Cramer's Rule

Q. Using Cramer's rule or otherwise, the solution of equations:
x + y + z = 6
x - y + z = 2 and
3x + 2y - 4z = -5
is .

1. x = 2, y = 1, z = 3
2. x = 2, y = 3, z = 1
3. x = 1, y = 3, z = 2
4. x = 1, y = 2, z = 3

Q.

There are three values of t for which the following system of equations has non-trivial solutions.

(a-t) x+by + cz=0

bx + (c - t) y +az=0

cx + ay + (b-t) z=0

We can express the product of the three values of t in the form of determainant as

1. 

2. 

3. 

4. D

Q.

If $D\ne 0$ and at least one of $D_1,D_2,D_3$ is not 0 then according to Cramer's rule the system of linear equations will have

1. non trivial solution

2. trivial solution

3. infinite solutions

4. no solutions

Q.

Let a, b, c, be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x= cy +bz, y=az + cx and z = bx +ay. Then , a² + b²+ c²+ 2abc is equal to

1. 2

2. -1

3. 0

4. 1

Q. The system of equations
$px+y+z=px+py+z=px+y+pz=p$
has no solution for

1. three distinct values of $p$
2. two distinct values of $p$
3. only one value of $p$
4. no such $p$ exists

Q. The system of equations $x+y+z=2,3x-y+2z=6$ and $3x+y+z=-18$ has

1. A unique solution
2. No solutions
3. An infinite number of solutions
4. Zero solution as the only solution

Q. If the system of linear equation $x+2ay+az=0,x+3by+bz=0,x+4cy+cz=0$ has a non zero solution, then a, b, c

1. Are in A.P.
2. Are in G. P.
3. Are in H. P.
4. Satisfy a +2b + 3c = 0

Q. If the system of linear equations
$x+y+z=5$
$x+2y+3z=9$
$x+3y+\alpha z=\beta$
has infinitely many solutions, then $\beta -\alpha$ equals:

1. $5$
2. $18$
3. $8$
4. $21$

Q.

If $D\ne 0$ and at least one of $D_1,D_2,D_3$ is not 0 then according to Cramer's rule the system of linear equations will have

1. non trivial solution

2. trivial solution

3. infinite solutions

4. no solutions

Q.

Total number of solutions to the system of linear equations 2x + 3y + z = 1, 4X + 6y + 2z = 3, 6x + 9y + 3z = 2 is ___

Q.

If $D=0$ and at least one of $D_1,D_2,D_3$ is not 0 then according to Cramer's rule the system of linear equations will have

1. non trivial solution

2. trivial solution

3. infinite solutions

4. no solutions

Q. If the system of linear equations
$x+y+z=5$
$x+2y+3z=9$
$x+3y+\alpha z=\beta$
has infinitely many solutions, then $\beta -\alpha$ equals:

1. $5$
2. $18$
3. $8$
4. $21$

Q. If a, b, c are non – zero real numbers and if the equations (a-1) x = y + z, (b -1)y = z + x, (c - 1)z = x + y has a non trivial solution, then ab + bc + ca is equal to

1. a +b + c
2. abc
3. 1
4. None of these

Q.

Which of the following cases will not lead to non trivial solutions in case of system of linear equations according to Cramer's rule Convention given D != 0 ?

1. D1 = 0, D2 = 0, D3 = 0

2. D1 = 0, D2 = 0, D3 != 0

3. D1 != 0, D2 != 0, D3 = 0

4. D1 != 0, D2 != 0, D3 != 0

Q. $x+ky-z=0,3x-ky-z=0$and$x-3y+z=0$ has non-zero solution for k =

1. -1
2. \N
3. 1
4. 2

Q. For which of the following ordered pairs $(\mu ,\delta ),$ the system of linear equations
$x+2y+3z=1$
$3x+4y+5z=\mu$
$4x+4y+4z=\delta$
is inconsistent?

1. $(4,6)$
2. $(3,4)$
3. $(1,0)$
4. $(4,3)$

Q.

If $D=0$ and at least one of $D_1,D_2,D_3$ is not 0 then according to Cramer's rule the system of linear equations will have

1. non trivial solution

2. trivial solution

3. infinite solutions

4. no solutions

Q. The value of k for which, the system of equations
kx + 3y - z = 1; x + 2y + z = 2 and -kx + y + 2z = -1 have no solution, is .

1. k = -7
2. k = 7
3. $k=-\frac{7}{2}$
4. $k=\frac{7}{2}$

Q.

The system of linear equations $x+y=0,x–y=0,z=0$ will have

1. non trivial solution

2. trivial solution

3. infinite solutions

4. no solutions

Q.

Total number of solutions to the system of linear equations 2x + 3y + z = 1, 4X + 6y + 2z = 3, 6x + 9y + 3z = 2 is 1

1. True

2. False

Q.

Total number of solutions to the system of linear equations x + y + 0.z = 0, X – y + 0.z = 0, X + 4y + 0.z = 0 is infinity.

1. True

2. False

Q. The number of values of k for which the system of equations $(k+1)x+8y=4k,kx+(k+3)y=3k-1$ has infinitely many solutions, is

1. \N
2. 1
3. 2
4. Infinite