Maths 2 Quiz 2: 31 October 2021, Set QPA (September 2021 term)

Choose the set of correct options.

Which of the following options is correct?

Which of the following options is correct?

Consider the following two statements:

Statement P:

If $V = \mathbb{R}^2$, with the operations:

Addition:

$(x_1, y_1) + (x_2, y_2) = (x_1x_2, y_1y_2); \ (x_1, y_1), (x_2, y_2) \in V$

and

Scalar multiplication:

$c(x, y) = (cx, cy); \ (x, y) \in V, \ c \in \mathbb{R}$

is a vector space.

Statement Q:

Let $V$ be a vector space w.r.t usual addition and scalar multiplication. If $u, v, w \in V$ are such that $au + bv + cw = 0$ for some scalars $a, b, c \in \mathbb{R}$ and $ac \neq 0$, then $\text{span}\{u, v\} = \text{span}\{v, w\}$.

Consider the following statements:

• Statement 1: P is true, but Q is false.
• Statement 2: Q is true, but P is false.
• Statement 3: Both P and Q are true.
• Statement 4: Both P and Q are false.

Which one of the above statements is correct? (If Statement 3 is correct, then enter 3 as your answer).

NOTE: Enter your answer to the nearest integer

The basis (in Column B) of the vector space in the row a of Column A is
[If the basis of the vector space (a) of Column A has the basis with the serial number 1 in column B then enter 1 (enter the serial number only) as your answer.]

The basis (in Column B) of the vector space in the row b of Column A is
[If the basis of the vector space (b) of Column A has the basis with the serial number 1 in column B then enter 1 (enter the serial number only) as your answer.]

The basis (in Column B) of the vector space in the row c of Column A is
[If the basis of the vector space (c) of Column A has the basis with the serial number 1 in column B then enter 1 (enter the serial number only) as your answer.]

Represent the system of linear equations given in terms of its matrix form $Ax = b$, where $x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$, and $b = \begin{bmatrix} 13 \\ 24 \\ 17 \end{bmatrix}$. Which one of the following options represent $A$?

Find the (2,1)-th cofactor of A.
NOTE: Enter your answer to the nearest integer

NOTE: Enter your answer to the nearest integer

The determinant of A^(T) is (A^(T) denotes the transpose of the matrix A)
NOTE: Enter your answer to the nearest integer