Maths 2 End Term: 3 April 2022 (January 2022 term)

Choose the correct options from the following:

Choose the correct options from the following:

Choose the set of correct options.

Choose the correct statements for the function

Consider the following system of linear equations

The number of solutions of the above system is

NOTE: Enter your answer to the nearest integer.

NOTE: Enter your answer to the nearest integer.

A potter wants to make a rectangular box of volume 17 cubic units such that its total surface area is minimum. He successfully creates one of dimensions x, y, and z units. Find the value of x³ + z³. [Note: If the dimensions of a rectangular box are l, b, and h units, then volume of the box is V = lbh cubic units and the total surface area of the rectangular box is S = 2(lb + bh + lh) square units.] NOTE: Enter your answer to the nearest integer.

From the list of given terms find out the best possible options for each of the given subquestions:

1. Rank
2. Limit
3. Determinant
4. Closure with respect to addition and scalar multiplication
5. Existence of zero element
6. Existence of additive inverse
7. Commutativity of addition
8. Associativity of addition
9. Elements
10. Local maxima
11. Local minima
12. Saddle points
13. Gradient
14. Directional derivative
15. Partial derivative
16. Set of orthonormal vectors
17. Standard ordered basis
18. Affine subspace.

Critical points can be __________. (Enter 3 best possible options. Enter only the serial numbers of those options in increasing order without adding any comma or space in between them.) [Suppose your answer is 7, 14 and 17, then you should enter 71417]

From the list of given terms find out the best possible options for each of the given subquestions:

1. Rank
2. Limit
3. Determinant
4. Closure with respect to addition and scalar multiplication
5. Existence of zero element
6. Existence of additive inverse
7. Commutativity of addition
8. Associativity of addition
9. Elements
10. Local maxima
11. Local minima
12. Saddle points
13. Gradient
14. Directional derivative
15. Partial derivative
16. Set of orthonormal vectors
17. Standard ordered basis
18. Affine subspace.

Similar matrices have the same __________. (Enter 2 best possible options. Enter only the serial numbers of those options in increasing order without adding any comma or space in between them.) [Suppose your answer is 7, 14 and 17, then you should enter 71417]

From the list of given terms find out the best possible options for each of the given subquestions:

1. Rank
2. Limit
3. Determinant
4. Closure with respect to addition and scalar multiplication
5. Existence of zero element
6. Existence of additive inverse
7. Commutativity of addition
8. Associativity of addition
9. Elements
10. Local maxima
11. Local minima
12. Saddle points
13. Gradient
14. Directional derivative
15. Partial derivative
16. Set of orthonormal vectors
17. Standard ordered basis
18. Affine subspace.

Gram-Schmidt algorithm generates a __________. (Enter the best possible option (only one). Enter only the serial number of that option.) [Suppose your answer is 7, 14 and 17, then you should enter 71417]

From the list of given terms find out the best possible options for each of the given subquestions:

1. Rank
2. Limit
3. Determinant
4. Closure with respect to addition and scalar multiplication
5. Existence of zero element
6. Existence of additive inverse
7. Commutativity of addition
8. Associativity of addition
9. Elements
10. Local maxima
11. Local minima
12. Saddle points
13. Gradient
14. Directional derivative
15. Partial derivative
16. Set of orthonormal vectors
17. Standard ordered basis
18. Affine subspace.

The conditions that need to be checked to identify a subspace W of a vector space V :___________. (Enter 2 best possible options. Enter only the serial numbers of those options in increasing order without adding any comma or space in between them.) [Suppose your answer is 7, 14 and 17, then you should enter 71417]

• Consider the function $f : \mathbb{R}^2 \to \mathbb{R}$ defined as

  $f(x, y) = x^2y + xy^2$

• The tangent plane to the function $f$ at the point (1,1) is given by $T_1$. Let $A_1$ denote the affine subspace of $\mathbb{R}^3$ formed by the set of points on $T_1$. Let $V_1$ denote the corresponding vector subspace associated with $A_1$.

• Let $A$ denote the Hessian matrix of the function $f$ at the point $(1, 1)$.

• Consider the affine subspace of $\mathbb{R}^3$ given by

  $A_2 = \{(x, y, z) \mid x - y + z = 1, \text{ where } x, y, z \in \mathbb{R}\}$

  Let $V_2$ denote the corresponding vector subspace associated with $A_2$.

Using the above information answer the given subquestions.

Which one of the following represents the equation of the tangent lines to the function ƒ at the point (1, 1) in the direction of the unit vector (1, 0)?

• Consider the function $f : \mathbb{R}^2 \to \mathbb{R}$ defined as

  $f(x, y) = x^2y + xy^2$

• The tangent plane to the function $f$ at the point (1,1) is given by $T_1$. Let $A_1$ denote the affine subspace of $\mathbb{R}^3$ formed by the set of points on $T_1$. Let $V_1$ denote the corresponding vector subspace associated with $A_1$.

• Let $A$ denote the Hessian matrix of the function $f$ at the point $(1, 1)$.

• Consider the affine subspace of $\mathbb{R}^3$ given by

  $A_2 = \{(x, y, z) \mid x - y + z = 1, \text{ where } x, y, z \in \mathbb{R}\}$

  Let $V_2$ denote the corresponding vector subspace associated with $A_2$.

Using the above information answer the given subquestions.

The unit vector along which the rate of change of the function ƒ at (1, 1) is the maximum is

• Consider the function $f : \mathbb{R}^2 \to \mathbb{R}$ defined as

  $f(x, y) = x^2y + xy^2$

• The tangent plane to the function $f$ at the point (1,1) is given by $T_1$. Let $A_1$ denote the affine subspace of $\mathbb{R}^3$ formed by the set of points on $T_1$. Let $V_1$ denote the corresponding vector subspace associated with $A_1$.

• Let $A$ denote the Hessian matrix of the function $f$ at the point $(1, 1)$.

• Consider the affine subspace of $\mathbb{R}^3$ given by

  $A_2 = \{(x, y, z) \mid x - y + z = 1, \text{ where } x, y, z \in \mathbb{R}\}$

  Let $V_2$ denote the corresponding vector subspace associated with $A_2$.

Using the above information answer the given subquestions.

The number of critical points of the function ƒ is _______________.
NOTE: Enter your answer to the nearest integer.

• Consider the function $f : \mathbb{R}^2 \to \mathbb{R}$ defined as

  $f(x, y) = x^2y + xy^2$

• The tangent plane to the function $f$ at the point (1,1) is given by $T_1$. Let $A_1$ denote the affine subspace of $\mathbb{R}^3$ formed by the set of points on $T_1$. Let $V_1$ denote the corresponding vector subspace associated with $A_1$.

• Let $A$ denote the Hessian matrix of the function $f$ at the point $(1, 1)$.

• Consider the affine subspace of $\mathbb{R}^3$ given by

  $A_2 = \{(x, y, z) \mid x - y + z = 1, \text{ where } x, y, z \in \mathbb{R}\}$

  Let $V_2$ denote the corresponding vector subspace associated with $A_2$.

Using the above information answer the given subquestions.

What is the rank of A?
NOTE: Enter your answer to the nearest integer.

• Consider the function $f : \mathbb{R}^2 \to \mathbb{R}$ defined as

  $f(x, y) = x^2y + xy^2$

• The tangent plane to the function $f$ at the point (1,1) is given by $T_1$. Let $A_1$ denote the affine subspace of $\mathbb{R}^3$ formed by the set of points on $T_1$. Let $V_1$ denote the corresponding vector subspace associated with $A_1$.

• Let $A$ denote the Hessian matrix of the function $f$ at the point $(1, 1)$.

• Consider the affine subspace of $\mathbb{R}^3$ given by

  $A_2 = \{(x, y, z) \mid x - y + z = 1, \text{ where } x, y, z \in \mathbb{R}\}$

  Let $V_2$ denote the corresponding vector subspace associated with $A_2$.

Using the above information answer the given subquestions.

Which of the following sets forms a basis β of V1?

• Consider the function $f : \mathbb{R}^2 \to \mathbb{R}$ defined as

  $f(x, y) = x^2y + xy^2$

• The tangent plane to the function $f$ at the point (1,1) is given by $T_1$. Let $A_1$ denote the affine subspace of $\mathbb{R}^3$ formed by the set of points on $T_1$. Let $V_1$ denote the corresponding vector subspace associated with $A_1$.

• Let $A$ denote the Hessian matrix of the function $f$ at the point $(1, 1)$.

• Consider the affine subspace of $\mathbb{R}^3$ given by

  $A_2 = \{(x, y, z) \mid x - y + z = 1, \text{ where } x, y, z \in \mathbb{R}\}$

  Let $V_2$ denote the corresponding vector subspace associated with $A_2$.

Using the above information answer the given subquestions.

Which of the following sets forms a basis γ of V2?

• Consider the function $f : \mathbb{R}^2 \to \mathbb{R}$ defined as

  $f(x, y) = x^2y + xy^2$

• The tangent plane to the function $f$ at the point (1,1) is given by $T_1$. Let $A_1$ denote the affine subspace of $\mathbb{R}^3$ formed by the set of points on $T_1$. Let $V_1$ denote the corresponding vector subspace associated with $A_1$.

• Let $A$ denote the Hessian matrix of the function $f$ at the point $(1, 1)$.

• Consider the affine subspace of $\mathbb{R}^3$ given by

  $A_2 = \{(x, y, z) \mid x - y + z = 1, \text{ where } x, y, z \in \mathbb{R}\}$

  Let $V_2$ denote the corresponding vector subspace associated with $A_2$.

Using the above information answer the given subquestions.

Let $T$ denote the linear transformation from $V_1$ to $V_2$, whose matrix representation with respect to the ordered bases $\beta$ and $\gamma$ (mentioned in the previous questions 19 & 20 ), respectively for $V_1$ and $V_2$ is the identity matrix, i.e.,$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Which of the following functions denotes an affine mapping from $A_1$ to $A_2$ corresponding to $T$?