Page 1 Exercise 3.5 In each of the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it: (1 -4) 1. 3 3 0 3 9 2 0 xy xy ? ? ? ? ? ? Sol: The given system of equations may be written as Page 2 Exercise 3.5 In each of the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it: (1 -4) 1. 3 3 0 3 9 2 0 xy xy ? ? ? ? ? ? Sol: The given system of equations may be written as 3 3 0 3 9 2 0 xy xy ? ? ? ? ? ? The given system of equations is of the form 1 1 1 2 2 2 0 0 a x b y c a x b y c ? ? ? ? ? ? Where, 1 1 1 1, 3, 3 a b c ? ? ? ? ? And 2 2 2 3, 9, 2 a b c ? ? ? ? ? We have, 1 2 1 2 1 3 31 93 a a b b ? ? ? ? ? And, 1 2 33 22 c c ? ? ? ? Clearly, 1 1 1 2 2 2 a b c a b c ?? So, the given system of equation has no solutions. 2. 2 5 0 4 2 10 0 xy xy ? ? ? ? ? ? Sol: The given system of equation may be written as 2 5 0 4 2 10 0 xy xy ? ? ? ? ? ? The given system of equations is of the form 1 1 1 2 2 2 0 0 a x b y c a x b y c ? ? ? ? ? ? Where, 1 1 1 2, 1, 5 a b c ? ? ? ? And 222 4, 2, 10 a b c ? ? ? ? We have, 1 2 1 2 21 42 1 2 a a b b ? ? ? And, 1 2 51 10 2 c c ? ? ? ? Page 3 Exercise 3.5 In each of the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it: (1 -4) 1. 3 3 0 3 9 2 0 xy xy ? ? ? ? ? ? Sol: The given system of equations may be written as 3 3 0 3 9 2 0 xy xy ? ? ? ? ? ? The given system of equations is of the form 1 1 1 2 2 2 0 0 a x b y c a x b y c ? ? ? ? ? ? Where, 1 1 1 1, 3, 3 a b c ? ? ? ? ? And 2 2 2 3, 9, 2 a b c ? ? ? ? ? We have, 1 2 1 2 1 3 31 93 a a b b ? ? ? ? ? And, 1 2 33 22 c c ? ? ? ? Clearly, 1 1 1 2 2 2 a b c a b c ?? So, the given system of equation has no solutions. 2. 2 5 0 4 2 10 0 xy xy ? ? ? ? ? ? Sol: The given system of equation may be written as 2 5 0 4 2 10 0 xy xy ? ? ? ? ? ? The given system of equations is of the form 1 1 1 2 2 2 0 0 a x b y c a x b y c ? ? ? ? ? ? Where, 1 1 1 2, 1, 5 a b c ? ? ? ? And 222 4, 2, 10 a b c ? ? ? ? We have, 1 2 1 2 21 42 1 2 a a b b ? ? ? And, 1 2 51 10 2 c c ? ? ? ? Clearly, 111 222 a b c a b c ?? So, the given system of equation has infinity many solutions. 3. 3 5 20 6 10 40 xy xy ? ? ? ? Sol: 3 5 20 6 10 40 xy xy ? ? ? ? Compare it with 1 1 1 1 2 2 0 0 a x by c a x by c ? ? ? ? ? ? We get 1 3, 1 5 1 20 2 6, 2 10 2 40 a b and c a b and c ? ? ? ? ? ? ? ? ? ? 1 1 1 2 2 2 3 5 20 , 6 10 40 a b c and a b c ? ? ? ? ? ?? Simplifying it we get 1 1 1 2 2 2 1 1 1 , 2 2 2 a b c and a b c ? ? ? Hence 111 222 a b c a b c ?? So both lines are coincident and overlap with each other So, it will have infinite or many solutions 4. 2 8 0 5 10 10 0 xy xy ? ? ? ? ? ? Sol: The given system of equation may be written as 2 8 0 5 10 10 0 xy xy ? ? ? ? ? ? The given system if equation is of the form 1 1 1 2 2 2 0 0 a x b y c a x b y c ? ? ? ? ? ? Where, 1 1 1 1, 2, 8 a b c ? ? ? ? ? And, 2 2 2 5, 10, 10 a b c ? ? ? ? ? Page 4 Exercise 3.5 In each of the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it: (1 -4) 1. 3 3 0 3 9 2 0 xy xy ? ? ? ? ? ? Sol: The given system of equations may be written as 3 3 0 3 9 2 0 xy xy ? ? ? ? ? ? The given system of equations is of the form 1 1 1 2 2 2 0 0 a x b y c a x b y c ? ? ? ? ? ? Where, 1 1 1 1, 3, 3 a b c ? ? ? ? ? And 2 2 2 3, 9, 2 a b c ? ? ? ? ? We have, 1 2 1 2 1 3 31 93 a a b b ? ? ? ? ? And, 1 2 33 22 c c ? ? ? ? Clearly, 1 1 1 2 2 2 a b c a b c ?? So, the given system of equation has no solutions. 2. 2 5 0 4 2 10 0 xy xy ? ? ? ? ? ? Sol: The given system of equation may be written as 2 5 0 4 2 10 0 xy xy ? ? ? ? ? ? The given system of equations is of the form 1 1 1 2 2 2 0 0 a x b y c a x b y c ? ? ? ? ? ? Where, 1 1 1 2, 1, 5 a b c ? ? ? ? And 222 4, 2, 10 a b c ? ? ? ? We have, 1 2 1 2 21 42 1 2 a a b b ? ? ? And, 1 2 51 10 2 c c ? ? ? ? Clearly, 111 222 a b c a b c ?? So, the given system of equation has infinity many solutions. 3. 3 5 20 6 10 40 xy xy ? ? ? ? Sol: 3 5 20 6 10 40 xy xy ? ? ? ? Compare it with 1 1 1 1 2 2 0 0 a x by c a x by c ? ? ? ? ? ? We get 1 3, 1 5 1 20 2 6, 2 10 2 40 a b and c a b and c ? ? ? ? ? ? ? ? ? ? 1 1 1 2 2 2 3 5 20 , 6 10 40 a b c and a b c ? ? ? ? ? ?? Simplifying it we get 1 1 1 2 2 2 1 1 1 , 2 2 2 a b c and a b c ? ? ? Hence 111 222 a b c a b c ?? So both lines are coincident and overlap with each other So, it will have infinite or many solutions 4. 2 8 0 5 10 10 0 xy xy ? ? ? ? ? ? Sol: The given system of equation may be written as 2 8 0 5 10 10 0 xy xy ? ? ? ? ? ? The given system if equation is of the form 1 1 1 2 2 2 0 0 a x b y c a x b y c ? ? ? ? ? ? Where, 1 1 1 1, 2, 8 a b c ? ? ? ? ? And, 2 2 2 5, 10, 10 a b c ? ? ? ? ? We have, 1 2 1 2 1 5 21 10 5 a a b b ? ? ?? ? And, 1 2 84 10 5 c c ? ? ? ? Clearly, 1 2 1 2 2 2 a b c a b c ?? So, the given system of equation has no solution. 5. 2 5 0 3 1 0 kx y xy ? ? ? ? ? ? Sol: The given system of equation is 2 5 0 3 1 0 kx y xy ? ? ? ? ? ? The system of equation is of the form 1 1 1 2 2 2 0 0 a x b y c a x b y c ? ? ? ? ? ? Where, 1 1 1 , 2, 5 a k b c ? ? ? ? And, 2 2 2 3, 1, 1 a b c ? ? ? ? For a unique solution, we must have 11 22 2 31 6 ab ab k k ? ? ? ?? So, the given system of equations will have a unique solution for all real values of k other than 6. 6. 4x + ky + 8 = 0 2x + 2y + 2 = 0 Sol: Here 1 2 1 2 4, , 2, 2 a a k b b ? ? ? ? Now for the given pair to have a unique solution: 11 22 ab ab ? Page 5 Exercise 3.5 In each of the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it: (1 -4) 1. 3 3 0 3 9 2 0 xy xy ? ? ? ? ? ? Sol: The given system of equations may be written as 3 3 0 3 9 2 0 xy xy ? ? ? ? ? ? The given system of equations is of the form 1 1 1 2 2 2 0 0 a x b y c a x b y c ? ? ? ? ? ? Where, 1 1 1 1, 3, 3 a b c ? ? ? ? ? And 2 2 2 3, 9, 2 a b c ? ? ? ? ? We have, 1 2 1 2 1 3 31 93 a a b b ? ? ? ? ? And, 1 2 33 22 c c ? ? ? ? Clearly, 1 1 1 2 2 2 a b c a b c ?? So, the given system of equation has no solutions. 2. 2 5 0 4 2 10 0 xy xy ? ? ? ? ? ? Sol: The given system of equation may be written as 2 5 0 4 2 10 0 xy xy ? ? ? ? ? ? The given system of equations is of the form 1 1 1 2 2 2 0 0 a x b y c a x b y c ? ? ? ? ? ? Where, 1 1 1 2, 1, 5 a b c ? ? ? ? And 222 4, 2, 10 a b c ? ? ? ? We have, 1 2 1 2 21 42 1 2 a a b b ? ? ? And, 1 2 51 10 2 c c ? ? ? ? Clearly, 111 222 a b c a b c ?? So, the given system of equation has infinity many solutions. 3. 3 5 20 6 10 40 xy xy ? ? ? ? Sol: 3 5 20 6 10 40 xy xy ? ? ? ? Compare it with 1 1 1 1 2 2 0 0 a x by c a x by c ? ? ? ? ? ? We get 1 3, 1 5 1 20 2 6, 2 10 2 40 a b and c a b and c ? ? ? ? ? ? ? ? ? ? 1 1 1 2 2 2 3 5 20 , 6 10 40 a b c and a b c ? ? ? ? ? ?? Simplifying it we get 1 1 1 2 2 2 1 1 1 , 2 2 2 a b c and a b c ? ? ? Hence 111 222 a b c a b c ?? So both lines are coincident and overlap with each other So, it will have infinite or many solutions 4. 2 8 0 5 10 10 0 xy xy ? ? ? ? ? ? Sol: The given system of equation may be written as 2 8 0 5 10 10 0 xy xy ? ? ? ? ? ? The given system if equation is of the form 1 1 1 2 2 2 0 0 a x b y c a x b y c ? ? ? ? ? ? Where, 1 1 1 1, 2, 8 a b c ? ? ? ? ? And, 2 2 2 5, 10, 10 a b c ? ? ? ? ? We have, 1 2 1 2 1 5 21 10 5 a a b b ? ? ?? ? And, 1 2 84 10 5 c c ? ? ? ? Clearly, 1 2 1 2 2 2 a b c a b c ?? So, the given system of equation has no solution. 5. 2 5 0 3 1 0 kx y xy ? ? ? ? ? ? Sol: The given system of equation is 2 5 0 3 1 0 kx y xy ? ? ? ? ? ? The system of equation is of the form 1 1 1 2 2 2 0 0 a x b y c a x b y c ? ? ? ? ? ? Where, 1 1 1 , 2, 5 a k b c ? ? ? ? And, 2 2 2 3, 1, 1 a b c ? ? ? ? For a unique solution, we must have 11 22 2 31 6 ab ab k k ? ? ? ?? So, the given system of equations will have a unique solution for all real values of k other than 6. 6. 4x + ky + 8 = 0 2x + 2y + 2 = 0 Sol: Here 1 2 1 2 4, , 2, 2 a a k b b ? ? ? ? Now for the given pair to have a unique solution: 11 22 ab ab ? i.e., 4 22 k ? i.e., 4 k ? Therefore, for all values of k, except 4, the given pair of equations will have a unique solution. 7. 45 2 3 12 x y k xy ?? ? ? Sol: The given system of equation is 4 5 0 2 3 12 0 x y k xy ? ? ? ? ? ? The system of equation is of the form 1 1 1 2 2 2 0 0 a x b y c a x b y c ? ? ? ? ? ? Where, 1 1 1 4, 5, a b c k ? ? ? ? ? And, 2 2 2 2, 3, 12 a b c ? ? ? ? ? For a unique solution, we must have 11 22 45 23 ab ab ? ? ? ? ? k ? is any real number. So, the given system of equations will have a unique solution for all real values of k. 8. 23 5 7 0 xy x ky ? ? ? ? ? Sol: The given system of equation is 2 3 0 5 7 0 xy x ky ? ? ? ? ? ? The system of equation is of the form 1 1 1 2 2 2 0 0 a x b y c a x b y c ? ? ? ? ? ? Where, 1 1 1 1, 2, 3 a b c ? ? ? ? And, 2 2 2 5, , 7 a b k c ? ? ? For a unique solution, we must haveRead More

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