Exam 2 11 October 2013, questions PDF

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Mathematics and Statistics SEMESTER 1, 2016 EXAMINATIONS MATH1001 Mathematical Methods 1 FAMILY NAME: ____________________________

STUDENT ID:

GIVEN NAMES: ______________________

SIGNATURE: ________________________ This Paper Contains: 32 pages (including title page) Time allowed: 2:00 hours

INSTRUCTIONS: The exam has 12 Multiple Choice questions (worth 4 marks each) and 10 short-answer questions (worth 8 marks each) meaning that the total marks for the paper is 48 + 80 = 128. Answers tot the multiple choice questions should be given only on the multiple choice answer sheet in dark pencil (not in the exam booklet). these questions have only one correct answer. For the remaining questions, write your solutions in the spcaes provided in the exam paper. Your answers must always be fully justified, in that you must include the working-out and anything else that will enable the markers to understand how your arrived at your answer.

PLEASE NOTE Examination candidates may only bring authorised materials into the examination room. If a supervisor finds, during the examination, that you have unauthorised material, in whatever form, in the vicinity of your desk or on your person, whether in the examination room or the toilets or en route to/from the toilets, the matter will be reported to the head of school and disciplinary action will normally be taken against you. This action may result in your being deprived of any credit for this examination or even, in some cases, for the whole unit. This will apply regardless of whether the material has been used at the time it is found. Therefore, any candidate who has brought any unauthorised material whatsoever into the examination room should declare it to the supervisor immediately. Candidates who are uncertain whether any material is authorised should ask the supervisor for clarification.

Supervisors Only - Student left at:

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3. MATH1001

Semester 1 Examinations June 2016

1. Consider the following augmented matrix for ⎡ 1 0 ⎢ ⎢ 0 −1 ⎢ ⎢ 0 0 ⎣ 0 0

a system of linear equations ⎤ 0 0 0 ⎥ 2 1 2 ⎥ ⎥ ℓ 1 1 ⎥ ⎦ 0 k 1

where ℓ, k are real numbers. Which of the following statements is correct ?

(A) This system has a unique solution when k = 0 and ℓ =  0. (B) This system always has exactly one solution. (C) This system never has infinitely many solutions. (D) This system is only feasible when k = 0 and ℓ = 0. (E) This system has infinitely many solutions when k = 0 and ℓ =  0.

2. Which of the following conditions implies that an n × n matrix A is invertible? (A) A has determinant 0. (B) The nullity of A is n. (C) A has determinant −1. (D) The rows of A are linearly dependent. (E) The column rank of A equals the row rank of A.

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4. MATH1001

Semester 1 Examinations June 2016

3. You are asked to determine whether or not a set S of three vectors in R4 is linearly independent or not. You place the vectors into the rows of a matrix and row-reduce to the following matrix in row echelon form.

⎡

1

2 0

4

⎤

⎢ ⎥ A = ⎣ 0 −1 0 0 ⎦ 0 0 0 −7

Which of the following is the correct deduction to make from this matrix? (A) S is linearly dependent because A has a non-basic column. (B) S is linearly dependent because there are infinitely many solutions. (C) S is linearly independent because no two rows of A are equal. (D) There is not enough information to deduce whether or not S is linearly independent. (E) S is linearly independent because there is no all-zero row.

4. Let A be a 4 × 8 matrix with nullity 2. Which of the following statements is correct ? (A) The rank of A is 6. (B) No such matrix A exists. (C) The rank of A is 2. (D) The rows of A are linearly independent. (E) The rank of A is 8.

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5. MATH1001

Semester 1 Examinations June 2016

5. Consider the function f (x, y) =



x2 + y 2 − 1 . ln(2 − x2 − y 2 )

From the following sets D, choose one that could be the domain of f . (A) D = {(x, y) ∈ R2 : x2 + y 2 < 2}. (B) D = R2 . (C) D = {(x, y) ∈ R2 : 1 < x2 + y 2 < 2}. (D) D = {(x, y) ∈ R2 : 1 ≤ x2 + y 2 < 2}. (E) D = {(x, y) ∈ R2 : 1 ≤ x2 + y 2 }.

6. Consider the function f (x, y) =

xy 2 . Which of the following statements is not correct? x2 + 2y 4

(A) limy→0 f (y, y) = 0. (B) lim(x,y)→(0,0) f (x, y) = 0 (C) limx→0 f (x, 0) = 0. (D) limy→0 f (y 2 , y) = 31. (E) limy→0 f (0, y) = 0.

7. Consider a function f : R2 → R of two variables x and y and the surface defined by z = f (x, y). Which of the following is not a correct statement about the gradient vector of f at the point (x, y) = (1, 2)? (A) It is perpendicular to the level curve for f passing through (1, 2). (B) It is the direction of the maximum rate of change of f at (1, 2). ∂f ∂f (1, 2), ∂y (1, 2)). (C) It is parallel to ( ∂x

(D) It is perpendicular to the surface given by the graph of f . (E) Its coordinates can be used to find a normal vector to the surface at (1, 2, f (1, 2))

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6. MATH1001

Semester 1 Examinations June 2016

8. Suppose that f : R2 → R is given by f (x, y) = (x + 2y)2 . What is ∇f at the point (1, 1)? (A) 6 (B) (6, 12) (C) 12 (D) (12, 6) (E) 18

9. Consider the infinite series

∞  (−1)n n=0

n+1

. Which of the following statements is correct ?

(A) The series is absolutely convergent. (B) The series is not convergent. ∞ ∞   1 (−1)n (C) The series is not convergent. is convergent, however n+1 n+1 n=0 n=0 (D) Both

∞  n=0

∞

 (−1)n 1 are convergent. and n+1 n+1 n=0

(E) The series is conditionally convergent.

10. Which of the following is the most appropriate choice to try for a particular solution to the differential equation dx d2 x + 4x = 2t + sin t. −4 2 dt dt (A) x(t) = At + B sin t (B) x(t) = A cos t + B sin t (C) x(t) = A + Bt + C sin t (D) x(t) = At + B sin(2t) (E) x(t) = A + Bt + C cos t + D sin t

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7. MATH1001

Semester 1 Examinations June 2016

11. Which of the following is the 4th Taylor polynomial T4,0 (x) for f (x) = cos(2x) about 0? 2x4 3 2 x4 x − (B) −x − 2 4 2 (C) x + 2x + 4x4

(A) 1 − 2x2 +

x2 x4 + 2 4 1 2 (E) 1 − x + x4 3

(D) 1 +

12. Consider the infinite sequence a1 , a2 , . . . , an , . . . defined by the formula an = (−1)n Which of the following statements is correct ? (A) The sequence is divergent. (B) lim an = π. n→∞

(C) lim an = 0. n→∞

(D) lim an = 1. n→∞

(E) lim an = ∞. n→∞

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cos n . n

8. MATH1001

Semester 1 Examinations June 2016

13. Consider the following system of linear equations (SLE): x1 + 2x2 − x3 = 0

x1 + 2x2 + x3 = 2 2x1 + 4x2 = 2 (a) What is the augmented matrix for this SLE?

[1 mark] (b) Use Gaussian elimination to reduce this matrix to row-echelon form, indicating which elementary row operations you are using at each step.

[3 marks] QUESTION 13 CONTINUES OVER THE PAGE

9. MATH1001

Semester 1 Examinations June 2016

13 (Continued) (c) Perform back-substitution on the reduced matrix to find the set of all solutions of this SLE.

[3 marks] (d) Verify that (x1 , x2 , x3 ) = (−1, 1, 1) is a solution to the SLE, and check that your solution set contains this vector.

[1 mark]

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10. MATH1001

Semester 1 Examinations June 2016

14. (a) Determine whether the set of vectors S = {(x, y, z) | x + yz ≥ 0} is a subspace of R3 or not. (Remember to fully explain your answer.)

[4 marks] QUESTION 14 CONTINUES OVER THE PAGE

11. MATH1001

Semester 1 Examinations June 2016

14 (Continued) (b) Suppose that V and W are both subspaces of R2 , and consider their union V ∪ W = {u | u ∈ V or u ∈ W }. Carefully explain whether or not V ∪ W is always a subspace. (Remember to give a proof if you think that it is always a subspace, and a counterexample otherwise.)

[4 marks]

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12. MATH1001

Semester 1 Examinations June 2016

15. Consider the following matrix which is in row echelon form. ⎡

1 0

0 1 2

⎤

⎢ ⎥ A = ⎣ 0 0 −1 1 3 ⎦ 0 0 0 0 2 (a) Find a basis for the row space of A.

[2 marks] (b) What is the rank of A?

[2 marks] QUESTION 15 CONTINUES OVER THE PAGE

13. MATH1001

Semester 1 Examinations June 2016 15 (Continued) (c) Find a basis for the null space of A.

[4 marks]

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Semester 1 Examinations June 2016

14. MATH1001

16. Consider the function f (x, y) = x3 + 3y − y 3 − 3x. (a) Find the critical points of f and determine the nature of each of them — local maximum, local minimum or saddle point.

[4 marks] QUESTION 16 CONTINUES OVER THE PAGE

15. MATH1001

Semester 1 Examinations June 2016

16 (Continued) (b) Find the absolute maximum and the absolute minimum of f on the closed triangle in the first quadrant in R2 (i.e. where x ≥ 0, y ≥ 0) bounded by the lines y = x, x = 0 and y = 0.

[4 marks]

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16. MATH1001

Semester 1 Examinations June 2016

17. A function f : R2 → R is defined by f (x, y) =

⎧ ⎨ x2 − y 2 , (x, y) = (0, 0), ⎩ 1,

(x, y) = (0, 0)

(a) Show that f is not continuous at (0, 0).

[2 marks] (b) Explain whether f (x, y) is differentiable at (0, 0) or otherwise.

[1 mark] QUESTION 17 CONTINUES OVER THE PAGE

17. MATH1001

Semester 1 Examinations June 2016

17 (Continued) (c) Determine the directional derivative Du f (x, y) for (x, y) = (0, 0), where u = (1, 1).

[2 marks] (d) Find an equation of the form ax + by + cz = d for the tangent plane to the surface z = f (x, y) at the point(1, 1, 0).

[3 marks]

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18. MATH1001

Semester 1 Examinations June 2016

18. Consider the second-order differential equation x′′ + 3x′ + 2x = 0 and set x1 = x and x2 = x′ . (a) Find a matrix A such that 

x′1 x′2



=A



x1 x2



is the matrix-vector form of this equation.

[3 marks] QUESTION 18 CONTINUES OVER THE PAGE

19. MATH1001

Semester 1 Examinations June 2016

18 (Continued) (b) The matrix from the previous part of the question has eigenvalues λ1 = −1 and λ2 = −2, with the corresponding eigenvectors:   1 v1 = , −1

v2 =



1 −2



.

Write down the general solution to this differential equation for both x1 (t) and x2 (t).

[3 marks] QUESTION 18 CONTINUES OVER THE PAGE

20. MATH1001

Semester 1 Examinations June 2016

18 (Continued) (c) Find the Jacobian matrix for the function g : R2 → R2 given by g(x, y) = (x2 + 3y, y 2 + xy)

[2 marks]

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21. MATH1001

Semester 1 Examinations June 2016

19. State whether each of the following statements is true or false, giving a brief reason for each of your choices. If you use theorems from the lectures as part of your reason, you should name the theorem you are using. (a) If {bn } is a decreasing sequence of non-negative real numbers, then limn→∞ bn exists.

[2 marks] (b) The series

is convergent.

∞  1 √3 n2 n=1

[2 marks] QUESTION 19 CONTINUES OVER THE PAGE

22. MATH1001

Semester 1 Examinations June 2016 19 (Continued) (c) The series

is convergent.

∞  cos2 n + 1 n2 n=1

[2 marks] (d) The series ∞  (−1)n 1000 i=1

is divergent.

n

[2 marks]

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23. MATH1001

Semester 1 Examinations June 2016

20. Consider the following non-homogeneous second order ordinary differential equation: dy d2 y − 7 + 12y = sin(3x). 2 dx dx (a) Find the complementary solution for this ODE

[3 marks] (b) What is the shape of a particular solution to this ODE (i.e. just express it using symbols for the unknown coefficients).

[1 mark] QUESTION 20 CONTINUES OVER THE PAGE

24. MATH1001

Semester 1 Examinations June 2016

20 (Continued) (c) Use the method of undetermined coefficients to find a particular solution to this ODE.

[3 marks] (d) What is the general solution to this ODE?

[1 mark]

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25. MATH1001

Semester 1 Examinations June 2016

21. Let A be the following symmetric matrix ⎡

0 1 1

⎤

⎥ ⎢ ⎣ 1 0 1 ⎦. 1 1 0 (a) Show that 2 is an eigenvalue of A by finding an eigenvector for A.

[2 marks] (b) You are told that the characteristic polynomial of A is λ3 − 3λ − 2. Using this, find all the eigenvalues of A.

[4 marks] QUESTION 21(b) CONTINUES OVER THE PAGE

26. MATH1001

Semester 1 Examinations June 2016 21(b) (Continued) (c) Find a diagonal matrix D that is similar to A.

[2 marks]

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27. MATH1001

Semester 1 Examinations June 2016

22. (a) Let f (x, y) be a continuously differentiable function in R2 and let g(s, t) = f (st, ln s + ln t) ,

(s, t) ∈ R2 for s > 0, t > 0 .

That is, g(s, t) = f (x(s, t), y(s, t)), where x(s, t) = st and y(s, t) = ln s + ln t. Use the Chain Rule to show that

∂g ∂g (s, t) − t (s, t) = 0 ∂s ∂t 2 for all (s, t) ∈ R with s > 0, t > 0. s

[4 marks] QUESTION 22 CONTINUES OVER THE PAGE

28. MATH1001

Semester 1 Examinations June 2016

22 (Continued) (b) Suppose that f : R2 → R is given by f (x, y) = y 2 − x2 . On the grid below, draw the level curves Lk for k = −2, −1, 0, 1, 2. Your drawing should be labelled, and to the correct scale, given that the range of the grid is −2 ≤ x ≤ 2 and −2 ≤ y ≤ 2. −2

−2

2

−2

[4 marks]

29. MATH1001

Semester 1 Examinations June 2016

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Semester 1 Examinations June 2016

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