In how many ways can 5 boys and 3 girls be seated in a row so that each girl is between 2 boys?
How many 5-digit numbers can be formed by using the digits 0, 1 and 2?
In how many ways can 4 letters be posted in 5 letter boxes?
Answer : Given: We have 4 letters and 5 letter boxes To Find: Number of ways of posting letters. One letter can be posted in any of 5 letter boxes. We have to assume that all the letters are...
How many words can be formed by the letters of the word ‘SUNDAY’?
How many permutations of the letters of the word ‘APPLE’ are there?
How many different words can be formed by using all the letters of the word ‘ALLAHABAD’?
In how many ways can the letters of the word ‘PERMUTATIONS’ be arranged if each word starts with P and ends with S?
In how many ways can the letters of the word ‘CHEESE’ be arranged?
How many numbers divisible by 5 and lying between 4000 and 5000 can be formed from the digits 4, 5, 6, 7, 8 if repetition of digits is allowed?
How many 3-digit numbers above 600 can be formed by using the digits 2, 3, 4, 5, 6, if repetition of digits is allowed?
. How many 3-digit numbers are there with no digit repeated?
find the value of x.
If (n + 1)! = 12 × [(n – 1)!], find the value of n.
Answer : To Find: Value of n Given: (n+1)! = 12× [(n-1)!] Formula Used: n! = (n) × (n-1) × (n-2) × (n-3)............. 3 × 2 × 1 Now, (n+1)! = 12× [(n-1)!] ⇒ (n+1) × (n) × [(n-1)!] = 12 × [(n-1)!] ⇒...
In how many different ways can a garland of 16 different flowers be made?
Answer : It is also in the form of a circle, So we need to arrange 16flowers in Circle 16 flowers can be arranged by 15! Now each flower have the same neighbour in the clockwise and anticlockwise...
In how many different ways can 20 different pearls be arranged to form a necklace?
Answer : We know that necklace in the form of a circle, So we need to arrange 20 pearls in Circle 20 pearls can be arranged by 19! Now each pearl have the same neighbour in the clockwise and...
In how many ways can 8 persons be seated at a round table so that all shall not have the same neighbour in any two arrangement?
Answer : By using the formula (n-1)! (Mention in Solution-1) So 8 persons can be arranged by 7! Now each person have the same neighbour in the clockwise and anticlockwise arrangement Total number of...
In how many ways can 11 members of a committee sit at a round table so that the secretary and the joint secretary are always the neighbour of the president?
There are 5 men and 5 ladies to dine at a round table. In how many ways can they sit so that no ladies are together?
In how many ways can 6 persons be arranged in
(i)a line,
(ii) a circle?
There are 4 candidates for the post of a chairman, and one is to be elected by votes of 5 men. In how many ways can the vote be given?
Answer : Let suppose 4 candidates be C1, C2, C3, C4 and 5 men be M1, M2, M3, M4, M5 Now M1 choose any one candidates from four (C1, C2, C3, C4) and give the vote to him by any 4 ways Similarly, M2...
In how many ways can 4 prizes be given to 3 boys when a boy is eligible for all prizes?
Answer : Let suppose 4 prizes be P1, P2, P3, P4 and 3 boys be B1, B2, B3 Now P1 can be distributed to 3 boys(B1, B2, B3) by 3 ways, Similarly, P2 can be distributed to 3 boys(B1, B2, B3) by 3 ways,...
How many 4-digit numbers can be formed with the digits 0, 2, 3, 4, 5 when a digit may be repeated any numbers of time in any arrangement?
How many 3-digit numbers are there when a digit may be repeated any numbers of time?
In how many ways can 3 letters can be posted in 2 letterboxes?
Answer : Let Suppose Letterbox be B1, B2 and letters are L1, L2, L3 So L1 can be posted in any 2 letterboxes (B1, B2) by 2 ways Similarly, L2 can be posted in any 2 letterbox (B1, B2) by 2 ways...
In how many ways can 5 bananas be distributed among 3 boys, there being no restriction to the number of bananas each boy may get?
Answer : As there is 5 banana, So suppose it as B1, B2, B3, B4, B5 And Let the Boy be A1, A2, A3 So B1 can Be distributed to 3 Boys (A1, A2, A3) by 3 ways, Similarly, B2, B3, B4, B5 Can be...
A child has 6 pockets. In how many ways, he can put 5 marbles in his pocket?
Answer : The first marble can be put into the pockets in 6 ways, i.e. Choose 1 Pocket From 6 by 6C1=6 Similarly second, third, Fourth, fifth & Sixth marble. Thus, the number of ways in which the...
The letters of the word ‘INDIA’ are arranged as in a dictionary. What are the 1st, 13th, 49th and 60th words?
How many 6 – digit numbers can be formed by using the digits 4, 5, 0, 3, 4, 5?
How many 7 – digit numbers can be formed by using the digits 1, 2, 0, 2, 4, 2, 4?
How many numbers can be formed with the digits 2, 3, 4, 5, 4, 3, 2 so that the odd digits occupy the odd places?
(i)Find the number of different words by using all the letters of the word, ‘INSTITUTION’. In how many of them
(ii) are the three T’ s together
(iii) are the first two letters the two N’ s?
In how many ways can the letters of the word ‘INTERMEDIATE’ be arranged so that: (i) The vowels always occupy even places? (ii) The relative orders of vowels and consonants do not change?
(i)How many arrangements can be made by using all the letters of the word ‘MATHEMATICS’?
(ii) How many of them begin with C?
(iii) How many of them begin with T?
In how many ways can the letters of the word ‘ASSASSINATION’ be arranged so that all S’s are together?
Answer : To find: number of ways letters can be arranged such that all S’s are together Let all S’s be represented by a single letter Z New word is AAINATIONZ Number of arrangements = Letters can be...
How many different words can be formed with the letters of the word ‘CAPTAIN’? In how many of these C and T are never together?
Answer : To find: number of words such that C and T are never together Number of words where C and T are never the together = Total numbers of words - Number of words where C and T are together...
In how many ways can the letters of the word ‘PARALLEL’ be arranged so that all L’s do not come together?
Answer : To find: number of words where L do not come together Let the three L’s be treated as a single letter say Z Number of words with L not the together = Total number of words - Words with L’s...
How many words can be formed from the letters of the word ‘SERIES’, which start with S and end with S?
Find the number of arrangements of the letters of the word ‘ALGEBRA’ without altering the relative positions of the vowels and the consonants.
How many words can be formed by arranging the letters of the word ‘INDIA’, so that the vowels are never together?
How many words can be formed by arranging the letters of the word ‘ARRANGEMENT’, so that the vowels remain together?
How many different signals can be transmitted by arranging 2 red, 3 yellow and 2 green flags on a pole, if all the seven flags are used to transmit a signal?
Answer : To find: Number of distinct signals possible Total number of fags = 7 2 are of 1 kind, 3 are of another kind, and 2 are of the 3rd kind ⇒ Number of distinct signals Hence 210 different...
A child has three plastic toys bearing the digits 3, 3, 5 respectively. How many 3 – digit numbers can he make using them?
Answer : To find: number of 3 digit numbers he can make If all were distinct, he could have made 3! = 6 numbers But 2 number are the same So the number of possibilities He can make 3 three - digit...
There are 3 blue balls, 4 red balls and 5 green balls. In how many ways can they are arranged in a row?
In how many ways can the letters of the expression x2y2z4 be arranged when written without using exponents?
Find the total number of permutations of the letters of each of the words given below:
(i) APPLE
(ii) ARRANGE
(iii) COMMERCE
(iv) INSTITUTE
(v) ENGINEERING
(vi) INTERMEDIATE
Find the number of ways in which m boys and n girls may be arranged in a row so that no two of the girls are together; it is given that m > n.
when a group photograph is taken, all the seven teachers should be in the first row, and all the twenty students should be in the second row. If the tow corners of the second row are reserved for the two tallest students, interchangeable only between them, and if the middle seat of the front row is reserved for the principal, how many arrangements are possible?
In how many ways can 5 children be arranged in a line such that
(i) two of them, Rajan and Tanvy, are always together?
(ii) two of them, Rajan and Tanvy, are never together,
Answer : (i) two of them, Rajan and Tanvy, are always together Consider Rajan and Tanvy as a group which can be arranged in 2! = 2 ways. The 3 children with this 1 group can be arranged in 4! = 24...
In an examination, there are 8 candidates out of which 3 candidates have to appear in mathematics and the rest in different subjects. In how many ways can they are seated in a row if candidates appearing in mathematics are not to sit together?
How many numbers divisible by 5 and lying between 3000 and 4000 can be formed by using the digits 3, 4, 5, 6, 7, 8 when no digit is repeated in any such number?
How many permutations can be formed by the letters of the word ‘VOWELS’, when (i) there is no restriction on letters; (ii) each word begins with E; (iii) each word begins with O and ends with L; (iv) all vowels come together; (v) all consonants come together?
Answer : (i) There is no restriction on letters The word VOWELS contain 6 letters. The permutation of letters of the word will be 6! = 720 words. Each word begins with Here the position of letter E...
Find the number of ways in which the letters of the word ‘MACHINE’ can be arranged such that the vowels may occupy only odd positions.
In how many arrangements of the word ‘GOLDEN’ will the vowels never occur together?
In how many ways can the letters of the word ‘FAILURE’ be arranged so that the consonants may occupy only odd positions?
How many words can be formed out of the letters of the word ‘ORIENTAL’ so that the vowels always occupy the odd places?
In how many ways can the letters of the word ‘HEXAGON’ be permuted? In how many words will the vowels be together?
Find the number of permutations of the letters of the word ‘ENGLISH’. How many of these begin with E and end with I?
How many words beginning with C and ending with Y can be formed by using the letters of the word ‘COURTESY’?
How many words can be formed from the letters of the word ‘SUNDAY’? How many of these begin with D?
Find the number of different 4-letter words (may be meaningless) that can be formed from the letters of the word ‘NUMBERS’,
Find the number of words formed (may be meaningless) by using all the letters of the word ‘EQUATION’, using each letter exactly once.
In how many ways can 6 pictures be hung from 4 picture nails on a wall?
If there are 6 periods on each working day of a school, in how many ways can one arrange 5 subjects such that each subject is allowed at least one period?
Ten students are participating in a race. In how many ways can the first three prizes be won?
Five letters F, K, R, R and V one in each were purchased from a plastic warehouse. How many ordered pairs of letters, to be used as initials, can be formed from them?
There are 6 items in column A and 6 items in column B. A student is asked to match each item in column A with an item in column B. How many possible, correct or incorrect answers are there to this question?
It is required to seat 5 men and 3 women in a row so that the women occupy the even places. How many such arrangements are possible?
Six students are contesting the election for the president ship of the students, union. In how many ways can their names be listed on the ballot papers?
In how many ways can 4 different books, one each in chemistry, physics, biology and mathematics, be arranged on a shelf?
In how many ways can 6 women draw water from 6 wells if no well remains unused?
In how many ways can 7 people line up at a ticket window of a cinema hall?
In how many ways can 5 persons occupy 3 vacant seats?
Find the number of permutations of 10 objects, taken 4 at a time.
Answer : To find: the number of permutations of 10 objects, taken 4 at a time. Formula Used: Total number of ways in which n objects can be arranged in r places (Such that no object is replaced) is...
Prove that 1 + 1. 1P1 + 2. 2P2 + 3. 3P3 + …. n. nPn = n+1Pn+1.
Prove that 9P3 + 3 × 9P2 = 10P3.
Evaluate:
Evaluate:
Evaluate:
A. Evaluate:
In how many ways can 3 prizes be distributed among 4 girls, when
(i)no girl gets more than one prize?
(ii)a girl may get any number of prizes?
(iii)no girl gets all the prizes?
Answer : (i)To distribute 3 prizes among 4 girls where no girl gets more than one prize the possible number of permutation possible are:4P3=24 To distribute 3 prizes among 4 girls where a girl may...
A customer forgets a four-digit code for an automated teller machine (ATM) in a bank. However, he remembers that this code consists of digits 3, 5, 6, 9. Find the largest possible number of trials necessary to obtain the correct code.
Answer : Given: code consists of digits 3, 5, 6, 9. To find: the largest possible number of trials necessary to obtain the correct code. The customer remembers that this 4 digit code consists of...
A number lock on a suitcase has three wheels each labeled with ten digits 0 to 9. if opening of the lock is a particular sequence of three digits with no repeats, how many such sequences will be possible? Also, find the number of unsuccessful attempts to open the lock.
In how many ways can three jobs, I, II and III be assigned to three persons A, B and C if one person is assigned only one job and all are capable of doing each job?
Answer : Given: three jobs, I, II and III to be assigned to three persons A, B and C. To find: In how many ways this can be done. Condition: one person is assigned only one job and all are capable...
How many 6-digit telephone numbers can be constructed using the digits 0 to 9, if each number starts with 67 and no digit appears more than once?
How many natural numbers less than 1000 can be formed from the digits 0, 1, 2, 3, 4, 5 when a digit may be repeated any number of times?
Answer : To find: number of natural numbers less than 1000 that can be formed from the digits 0, 1, 2, 3, 4, 5 when a digit may be repeated any number of times For forming a 3 digit number less than...
How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7, 9 when no digit is repeated? How many of them are divisible by 10?
How many 3-digit numbers can be formed by using the digits 0, 1, 3, 5, 7 while each digit may be repeated any number of times?
How many 3-digit numbers are there with no digit repeated?
How many numbers can be formed from the digits 1, 3, 5, 9 if repetition of digits is not allowed?
Answer : To find: number of numbers that can be formed from the digits 1, 3, 5, 9 if repetition of digits is not allowed Forming a 4 digit number:4! Forming a 3 digit number:4C3 × 3! Forming a 2...
How many 4-digit numbers are there, when a digit may be repeated any number of times?
Answer : To find: Number of 4 digit numbers when a digit may be repeated any number of times The first place has possibilities of any of 9 digits. (0 not included because 0 in starting would make...
How many 3-letters words can be formed using a, b, c, d, e if (i) Repetition of letters is not allowed? (ii) Repetition of letters is allowed
In how many ways can 5 letters be posted in 4 letter boxes?
Answer : Each letter has 4 possible letter boxes option. So the number of ways in which 5 letters can be posted in 4 letter boxes =4 × 4 × 4 × 4 × 4=45 (Each 4 for each letter.)
In how many ways 6 rings of different types can be worn in 4 fingers?
Answer : Given:6 rings and 4 fingers. Each ring has 4 different fingers that they can be worn. So total number of ways in which 6 rings of different types can be worn in 4 fingers =4 × 4 × 4 × 4 × 4...
A gentleman has 6 friends to invite. In how many ways can be send invitation cards to them, if he has 3 servants to carry the cards?
Answer : Given: A gentleman has 6 friends to invite. He has 3 servants to carry the cards. Each friend can be invited by 3 possible number of servants. So the number of ways of inviting 6 friends...
Find the total number of ways of answering 5 objective-type question, each question having 4 choices.
Answer : Given: 5 objective-type question, each question having 4 choices. To find: the number of ways of answering them. Each objective-type question has 4 choices. So the total number of ways of...
In how many ways can the following prizes be given away to a class of 20 students : first and second in mathematics; first and second in chemistry; first in physics and first in English?
Answer : Given: 20 students. The number of ways of giving first and second prizes in mathematics to a class of 20 students=20 × 19. (First prize can be given to any one of the 20 students but the...
For a set of five true or false questions, no student has written the all correct answer and no two students have given the same sequence of answers. What is the maximum number of students in the class for this to be possible?
Answer : Given: a set of five true – false questions. To find: the maximum number of students in the class. Condition: no student has written the all correct answer and no two students have given...
A sample of 3 bulbs is tested. A bulb is labeled as G if it is good and D if it is defective. Find the number of all possible outcomes.
Answer : A bulb can be good or defective, so there are 2 different possibilities of a bulb. So number of all possible outcomes (of all bulbs)=2 × 2 × 2=8
From among the 36 teachers in a school, one principal and one vice- principal are to be appointed. In how many ways can this be done?
Answer : Given: 36 teachers are there in a school. To find: Number of ways in which one principal and one vice-principal can be appointed. There are 36 options of appointing principal and 35 option...
A mint prepares metallic calendars specifying months, dates and days in the form of monthly sheets (one plate for each month). How many types of February calendars should it prepare to serve for all the possibilities in the future years?
Answer : To find: types of February calendars that can be prepared. There are two factors to develop FEBRUARY metallic calendars The day on the start of the year of which possibility=7 Whether the...
There are 6 items in column A and 6 items in column B. A student is asked to match each item in column A with an item in column B. How many possible (correct or incorrect) answers are there to this question?
How many arithmetic progressions with 10 terms are there whose first term in the set {1, 2, 3} and whose common difference is in the set {2, 3, 4}?
Answer : Given: Two sets: {1, 2, 3} & {2, 3, 4} To find: number of A.P. with 10n terms whose first term is in the set {1, 2, 3} and whose common difference is in the set {2, 3, 4} Number of...
Given, A = {2, 3, 5} and B = {0, 1}. Find the number of different ordered pairs in which the first entry is an element of A and the second is an element of B.
Answer : This is the example of Cartesian product of two sets. The pairs in which the first entry is an element of A and the second is an element of B are : (2,0),(2,1),(3,0),(3,1),(5,0),(5,1) ⇒ 3 ×...
How many 4-letter codes can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?
Answer : Given: first 10 letters of the English alphabet. In 4 letter code for first position there are 10 possibilities for second position there are 9 possibilities, for third position there are 8...
Find the number of different signals that can be generated by arranging at least 2 flags in order (one below the other) on a vertical staff, if five different flags are available.
(ac and the outcomes are recorded. How many possible outcomes are there?
(b) How many possible outcomes if the coin is tossed.
(i)four times?
(ii) five times?
(iii) n times?
Answer : (a) A coin is tossed three times So possible number of outcomes=23=8 (HHH,HHT,HTH,HTT,THH,THT,TTH,TTT) (b) i) A coin is tossed four times So possible number of outcomes=24=16...
How many 8-digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 270 and n digit appears more than once?
In how many ways can a vowel, a consonant and a digit be chosen out of the 26 letters of the English alphabet and the 10 digits?
In a school, there are four sections of 40 students each in XI standard. In how many ways can a set of 4 student representatives be chosen, one from each section?
In a textbook on mathematics there are three exercises A, B and C consisting of 12, 18 and 10 questions respectively. In how many ways can three questions be selected choosing one from each exercise?
In How many ways can 5 ladies draw water from 5 taps, assuming the no tap remains unused?
Answer : To find: number of ways in which 5 ladies draw water from 5 taps. Condition: no tap remains unused The condition given is that no well should remain unused. So possible number of ways are:...
In How many ways can 4 people be seated in a row containing 5 seats?
There are 12 steamers plying between A and B. In how many ways could the round trip from A be made if the return was made on (i) the same steamer? (ii) a different steamer?
Answer : Given: 12 steamers plying between A and B. To find: number of ways the round trip from A can be made. The steamer which will go from A to B will be returning back, since the given condition...
A, B and C are three cities. There are 5 routes from A to B and 3 routes from B to C. How many different routes are there from A to C via B?
Answer : Given: 5 routes from A to B and 3 routes from B to C. To find: number of different routes from A to C via B. Let E1 be the event : 5 routes from A to B Let E2 be the event : 3 routes from B...
There are 10 buses running between Delhi and Agra. In how many ways can a man go from Delhi to Agra and return by a different bus?
Answer : Given: 10 buses running between Delhi and Agra. To Find: Number of ways a man can go from Delhi to Agra and return by a different bus. There are 10 buses running between Delhi and Agra so...
Prove that
Prove that (n + 2) × (n!) + (n + 1) ! = (n!). (2n + 3)
Evaluate when n = 15 and r = 12.
find the value of n.
find the value of n.
If (n + 3) ! = 56 × (n + 1) !, find the value of n.
If (n + 2) ! = 2550 × n!, find the value of n.
If (n + 1) ! = 12 × (n – 1) !, find the value of n.
Which of the following are true of false?
(i) (2 + 3) ! = 2 ! + 3!
(ii) (2 × 3)! = (2!) × (3!)
Write the following products in factorial notation:
(i) 6 × 7 × 8 × 9 × 10 × 11 × 12
(ii) 3 × 6 × 9 × 12 × 15
find the value of x.
Prove that
Prove that LCM {6!, 7!, 8!} = 8!
Compute:
Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school A wants to award ₹ x each, ₹ y each and ₹ z each for the three respective values to 3, 2 and 1 students respectively with total award money of ₹ 1,600. School B wants to spend ₹ 2,300 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is ₹ 900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award. HINT: By the given data, we have
Solution: Assume the amount considered for sincerity, truthfulness and helpfulness are $x, y$ and $z$ respectively. As per the questions, $3 x+2 y+z=1600$ $\begin{array}{l} 4 x+y+3 z=2300 \\...
An amount of ₹ 5000 is put into three investments at 6%, 7% and 8% per annum respectively. The total annual income from these investments is ₹358. If the total annual income from first two investments is ₹70 more than the income from the third, find the amount of each investment by the matrix method. HINT: Let these investments be ₹x, ₹y and ₹z, respectively. Then, (i) And,
Solution: Suppose the investments are $\mathrm{x} \mathrm{x}$, Fy and $\mathrm{F} \mathrm{z}$, respectively. Therefore, $x+y+z=5000$ $\begin{array}{l} \frac{6 x}{100}+\frac{7 y}{100}+\frac{8...
The cost of 4 kg potato, 3 kg wheat and 2 kg of rice is ₹ 60. The cost of 1 kg potato, 2 kg wheat and 3 kg of rice is ₹45. The cost of 6 kg potato, 2 kg wheat and 3 kg of rice is ₹70. Find the cost of each item per kg by matrix method.
Solution: Suppose the price of 1kg potato, wheat and rice is $x$, $y$ and $z$ respectively. As per the question, $4x + 3y + 2z = 60$ $x+ 2y + 3z = 45$ $6x + 2y + 3z = 70$ Now converting the...
The sum of three numbers is 2. If twice the second number is added to the sum of first and third, we get 1. On adding the sum of second and third numbers to five times the first, we get 6. Find the three numbers by using matrices.
Solution: Assume the numbers are $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$. As per the question, $\begin{array}{l} x+y+z=2 \\ x+2 y+z=1 \\ 5 x+y+z=6 \end{array}$ Now converting the following...
Solution: We need to find: $-x, y, z$, The given set of lines are : $\begin{array}{l} \frac{1}{x}-\frac{1}{y}+\frac{1}{z}=4 \\ \frac{2}{x}+\frac{1}{y}-\frac{3}{z}=0 \\...
Solution: We need to find: $-x, y, z$ The given set of lines are : - $\begin{array}{l} \frac{2}{x}-\frac{3}{y}+\frac{3}{z}=10 \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=10 \\...
If and, find Hence, solve the system of equations:
and
HINT:
Solution: It is given, $\begin{array}{l} A=\left[\begin{array}{ccc} 1 & -2 & 0 \\ 2 & 1 & 3 \\ 0 & -2 & 1 \end{array}\right], B=\left[\begin{array}{ccc} 7 & 2 & -6 \\...
If , find Using , solve the following system of linear equations:
HINT: Here
Solution: It is given, $\begin{array}{l} A=\left[\begin{array}{ccc} 2 & 1 & 1 \\ 1 & -2 & -1 \\ 0 & 3 & -5 \end{array}\right] \\ A^{-1}=\frac{1}{|A|} \operatorname{adj}(A)...
If , find . Using , solve the following system of equations:
Solution: It is given, $A=\left[\begin{array}{ccc} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{array}\right]$ $\mathrm{A}^{-1}=\frac{1}{|A|} \operatorname{adj}(A)$...
Solve each of the following systems of equations using matrix method.
;
;
.
Solution: We need to find: - $x , y , z$ The given set of lines are : $\begin{array}{l} 4 x+3 y+2 z=60 \\ x+2 y+3 z=45 \\ 6 x+2 y+3 z=70 \end{array}$ Now converting the following equations in matrix...
Solve each of the following systems of equations using matrix method.
;
;
.
Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-y=3 \\ 2 x+3 y+4 z=17 \\ y+2 z=7 \end{array}$ Now, converting the following...
Solve each of the following systems of equations using matrix method.
;
;
.
Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-2 y+z=0 \\ y-z=2 \\ 2 x-3 z=10 \end{array}$ Now converting the following equations...
Solve each of the following systems of equations using matrix method.
;
;
.
Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} 5 x-y=-7 \\ 2 x+3 z=1 \\ 3 y-z=5 \end{array}$ Now, converting the following...
Solve each of the following systems of equations using matrix method.
;
;
.
Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-y-2 z=3 \\ x+y=1 \\ x+z=-6 \end{array}$ Now converting the following equations in...
Solve each of the following systems of equations using matrix method.
;
;
.
Solution: We need to find: $-x, y, z$ The given set of lines are : - $\begin{array}{l} x+2 y+z=4 \\ -x+y+z=0 \\ x-3 y+z=4 \end{array}$ Now converting the following equations in matrix form,...
Solve each of the following systems of equations using matrix method.
;
;
.
Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} 2 x+y-z=1 \\ x-y+z=2 \\ 3 x+y-2 z=-1 \end{array}$ Now, converting the following...
Solve each of the following systems of equations using matrix method.
;
;
.
Solution: We need to find: - $x , y , z$ The given set of lines are : - $x + y - z = 1$ $\begin{array}{l} 3 x+y-2 z=3 \\ x-y-z=-1 \end{array}$ Now, converting the following equations in matrix form,...
Solve each of the following systems of equations using matrix method.
;
;
.
Solution: We need to find: - $x , y , z$ The given set of lines are : - $\begin{array}{l} 3 x-4 y+2 z=-1 \\ 2 x+3 y+5 z=7 \\ x+z=2 \end{array}$ Now, converting the following equations in matrix...
Solve each of the following systems of equations using matrix method.
;
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Solution: We need to find: $-\mathrm{x}, \mathrm{y}, z$ The given set of lines are : - $\begin{array}{l} 6 x-9 y-20 z=-4 \\ 4 x-15 y+10 z=-1 \\ 2 x-3 y-5 z=-1 \end{array}$ Now, converting the...
Solve each of the following systems of equations using matrix method.
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Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x-y+2 z=7 \\ 3 x+4 y-5 z=-5 \\ 2 x-y+3 z=12 \end{array}$ Now, converting the...
Solve each of the following systems of equations using matrix method.
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Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are: $\begin{array}{l} 4 x-5 y-11 z=12 \\ x-3 y+z=1 \\ 2 x+3 y-7 z=2 \end{array}$ Now, converting the...
Solve each of the following systems of equations using matrix method.
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Solution: We need to find: - $x, y, z$ The given set of lines are : - $\begin{array}{l} 2 x+3 y+3 z=5 \\ x-2 y+z=-4 \\ 3 x-y-2 z=3 \end{array}$ Now, converting the following equations in matrix...
Solve each of the following systems of equations using matrix method.
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Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $x+y+z=6$ $\begin{array}{l} x+2 z=7 \\ 3 x+y+z=12 \end{array}$ Now converting following equations in...
Solve each of the following systems of equations using matrix method.
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Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x+y+z=1 \\ x-2 y+3 z=2 \\ 5 x-3 y+z=3 \end{array}$ Now converting the following...
Solve each of the following systems of equations using matrix method.
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Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} 2 x-3 y+5 z=11 \\ 3 x+2 y-4 z=-5 \\ x+y-2 z=-3 \end{array}$ Now converting the...
Solve each of the following systems of equations using matrix method.
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Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x+y+z=4 \\ 2 x-y+z=-1 \\ 2 x+y-3 z=-9 \end{array}$ Now, converting the following...
Solve each of the following systems of equations using matrix method.
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Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} 2 x-3 y+5 z=16 \\ 3 x+2 y-4 z=-4 \\ x+y-2 z=-3 \end{array}$ Now converting the...
Solve each of the following systems of equations using matrix method.
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Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} x+2 y+z=7 \\ x+3 z=11 \\ 2 x-3 y=1 \end{array}$ Now converting following equations...
Solve each of the following systems of equations using matrix method.
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Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are : - $\begin{array}{l} 3 x+4 y+7 z=4 \\ 2 x-y+3 z=-3 \\ x+2 y-3 z=8 \end{array}$ Now converting the...
Solve each of the following systems of equations using matrix method.
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Solution: We need to find: $-x, y, z$ The given set of lines are : - $\begin{array}{l} x-y+z=1 \\ 2 x+y-z=2 \\ x-2 y-z=4 \end{array}$ Now converting following equations in matrix form,...
Solve each of the following systems of equations using matrix method.
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Solution: We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$ The given set of lines are: $\begin{array}{l} 2 x+8 y+5 z=5 \\ x+y+z=-2 \\ x+2 y-z=2 \end{array}$ Now, converting the following...
Solve each of the following systems of equations using matrix method.
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Solution: We need to find: $-\mathrm{x}, \mathrm{y}$ The given set of lines are : - $\begin{array}{l} 4 x-3 y=3 \\ 3 x-5 y=7 \end{array}$ Now, converting the following equations in matrix form,...
Solve each of the following systems of equations using matrix method.
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Solution: We need to find: - $x, y$ The given set of lines are : - $\begin{array}{l} 2 \mathrm{x}-3 \mathrm{y}+1=0 \\ \mathrm{x}+4 y+3=0 \end{array}$ On converting the following equations in matrix...
Solve each of the following systems of equations using matrix method.
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Solution: We need to find: $-\mathrm{x}, \mathrm{y}$ The given set of lines are : - $\begin{array}{l} 5 \mathrm{x}+7 \mathrm{y}+2=0 \\ 4 \mathrm{x}+6 \mathrm{y}+3=0 \end{array}$ On converting the...
Solve each of the following systems of equations using matrix method.
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Solution: We need to find: $-\mathrm{x}, \mathrm{y}$ The given set of lines are : - $\begin{array}{l} x+2 y=1 \\ 3 x+y=4 \end{array}$ Now converting the following equations in matrix form,...
Solve each of the following systems of equations using matrix method.
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Solution: We need to find: $-\mathrm{x}, \mathrm{y}$ The given set of lines are : - $\begin{array}{l} 3 x+4 y-5=0 \\ x-y+3=0 \end{array}$ On converting the following equations in matrix form,...
Solve each of the following systems of equations using matrix method.
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Solution: We need to find: $-\mathrm{x}, \mathrm{y}$ The given set of lines are : - $\begin{array}{l} 5 x+2 y=4 \\ 7 x+3 y=5 \end{array}$ On converting the following equations in matrix form,...
Show that each one of the following systems of equations is inconsistent.
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Solution: We need to prove: Set of given lines are inconsistent. The given set of lines are : - $\begin{array}{l} 3 x-y-2 z=2 \\ 2 y-z=-1 \\ 3 x-5 y=3 \end{array}$ Now, converting the following...
Show that each one of the following systems of equations is inconsistent.
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Solution: We need to prove: let of given lines are inconsistent. The given set of lines are : - $\begin{array}{l} x+2 y+4 z=12 \\ y+2 z=-1 \\ 3 x+2 y+4 z=4 \end{array}$ Now, converting the following...
Show that each one of the following systems of equations is inconsistent.
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Solution: We need to prove: Set of given lines are inconsistent. The given set of lines are : - $\begin{array}{l} 2 x-y+3 z=1 \\ 3 x-2 y+5 z=-4 \\ 5 x-4 y+9 z=14 \end{array}$ On converting the...
Show that each one of the following systems of equations is inconsistent.
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Solution: We need to prove: Set of given lines are inconsistent. The given set of lines are : - $\begin{array}{l} x+y-2 z=5 \\ x-2 y+z=-2 \\ -2 x+y+z=4 \end{array}$ Now, converting the following...
Show that each one of the following systems of equations is inconsistent.
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Solution: We need to prove: Set of given lines are inconsistent. The given set of lines are : - $\begin{array}{l} 6 x+4 y=5 \\ 9 x+6 y=8 \end{array}$ Now converting the following equations in matrix...
Show that each one of the following systems of equations is inconsistent.
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Solution: We need to prove: Set of given lines are inconsistent. The given set of lines are : - $\begin{array}{l} 4 x-2 y=3 \\ 6 x-3 y=5 \end{array}$ Now, converting the following equations in...
Show that each one of the following systems of equations is inconsistent,
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Solution: We need to prove: Set of given lines are inconsistent. The given set of lines are: $\begin{array}{l} 2 x+3 y=5 \\ 6 x+9 y=10 \end{array}$ Now, converting the following equations in matrix...
Show that each one of the following systems of equations is inconsistent,
Solution: We need to prove: Set of given lines are inconsistent. The given set of lines are: - $\begin{array}{l} x+2 y=9 \\ 2 x+4 y=7 \end{array}$ Now, converting the following equations in matrix...
Find the greatest possible length which can be used to measure exactly the lengths 7m, 3m 85cm and 12m 95cm.
Answer: Given lengths are 7m (700cm), 3m 85cm (385cm) and 12m 95m (1295cm). The required length = HCF (700, 385, 1295) Using prime factorization, 700 = 2 × 2 × 5 × 5 × 7 = 22 × 52 × 7 385 = 5 × 7 ×...
Three pieces of timber 42m, 49m and 63m long have to be divided into planks of the same length. What is the greatest possible length of each plank? How many planks are formed?
Answer: The lengths of three pieces of timber are 42m, 49m and 63m Divide the timber into equal length of planks. ∴ Greatest possible length of each plank = HCF (42, 49, 63) Using prime...