A linear equation in three variables always represent a ___________.

• A) Plane in $$R^2$$ (2 - dimension)
• B) Line in $$R^2$$ (2 - dimension)
• C) Line in $$R^3$$ (3 - dimension)
• D) Plane in $$R^3$$ (3 - dimension)

An n×n real matrix is invertible if and only if the span of the rows of A is ${R^n}$

• A) True
• B) False
• C)
• D)

Which of the following will be the Matrix Product corresponding to Linear Combination $$\begin{gathered} \left( {\begin{array}{*{20}{c}} { - 2} \\ 5 \end{array}} \right)x + \left( {\begin{array}{*{20}{c}} 3 \\ 1 \end{array}} \right)y? \\ \end{gathered}$$

• A) $$\left( {\begin{array}{*{20}{c}} 1&{ - 3} \\ { - 5}&{ - 2} \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right)$$
• B) $$\left( {\begin{array}{*{20}{c}} 3&{ - 2} \\ 1&5 \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right)$$
• C) $$\left( {\begin{array}{*{20}{c}} { - 2}&5 \\ 3&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right)$$
• D) $$\left( {\begin{array}{*{20}{c}} { - 2}&3 \\ 5&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right)$$

$${\text{Which of the following is an example of Matrix in reduced Echelon form?}}$$

• A) $$\left( {\begin{array}{*{20}{c}} 0&0 \\ 1&1 \end{array}} \right)$$
• B) $$\left( {\begin{array}{*{20}{c}} 0&1 \\ 0&1 \end{array}} \right)$$
• C) $$\left( {\begin{array}{*{20}{c}} 0&1 \\ 1&0 \end{array}} \right)$$
• D) $$\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right)$$

A set of linear equations is represented by matrix equation $Ax=b$. The necessary condition for the existence of a solution for this system is ------

• A) A must be invertible,
• B) b must be linearly dependent on the column of A,
• C) b must be linearly independent on the columns of A,
• D) none of the above.

$${\text{If}} ~~{A^t} = {A^{ - 1}},then~~ \left| A \right| = - - - - .$$

• A) 1
• B) -1
• C) $$\pm 1$$
• D) 0

In ____ equation, all variables occur only to the first power and do not appear, as arguments of trigonometric, logarithmic, or exponential functions.

• A) linear
• B) nonlinear
• C)
• D)

$${\left( {\begin{array}{*{20}{c}} {\cos\theta }&{\sin\theta } \\ { - \sin\theta }&{\cos\theta } \end{array}} \right)^{ - 1}}$$=_______

• A) $$\left( {\begin{array}{*{20}{c}} {\cos\theta }&{\sin\theta } \\ {\sin\theta }&{ - \cos\theta } \end{array}} \right)$$
• B) $$\left( {\begin{array}{*{20}{c}} {\cos\theta }&{ - \sin\theta } \\ {\sin\theta }&{\cos\theta } \end{array}} \right)$$
• C) $$\left( {\begin{array}{*{20}{c}} {\cos\theta }&{\sin\theta } \\ { - \sin\theta }&{\cos\theta } \end{array}} \right)$$
• D) $$\left( {\begin{array}{*{20}{c}} { - \cos\theta }&{\sin\theta } \\ {\sin\theta }&{\cos\theta } \end{array}} \right)$$

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

• A) They do not guarantee the Consistency.
• B) They guarantee the Consistency.
• C) None of above all
• D) They guarantee the Inconsistency.

$$Siny + Sinx = 1,{\text{ is an example of - - - Equation}}{\text{.}}$$

• A) $${\text{Linear}}$$
• B) $${\text{Non - Linear}}$$*
• C) $${\text{Homogeneous}}$$
• D) $${\text{Quadratic}}$$

Every system of linear equations has _____ solutions; there are no other possibilities.

• A) (c) infinitely many
• B) (d) all (a) to (c).
• C) (a) zero
• D) (b) one

Under which of the following condition, a system of Linear Equations whose Row - Reduce form is $$\begin{gathered} \left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 1 \\ 0 \end{array}}&{\begin{array}{*{20}{c}} 2 \\ 0 \end{array}}&{\begin{array}{*{20}{c}} | \\ | \end{array}}&{\begin{array}{*{20}{c}} { - 1} \\ {h - 3k} \end{array}} \end{array}} \right) \end{gathered}$$ has NO solution?

• A) $$h \ne 3k$$
• B) $$\left( {h,k} \right) \ne \left( {0,0} \right)$$
• C) $$h = 3k$$
• D) $$\left( {h,k} \right) = \left( {0,0} \right)$$

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

• A) $$h \ne 3k$$
• B) $$h = 3k$$
• C) $$\left( {h,k} \right) \ne \left( {0,0} \right)$$
• D) $$\left( {h,k} \right) = \left( {0,0} \right)$$

Whenever a solution set is described explicitly with vectors, we say that the solution is in parametric vector form.The equation x=su+tv (s, t in R) is called a

• A) parametric vector equation of the line
• B) parametric vector equation of the plane*
• C)
• D)

The Equation $$2x - 3y + kz = 4$$ is linear in _________.

• A) $$x,y ~{\text{and}}~ z$$
• B) $${\text{any one variable}}$$
• C) $$x, y, z ~{\text{and}}~ k$$
• D) $$x ~{\text{and}}~ y ~{\text{only}}$$

Which of the following would be the Augmented Matrix associated with the system $$\begin{gathered} \begin{array}{*{20}{c}} {2x = 1} \\ { - 2y = x ?} \end{array} \\ \end{gathered}$$

• A) $$\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 2 \\ { - 1} \end{array}}&{\begin{array}{*{20}{c}} 0 \\ { - 2} \end{array}}&{\begin{array}{*{20}{c}} | \\ | \end{array}}&{\begin{array}{*{20}{c}} 1 \\ 1 \end{array}} \end{array}} \right)$$
• B) $$\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 2 \\ { - 1} \end{array}}&{\begin{array}{*{20}{c}} 0 \\ { - 2} \end{array}}&{\begin{array}{*{20}{c}} | \\ | \end{array}}&{\begin{array}{*{20}{c}} 1 \\ 0 \end{array}} \end{array}} \right)$$
• C) $$\left( {\begin{array}{*{20}{c}} 2&1 \\ { - 2}&1 \end{array}} \right)$$
• D) $$\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 2 \\ 1 \end{array}}&{\begin{array}{*{20}{c}} 0 \\ { - 2} \end{array}}&{\begin{array}{*{20}{c}} | \\ | \end{array}}&{\begin{array}{*{20}{c}} 1 \\ 0 \end{array}} \end{array}} \right)$$

$${\text{Which of the following system will have the non - trivial solution?}}$$

• A) $$x = y = 0$$
• B) $$x = y = 1$$
• C)
• D)

The equation $Ax=b$ has a solution if and only if

• A) b is the linear combination of the columns of A
• B) b is a non-linear combination of the columns of A
• C)
• D)

$${\text{Slope and }}y{\text{ - intercept of }}y{\text{ - axis are - - - - respectively}}{\text{.}}$$

• A) $${\text{both undefined}}$$*
• B) $${\text{undefined and zero}}$$
• C) $${\text{both zeros}}$$
• D) $${\text{zero and undefined}}$$

$$\begin{gathered} {\text{Under which of the following condition, a system of Linear Equations whose }} \ {\text{Row - Reduce form is }}\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} 1 \\\ 0 \end{array}}&{\begin{array}{*{20}{c}} 2 \\\ 0 \end{array}}&{\begin{array}{*{20}{c}} | \\\ | \end{array}}&{\begin{array}{*{20}{c}} { - 1} \\\ {h - 3k} \end{array}} \end{array}} \right){\text{ has NO solution?}} \\\ \end{gathered}$$

• A) $$h \ne 3k$$*
• B) $$h = 3k$$
• C) $$\left( {h,k} \right) \ne \left( {0,0} \right)$$
• D) $$\left( {h,k} \right) = \left( {0,0} \right)$$