COURSE CODE:MTH 211COURSE TITLE:LOGIC AND LINEAR ALGEBRACLASS:ND2 CIVIL , MECHANICAL, ELECTRICAL AAND MECHANICALENGINEERINGDefine:
(i) premises (ii) contradiction (iii) tautology
What is a valid argument, hence establish the validity of the premises
P1: Alll engineering student are intelligent.
P2: Emmanuel is an engineering student
Let P: Mariam is an intelligent girl
Q: Mariam is able to provide solution to complicated problem.
S: Either mariam is an intelligent girl or she is not able to provide solution to the problem. Draw the truth table for S.
SOLUTION
PREMISES : Premises are proposition affirmed as providing grounds or reasons for accepting the conclusion.
CONTRACTION (absurdity): A statement form that only false substitution instances is called contradiction. A statement or proposition that contradicts or denies another or itself and is logically incongruous.
Is a situation or ideas in operation to one another. A contradiction is a sentence together with its negation, and a theory is inconsistent if it include a contradiction OR a statement or proposition that contradict or denies another or itself and is logically incongruous.
TAUTOLOGY : Is a formular that is true in all possible state.
Tautology is a formular which is always true. It opposite is contradiction. Tautology can also be define as a logical statement in which the conclusion is equivalent to the premise.
Examples of tautology are;
Either it will rain tomorrow, or it won’t rain.
Bill will win the election, or he will not win the election.
She is brave, or she is not brave.
[i]What is the valid argument? An argument is valid if the truth value of the premises logically guarantees the truth of the conclusion.
[ii] Hence use the valid argument to establish the validity of the premises.
The answer for it is;
If p2: All engineering student are intelligent
P2: Emmanuel is an engineering student.
Therefore, Emmanuel is intelligent.
The answer is: EMMANUEL IS INTELLIGEENT
THE TRUTH TABLE FOR SMariam is an intelligent girl =P
And Mariam is able to provide solution to complicated problem =Q
S = PvQ]
PQP^QPvQPvQ]TTFFTTTTFFTFTTFTTFFTFFFTTFFT
(a) Derive truth table for p^q→p is a tautology
SOLUTION
PQP^QPvQTTFFTTTTFFTFTFFTTFFTTFFTTFFT
PQP^QPvQP^Q→PTTFFTTTTTFFTFTFTFTTFFTTTFFTTFFTTCONDITIONAL STATEMENTS ARE ONLY FALSE WHEN P IS TRUE AND Q IS FALSE FOR ALL OTHER CONDITIONS IT’S TUE
According to the definition of tautology being true any state given as the premises.
YOU CAN MESSAGE EMPERORELECTRICALWORKS@GMAIL.COM FOR MORE ENLIGHTMENT.
(b) Use the truth table to show the Morgan’s Law
DE Morgan’s law
THERE ARE TWO WAYS OF SOLVING DE MORGAN’S LAW ON TRUTH TABLE
EITHER [pVq] p^q
OR [p^q] pvq
USING THE FIRST ONE TO PROVE DE MORGAN’S LAW i.e [pVq] p^q
PQpPVq[pVq]pqTTFFTFFTFFTTFFFTTFTFFFFTTFTT
OR [p^q] pvq
PQpP^q[p^q]pvqTTFFTFFTFFTFTTFTTFFTTFFTTFTT
It is logically equivalent.
NOTE: IN EIITHER WAY IT MUST PROVE THAT THE LEFT HAND SIDE IS LOGICALLY EQUIVALENT WITH THE RIGHT HAND SIDE.
(c) Use the truth table to establish the proposition P↔Q=(P↔Q)^(Q↔P)
SOLUTION
PQP^QPvQQ↔PP↔Q[P↔Q^ Q↔P]TTFFTTTTTTFFTFTFFFFTTFFTFFFFFTTFFTTT
(a) Let P: Emmanuel is hardworking
Q: Emmanuel is happy
R: Emmanuel is successful
Make a reasonable conclusion from the statements
ANSWER
EMMANUEL IS HAPPY AND SUCCESSFUL IF AND ONLY IF HE IS HARDWORKING
[Q^R]↔P
(b) Write each of the following in symbolic form
If P: The temperature is 400c
Q:The weather is hot
If the temperature is 400c then the weather is hot
The temperature is not up to 400c and the weather is not hot.
The temperature is 400c or the weather is cold
The weather is hot if and only the temperature is 400c
If the temperature is 400c then the weather is hot
ANSWER
If the temperature is 400c then = P
Weather is hot =Q
FINAL ANSWER = P→Q
The temperature is not up to 400c and the weather is not hot.
ANSWER
The temperature is not up to 400c =
And = ^
Weather is not hot =
FINAL ANSWER=
The temperature is 400c or the weather is cold
ANSWER
The temperature is 400c= P
OR = v;
Weather is cold=
The temperature is 400c or the weather is cold = Pv
The weather is hot if and only the temperature is 400c
ANSWER
Weather is hot = Q
If and only = ↔
The temperature is 400c = P
The weather is hot if and only the temperature is 400c = Q↔ P
(a) Define (i) Adjunct matrix (ii) identity matrix (iii) transpose matrix (iv) singular matrix
SOLUTION
ADJUNCT MATRIX: ADJUNCT MATRIX ( ALSO CALLED ADJUGATE MATRIX) is defined as the transpose of the cofactor matrix of that particular matrix.
IDENTITY MATRIX : is a n×n matrix or a square matrix in which all the elements of the principal diagonal are one’s [1’s] and all other elements are zero’s [0’s].
TRANSPOSE MATRIX: This is an operator which flips a matrix over its diagonal, that is it switches the row and column matrix by producing another matrix denoted as AT
SINGULAR MATRIX: Is a square matrix which is not invertible. Alternatively, a matrix is singular if and only the determinant is equal to zero[0].
(b) find the value of x for which IS A SINGULAR MATRIX
SOLUTION
A SINGULAR MATRIX IS A SAID TO BE A MATRIX IF THE DETERMINANT OF THAT PARTICULAR MATRIX IS EQUAL TO ZERO
FIRST FIND THE DETERMINANT OF THE MATRIX
= X + 3+2
X+3[X2 – 1] -0[X – 0] +2[1 – 0] = 0
X3 -X + 3X2 -3-0+2 =0
X3+ 3X2 -X-3+2 =0
X3 + 3X2 -X-1 =0
USING POLYNOMIAL,
F[X] = X3 + 3X2 -X-1 =0
We can only use a number ranging from 1 to 3 and -1 to -3
Reason is because they won’t give us a problem to solve exceeding this limit in other not to waste our time.
So solving for when X = 1 [ the main reason of substituting x as 1, is to get zero as our result after solving
F[X] = X3 + 3X2 -X-1
F[1] = -13 + 3[-1]2 –[-1]-1
= -1+3+1-1
=2
When x = 2
F[X] = X3 + 3X2 -X-1
F[2] = -23 + 3[-2]2 –[-2]-1
=5
When x = -3
F[3] = -33 + 3[-3]2 –[-3]-1
=2
So also applies to when x = -1,-2,-3……….
Note: no value of x gives us zero[0] therefore the problem can be solve only by remaining with no definite answer OR the problem has no answer
(a) If A= find AT OR A-1
Find the transpose of A
SOLUTION
IF A =
Then the transpose will be
DUE TO THE DEFINITION OF TRANSPOSE THAT SATS; A MATRIX IS SAID TO BE TRANSPOSE IF THE ROW OR ROWS AND COLUMN OR COLUMNS ARE INTERCHANGED.
(a) A committee of 9 is to be formed from 12 males and 9 females, the committee is to consist of five males and four females. In how many ways can this be done if;
no restriction
four particular male and female must not be included
SOLUTION
NOTE: No restriction simply not removing or adding any person in the committee, but solving it in the original way the committee appears.
So therefore male will go with male and female will go with female. n
HENCE
NO RESTRICTION = ×
=
=
=
= 249,480 Ways
four particular male and female must not be included
in combination, whenever you see the word must not be included means to remove that certain number from only
but when the question says that certain man or woman is included then you remove from both
=
=
(b) Write down the expansion (1 +x)5 using pascal triangle evaluate (1.85)5
1
1
1 2 1
3 3 1
1 4 6 4 1
1 5 10 10 5 1
USING DIRECT SUBSTITUTION, From pascal triangle we have,
=1[15 X0 ] + 5[14 X1] + 10[13 X2] + 10[12 X3] + 5[11 X4] + 1[10 X5]
=1 + 5X + 10X2 + 10 X3 + 5 X4 + 1 X5
SINCE [1+X}5 = [1.85]5
i.e [1.85]5 = [1+0.85]5
HENCE, X = 0.85
=1 + 5[0.85] +10[0.85]2 + 10 [0.85]3 + 5 [0.85]4 + 1[0.85]5
=1 + 4.25 + 7.225 + 6.14125 + 2.21853 + 0.443705
APPROXIMATELTY TO FIVE[5] DECIMAL POINT
=21.22785
OR CAN STILL BE SOLVED USING COMBINATION METHOD
=[15 X0 ] + [14 X1] + [13 X2] + [12 X3] + [11 X4] + [10 X5]
= [15 X0 ] + [14 X1] + [13 X2] + [12 X3] + [11 X4] + [10 X5]
= [15 X0 ] + [14 X1] + [13 X2] + [12 X3] + [11 X4] + [10 X5]
=1 + 5X + 10X2 + 10 X3 + 5 X4 + 1 X5
SINCE [1+X}5 = [1.85 5]5
i.e [1.85]5 = [1+0.85]5
HENCE, X = 0.85
=1 + 5[0.85] +10[0.85]2 + 10 [0.85]3 + 5 [0.85]4 + 1[0.85]5
=1 + 4.25 + 7.225 + 6.14125 + 2.21853 + 0.443705
APPROXIMATELTY TO FIVE[5] DECIMAL POINT
=21.22785
DEPARTMENT OF MATHEMATICS AND STATISTICS
CLASS: NDII FT/DPP ENGINEEERING [MECH, CIVIL, ELECT/ELECT, COMPUTER] & NDII PT [COMPUTER]
YEAR: 2016/2017
[a] i. Define proposition and argument.
PROPOSITION: A proposition or a statement is a declarative sentence that is either true or false, but not both.
ARGUMENT: Is any group of statement of which one is claimed to follow from the others, which are alleged to provide grounds for the truth of that one.
[Ii] Let P and Q be two statements
P: He is lazy.
Q: He will be a successful business man.
Write the symbol and construct truth table of the conclusion “He will be a successful business man if and only if he is hardworking”
SOLUTION
He will be a successful business man =Q
IF AND ONLY IF =
He is hardworking =
Conclusion=Q
PQQTTFFFTFFTTFTTFTFFTTF
1b. Define the following with illustration
Conjunction
Disjunction
Conditional statement
Bi-conditional statement
[1] CONJUNCTION: Is a compound statement formed by inserting the word and’ between two statements. Two statements so combined are called conjunctions. Symbolically P^Q denotes the conjunction of the statement. [2] DISJUNCTION : IS a compound statement formed by inserting the word ‘or between two statement . The two statement so combined are called disjunts or alternative symbolically PVa denotes the disjunction of statement p and a and is read as( p or a).
[3] condition statement : are made up of two parts a hypothesis[represented by p] and a conclusion statement is false if hypothesis is true and
the conclusion is false .The conditional is defined to be true unless a true hypothesis leads to a false conclusion.
[4] Bi-condition statement :is a combination of a conditional statemernt and it converse written in the if and only if form . A biconditional is true and only if both the conditions are true .
CONTRACTION (absurdity): A statement form that only false substitution instances is called contradiction. A statement or proposition that contradicts or denies another or itself and is logically incongruous.
Is a situation or ideas in operation to one another. A contradiction is a sentence together with its negation, and a theory is inconsistent if it include a contradiction OR a statement or proposition that contradict or denies another or itself and is logically incongruous.
TAUTOLOGY : Is a formular that is true in all possible state.
Tautology is a formular which is always true. It opposite is contradiction. Tautology can also be define as a logical statement in which the conclusion is equivalent to the premise.
Examples of tautology are;
Either it will rain tomorrow, or it won’t rain.
Bill will win the election, or he will not win the election.
She is brave, or she is not brave.
PQP→QQ→P[P→Q]vPTTFFTTTTFFTFTTFTTFTFTFFTTTTT
SINCE ALL ITS CONCLUSIONS IS TRUE THEN IT’S A TAUTOLOGY
THE ANSWER: TAUTOLOGY.
EXAMPLE
Statement Equivalent statementIf it is blue,then it is the skyIt is not blue or it is sky Show the equivalent of the following relations
P^[Q^R]
P^[QvR] ]
PQR[Q^R]P^[Q^R]TTTTTTTTTFFTFFTFTFFFFTFFFFFFFTTFFFFFTFFFFFFFTFFFFFFFFFFF
Since the truth values of P^[Q^R] and are the same then they are logically equivalent
PQRQvRP^QP^RP^[QvR]]TTTTTTTTTTFTTFTTTFTTFTTTTFFFFFFFFTTTFFFFFTFTFFFFFFTTFFFFFFFFFFFF
Since the truth values of P^[QvR] and ] are the same then they are logically equivalent
3a. Define the following with illustrations
Identity matrix
Transpose matrix
Symmetrical matrix
Singular matrix
Triangular matrix
IDENTITY MATRIX: is a n×n matrix or a square matrix in which all the elements of the principal diagonal are one’s [1’s] and all other elements are zero’s [0’s].
TRANSPOSE OF A MATRIX: This is an operator which flips a matrix over its diagonal, that is it switches the row and column matrix by producing another matrix denoted as AT
SYMMETRICAL MATRIX: Is a square matrix that is equal to its transpose.
If the transpose of a matrix is equal to itself, that matrix is said to be symmetric.
SINGULAR MATRIX: Is a square matrix which is not invertible. Alternatively, a matrix is singular if and only the determinant is equal to zero[0].
A TRIANGULAR MATRIX: Is a special kind of square matrix. A square matrix is called a lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangle if all the entries below the diagonal are zero.
3b.Find the value of X for which AIS a singular matrixSOLUTION
SINGULAR MATRIX Is a square matrix which is not invertible. Alternatively, a matrix is singular if and only the determinant is equal to zero[0].
=0
FIND THE DETERMINANT
1 –[-3] +X
1[4-0] +3[2-0] +X[2-(-8)]
4+6+10X=0
10X= -10
X=-1
4a. What is the orthogonal matrix? Hence compute A-1 IF A =
ORTHOGONAL MATRIX: is said to be a matrix if the inverse of a particular matrix is equal to the transpose of itself i.e A- = AT
A =
There are four methods to use in solving inverse of A i.e A- which is denoted as DCTA
D stands for determinant
C stands for cofactor of the matrix
T stands for transpose of the cofactor matrix
A stands for the ADJUNT of the matrix
FINDING THE DETERMINANT,
= 1 -2 -7
1[2—7] -2[-6-14] -7[3+2]
-5+40-35 = 0
Therefore is a singular matrix because the determinant is equal to zero and hence invertible
Finding the cofactor of A
=
FINDING THE TRANSPOSE OF THE COFACTOR; it’s the simplest method of them all, just interchange column or columns to row or rows.
Cofactor=
TRANSPOSE, =
FINDING THE INVERSE OF A, A- =
= = maths error
it’s invertible, because is a singular matrix therefore it has no definite answer.
4b. Use your result in 4[a] above to solve the system of the linear equation.
X + 2y -7z = -10
3x –y +7x = 13
2x + y -2z =3
Since the inverse of A has no definite answer t cannot be solved, only could be solved using crammers rule and the question didn’t ask for it.
5a. A committee of eleven is to be formed from twelve males and eight females, the committee is to consist of six males and five females. In how many ways can this be done if;
[i] No restriction
[ii] Three particular males and females must not be included
SOLUTION
NOTE: No restriction simply not removing or adding any person in the committee, but solving it in the original way the committee appears.
So therefore male will go with male and female will go with female. n
HENCE
NO RESTRICTION × =
×
=
=
=
= 51,744 Ways
Three particular male and female must not be included in combination, whenever you see the word must not be included means to remove that certain number from only
but when the question says that certain man or woman is included then you remove from both
×
=
=
= 84 Ways
5b. Write down the expansion of [1+x]7, using pascal triangle evaluate [1.85]7
SOLUTION
[1+x]7
1
1
1 2 1
3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
USING DIRECT SUBSTITUTION, From pascal triangle we have,
=1[17 X0 ] + 7[16 X1] + 21[15 X2] + 35[14 X3] + 35[13 X4] + 21[12 X5] + 7[11 X6] + 1[10X7]
=1 + 7X + 21X2 + 35X3 + 35X4 + 21X5 + 7X6 + X7
SINCE [1+X}7 W= [1.85]7
i.e [1.85]7 = [1+0.85]7
HENCE, X = 0.85
=1 + 7[0.85] + 21[0.85]2 + 35[0.85]3 + 35[0.85]4 + 21[0.85]5 + 7[0.85]6 + 0.857
=1 + 5.95 + 15.1725 + 21.494375 + 18.2701875 + 9.317811563 + 2.640046609 + 0.320577088
=74.16549776
APPROXIMATELTY TO FIVE[5] DECIMAL POINT
=74.16550
OR CAN STILL BE SOLVED USING COMBINATION METHOD
[17 X0 ] + [16 X1] + [15 X2] + [14 X3] + [13 X4] + [12 X5] + [11 X6] + [10X7]
[17 X0 ] + [16 X1] + [15 X2] + [14 X3] + [13 X4] + [12 X5] + [11 X6] + [10X7]
=1[17 X0 ] + 7[16 X1] + 21[15 X2] + 35[14 X3] + 35[13 X4] + 21[12 X5] + 7[11 X6] + 1[10X7]
=1 + 7X + 21X2 + 35X3 + 35X4 + 21X5 + 7X6 + X7
SINCE [1+X}7 W= [1.85]7
i.e [1.85]7 = [1+0.85]7
HENCE, X = 0.85
=1 + 7[0.85] + 21[0.85]2 + 35[0.85]3 + 35[0.85]4 + 21[0.85]5 + 7[0.85]6 + 0.857
=1 + 5.95 + 15.1725 + 21.494375 + 18.2701875 + 9.317811563 + 2.640046609 + 0.320577088
=74.16549776
APPROXIMATELTY TO FIVE[5] DECIMAL POINT
=74.16550
MATHEMATICS [MTH211 ND2]COURSE TITLE: LOGIC AND LINEER ALGEBRA
COURSE CODE: MTH 211
YEAR: 2017/2018
CLASS: ND2 CIVIL, MECHANICAL, ELECTRICAL, COMPUTER ENGINEERING [FT/DPP]
(a) Explain the term valid argument and give practical example
SOLUTION
What is the valid argument? An argument is valid if the truth value of the premises logically guarantees the truth of the conclusion.
HENCE A PRATICAL EXAMPLE IS GIVEN BELOW;
If p2: All engineering student are intelligent
P2: Emmanuel is an engineering student.
Therefore, Emmanuel is intelligent.
The answer is: EMMANUEL IS INTELLIGEENT
(b) Define (i) Singular (ii) Identity matrix (iii) Tautology
[I] SINGULAR MATRIX: Is a square matrix which is not invertible. Alternatively, a matrix is singular if and only the determinant is equal to zero[0].
[II] IDENTITY MATRIX: is a n×n matrix or a square matrix in which all the elements of the principal diagonal are one’s [1’s] and all other elements are zero’s [0’s].
[III] TAUTOLOGY : Is a formular that is true in all possible state.
Tautology is a formular which is always true. It opposite is contradiction. Tautology can also be define as a logical statement in which the conclusion is equivalent to the premise.
(c) Explain the term mathematical induction
ANSWER
What’s mathematical induction? It’s a mathematical technique that proves a statement form or theorem is true for all natural numbers. It make use of two mathematical technique which one is the base step i.e [prove a statement K to be in it initial value is true] and an inductive step where K+1 is true for all natural numbers.
(d) Use the truth table to show demorgans law .
SOLUTION
THERE ARE TWO WAYS OF SOLVING DE MORGAN’S LAW ON TRUTH TABLE
EITHER [pVq] p^q
OR [p^q] pvq
USING THE FIRST ONE TO PROVE DE MORGAN’S LAW i.e [pVq] p^q
PQpPVq[pVq]pqTTFFTFFTFFTTFFFTTFTFFFFTTFTT
OR [p^q] pvq
PQpP^q[p^q]pvqTTFFTFFTFFTFTTFTTFFTTFFTTFTT
It is logically equivalent.
NOTE: IN EIITHER WAY IT MUST PROVE THAT THE LEFT HAND SIDE IS LOGICALLY EQUIVALENT WITH THE RIGHT HAND SIDE.
(A) write down the binominal expressions(1+2x)7 in ascending powers of x. Use this expansion to calculate the value of (0.98)7 to four decimal points.
SOLUTION
[1+2x]7
1
1
1 2 1
3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
USING DIRECT SUBSTITUTION, From pascal triangle we have,
=1[17 X0 ] + 7[16 X1] + 21[15 X2] + 35[14 X3] + 35[13 X4] + 21[12 X5] + 7[11 X6] + 1[10X7]
=1 + 7X + 21X2 + 35X3 + 35X4 + 21X5 + 7X6 + X7
SINCE [1+2X}7 = [0.98]7
i.e [1+2X] = [0.98]
HENCE,2X = 0.98-1
2X=–0.02 ; X=-0.01
=1 + 7[0.01] + 21[0.01]2 + 35[0.01]3 + 35[0.01]4 + 21[0.01]5 + 7[0.01]6 + 0.017
=1 + 0.07 + 0.0021 + + 0.000035 + 0.00000035 + 0.00000000021 + 0.000000000007 + 0.000000000000001
=1.0721[ to four decimal place]
OR CAN STILL BE SOLVED USING COMBINATION METHOD
[17 X0 ] + [16 X1] + [15 X2] + [14 X3] + [13 X4] + [12 X5] + [11 X6] + [10X7]
[17 X0 ] + [16 X1] + [15 X2] + [14 X3] + [13 X4] + [12 X5] + [11 X6] + [10X7]
=1[17 X0 ] + 7[16 X1] + 21[15 X2] + 35[14 X3] + 35[13 X4] + 21[12 X5] + 7[11 X6] + 1[10X7]
=1 + 7X + 21X2 + 35X3 + 35X4 + 21X5 + 7X6 + X7
SINCE [1+2X}7 = [0.98]7
i.e [1+2X] = [0.98]
HENCE,2X = 0.98-1
2X=–0.02 ; X=-0.01
=1 + 7[0.01] + 21[0.01]2 + 35[0.01]3 + 35[0.01]4 + 21[0.01]5 + 7[0.01]6 + 0.017
=1 + 0.07 + 0.0021 + + 0.000035 + 0.00000035 + 0.00000000021 + 0.000000000007 + 0.000000000000001
=1.0721[ to four decimal place]
(b) in how many ways can ADMINISTRATION be arranged.
SOLUTION
This can be solved using permutation
How many times do each letter repeat itself?
3222
(A) let P = it rains, Q = the meeting will be postponed , S = if it rains the meeting will not hold also if it does not rain the meeting will hold. Draw the truth table for s. (b) show that (pvq)^ →q ) → p is a tautology.
SOLUTION
P= It rains
Q = the meeting will be postponed
S= if it rains the meeting will not hold also if it does not rain the meeting will hold.
Those are two different sentence
S1= if it rains the meeting will not hold
S2= if it does not rain the meeting will hold
S1=P→Q
S2=P→Q
ALSO is a conjunction that means AND; its symbol ^
i.e S= P→QP→Q
PQpP→QP→QP→QP→QTTFFTTTTFFTFTFFTTFTFFFFTTTTT
(A) if A= B= find where I is the identity matrix
SOLUTION
This is a scaler matrix
(b) form the 3×3 matrix of A of variable x,y,z if a11=1, a12=1, a13 =1 C1=4, a21=2, a22=3, a23=1, C2=8, a31=3, a32=2, a33=2, C3=6. Compute the inverse A, hence solve the system using matrix method.
SOLUTION
There are four methods to use in solving inverse of A i.e A- which is denoted as DCTA
D stands for determinant
C stands for cofactor of the matrix
T stands for transpose of the cofactor matrix
A stands for the ADJUNT of the matrix
FINDING THE DETERMINANT,
= 1 -1 +4
1[6—2] -1[4-3] +4[4-9]
4-1-20= -17
DET=-17
Finding the cofactor of A
=
FINDING THE TRANSPOSE OF THE COFACTOR; it’s the simplest method of them all, just interchange column or columns to row or rows.
Cofactor=
TRANSPOSE, =
FINDING THE INVERSE OF A, A- =
This is a scalar matrix therefore
=
Inverse of A =
(A) Explain the principles of mathematical induction.
ANSWER
What’s mathematical induction? It’s a mathematical technique that proves a statement form or theorem is true for all natural numbers. It make use of two mathematical technique which one is the base step i.e [prove a statement K to be in it initial value is true] and an inductive step where K+1 is true for all natural numbers.
(b) Show that
SOLUTION
We will be teaching you two different methods, you can use in solving as true for all natural numbers. So you can use any method, you think it seems best for you.
3 =
First method
13+23+33…………..13 = ; 13 =
Let n = 1
13 = ; 13 =
13 = 1 =1
Therefore it is true for all natural numbers
Using base method
When n = k
13+23+33…………..k3 = (1)
Let n = k+1
13+23+33…………..k3+ [k+1]3 = …..equatn (2)
Subtract equation (1) from equation (2)
k3+ [k+1]3 – k3= –
collect like terms
k3- k3 +[k+1]3 = –
[k+1]3 = –
[k+1]3 = –
Factor out
[k+1]3 =
[k+1]3 =
[k+1]3 =
[k+1]3 =
[k+1]3 =
[k+1]3 =
[k+1]3 =
[k+1]3 =
Cancel them out
1=1
Since the left hand side is equal to the right hand side then the statement is true for all natural numbers.
For second method (substitutional method)
13+23+33…………..13 = ; 13 =
Let n = 1
13 = ; 13 =
13 = 1 =1
Therefore it is true for all natural numbers
Using base method
When n = k
13+23+33…………..k3 = (1)
Let n = k+1
13+23+33…………..k3+ [k+1]3 = …..equatn (2)
Substitute equation (1) into equation (2)
k3+ [k+1]3 =
- [k+1]3 =
=
The both L.c.m will cancel out,
=
2=
You can observe that the L.H.S=R.H.S, therefore cancel them out.
Since the left hand side is equal to the right hand side then the statement is true for all natural numbers.
Solve the system using crammer method
3x-2y=8
5x+4y=6
SOLUTION
Find the determinant
[3=2 [8= 32 – 12
= 20
X ==
= 10 [318 – 40
= -22
Y ==
= -11
[X,Y] = [10,-11]
6)(A) Showthat SOLUTION
- =
Now solve to prove that the L.H.S =R.H.S
+
Factor out
Recall in permutation and combination n
Therefore,
Factor out and cancel it out from the L.C.M
ǃ+ ( +1− )ǃ ǃ
Recall that [n+1]
[n+1]n I.e
So therefore=
Since,
(b) A committee of 5 members is to be formed from 8 men and 9 women. How manyCommittees, of 5 members are possible if there will be (i) no restrictions (ii) 1 manand 4 women (iii) 3 men and 2 women such that a certain man must be on eachof the committee.SOLUTION
NOTE: No restriction simply not removing or adding any person in the committee, but solving it in the original way the committee appears.
So therefore male will go with male and female will go with female. n
HENCE
NO RESTRICTION × =
×
=
=
=
= 7,056 WAYS
(ii) 1 man and 4 women
×
=
=
= 756 WAYS
(iii) 1 man and 4 women
×
=
=
= 1008 WAYS
