Solving System of Linear Equations Using Inverse

Q.

For which value(s) of $\lambda$ , do the pair of linear equations $\lambda x+y={\lambda }^2$ and $x+\lambda y=1$ have infinitely many solutions?

Q. If the system of the equations
$2x+ay+6z=8,x+2y+bz=5,x+y+3z=4$
has a unique solution, then which of the following is true regarding $a$ and $b$?

1. $a=2,b=3$
2. $a\ne 2,b=3$
3. $a=2,b\ne 3$
4. $a\ne 2,b\ne 3$

Q.

Using properties of proportion, solve each of the following for $x$:

(ii) $\frac{a+\sqrt{a^2-2ax}}{a-\sqrt{a^2-2ax}}=b$.

Q. Solve the following system of equations by matrix method.
$3x-2y+3z=8$
$2x+y-z=1$
$4x-3y+2z=4$

Q. The solution of the systems of linear equations $x-y+2z=7;3x+4y-5z=-5;2x-y+3z=12$ is:

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

Q. The system of equations x + ky + 3z = 0, 3x + ky - 2z = 0, 2x + 3y - 4z = 0 possess a non-trivial solution over the set of rationals, then 2k is an integral element of the interval

1. (20, 30)
2. (40, 50)
3. [30, 40]
4. [10, 20]

Q. If the system of equations $x+ay=0,az+y=0,ax+z=0$ has
infinite number of solutions then the value of $a$ is

1. $-1$
2. $1$
3. $0$
4. No real value

Q. 3. Using matrix method solve the equations (a) x + 2y = 5 (b) 3x – 2y = –1

Q. The system of equations $x+4y-2z=3,3x+y+5z=7,2x+3y+z=5$ will have

1. a unique solution
2. infinite number of solutions
3. no solution
4. two solutions

Q.

For a system of n linear equations in n variables equations let A, B and X represent the coefficient, constant and variable matrices.If $\left|A\right|=0$, and (adj A). B $\ne$ 0 the system of equations is inconsistent and has no solution.

1. True

2. False

Q. If the system of linear equations
$x+y+3z=0$
$x+3y+k^{2}z=0$
$3x+y+3z=0$
has a non-zero solution $(x,y,z)$ for some $k\in R,$ then $x+\left(\frac{y}{z}\right)$ is equal to:

1. $-9$
2. $9$
3. $-3$
4. $3$

Q. If , find A −1 . Using A −1 solve the system of equations

Q.

Solve system of linear equations, using matrix method.
$x-y+2z=7$
$3x+4y-5z=-5$
$2x-y+3z=12$

Q. Solve the following system of inequalities graphically:
$4x+3y\le 60,y\ge 2x,x\ge 3,x\ge 0,y\ge 0$

Q.

Consider a LPP given by
Min Z = 6x + 10y
Subjected to x ≥ 6 ; y ≥ 2;
2x + y ≥ 10; x, y ≥ 0
Redundant constraints in this LPP are

1. x ≥ 6, 2x + y ≥ 10

2. x > 0

3. 2x + y ≥ 10

4. x ≥ 0, y ≥ 0

Q. Using the properties of determinants, show that:
(i)$\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|=(a-b)(b-c)(c-a)$
(ii)$\left|\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^3& b^3& c^3\end{array}\right|=(a-b)(b-c)(c-a)(a+b+c)$

Q. The value of ${}^{\prime }{a}^{\prime }$ for which the system of equations $(a+1{)}^{3}x+(a+2{)}^{3}y=(a+3{)}^3,(a+1)x+(a+2)y=a+3,x+y=1$ is consistent, is

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

Q. Solve the system of the following equations

Q. If the system of linear equations $x+y+z=6,x+2y+3z=14and2x+5y+\lambda z=\mu ,(\lambda ,\mu ϵR)$ has no solution, then

1. $Noneofthese$
2. $\lambda \ne 8$
3. $\lambda =8,\mu \ne 36$
4. $\lambda =8,\mu =36$

Q. Show that the differential equation
$ydx+x\log \left(\frac{y}{x}\right)dy-2xdy=0$
is homogenous and solve it.

Q.

The given set of equations has at least one non trivial solution.

x + 3y + 2z =0

2x + y + 4z = 0

x + 11y + 14z =0

1. True

2. False

Q. The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix method.

Q.

The solution of the equation is

1. 
   

2. 
   

3. 
   

4. None of these

Q. 3. Using matrix method solve the equations (a) x + 2y = 5 (b) 3x – 2y = –1

Q.

For a system of n non homogeneous equations in n variables equations let A, B and X represent the coefficient, constant and variable matrices. Then,

If $\left|A\right|\ne$ 0, then X = $A^{-1}$ B gives a trivial solution.

1. True

2. False

Q. 14. x-y+2z = 73x + 4y-5z =-52x-y+3z = 12

Q.

The Equation $y=2x+3$ has

Q. Consider the system of equations,
$AX=B,$ where
$A=\left[\begin{array}{ccc}-2& 1& 1\\ 1& -2& 1\\ 1& 1& -2\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],B=\left[\begin{array}{c}a\\ 0\\ b\end{array}\right]$.

Then which of the following(s) is/are always correct?
$(O$ be the null matrix and $a,b,x,y,z\in R)$

1. If $\left(\text{adj}A\right)B\ne O$, then the system of equations is always inconsistent.
2. If $\left(\text{adj}A\right)B\ne O$, then the system of equations may be either consistent or inconsistent depending on $a$ and $b$.
3. If $\left(\text{adj}A\right)B=O$, then the system of equations is always inconsistent.
4. If $\left(\text{adj}A\right)B=O$, then the system of equations may be either consistent or inconsistent depending on $a$ and $b$.

Q. For a non-trivial solution $|A|$ is

1. $|A|>0$
2. $|A|<0$
3. $|A|=0$
4. $|A|\ne 0$

Q.

Solve system of linear equations, using matrix method.