Maths 2 End Term: 8 August 2021 (May 2021 term)

Choose the set of correct options about estimating the area of the region bounded by the graph of function f, above the interval [0,10] using Riemann sums.

Match the properties of the function mentioned in Column A with the appropriate numerical values given in Column B:

Table: M2ES1

Consider the system of linear equations given by $Ax = b$,

$\text{where } A = \begin{bmatrix} 1 & 0 & 3 \\ 2 & 4 & 5 \\ 7 & 8 & 19 \end{bmatrix}, \quad x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, \text{ and } b = \begin{bmatrix} 4 \\ 11 \\ 34 \end{bmatrix}.$

The row echelon form of the matrix $A$ is given by $R = \begin{bmatrix} 1 & 0 & 3 \\ 0 & 1 & -\frac{1}{4} \\ 0 & 0 & 0 \end{bmatrix}$.

Answer the subquestions using the given information.

Consider the system of linear equations given by $Ax = b$,

$\text{where } A = \begin{bmatrix} 1 & 0 & 3 \\ 2 & 4 & 5 \\ 7 & 8 & 19 \end{bmatrix}, \quad x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, \text{ and } b = \begin{bmatrix} 4 \\ 11 \\ 34 \end{bmatrix}.$

The row echelon form of the matrix $A$ is given by $R = \begin{bmatrix} 1 & 0 & 3 \\ 0 & 1 & -\frac{1}{4} \\ 0 & 0 & 0 \end{bmatrix}$.

Answer the subquestions using the given information.

Choose the set of correct options.

Match the vector subspaces of $\mathbb{R}^3$ (with the usual scalar multiplication and vector addition) in column A with their bases in column B and the dimensions of the vector spaces in column C in Table : M2ES2.

Table : M2ES2

Choose the set of correct options.

NOTE: Enter your answer to the nearest integer.

Find out the directional derivative of f at the point (1, 1) in the direction of the vector (1, 1).

Suppose the temperature ($T$) at a point $(x, y, z)$ in $\mathbb{R}^3$ is given by the function

$T(x, y, z) = e^{x+y+z}$

Let $T_1$, $T_2$, and $T_3$ be the tangent planes to the function $f(x, y) = \sqrt{9 - x^2 - y^2}$ at the points $(1, 2)$, $(2, 1)$, and $(2, 2)$, respectively. The set of points on each plane $T_1$, $T_2$, and $T_3$ form affine subspaces $A_1$, $A_2$, and $A_3$, respectively, of the vector space $\mathbb{R}^3$, with respect to usual addition and scalar multiplication. Let $V_1$, $V_2$, and $V_3$ denote the vector subspaces corresponding to the affine subspaces $A_1$, $A_2$, and $A_3$, respectively.

Answer the subquestions using the given information.

Suppose the temperature ($T$) at a point $(x, y, z)$ in $\mathbb{R}^3$ is given by the function

$T(x, y, z) = e^{x+y+z}$

Let $T_1$, $T_2$, and $T_3$ be the tangent planes to the function $f(x, y) = \sqrt{9 - x^2 - y^2}$ at the points $(1, 2)$, $(2, 1)$, and $(2, 2)$, respectively. The set of points on each plane $T_1$, $T_2$, and $T_3$ form affine subspaces $A_1$, $A_2$, and $A_3$, respectively, of the vector space $\mathbb{R}^3$, with respect to usual addition and scalar multiplication. Let $V_1$, $V_2$, and $V_3$ denote the vector subspaces corresponding to the affine subspaces $A_1$, $A_2$, and $A_3$, respectively.

Answer the subquestions using the given information.

Which of the following denotes the maximum rate of change of temperature (T) at the point of intersection of the tangent planes T1, T2, and T3?

Suppose the temperature ($T$) at a point $(x, y, z)$ in $\mathbb{R}^3$ is given by the function

$T(x, y, z) = e^{x+y+z}$

Let $T_1$, $T_2$, and $T_3$ be the tangent planes to the function $f(x, y) = \sqrt{9 - x^2 - y^2}$ at the points $(1, 2)$, $(2, 1)$, and $(2, 2)$, respectively. The set of points on each plane $T_1$, $T_2$, and $T_3$ form affine subspaces $A_1$, $A_2$, and $A_3$, respectively, of the vector space $\mathbb{R}^3$, with respect to usual addition and scalar multiplication. Let $V_1$, $V_2$, and $V_3$ denote the vector subspaces corresponding to the affine subspaces $A_1$, $A_2$, and $A_3$, respectively.

Answer the subquestions using the given information.

Let $T$ be a linear transformation between the vector subspaces $V_1$ and $V_3$. Suppose the matrix representation of $T$ with respect to the ordered bases $\beta = \{(-2,1,0),(-2,0,1)\}$ for $V_1$ and $\gamma = \{(1,0,-2),(0,1,-2)\}$ for $V_3$ is given by $\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$. Then which of the following mappings can be an affine mapping $f$ between the affine subspaces $A_1$ and $A_3$, corresponding to $T$ ?