Exam October 2014, questions PDF

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School of Mathematics and Statistics SEMESTER 1, 2014 EXAMINATIONS MATH1001 Mathematical Methods 1 FAMILY NAME: ____________________________

STUDENT ID:

GIVEN NAMES: ______________________

SIGNATURE: ________________________ This Paper Contains: 34 pages (including title page) Time allowed: 3 hours 10 minutes

INSTRUCTIONS:

The exam has 30 questions giving a total of 120 marks for the paper. Answers to the 15 multiple choice questions should be given on the multiple choice answer sheet. Answers for the multiple choice questions on the exam paper WILL NOT be marked. There is only one correct answer to each multiple choice question. Full marks to the non-multiple choice questions will be awarded for complete fully justified answers. Only calculators with an appropriate approval sticker may be used.

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:

Semester 1 Examinations June 2014

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2. MATH1001

3. MATH1001

Semester 1 Examinations June 2014

1. Consider the following augmented matrix for a ⎡ 1 1 1 1 ⎢ ⎢ 0 6 3 1 ⎢ ⎢ 0 0 0 k−2 ⎣ 0 0 0 2

system of linear equations ⎤ 0 ⎥ π ⎥ ⎥ 0 ⎥ ⎦ 0

where k is some real number. Which of the following statements is correct? [2 marks] (A) The only value of k for which there are infinitely many solutions is k = 2. (B) When k = 2 there are no solutions. (C) There are never any solutions. (D) When k = 4 there no solutions. (E) For all values of k there are infinitely many solutions.

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4. MATH1001

Semester 1 Examinations June 2014

The next two questions concern the following ⎡ 1 2 3 4 ⎢ ⎢4 5 6 7 ⎢ ⎢5 7 9 11 ⎢ A=⎢ ⎢3 3 3 3 ⎢ ⎢2 4 6 8 ⎣

matrix ⎤ 5 ⎥ 8⎥ ⎥ 12 ⎥ ⎥ ⎥ 3⎥ ⎥ 10 ⎥ ⎦ 9 12 15 18 20

After performing elementary row operations you are able to reduce A to the following matrix: ⎡

1

2

3

4

5

⎤

⎢ ⎥ ⎢0 −3 −6 −9 −12 ⎥ ⎢ ⎥ ⎢0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢0 0 0 0 0 ⎥ ⎢ ⎥ ⎢0 0 0 0 0 ⎥ ⎣ ⎦ 0 0 0 0 0

2. Which of the following is the nullity of A? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4

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[2 marks]

5. MATH1001

Semester 1 Examinations June 2014

3. Which of the following is a basis for the column space of A?

[2 marks]

(A) {(1, 2, 3, 4, 5), (0, −3, −6, −9, −12)} (B) {(1, 0, 0, 0, 0, 0), (2, −3, 0, 0, 0, 0), (−3, −6, 0, 0, 0, 0), (4, −9, 0, 0, 0, 0), (5, −12, 0, 0, 0, 0)} (C) {(1, 0, 0, 0, 0, 0), (2, −3, 0, 0, 0, 0)} (D) {(1, 4, 5, 3, 2, 9), (2, 5, 7, 3, 4, 12), (3, 6, 9, 3, 6, 15)} (E) {(1, 4, 5, 3, 2, 9), (3, 6, 9, 3, 6, 15)}

4. You are asked to determine whether or not the set of vectors S = {(1, 2, 3, 4), (5, 7, 7, 8), (1, 5, 7, 9)} is linearly independent or not. You place the vectors as the rows of a matrix and row reduce to the following matrix. ⎡ ⎤ 1 2 3 4 ⎢ ⎥ ⎣ 0 −4 −8 −12 ⎦ 0

0

−4

−7

Which of the following is the correct deduction to make from this matrix? [2 marks] (A) S is linearly dependent because there is a free variable. (B) S is linearly dependent because there are infinitely many solutions. (C) S is linearly independent because there is no all zero row. (D) S is linearly independent because no row in the new matrix is a scalar multiple of another. (E) There is not enough information to deduce whether or not S is linearly independent.

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6. MATH1001

Semester 1 Examinations June 2014

5. Let A = {v 1 , v 2 , v 3 , v 4 , v 5 } be a set of linearly independent vectors in R6 . Which of the following is not a linearly independent set of vectors? [2 marks] (A) {v 1 , v 2 , v 3 , v 4 , v 5 , w} for any w ∈ R6 \span(A). (B) {v 2 , v 3 , v 4 , v 5 } (C) {v 1 + v 2 , v 3 + v4 , v 5 } (D) {v 1 , v 2 , v 3 + v 2 } (E) {v 1 , v 2 , v 3 , 4v 1 + 2v 2 }

6. Which of the following statements is NOT CORRECT?

[2 marks]

(A) A k-dimensional subspace of Rn contains k − 1 linearly independent vectors. (B) A k-dimensional subspace of Rn contains k − 1 dependent vectors. (C) The span of a set of k vectors in Rn has dimension k. (D) For each integer k such that 0 ≤ k ≤ n there is a subspace of Rn of dimension k.

(E) Rn has a spanning set containing n + 1 vectors.

7. Which of the following implies that the determinant of a matrix A is 0? [2 marks] (A) A is invertible. (B) The set of rows of A are linearly independent. (C) A has two equal columns. (D) A does not have 0 as an eigenvalue. (E) The nullity of A is 0.

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7. MATH1001

Semester 1 Examinations June 2014

8. Let A be a 15 × 23 matrix with nullity 5. Which of the following statements holds?

[2 marks]

(A) The rank of A is 18. (B) The rank of A is 10. (C) A is invertible. (D) No such matrix A exists. (E) The set of columns of A is linearly independent.

9. Let A be a 3 × 3 matrix, let λ be an eigenvalue for A with corresponding eigenvector v, and let AT denote the transpose of A. Which of the following assertions is NOT CORRECT?

[2 marks]

(A) det(A − λI) = 0. (B) λ2 is an eigenvalue of A2 . (C) If A is invertible then λ = 0 and A−1 v = λ1 v. (D) det(AT − λI) = 0. (E) For all positive real numbers a, the number a3 λ is an eigenvalue of aA.

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8. MATH1001

Semester 1 Examinations June 2014

10. Which of the following is the curve defined by r(t) = (t, cos(t), sin(t))? [2 marks] (A)

(B)

(C) x

y

z

z

z

x y

x

(D)

y

(E) y

y

x

z

z

x

11. Suppose that f (x) and g(x) are two functions such that limf (x) = 0 and x→0 [2 marks] lim g(x) = ∞. Which of the following deductions is correct? x→0

(A) lim f (x)g(x) = 2 x→0

(B) lim f (x)g(x) = ∞ x→0

f (x) =0 x→0 x (D) lim f (cos(x)) = 0 x→π/2   1 =0 (E) lim g x→∞ x (C) lim

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9. MATH1001

Semester 1 Examinations June 2014

12. Which of the following implies that a function f (x, y) of two variables is differentiable at (a, b)?

[2 marks]

(A) The partial derivatives exist at (a, b). (B) The partial derivatives equal 0. (C) The partial derivatives exist on a disc centred at (a, b) and are continuous at (a, b). (D) ∇f (a, b) exists. (E) f (x, y) is continuous at (a, b).

13. On which of the following domains is the function  2  t + t + 1 sin(t)−1 t + 1 ,e , r(t) = √ 2 t+5 t −4 defined and continuous?

[2 marks]

(A) R\{2, −5} (B) [2, ∞) (C) (−∞, −2) (D) (−2, 2) (E) (10, ∞)

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10. MATH1001

Semester 1 Examinations June 2014

∞ 

∞ 

bn . Which of the following statements

bn is a convergent series?

[2 marks]

14. Consider the two series

an and

n=1

imply that

∞ 

n=1

n=1

(A) 0  an  bn for all n and



n

an is a convergent series.

 (B) 0  an  bn for all n and n an is a divergent series.  an = 5 and n an is a convergent series. (C) lim n→∞ bn (D) lim bn = 0. n→∞

 an = 0 and n an is a convergent series. n→∞ bn

(E) lim

∞  (−2)n . Which of the following statements is 15. Consider the infinite series n2 + 1 n=1 correct? [2 marks]

(A) The series is absolutely convergent. (B) The series is not convergent. ∞ ∞   2n (−2)n (C) The series is convergent, but is not convergent. n2 + 1 n2 + 1 n=1 n=1 (D) The series

∞ ∞   (−2)n 2n is convergent, but is not convergent. n2 + 1 n2 + 1 n=1 n=1

(E) The series is conditionally convergent.

END OF THE MULTIPLE CHOICE QUESTIONS

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11. MATH1001

Semester 1 Examinations June 2014

16. Solve the following system of linear equations x1 + x1

x2 + x3 + + 4x3 +

2x1 + 2x2 +

x3 +

[5 marks] x4 = 1 x4 = 4 x4 = 6

3x1 + 3x2 + 2x3 + 2x4 = 7

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12. MATH1001

Semester 1 Examinations June 2014

17. Let U and V be subspaces of Rn . Show that U ∩ V is a subspace.

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[4 marks]

13. MATH1001

Semester 1 Examinations June 2014

18. Let S = {(1, 1, 1, 1), (2, 1, 2, 3), (0, −1, 0, 1)} be a set of three vectors in R4 . (a) Determine the dimension of the subspace of R4 that is spanned by S. [3 marks] (b) Is it possible to extend S to a basis for R4 ? If yes, find such a basis. If no, justify your answer.

[2 marks]

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14. MATH1001

Semester 1 Examinations June 2014

19. Suppose that A is an n × n matrix such that A2 − 3A + I = 0. Show that A is invertible. [3 marks]

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15. MATH1001

Semester 1 Examinations June 2014

20. Let T : R2 → R2 be the linear transformation given by the reflection in the line x + y = 0. (a) Determine the standard matrix for T .

[2 marks]

(b) Determine the matrix ABB for T where B = {(1, 1), (1, −1)}.

[3 marks]

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16. MATH1001

Semester 1 Examinations June 2014

21. Consider the function f (x, y) = ex sin(y ). (a) Find the gradient vector of f at the point (x, y).

[2 marks]

(b) Find the directional derivative of f in the direction (−1, 2) at the point (1, π/4).

[3 marks]

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17. MATH1001

Semester 1 Examinations June 2014

(c) What is the maximum rate of change of f at the point (1, π/4) and in what direction does it occur?

[2 marks]

(d) Find an equation of the tangent plane to the surface z = f (x, y) at the point √ [3 marks] (1, π/4, e/ 2).

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18. MATH1001

Semester 1 Examinations June 2014

22. Consider the function f (x, y) = x2 + y 2 (1 + 2x). (a) Find the critical points of f and determine their nature.

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[6 marks]

19. MATH1001

Semester 1 Examinations June 2014

(b) Find the absolute maximum of f on the region D = {(x, y) | x2 + y 2 ≤ 3}. [4 marks]

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20. MATH1001

Semester 1 Examinations June 2014

23. Consider the ellipsoid E defined by 3x2 + 2y 2 + z 2 = 9 Determine the constants a, b and c such that the sphere x2 + y 2 + z 2 + ax + by + cz + 42 = 0 has a common tangent plane with E at the point (1, 1, 2).

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[5 marks]

21. MATH1001

Semester 1 Examinations June 2014

24. Consider the following two functions f (x, y) = (2x8 + 3y 7 , 5x4 + 5y 8 ) and r(t) = (e5t , cos(6t)) (a) Find the derivative of r(t).

[2 marks]

(b) Determine the Jacobian matrix of f .

[2 marks]

(c) Use the Chain Rule to determine Df (r(0)).

[3 marks]

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22. MATH1001

Semester 1 Examinations June 2014

25. Recall that given a function f : R → R with continuous nth order derivatives we have that f (x) = Tn,a(x) + Rn,a (x) where Tn,a(x) = f (a) + f ′ (a)(x − a) +

f (n) (a) f ′′(a) (x − a)n (x − a)2 + · · · + 2! n!

is the nth Taylor Polynomial for f centred at a, and Rn,a(x) =

f (n+1)(z) (x − a)n+1 (n + 1)!

for some z between x and a. (a) Determine T4,π/4 (x) for f (x) = sin(2x).

QUESTION 25 CONTINUES OVER THE PAGE

[3 marks]

23. MATH1001

Semester 1 Examinations June 2014

25 (Continued) (b) Determine the maximum error in the approximation of sin(2x) on the interval [0, π/2] given by your answer in part (a).

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[3 marks]

24. MATH1001

Semester 1 Examinations June 2014

26. Solve

dy (x − 1)(y + 1) = xy dx

subject to the initial condition y(1) = 2.

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[4 marks]

25. MATH1001

Semester 1 Examinations June 2014

27. Find the general solution of the differential equation d2 y − y = sin 2 (x) + x2 dx2 (Hint: The double angle formula cos(2A) = 1−2 sin2 (A) may be useful.) [6 marks]

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26. MATH1001

Semester 1 Examinations June 2014

28. Let A be the matrix

⎤ 1 0 0 ⎢ ⎥ ⎣ 4 2 2⎦. 4 0 3 ⎡

(a) Find the characteristic equation of A.

(b) Find the eigenvalues of A.

[2 marks]

[1 mark]

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27. MATH1001

Semester 1 Examinations June 2014

(c) For each eigenvalue of A, find an eigenvector.

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[3 marks]

28. MATH1001

Semester 1 Examinations June 2014

(d) Find 3 × 3 matrices D, P such that D is a diagonal matrix and D = P −1 AP . [3 marks]

(e) Consider the homogeneous system of linear differential equations x′ (t) = Ax(t) ⎡ ⎤ x1 (t) ⎢ ⎥ where x(t) = ⎣x2 (t)⎦. Find the general solution for this system. x3 (t)

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[2 marks]

29. MATH1001

Semester 1 Examinations June 2014

29. Determine all the values of x for which the series ∞  (−1)n xn √ n n=1

is convergent.

[5 marks]

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30. MATH1001

Semester 1 Examinations June 2014

30. Let A be an n × m matrix and B be an m × t matrix. Show that rank(AB)  rank(B) [4 marks]

Semester 1 Examinations June 2014

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34. MATH1001

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Marking Table : For markers only

Q16

Q17

Q18

Q19

Q20

Q21

Q22

Q24

Q25

Q26

Q27

Q28

Q29

Q30

Q23...