Determinants

If the system of linear equations : x+y+z=6, x+2y+5z=10, 2x+3y+ z= has infinitely many solutions, then the value of + equals :

1. 12
2. 16
3. 22
4. 28

The sum of all possible values of [0, 2 ], for which the system of equations : x 3 -8y-12z=0 x 2 +3y+3z=0 x+y+3z=0 has a non-trivial solution, is equal to :

2. 2
3. 3
4. 4

Consider the system of linear equations in x, y, z : x+2y+tz=0 6x+y+5tz=0 3x+t2y+f(t)z=0 where f : R R is a differentiable function. If this system has infinitely many solutions for all t R, then f

1. is a constant function
2. is strictly increasing on R
3. is strictly decreasing on R
4. has two critical points

If the system of equations : x+y+z=5 x+2y+3z=9 x+3y+ z= has infinitely many solutions, then the value of + is :

1. 16
2. 18
3. 19
4. 21

Let , R be such that the system of linear equations x+2y+z=5 2x+y+ z=5 8x+4y+ z=18 has no solution. Then is equal to :

1. -4
2. 4
3. 8
4. -8

If the system of equations x+5y+6z=4, 2x+3y+4z=7, x+6y+az=b has infinitely many solutions, then the point (a, b) lies on the line

1. y-x=3
2. x-y=3
3. x+y=11
4. x+y=12

The system of linear equations x + y + z = 6 2x + 5y + az = 36 x + 2y + 3z = b has :

1. unique solution for a = 8 and b = 16
2. infinitely many solutions for a = 8 and b = 14
3. infinitely many solutions for a = 8 and b = 16
4. unique solution for a = 8 and b = 14

Let n be the number obtained on rolling a fair die. If the probability that the system x - ny + z = 6 x + (n - 2)y + (n + 1)z = 8 (n - 1)y + z = 1 has a unique solution is k6, then the sum of k and all possible values of n is :

1. 22
2. 20
3. 24
4. 21

If the system of equations 3x + y + 4z = 3 2x + y - z = -3 x + 2y + z = 4 has no solution, then the value of is equal to:

1. 13
2. 4
3. 19
4. 23

Let the system of equations : 2x + 3y + 5z = 9 7x + 3y - 2z = 8 12x + 3y - (4 + )z = 16 - have infinitely many solutions. Then the radius of the circle centred at ( , ) and touching the line 4x = 3y is :

1. 75
2. 215
3. 7
4. 175

Let the system of equations x + 5y - z = 1 4x + 3y - 3z = 7 24x + y + z = , R, have infinitely many solutions. Then the number of the solutions of this system, if x, y, z are integers and satisfy 7 x + y + z 77, is :

1. 4
2. 5
3. 3
4. 6

If the system of linear equations 3x + y + z = 3 2x + y - z = -3 x + 2y + z = 4 has infinitely many solutions, then the value of 22 - 9 is :

1. 31
2. 37
3. 43
4. 49

If the system of equations 2x + y + 3z = 5 3x + 2y - z = 7 4x + 5y + z = 9 has infinitely many solutions, then ( 2 + 2) is equal to :

1. 30
2. 26
3. 22
4. 18

Let , ( ) be the values of m, for which the equations x + y + z = 1, x + 2y + 4z = m and x + 4y + 10z = m2 have infinitely many solutions. Then the value of n=110(n + n ) is equal to :

1. 3410
2. 560
3. 3080
4. 440

If the system of equations 2x-y+z=4 5x+ y+3z=12 100x-47y+ z=212 has infinitely many solutions, then -2 is equal to

1. 56
2. 59
3. 57
4. 55

If the system of equations x+2y-3z=2 2x+ y+5z=5 14x+3y+ z=33 has infinitely many solutions, then + is equal to :

1. 13
2. 10
3. 12
4. 11

If the system of equations ( -1)x+( -4)y+ z=5 x+( -1)y+( -4)z=7 ( +1)x+( +2)y-( +2)z=9 has infinitely many solutions, then 2+ is equal to

1. 20
2. 10
3. 6
4. 12

The system of equations x+y+z=6, x+2y+5z=9, x+5y+ z= , has no solution if

1. =17, =18
2. =17, 18
3. =15, 17
4. 17, 18

If the system of linear equations : x+y+2z=6 2x+3y+az=a+1 -x-3y+bz=2b where a,b , has infinitely many solutions, then 7a+3b is equal to :

1. 12
2. 9
3. 22
4. 16

Let , R. If the system of equations 3x + 5y + z = 3 7x + 11y - 9z = 2 97x + 155y - 189z = has infinitely many solutions, then + 2 is equal to :

1. 24
2. 25
3. 27
4. 22

Consider the matrices : A = bmatrix 2 & -5 \ 3 & m bmatrix, B = bmatrix 20 \ m bmatrix and X = bmatrix x \ y bmatrix. Let the set of all m, for which the system of equations AX = B has a negative solution (i.e., x < 0 and y < 0), be the interval (a, b). Then 8 ab |A| dm is equal to .

If the system of equations x + 4y - z = , 7x + 9y + z = -3, 5x + y + 2z = -1 has infinitely many solutions, then (2 + 3 ) is equal to :

1. -2
2. 2
3. 3
4. -3

Let = 45; , , R. If x( , 1, 2) + y(1, , 2) + z(2, 3, ) = (0, 0, 0) for some x, y, z R, xyz 0, then 6 + 4 + is equal to .

If the system of equations 2x + 7y + z = 3 3x + 2y + 5z = 4 x + y + 32z = -1 has infinitely many solutions, then ( - ) is equal to :

If the system of equations 11x + y + z = -5 2x + 3y + 5z = 3 8x - 19y - 39z = has infinitely many solutions, then 4 - is equal to :

1. 51
2. 45
3. 47
4. 49

The values of m, n, for which the system of equations x + y + z = 4, 2x + 5y + 5z = 17, x + 2y + mz = n has infinitely many solutions, satisfy the equation :

1. m2 + n2 - m - n = 46
2. m2 + n2 + mn = 68
3. m2 + n2 - mn = 39
4. m2 + n2 + m + n = 64

If the system of equations x+( 2 )y+( 2 )z=0 x+( )y+( )z=0 x+( )y-( )z=0 has a non-trivial solution, then (0, 2 ) is equal to :

1. 5 24
2. 11 24
3. 7 24
4. 3 4

If the system of equations 2x+3y-z=5 x+ y+3z=-4 3x-y+ z=7 has infinitely many solutions, then 13 is equal to :

1. 1110
2. 1120
3. 1210
4. 1220

Let the system of equations x+2 y+3 z=5, 2 x+3 y+z=9, 4 x+3 y+ z= have infinite number of solutions. Then +2 is equal to :

1. 22
2. 17
3. 15
4. 28

If the system of linear equations x-2y+z=-4 2x+ y+3z=5 3x-y+ z=3 has infinitely many solutions, then 12 +13 is equal to

1. 60
2. 54
3. 64
4. 58