrete random variables include the number of children in a family, the Friday night attendance at a cinema, the number of patients in a doctor’s surgery, the number of defective light bulbs in a box of ten. “

“● From the matrix property : If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.
● Consider matrix order is 3, there are 3 rows and each row is multiplied by 2 means we need to multiply 8 to the existing determinant.
● The given existing determinant is 5 and each row multiplied by 2 means 8 *5 =40. “

“● Given the recurrence relation a​ r​ =a​ r-1​ +2a​ r-2 and a​ 0​ =2,a​ 1​ =7.
● For r =2, a​ 2=​ a​ 2-1​ +2a​ 2-2​ =a​ 1​ +2a​ 0​ =7+2*2=7+4=11 ● For r=3,a​ 3=​ a​ 3-1​ +2a​ 3-2​ =a​ 2​ +2a​ 1​ =11+2*7=11+14=25
● From the above options ,Substitute the r values 0,1,2,3 then option -D gives the solution to recurrence relation. “