Maths 2 End Term: 30 April 2023, Set QPF1-S1 (January 2023 term)

What is the maximum product of three non-negative numbers whose sum is 6?

Let $U$ denote the set of all $2 \times 2$ upper triangular matrices. Consider an ordered basis $\beta = \left\{\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\right\}$. Let $T : U \to \mathbb{R}^2$ be a linear transformation defined as $T\begin{pmatrix} a & b \\ 0 & c \end{pmatrix} = (a+b, c)$. Let matrix $A$ be the matrix representation of $T$ with respect to the ordered basis $\beta$ for $U$ and the standard ordered basis for the co-domain $\mathbb{R}^2$. Answer the given subquestions.

Which of the following matrix is A?

Let $U$ denote the set of all $2 \times 2$ upper triangular matrices. Consider an ordered basis $\beta = \left\{\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\right\}$. Let $T : U \to \mathbb{R}^2$ be a linear transformation defined as $T\begin{pmatrix} a & b \\ 0 & c \end{pmatrix} = (a+b, c)$. Let matrix $A$ be the matrix representation of $T$ with respect to the ordered basis $\beta$ for $U$ and the standard ordered basis for the co-domain $\mathbb{R}^2$. Answer the given subquestions.

Which of the following is/are true about T?

Let $U$ denote the set of all $2 \times 2$ upper triangular matrices. Consider an ordered basis $\beta = \left\{\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\right\}$. Let $T : U \to \mathbb{R}^2$ be a linear transformation defined as $T\begin{pmatrix} a & b \\ 0 & c \end{pmatrix} = (a+b, c)$. Let matrix $A$ be the matrix representation of $T$ with respect to the ordered basis $\beta$ for $U$ and the standard ordered basis for the co-domain $\mathbb{R}^2$. Answer the given subquestions.

If $\gamma$ is the orthonormal basis of $W$ obtained from $\beta$ (from the previous question) by using the Gram Schmidt process with respect to the usual inner product, and $(a, b, c, d)$ is the projection of $\left(0, \sqrt{\frac{11}{2}}, 0, \sqrt{\frac{11}{2}}\right)$ onto $W$, then what is $\sqrt{22}(a + b + c + d)$?

Based on the above data, answer the given subquestions.

Based on the above data, answer the given subquestions.

Based on the above data, answer the given subquestions.

Based on the above data, answer the given subquestions.

Based on the above data, answer the given subquestions.

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

1. Rank 2
2. Non-zero nullity
3. Non-zero 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. Global maxima
11. Global minima
12. A critical point
13. Gradient exists
14. Directional derivative exists in any direction
15. Partial derivatives exist
16. Orthonormal columns
17. Standard ordered basis
18. Affine subspace.
19. Limit exists

A local maxima is/can be _______________ . (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 and 17, then you should enter 717]

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

1. Rank 2
2. Non-zero nullity
3. Non-zero 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. Global maxima
11. Global minima
12. A critical point
13. Gradient exists
14. Directional derivative exists in any direction
15. Partial derivatives exist
16. Orthonormal columns
17. Standard ordered basis
18. Affine subspace.
19. Limit exists

Orthogonal matrix of order 3 has ______________ . (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 and 17, then you should enter 717]

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

1. Rank 2
2. Non-zero nullity
3. Non-zero 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. Global maxima
11. Global minima
12. A critical point
13. Gradient exists
14. Directional derivative exists in any direction
15. Partial derivatives exist
16. Orthonormal columns
17. Standard ordered basis
18. Affine subspace.
19. Limit exists

If the tangent plane exists for a function at a point then at that point ________________ . (Enter 4 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, 15 and 17, then you should enter 7141517]

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

1. Rank 2
2. Non-zero nullity
3. Non-zero 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. Global maxima
11. Global minima
12. A critical point
13. Gradient exists
14. Directional derivative exists in any direction
15. Partial derivatives exist
16. Orthonormal columns
17. Standard ordered basis
18. Affine subspace.
19. Limit exists

Some of the conditions that need to be checked to identify a vector space V are _______________ . (Enter 5 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, 11, 15 and 17, then you should enter 714111517]

Consider the system of linear equations formed by the equations in column B of Table M2ES2 and let A be its coefficient matrix. Answer the related subquestions.

Which of the following denotes the solution space of this system of linear equations?

Consider the system of linear equations formed by the equations in column B of Table M2ES2 and let A be its coefficient matrix. Answer the related subquestions.

Consider the system of linear equations formed by the equations in column B of Table M2ES3 and let A be its coefficient matrix. Answer the related subquestions.

Find Rank(A).

Consider the system of linear equations formed by the equations in column B of Table M2ES3 and let A be its coefficient matrix. Answer the related subquestions.

Which of the following denotes the column space of A?(More than one option may be correct)