Linear System of Equations

Q.

If the system of equations $2x+3y=7$and $2ax+(a+b)y=28$ has infinitely many solutions, then ______.

1. $a=2b$

2. $b=2a$

3. $a+2b=0$

4. $2a+b=0$

Q. An ordered pair $(\alpha ,\beta )$ for which the system of linear equations
$(1+\alpha )x+\beta y+z=2\alpha x+(1+\beta )y+z=3\alpha x+\beta y+2z=2$
has a unique solution, is:

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

Q. The system of equations,
$ax-y-z=\alpha -1,x-ay-z=\alpha -1x-y-az=\alpha -1$
has no solution if $\alpha$ is

1. Either $-2$ or $1$
2. $-2$
3. $1$
4. Not $-2$

Q. Let each of the equations $x^2+2xy+a{y}^2=0$ & $a{x}^2+2xy+y^2=0$ represent two straight lines passing through the origin. If they have a common line, then the other two lines are given by

1. $x+3y=0,3x+y=0$
2. $3x+y=0,3x-y=0$
3. $x-y=0,x-3y=0$
4. $(3x-2y)=0,x+y=0$

Q.

Let $S$ be the set of all $\lambda \in R$ for which the system of linear equations

$$\begin{gathered} 2x-y+2z=0 \\ x-2y+\lambda z=-4 \\ x+\lambda y+z=4 \end{gathered}$$

has no solution. Then the set $S$ is.

1. is an empty set

2. is a singleton

3. contains more than two elements

4. contains exactly two elements

Q. The greatest value of $c\in R$ for which the system of linear equations
$x-cy-cz=0cx-y+cz=0cx+cy-z=0$
has a non-trivial solution, is :

1. $0$
2. $\frac{1}{2}$
3. $2$
4. $-1$

Q. For the system of equations given by, $2x+py+6z=8,x+2y+qz=5,x+y+3z=4$, which of the following is/are true:

1. For $p\ne 2$, $q\ne 3$, the system has unique solution.
2. For $p=2,q\in R,$ the system has infinitely many solutions.
3. For $p\ne 2,q=3,$ the system has no solution.
4. Can't be determined.

Q. If the system of linear equations
$x+y+z=5x+2y+2z=6x+3y+\lambda z=\mu ,$
$(\lambda ,\mu \in R),$ has infinitey many solutions, then the value of $\lambda +\mu$ is :

1. $12$
2. $10$
3. $9$
4. $7$

Q. The system of equations
$kx+(k+1)y+(k-1)z=0$
$(k+1)x+ky+(k+2)z=0$
$(k-1)x+(k+2)y+kz=0$ has a non-trivial solution for

1. Exactly three values of k
2. Exactly two real values of k
3. Exactly one real value of k
4. Infinite real values of k

Q. The value of $\lambda$ for which the system of equations $x+y-2z=0,2x-3y+z=0,x-5y+4z=\lambda$ is consistent is

Q. Let $a,b$ and $c$ be positive real numbers. The following system of equations in $x,y$ and $z$
$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1,\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1,-\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ has

1. no solution
2. unique solution
3. infinitely many solutions
4. finitely many solutions

Q. Let $\left[a\right]$ denotes the integral part of $a$ and $x=a_{3}y+a_{2}z,y=a_{1}z+a_{3}x$ and $z=a_{2}x+a_{1}y$, where $x,y,z$ are not all zero. If $a_1=m-\left[m\right],m$ being a non-integral constant, then the least integral value of $|a_1{a}_2{a}_3|$ is

Q.

If the system of equations $x-ky-z=0,kx-y-z=0,x+y-z=0$ has a non-zero solution, then the possible values of $k$ are

1. $-1,2$

2. $1,2$

3. $0,1$

4. $-1,1$

Q.

Let the system of linear equations$4x+\lambda y+2z=0$, $2x-y+z=0$ and$\mu x+2y+3z=0$, $\lambda$ $\mu \in R$ has a non-trivial solution. Then which of the following is true?

1. $\mu =6,\lambda \in R$

2. $\lambda =2,\mu \in R$

3. $\lambda =3,\mu \in R$

4. $\mu =-6,\lambda \in R$

Q. Solution of the system of equations $x+18=2y,y=2x-9$ is:

1. $x=19,y=14$
2. $x=13,y=12$
3. $x=12,y=15$
4. $x=17,y=10$

Q. The system of equations
$-2x+y+z=a,x-2y+z=b,x+y-2z=c$ has:

1. no solution if $a+b+c\ne 0$
2. unique solution if $a+b+c=0$
3. infinte number of solutions if $a+b+c=0$
4. infinte solution if $a+b+c\ne 0$

Q. The number of values of $k$ for which the system of linear equations,
$(k+2)x+10y=kkx+(k+3)y=k–1$
has no soution, is :

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

Q. The system has infinite solutions if:

1. $\lambda \ne 3$
2. $\lambda =3,\mu =10$
3. $\lambda =3,\mu \ne 10$
4. none of these

Q. If $A=\left[\begin{array}{cc}a& b\\ 0& c\end{array}\right],$ then $A^{-1}+(A-aI)(A-cI)=$

1. $\frac{1}{ac}\left[\begin{array}{cc}a& b\\ 0& -c\end{array}\right]$
2. $\frac{1}{ac}\left[\begin{array}{cc}-a& b\\ 0& c\end{array}\right]$
3. $\frac{1}{ac}\left[\begin{array}{cc}c& -b\\ 0& a\end{array}\right]$
4. $\frac{1}{ac}\left[\begin{array}{cc}c& b\\ 0& a\end{array}\right]$

Q. For what values of k, the system of linear equations

Q. lf the system of equations $x+y+z=6,x+2y+\lambda z=0,x+2y+3z=10$ has no solution,
then $\lambda$ is equal to

1. $2$
2. $3$
3. $4$
4. $5$

Q. Let $S$ be the set of all column matrices $\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]$
such that $b_1,b_2,b_3\in R$ and the system of equations (in real variables)
$$\begin{array}{c}-x+2y+5z=b_1\\ 2x-4y+3z=b_2\\ x-2y+2z=b_3\end{array}$$
has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]\in S$?

1. $x+2y+3z=b_1,4y+5z=b_2\text{and}x+2y+6z=b_3$
2. $x+y+3z=b_1,5x+2y+6z=b_2\text{and}-2x-y-3z=b_3$
3. $-x+2y-5z=b_1,2x-4y+10z=b_2\text{and}x-2y+5z=b_3$
4. $x+2y+5z=b_1,2x+3z=b_2\text{and}x+4y-5z=b_3$

Q. Find the value of $k$ for which each of the following systems of equations have infinitely many solution:
$4x+5y=3$
$kx+15y=9$

Q. The system of equation $\lambda x+y+z=0,-x+\lambda y+z=0,-x-y+\lambda z=0$ will have a non-zero solution if real values of $\lambda$ are given by ................

Q. If the system of equations
$x-2y+3z=9$
$2x+y+z=b$
$x-7y+az=24,$ has infinitely many solutions, then $a-b$ is equal to

Q. Consider the system of equations
$x+y+z=6,$
$x+2y+3z=10$ and
$x+2y+\lambda z=\mu$
Statement $1$: If the system has infinite number of solutions, then $\mu =10.$
Statement $2$: The value of $\left|\begin{array}{ccc}1& 1& 6\\ 1& 2& 10\\ 1& 2& \mu \end{array}\right|=0\text{for}\mu =10.$

1. Statement $1$ is false and Statement $2$ is true.
2. Both the statements are true and Statement $2$ is the correct explanation of Statement $1$.
3. Statement $1$ is true and Statement $2$ is false.
4. Both the statements are true but Statement $2$ is not the correct explanation of Statement $1$.

Q. Match the equations in List 1 with the solutions in List 2

	List I		List II
A.	${\log }_{0.25}{(x^2+2x-8)}^2$ $-{\log }_{0.5}(10+3x-x^2)=1$	1.	$\{-3\}$
B.	${\log }_2(x^2+7)$ $=5+{\log }_{2}x$ $-\frac{6}{{\log }_2(x+\frac{7}{x})}$	2.	$\left\{\frac{1}{4}\right\}$
C.	${\log }_{(1-2x)}(6x^2-5x+1)$$-{\log }_{(1-3x)}(4x^2-4x+1)=2$	3.	$\{\frac{1}{6}(\sqrt{313}-1),\frac{1}{2}(\sqrt{73}-7)\}$
D.	${\log }_{10}(1{+x}^2-2x)+1-{\log }_{10}(1+x^2)=$ $2{\log }_{10}(1-x)$	4.	$\{1,7\}$

1. A- 3, B- 2, C- 1, D-4
2. A- 3, B- 4, C- 1, D-2
3. A- 3, B- 4, C- 2, D-1
4. A- 3, B- 1, C- 2, D-4

Q. If the system of equations $ax+hy+g=0,hx+by+f=0$ and $gx+fy+c=k$
is consistent and
$k=\frac{1}{\lambda }\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|$, then $\lambda$ is equal to

1. $h^2-ab$
2. $ab-h^2$
3. $h^2+ab$
4. $g^2-ac$

Q. The values of $k\in R$ for which the system of equations
$x+ky+3z=0,$
$kx+2y+2z=0$
$2x+3y+4z=0$
admits a non-trivial solution is

1. $2$
   
2. $\frac{5}{2}$
   
3. $3$
   
4. $\frac{5}{4}$

Q. If $t$ is a real number and $k=\frac{t^2-t+1}{t^2+t+1},$ then the system of equations
$3x-y+4z=3$
$x+2y-3z=-2$
$6x+5y+kz=-3$
For any allowable value of $k$, has