MTH501 Quiz 1 Set 2

Q1. Which of the following is true about the existence of free variables (parameter) in a system of Linear Equations?

1. They guarantee the Inconsistency.
2. None of above all
3. They do not guarantee the Consistency.
4. They guarantee the Consistency.

Correct answer: They do not guarantee the Consistency.

Q2. $$\begin{gathered} The ~values~ of~ x~ and~ y~ which~ satisfy~ the ~matrix~ equation: \ \left( {\begin{array}{*{20}{c}} 4 \\ y \end{array}} \right) = x\left( {\begin{array}{*{20}{c}} 2 \\ 3 \end{array}} \right) are - - - - . \\ \end{gathered}$$

1. $$( {{array}{*{20}{c}} x \\ y {array}} ) = ( {{array}{*{20}{c}} 3 \\ 2 {array}} )$$
2. $$( {{array}{*{20}{c}} x \\ y {array}} ) = ( {{array}{*{20}{c}} 2 \\ 6 {array}} )$$
3. $$( {{array}{*{20}{c}} x \\ y {array}} ) = ( {{array}{*{20}{c}} 2 \\ 3 {array}} )$$
4. $$( {{array}{*{20}{c}} x \\ y {array}} ) = ( {{array}{*{20}{c}} 6 \\ 2 {array}} )$$

Correct answer: $$( {{array}{*{20}{c}} x \\ y {array}} ) = ( {{array}{*{20}{c}} 2 \\ 6 {array}} )$$

Q3. If a matrix is in reduced row echelon form, then it is also in row echelon form.

1. False
2. None of the above
3. May be
4. True

Correct answer: True

Q4. $$\begin{gathered} {\text{Which of the following is corresponding Matrix form of }} \ {\text{the Linear equation }}x + y = 3? \\ \end{gathered}$$

1. $$x( {{array}{*{20}{c}} 1 \\ 0 {array}} ) + y( {{array}{*{20}{c}} 0 \\ 1 {array}} ) = 3( {{array}{*{20}{c}} 0 \\ 1 {array}}
2. $$x( {{array}{*{20}{c}} 1 \\ 0 {array}} ) + y( {{array}{*{20}{c}} 0 \\ 1 {array}} ) = 3( {{array}{*{20}{c}} 1 \\ 0 {array}}
3. $$x[ 1 ] + y[ 1 ] = 3[ 1 ]$$
4. Cant be expressed in Matrix form.

Correct answer: $$x[ 1 ] + y[ 1 ] = 3[ 1 ]$$

Q5. $${\text{Which of the following would be the value of t, if}} \left[ {\begin{array}{*{20}{c}} {2{e^{ - 3t}}} \\ {5{e^{ - 3t}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 2 \\ 5 \end{array}} \right]?$$

1. $${{[ {{array}{*{20}{c}} 2 \\ 5 {array}} ]}}{{[ {{array}{*{20}{c}} 2 \\ 5 {array}} ]}} = 1$$
2. $$ {1}{3}$$
3. zero
4. $$[ {{array}{*{20}{c}} 1 \\ 1 {array}} ]$$

Correct answer: zero

Q6. $$\begin{gathered} {\text{For the}} ~{\text{Linear Equation}}: {x_1} + {x_2} + {x_3} + {x_4} = 1,{\text{how many Parametres}} ~{t_{\text{i}}}{\text{'s }} \ {\text{would have to introduce in order to find its solution?}} \\ \end{gathered}$$

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

Correct answer: 3

Q7. $$\begin{array}{*{20}{c}} {{\text{For the following system of Equations:}}} \\ {x + 2y = 0,{\text{ }}2x - y = 0{\text{;}}} \\ {{\text{the solution set is - - - - - }}{\text{.}}} \end{array}$$

1. $${{Real line}}~ {R}$$
2. $${{Line through origin in}} ~{{R}^3}$$
3. $${{Line through origin in}} ~{{R}^2}$$
4. $${{Any arbitrary line in}} ~{{R}^2}$$

Correct answer: $${{Line through origin in}} ~{{R}^2}$$

Q8. $If{\text{ }}A = \,\left( {\begin{array}{*{20}{c}} a&b \\\ c&d \end{array}} \right),{\text{ }}then{\text{ }}which{\text{ }}of{\text{ }}the{\text{ }}following{\text{ }}is\,\,\left| A \right|$

1. bc+ad
2. ad-bc
3. ad+bc
4. bc-ad

Correct answer: ad-bc

Q9. $$\begin{gathered} {\text{Under which of the following condition, a system of Linear Equations whose }} \ {\text{Row - Reduce form is }}\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 1 \\ 0 \end{array}}&{\begin{array}{*{20}{c}} 2 \\ 0 \end{array}}&{\begin{array}{*{20}{c}} | \\ | \end{array}}&{\begin{array}{*{20}{c}} { - 1} \\ {h - 3k} \end{array}} \end{array}} \right){\text{ has }}~Infinite ~{\text{many}} ~{\text{solutions?}} \\ \end{gathered}$$

1. $$h 3k$$
2. $$( {h,k} ) ( {0,0} )$$
3. $$h = 3k$$
4. $$( {h,k} ) = ( {0,0} )$$

Correct answer: $$h = 3k$$

Q10. $$How ~many ~Pivot ~positions ~the ~matrix:\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 2&3&1 \end{array}} \\ {\begin{array}{*{20}{c}} 4&6&2 \end{array}} \end{array}} \right) will~ have?$$

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

Correct answer: 1