chapter1 · mas111: mathematics coreii problems booklet – chapters 1–4 chapter1 1. find the...

MAS111: Mathematics Core II

Problems booklet – Chapters 1–4

Chapter 1

1. Find the equation of the line through (−3, 1) and (1, 2). Use your result to showthat the points (−3, 1), (1, 2) and (9, 4) are collinear.

2. Find the equation of the line parallel to the line 7x − y = 6 which passes throughthe point (−2, 3). Write down the equation of the perpendicular line which passesthrough the origin.

3. Find the point at which the line y = 5x− 7 intersects the line y = −3x+ 1.

4. Let A = (−1, 0), B = (1, 0), and let P = (a, b) be a point on x2+y2 = 1. Show thatAP and BP are perpendicular, i.e., that the angle in a semicircle is a right angle.

5. The point (x, y) is equidistant from the points P = (−1, 2) and Q = (2,−3). Findthe relation between x and y.

6. Let A = (−1, 2) and B = (3, 1). If P = (x, y) satisfies 2|AP | = 3|PB|, find therelation between x and y.

7. At time t, the position of two aircraft are given by (500t + 10,−312t − 12) and(496t + 22,−310t − 23), where t is measured in hours, and distance is measuredin kilometres. Find the closest the aircraft approach, and the time at which thisoccurs.

8. Consider the line ℓ given by ax + by + c = 0 in the Cartesian plane, and the pointP = (x0, y0).

(a) What is the gradient of ℓ?

(b) Find the line ℓ′ through P which is perpendicular to ℓ.

(c) Find the point Q of intersection of ℓ and ℓ′.

(d) Convince yourself that Q is the closest point of ℓ to P , and show that thedistance d of P from the line (the same as the distance between P and Q) is

d =|ax0 + by0 + c|√

a2 + b2.

9. Let A, B, C be the points (x1, y1), (x2, y2) and (x3, y3) respectively. Determine thelength of AB, the equation of the line AB and the distance of the point C from theline AB. Hence show that the area of the triangle ABC is given by

1

2|(x2y3 − x3y2)− (x1y3 − x3y1) + (x1y2 − x2y1)| .

10. (a) Given a point (ρ, θ, z) in cylindrical polar coordinates, express it in sphericalpolar coordinates.

MAS111 1 Spring 2016-17

(b) Given a point (r, θ, φ) in spherical polar coordinates, express it in cylidricalpolar coordinates.

11. Find the plane containing the point (2, 3, 5) and the vectors (1, 1, 0) and (0, 1, 1), ina form without any parameters.

12. Find the distance of the point (x0, y0, z0) from the plane ax+ by + cz + d = 0.

[Hint: find the normal to the plane that passes through the given point in parametric

form; find the intersection of this normal with the plane, and then work out the

distance between this intersection and the original point.]

13. What is the normal to x+ y+ z = 0 which passes through (1, 1, 1)? Use your resultfrom the previous question to find the distance from (1, 1, 1) to the plane.

What is the nearest point on the plane to (1, 1, 1)?

Chapter 2

1. Consider the three planes

2x+ y + z = 1

x+ y + z = 2

x+ 3y + z = 3

Show that these intersect in a single point. What can you deduce about the vectors(2, 1, 1), (1, 1, 1) and (1, 3, 1)?

2. Are the following sets of vectors linearly dependent or independent?

(a)

3−21

,

213

,

0−7−7

.

(b)

3−21

,

213

,

14−3

.

3. Reduce the matrix

2 1 11 1 11 3 1

to row echelon form.

4. For the following sets of equations

(a) form the augmented matrix (A|b)(b) reduce (A|b) to row echelon form (H|c), where H is upper triangular

(c) find all the solutions of the equations:

i.

x+ 2y + z = 2

2x+ 3y + z = 4

x+ y − z = 3

MAS111 2 Spring 2016-17

ii.

x+ y + z = 2

x− y + 2z = 4

2y − z = 2

5. For each of the following systems of linear equations, write down the augmented ma-trix (A|b), and reduce it to row echelon form. Hence determine whether a solutionexists. If it does find the full solution:

(a)

x+ y = 2

y = 3

(b)

x+ y + z = 1

y + z = 1

x = 1

(c)

x+ y + z = 3

y + z = 2

x = 1

(d)

x+ y = 2

x+ z + w = 1

2x+ y + z + w = 2

(e)

x+ y = 1

x+ z + w = 1

2x+ y + z + w = 2

6. Consider the three planes:

x+ y + z = 1

x− y + 2z = 0

2x+ 4y + z = 4

Show that these planes have no common point of intersection, but that all choicesof pairs of these equations intersect in lines, and these lines are all parallel. Find arelationship between the normal vectors to the planes.

MAS111 3 Spring 2016-17

7. Find all values of λ such that the vector b is a linear combination of the vectors a1,a2, a3, where

(a) a1 = (2, 3, 5), a2 = (3, 7, 8), a3 = (1,−6, 1), b = (7,−2, λ);

(b) a1 = (4, 4, 3), a2 = (7, 2, 1), a3 = (4, 1, 6), b = (5, 9, λ);

(c) a1 = (3, 2, 6), a2 = (5, 1, 3), a3 = (7, 3, 9), b = (λ, 2, 5).

8. Perform a complete elimination to reduce the augmented matrix

1 2 1 3 04 −1 −5 −6 91 −3 −4 −7 52 1 −1 0 3

to reduced row echelon form. Hence give the general solution to the system ofequations

x+ 2y + z + 3u = 0

4x− y − 5z − 6u = 9

x− 3y − 4z − 7u = 5

2x+ y − z = 3.

9. A Pythagorean triple is a triple (p, q, r) of positive integers such that p2 + q2 = r2.

Let P =

−1 −2 2−2 −1 2−2 −2 3

.

Suppose that (p, q, r) is a Pythagorean triple. Write down P

−pqr

, P

p−qr

and

P

−p−qr

, and show that they all form Pythagorean triples.

Chapter 3

1. Consider the following graph:

x

y

MAS111

Sketch the effect on this picture of the transformations

MAS111 4 Spring 2016-17

(a)

(

2 00 1/2

)

;

(b)

(

−1 00 −2

)

;

(c)

(

1 01 1

)

;

(d)

(√3/2 1/2

−1/2√3/2

)

;

2. Let A =

(

0 1−1 0

)

, B =

(

0 10 0

)

, C =

(

1 11 −1

)

, D =

(

1 −1−1 −1

)

, E =

(

1 11 1

)

and F =

(

1 1−1 −1

)

. Show that

(a) A2 = −I (so A is like a square root of −1)

(b) B2 = 0 (but B 6= 0)

(c) CD = −DC (but CD 6= 0)

(d) EF = 0 (but E 6= 0 and F 6= 0)

3. Given the matrices

A =

1 −2−1 10 2

, B =

(

3 26 −4

)

, C =

(

5 −41 −2

)

,

evaluate each of the following, where possible. If evaluation is not possible explainwhy not:

(a) C − 3A,

(b) C − 3B,

(c) AB,

(d) BA,

(e) BC,

(f) CB.

4. Compute(

cosα sinα− sinα cosα

)(

cos β sin β− sin β cos β

)

,

and simplify your answer using addition formulae for sin and cos.

Give a geometrical interpretation of this result.

What is

(


)n

?

5. Compute(

1 a0 1

)(

1 b0 1

)

;

(

a 00 1

)(

b 00 1

)

.

MAS111 5 Spring 2016-17

6. Let Eij denote the 3×3 matrix with all entries equal to 0, except for a 1 in the ijth

place. Suppose that A =

a b cd e fg h i

. Calculate

(a) E11A;

(b) E12A;

(c) E21A;

(d) AE11;

(e) AE12;

(f) AE21.

7. Given two 2× 2 matrices, A = (aij) and B = (bij), compute AB and BA.

The trace of a square matrix is defined to be the sum of the entries down the maindiagonal. Verify that the trace of AB is the same as the trace of BA.

Harder: In fact this holds for any m × n matrix A and n × m matrix B. Try toprove this by using the formula for the entries of a matrix product.

8. Given the symmetric matrices A =

(

1 22 3

)

and B =

(

3 55 7

)

, show that is AB is

not symmetric.

9. For the matrices A and C in Q3, show that (AC)T = CTAT .

10. Deduce from Q7 that if A and B are square matrices of the same size, with Ainvertible, that the trace of B is the same as the trace of A−1BA.

11. Solve the matrix equations

(

1 31 2

)

X =

(

1 11 1

)

and

X

(

−1 13 −4

)

=

(

−2 −13 4

)

.

12. (a) Solve the linear equations

5x− 4y = 2

x− 2y = 4

by inverting a matrix.

(b) Again using matrix techniques, explain why the equations

2x− 4y = 8

x− 2y = 6

have no solution. Give a geometrical interpretation of your result.

MAS111 6 Spring 2016-17

13. Consider the matrix A =

(

2 31 2

)

. Give a sequence of row operations to transform

A into the identity matrix. By interpreting row operations as multiplications byelementary matrices, write A as a product of elementary matrices.

14. Use the Gauss-Jordan procedure to determine the inverse of

A =

1 3 −1−2 −5 14 11 −2

.


(

1 −1 2 52 −2 5 12

)

. Find a sequence of row operations to

transform this matrix into reduced row echelon form. Hence give a matrix P suchthat PA is in reduced row echelon form.

Now find a sequence of column operations to transform the matrix PA into

(

1 0 0 00 1 0 0

)

.

Interpret your column operations as postmultiplying by elementary matrices, and

hence find a matrix Q so that (PA)Q =

(

1 0 0 00 1 0 0

)

.

Chapter 4

1. Recall the definition of the trace of a matrix from Q7 of Chapter 3. Let A =

(

a bc d

)

.

Verify thatA2 − trace(A).A+ det(A).I = 0.

2. Calculate∣

∣

∣

∣

5 33 5

∣

∣

∣

∣

;

∣

∣

∣

∣

a bλa λb

∣

∣

∣

∣

;

∣

∣

∣

∣


∣

∣

∣

∣

;

∣

∣

∣

∣

a + bi c+ di−c + di a− bi

∣

∣

∣

∣

.

3. Evaluate the determinant

∣

∣

∣

∣

∣

∣

4 1 00 2 −12 3 1

∣

∣

∣

∣

∣

∣

by (i) expanding along the first row, and (ii)

expanding down the first column.

4. Evaluate

(a)

∣

∣

∣

∣

∣

∣

1 2 34 5 67 8 9

∣

∣

∣

∣

∣

∣

;

(b)

∣

∣

∣

∣

∣

∣

1 2 30 4 50 0 6

∣

∣

∣

∣

∣

∣

.

5. Compute

∣

∣

∣

∣

∣

∣

∣

∣

a 3 0 50 b 0 21 2 c 30 0 0 d

∣

∣

∣

∣

∣

∣

∣

∣

.

MAS111 7 Spring 2016-17

6. Show that

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

a b c d ef g h i jk l 0 0 0m n 0 0 0p q 0 0 0

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

= 0.

7. Show that∣

∣

∣

∣

∣

∣

1 1 1x y zx2 y2 z2

∣

∣

∣

∣

∣

∣

= (y − x)(z − x)(z − y).

8. Given that the integers 123, 456 and 753 are all divisible by 3, explain why the

determinant

∣

∣

∣

∣

∣

∣

1 2 34 5 67 5 3

∣

∣

∣

∣

∣

∣

is divisible by 3.

9. Let z and w denote two complex numbers. Consider the determinant D =

∣

∣

∣

∣

z wz w

∣

∣

∣

∣

.

Preferably without writing z and w in the form x+ iy, show that D = −D. Deducethat D has real part 0.

10. Calculate

∣

∣

∣

∣

∣

∣

2 1 30 2 13 1 6

∣

∣

∣

∣

∣

∣

. Using this, write down the values of

(a)

∣

∣

∣

∣

∣

∣

2 1 33 1 60 2 1

∣

∣

∣

∣

∣

∣

;

(b)

∣

∣

∣

∣

∣

∣

2 0 31 2 13 1 6

∣

∣

∣

∣

∣

∣

;

(c)

∣

∣

∣

∣

∣

∣

2 3 33 3 60 6 1

∣

∣

∣

∣

∣

∣

.

11. What will happen with a 3× 3 determinant if

(a) all its entries are multiplied by −1?

(b) all the entries on the middle row are doubled?

(c) all the entries are doubled?

(d) the matrix is transposed?

12. Without doing any expansion, explain why the determinant

∣

∣

∣

∣

∣

∣

∣

∣

x y z 1y z x 1z x y 1

z+x2

x+y2

y+z2

1

∣

∣

∣

∣

∣

∣

∣

∣

van-

ishes.

13. By computing a determinant, show that the vectors

123

,

235

and

357

are

linearly independent.

MAS111 8 Spring 2016-17

14. Let A =

a b c d−b a d −c−c −d a b−d c −b a

. Compute AAT , and hence deduce that detA =

±(a2 + b2 + c2 + d2)2. (For a bonus, can you work out the sign?)

15. Let A =

1 2 32 3 57 11 13

. Compute the adjoint matrix adjA, and work out A.adjA.

What is detA?

16. Consider the system of equations

2x+ y − z = 0

x+ 2y + z = 3

3x− z = 3.

Solve this system using Cramer’s formula.

17. Consider matrices A =

(

a b−b a

)

and B =

(

c d−d c

)

.

(a) Write down their determinants.

(b) Show that AB also has the form

(

α β−β α

)

for some α and β.

(c) Deduce that if m and n can be written as the sum of two squares, so can mn.

(d) Finally, calculate the product of the two complex numbers a + ib and c + id.What do you notice?

18. Let z1 = a+ ib and w1 = c+ id be two complex numbers.

(a) Let

A =

(

z1 w1

−w1 z1

)

=

(

a+ ib c+ id−c+ id a− ib

)

.

Write down detA in terms of a, b, c and d.

(b) If B =

(

z2 w2

−w2 z2

)

, where z2 = e+if , w2 = g+ih, show that AB =

(

z w−w z

)

for certain complex numbers z and w.

(c) It is a fact that every prime number can be written as the sum of four squarenumbers. Deduce that all positive integers can be written as the sum of foursquare numbers.

19. William Hamilton (1805–1865) invented an extension of the complex numbers, calledthe quaternions. In this number system, there are three new square roots of −1,called i, j and k, and so a quaternion is a number of the form a+ bi+ cj+dk, wherea, b, c and d are real.

One new feature is that multiplication is not commutative – we have ij = k, but ji =−k. This may seem weird at first! Let’s make it appear more natural (hopefully),using the results of the last exercise.

MAS111 9 Spring 2016-17

(a) From the statement that ij = k and ji = −k and the facts that these are allsquare roots of −1, deduce that:

ijk = −1, jk = i, kj = −i, ki = j, ik = −j.

(b) Calculate (a+ bi+ cj + dk)(e+ fi+ gj + hk).

(c) Compare this with your calculation of the product AB in question 18. Whatdo you notice?

(d) Suggest a bijection between the quaternions and complex matrices of the form(

z w−w z

)

. What do i, j and k correspond to?

(e) Verify that the matrices corresponding to i, j and k, actually satisfy the samerelations as those in the first part of the question.

20. (a) Show that∣

∣

∣

∣

∣

∣

x y zz x yy z x

∣

∣

∣

∣

∣

∣

= x3 + y3 + z3 − 3xyz.

(b) Show that the product of two matrices of this form is another matrix of thesame form.

(c) Deduce that if a and b are two integers which can be expressed in the formx3 + y3 + z3 − 3xyz for integers x, y and z, then so can ab.

21. Recall that the Fibonacci numbers Fn are defined by setting F1 = 1, F2 = 2, anddefining Fn = Fn−1 + Fn−2 for n ≥ 3. Show by induction (expanding down the firstcolumn) that

Fn =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

1 1 0 0 . . . 0 0−1 1 1 0 . . . 0 00 −1 1 1 . . . 0 0...

......

.... . .

......

0 0 0 0 . . . −1 1

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

,

where the matrix is an n× n matrix.

22. Show that for any a, b, c, the determinant of the antisymmetric matrix

0 a b−a 0 c−b −c 0

vanishes.

Harder: Deduce the same result for the determinant of any antisymmetric matrix(one where AT = −A) of odd order vanishes.

23. [For students taking MAS114] Write down the 6 permutations in S3 in the form(

1 2 3i1 i2 i3

)

. For each one, form a 3× 3-matrix A = (ars), where ars = 1 for s = ir

and ars = 0 otherwise. Verify in each case that the determinant of A is equal to thesign of the permutation (i.e., is 1 if the permutation is even and −1 if it is odd).

Recall that S3 is generated by transpositions τ1 = (1 2) and τ2 = (1 3), which satisfythe relations τ 21 = τ 22 = (τ1τ2)

3 = id. If A1 corresponds to τ1 and A2 to τ2, showthat A2

1 = A22 = (A1A2)

3 = I.

MAS111 10 Spring 2016-17



Chapter 5

1. Find the eigenvalues and eigenvectors of the following matrices:

(a)

(

2 00 3

)

;

(b)

(

2 10 1

)

;

(c)

(

2 32 1

)

;

(d)

(

2 10 2

)

;

(e)

(

1 11 0

)

.

2. Find the eigenvalues and eigenvectors of the following matrices:

(a)

2 0 00 −1 00 0 2

;

(b)

1 0 02 3 01 0 −7

.

3. Find all eigenvalues and eigenvectors for

0 1 0−4 4 0−2 1 2

.

4. Find the eigenvalues and eigenvectors of

4 −5 71 −4 9−4 0 5

. Note that the eigenvalues

are not all real, and nor are all the eigenvectors.

5. Let A =

(

1 22 1

)

.

(a) What are its eigenvectors and eigenvalues?

(b) Find the eigenvalues of A2, A3 −A and A2 − 6A+ 2I.

(c) Take a general vector

(

ab

)

. Write it as αx1 + βx2, where x1 and x2 are the

eigenvectors for A you found earlier, and α and β are expressions involving aand b.

(d) Hence give a formula for An

(

ab

)

.

MAS111 11 Spring 2016-17

6. Suppose that if u is an eigenvector of A, so that Au = λu for some λ. Suppose alsothat AB = BA for another matrix B. Show that Bu is also an eigenvector of A.

7. Show that the eigenvalues of the matrix

A =

7 −9 6−3 13 −6−3 9 −2

are roots of λ3 − 18λ2 + 96λ − 160 = 0. Show that λ = 10 is a solution of thisequation, and hence find the other eigenvalues of A.

8. Let A = (x1, y1), B = (x2, y2) and C = (x3, y3) be points in the plane. Determinethe length of AB, and the distance of the point C from AB. Hence give a formulafor the area ∆ of the triangle ABC, and show that it is equal to the modulus of

1

2

∣

∣

∣

∣

∣

∣

1 x1 y11 x2 y21 x3 y3

∣

∣

∣

∣

∣

∣

.

(See Chapter 1, question 9.)

9. Suppose that A =

(

1 23 4

)

. Write e1 =

(

10

)

, e2 =

(

01

)

, and f1 =

(

11

)

, f2 =

(

1−1

)

.

(a) Write Ae1 and Ae2 as linear combinations of e1 and e2.

(b) Write Ae1 and Ae2 as linear combinations of f1 and f2.

(c) Write Af1 and Af2 as linear combinations of e1 and e2.

(d) Write Af1 and Af2 as linear combinations of f1 and f2.

(e) Hence give a matrix which represents A using the vectors f1 and f2.

(f) Write M =

(

1 11 −1

)

for the matrix with f1 and f2 as columns. Compute

M−1AM .

10. Let A =

(

2 33 2

)

. Find a matrix M such that M−1AM is diagonal.

11. Let A =

(

1 21 0

)

.

(a) Show that the eigenvalues are λ1 = 2, λ2 = −1 and work out the correspondingeigenvectors v1 and v2.

(b) Choose any vector, for example x =

(

11

)

. Work out Ax, A2x = A(Ax), A3x

and A4x, and plot them on a graph.

(c) Suppose that we can write x = αv1 + βv2. Show that

Ax = 2αv1 − βv2,

and give similar expressions for Anx for any n. Why does this vector pointmore and more closely in the direction of v1?

MAS111 12 Spring 2016-17

(d) Verify this on your graph.

12. (a) Show that (M−1AM)2 = M−1A2M for any invertible matrix M and matrix A.

(b) Find the eigenvalues and eigenvectors of A =

(

0 1−3 4

)

.

(c) Denote the eigenvectors by x1, x2, with eigenvalues λ1 and λ2. Consider the2 × 2 matrix M whose first column is x1 and whose second column is x2.Compute M−1AM .

(d) What is A10?

13. Let A =

3 0 0−2 2 02 −2 1

. Verify that A satisfies the Cayley-Hamilton Theorem.

14. Let A =

(

0 10 0

)

, and B =

(

0 01 0

)

. Using the series definition, compute eA and

eB. Convince yourself that eA+B 6= eAeB.

15. Suppose that D =

(

1 00 2

)

. Write down eD. Given that

(

0 −12 3

)

=

(

2 11 1

)−1(1 00 2

)(

2 11 1

)

,

compute eA where A =

(

0 −12 3

)

.

16. If A is a diagonal matrix, show that det(eA) = etrace(A).

17. Consider two functions x = x(t) and y = y(t) of a parameter t which satisfy thecoupled ordinary differential equations:

x = x+ 2y

y = 4x+ 3y.

At time t = 0, we have x(0) = 1 and y(0) = 2. Solve these equations.


(

1 11 0

)

.

(a) What is A

(

xy

)

?

(b) What are A

(

11

)

, A2

(

11

)

, A3

(

11

)

and A4

(

11

)

?

(c) Recall that the Fibonacci numbers are described by F1 = F2 = 1, and Fn+2 =

Fn + Fn+1 for n ≥ 0. Guess a formula for An

(

11

)

, and prove it by induction.

(d) What are the eigenvalues λ1 and λ2, and corresponding eigenvectors x1 and x2

of A?

MAS111 13 Spring 2016-17

(e) Find values α and β such that

(

11

)

= αx1 + βx2.

(f) What are Anx1 and Anx2?

(g) Deduce a formula for Fn.

(h) The Lucas numbers are defined similarly to the Fibonacci numbers, exceptthat L1 = 1 and L2 = 3. By repeating the above analysis, give a formula forLn, and show that Ln/Fn →

√5 as n → ∞.

Chapter 6

1. Sketch some level curves of each of the following functions:

(a) f(x, y) = x− y2;

(b) f(x, y) = x2 + 2y2;

(c) f(x, y) = xy.

2. Sketch some level curves of the function f(x, y) = (1/3)y2 + x. At the point (1, 1),according to your picture does f appear to be increasing more rapidly in the xdirection or in the y direction? Confirm your answer algebraically.

3. Let f(x, y) = x3y2 − 5exy. Calculate ∂f∂x

and ∂f∂y.

4. Let f(x, y) = x2y + ex cos y + sin x ln y. Write down ∂f∂x

and ∂f∂y.

5. I walk along the surface 4z = x2 + (1/3)y2 in such a way that x increases and ystays constant. As I pass through the point (1, 1, 1/3), how steep is my path?

6. The radius of a circular cylinder goes up by 0.3% while its height goes down by0.2%. Estimate the percentage change in volume.

7. Consider the surface z = x2 + y2, and imagine using the transformation into polarcoordinates x = r cos θ, y = r sin θ. Write z in terms of r and θ, and compute ∂z

∂r,

∂z∂θ, ∂z

∂x, ∂z

∂y, ∂x

∂r, ∂x

∂θ, ∂y

∂rand ∂y

∂θ. Show that

∂z

∂r=

∂z

∂x

∂x

∂r+

∂z

∂y

∂y

∂r,

and verify the same result with r replaced everywhere by θ.

8. Recall the change of variables between spherical and cylindrical polar coordinates(Q10 of Chapter 1). Compute the Jacobians

∂(r, φ)

∂(ρ, z)and

∂(ρ, z)

∂(r, φ),

and verify that they are inverse to each other.

MAS111 14 Spring 2016-17

9. Recall that if (x, y, z) are the coordinates of a point in Cartesian coordinates, thenits coordinates in spherical polars is (r, θ, φ), where

x = r cos θ sinφ

y = r sin θ sinφ

z = r cosφ.

Compute the Jacobian

∂(x, y, z)

∂(r, θ, φ)=

∣

∣

∣

∣

∣

∣

∣

∂x∂r

∂x∂θ

∂x∂φ

∂y∂r

∂y∂θ

∂y∂φ

∂z∂r

∂z∂θ

∂z∂φ

∣

∣

∣

∣

∣

∣

∣

.

10. Find an equation of the tangent plane to the surface z = x2+ y2 at the point wherex = 1 and y = 1.

11. Find the equation of the tangent plane to z = ln(2x+ y) at (−1, 3).

12. Find the stationary points of:

(a) f1(x, y) = x2 + y2;

(b) f2(x, y) = xy − x− y;

(c) f3(x, y) = x2 − y2;

(d) f4(x, y) = (x+ y)e−xy.

13. Let f(x, y) = x2y + xy2. Write down ∂2f∂x2 ,

∂2f∂x ∂y

and ∂2f∂y2

.

14. Let φ(x, y) = x3y + exy2

. Find

∂φ

∂x,∂φ

∂y,∂2φ

∂x2,

∂2φ

∂x∂y,∂2φ

∂y2.

15. Let f(x, y) = xyex+y. Write down the Taylor series of f as far as the degree 2 terms.

16. In this question, we will find the Taylor series of a function of two variables up todegree 3.

Let f(x, y) be given, and choose a point (a, b) about which we can find the Taylorseries. Recall that

f(a+ h, b) = f(a, b) +∂f

∂x(a, b)h+

∂2f

∂2x(a, b)

h2

2!+ · · ·

f(a, b+ k) = f(a, b) +∂f

∂y(a, b)k +

∂2f

∂2y(a, b)

k2

2!+ · · · .

(a) Give the next terms in these expansions.

(b) Write down the Taylor series for f(a+ h, b+ k) around (a, b+ k), regarded asa function of x, with y = b + k as constant, as far as the term in h3. (So theright-hand side of your expression should involve terms evaluated at (a, b+k).)

(c) Write down the Taylor series for fx(a, b + k) up to degree 2, the Taylor seriesfor fxx(a, b + k) to degree 1, and for fxxx(a, b + k) to degree 0, around (a, b)(where we are writing fx for ∂f

∂xfor example).

MAS111 15 Spring 2016-17

(d) Hence give the 2-variable Taylor series for f(a + h, b + k) around (a, b) up toterms of degree 3.

17. For each example in Q12, write down the Taylor series around the stationary pointas far as the degree 2 terms.

18. Show that f(x, y) = e−y cosx is harmonic, i.e., ∂2f∂x2 +

∂2f∂y2

= 0.

19. Show that if f has equal mixed second order partial derivatives and g(x, y) satisfies

∂g

∂x=

∂f

∂yand

∂g

∂y= −∂f

∂x,

then g is harmonic. Find a function g that is related in this way to the function fin Problem 18.

Chapter 7

1. Find the circle centre (−1, 2) with 4x− 3y = 15 as a tangent.

2. Find the equations of the circles with centre (0, 0) which touch the circle withequation x2 + y2 − 8x− 6y + 24 = 0.

3. Find the equation of each of the following parabolas:

(a) Directrix x = 0, focus (6, 0);

(b) Vertex (0, 4), focus (0, 2).

4. Show that

x = a1− t2

1 + t2, y = b

2t

1 + t2

is a parametric representation of the ellipse x2/a2 + y2/b2 = 1.

5. Consider the special case of Q4 with a = b = 1. Then this gives a way to parametriseall points on the circle. Convince yourself that if t ∈ Q, then so is (x, y). Try thefollowing values of t: t = 1

2, t = 2

3, t = 1

4, and find some well-known Pythagorean

triangles.

6. Find the equation of the tangent and normal to the hyperbola x2/4 − y2 = 1 at(5/2, 3/4).

7. Classify the following quadratic curves:

(a) 2x2 + 4x+ y2 − 2y + 2 = 0;

(b) 2x2 − 4x− y2 + 2y = 3.

8. Using the definitions of cosh x and sinh x, prove that 2 sinh x cosh x = sinh 2x, andthat cosh2 x + sinh2 x = cosh 2x. Hence express cosh2 x and sinh2 x in terms ofcosh 2x, and work out

∫

cosh2 x dx.

MAS111 16 Spring 2016-17

9. Consider the curve y = cosh x for 0 ≤ x ≤ a. Find the area between the curveand the x-axis (bounded by x = 0 and x = a), and the volume,

∫ a

0πy2 dx, of the

solid obtained by rotating this area through a full turn about the x-axis. Find also

the length∫ a

0

√

1 + ( dydx)2dx of the curve, and the area

∫ a

02πy

√

1 + ( dydx)2dx of the

surface obtained by rotating it through a full turn about the x-axis.

10. If y = cosh x, and z = ex, show that 2y = z + 1/z. Express z in terms of y, andhence show that

cosh−1 y = ln(y ±√

y2 − 1).

Show that (y +√

y2 − 1)(y −√

y2 − 1) = 1, and deduce that cosh−1 y = ± ln(y +√

y2 − 1).

Similarly, show that sinh−1 y = ln(y +√

y2 + 1). Explain, perhaps with referenceto the graph of sinh, why there is only one value.

11. Show that∫

1√1+x2

dx = sinh−1 x+C by making the substitution x = sinh u. Prove

also that∫

1√a2+x2

dx = sinh−1(x/a) + C.

12. Calculate∫ 1

01√

x2+x+1dx.

13. What are the domain and range of the inverse hyperbolic tangent function tanh−1?Prove that d

dxtanh−1 x = 1

1−x2 . Deduce that∫ x

01

1−t2dt = tanh−1 x. By evaluating

this integral using partial fractions, deduce that tanh−1 x = 12ln(

1+x1−x

)

. Now prove

this formula a different way, by putting y = tanh−1 x and solving for e2y.

Chapter 8

1. Work out the eigenvalues and eigenvectors of

1 0 10 1 21 2 5

. Verify that the eigenvec-

tors are orthogonal.

2. If A is any matrix, show that (A + AT )/2 is symmetric. Show also that A can bewritten as the sum of a symmetric matrix and an antisymmetric matrix.

3. Suppose that A and B are both orthogonal matrices. Show that AB is orthogonal.

4. For each of the following real symmetric matrices, find an orthogonal matrix Msuch that MTAM is diagonal.

(a) A =

(

1 22 1

)

;

(b) A =

(

4 22 1

)

.

5. Write x2 +2xy+ y2+4xz− 2yz+2z2 in the form uTAu for a symmetric matrix A,

where u =

xyz

.

MAS111 17 Spring 2016-17

6. Diagonalise the quadratic form 3x2 + 2xy + 3y2 by working out the eigenvectors ofthe matrix of the quadratic form.

7. For which values of λ is the quadratic form x2 + 2y2 + 2z2 + 2λxy + 2xz positivedefinite?

8. Classify the stationary points on the surfaces

(a) f(x, y) = x2 + y2;

(b) f(x, y) = xy − x− y;

(c) f(x, y) = x2 − y2;

(d) f(x, y) = (x+ y)e−xy.

(See Chapter 6, Q12.)

MAS111 18 Spring 2016-17



Chapter 9

1. Find the sum of the geometric series

∞∑

n=1

2

3n.

2. Find the sum of the series

1 +2

3+

4

9+

8

27+

16

81+ · · ·

3. Use the ratio test to show that the series

∞∑

n=1

2

3n

converges.

4. Use log(n) > 1 for n ≥ 2 and the comparison test to show that the series

∞∑

n=2

1

n2 log(n)

converges.

5. Show that the series∞∑

n=1

2n

n2

diverges.


n=1

1

n23n

converges.

7. In the notes, you saw that the infinite series

∞∑

n=1

1

n2

converges. Calculate the first seven partial sums (use a calculator). How big is thenext term to be added on to get the eighth partial sum? Does this mean that weknow

∑∞n=1

1n2 correct to 1 decimal place? Work out a few more partial sums and

reconsider your answer.

MAS111 19 Spring 2016-17

8. Why does the series∞∑

n=0

(

2

1 + i

)n

not converge?

9. Let sN be the sum of the first N terms of the series

1− 1

2+

1

3− 1

4+

1

5− 1

6+ · · · .

Calculate the first ten of these partial sums, to 4 decimal places. What do younotice about the sequence s1, s3, s5, . . ., and what do you notice about the sequences2, s4, s6, . . .? Can you explain this? What do you notice if you compare the ‘odd’partial sums with the ‘even’ partial sums? Do you think they converge to a limit?If so, how many terms of the infinite series would you have to add up to know thatyou had the sum correct to 4 decimal places? Are you sure?

10. A series∑

an is said to converge absolutely if∑

|an| converges. It is a fact that if aseries converges absolutely then it converges (you are invited to find a proof of thisresult).

(a) Prove that the infinite series∑∞

n=1cos(n)n2 converges.

(b) You have been told that if a series converges absolutely then the series con-verges. Use the comparison test on the alternating harmonic series

∞∑

n=1

(−1)n+1

n

to show that the series converges, and therefore that convergence of a seriesdoes not imply absolute convergence of the series.Hint: Look at the sum of two consecutive terms in the series; does this looklike a series that you can use the comparison test on?

11. Reread the arguments in your notes that show

∞∑

n=1

1

n2

converges and∞∑

n=1

1

n

diverges. Use similar methods to show that

∞∑

n=1

1

nk

converges if and only if k > 1.

12. Does the series∞∑

n=1

5n

n

converge?

MAS111 20 Spring 2016-17


n=1

n3

n!

converge?

14. You have been told that∞∑

n=1

1

n2=

π2

6.

Use this fact to find an exact value for∞∑

n=0

1

(2n+ 1)2.


n=2

1

n(log(n))2

converge?

16. Calculate the radius of convergence for the power series

∞∑

n=1

2n

n!xn.

17. Determine the interval on which the power series

∞∑

n=1

xn

n

converges.

18. Determine the interval on which the power series

∞∑

n=1

10n

n!xn

converges.

19. Determine the interval on which the series∞∑

n=1

(−1)n(x− 1)2n+1

n3n

converges.

Hint: For x fixed, we can use the ratio test.

20. Calculate the radius of convergence of the series

∞∑

n=1

(n!)2

(2n)!xn.

Can you determine whether the series converges on the endpoints of the interval ofconvergence?

MAS111 21 Spring 2016-17

21. Determine the interval on which the series

∞∑

n=1

x(x+ 1)n

2n

converges.

22. Use the first three terms of the Maclaurin series of sin x to a obtain an approximatevalue for

∫ 1

−1sin(x)

xdx. Repeat the process with the first four terms in the Maclaurin

series. Can you say anything about how accurate your approximation is?

23. Find the Maclaurin series for the function f(x) = xe−x2

.

24. Calculate the Maclaurin series for ln(1+x) and find its radius of convergence. Doesthe series converge on the endpoints of the interval of convergence?

Remark: You can do this by calculating the individual terms, or by integratingthe function f(x) = 1

1+x.

25. Find an exact value for the series

1

2!+

2

3!+

3

4!+

4

5!+ . . . .

Hint: You might find it helpful to consider ex

xas a series.


n=1

n

2n

converges and find its exact value.

Chapter 10

1. Calculate the derivative of the function

f(x) =

x∫

t=0

esin(t3) dt.


f(x) =

x∫

t=0

et2

1 + tdt

at the point x = 0.


f(x) =

x3

∫

t=0

1

1 + t3dt.

MAS111 22 Spring 2016-17

4. Find the derivative of the function

g(x) = x

x∫

t=1

sin(t3) dt

at the point x = 1.

5. Find the unique function f(x) that satisfies the equation

f(x) = 1 +

x∫

t=0

f(t) dt.

Hint: What happens if you differentiate the displayed equation? If your answer de-pends on an unknown constant then you can determine this constant by consideringwhat happens at any value of x = 0.

6. The following observations imply that −2 > 0:

• The function f(x) = x−2 is never negative, so when we integrate over aninterval we should always get a well defined positive number. i.e.,

b∫

a

f(x) dx > 0.

• We know thatd

dx

(

−x−1)

= x−2.

Therefore, the Fundamental Theorem of Calculus tells us that

1∫

−1

x−2dx =

[

−1

x

]x=1

x=−1

= −1− 1 = −2.

Setting a = −1 and b = 1 in the first statement, and the calculation in the secondstatement implies that −2 > 0. Both of the highlighted statements are incorrect;what is wrong in each case?

7. Use the substitution x = sin(u) to evaluate the integral

1∫

0

1√1− x2

dx.

Hint: Remember that cos2(u) = 1− sin2(u).

8. Use the substitution x = sinh(u) to evaluate the integral

1∫

0

1√1 + x2

dx

Hint: Remember that cosh2(u) = 1 + sinh2(u).

MAS111 23 Spring 2016-17

9. Use the substitution x = tan(u) to evaluate the integral

1∫

0

1√1 + x2

dx

and make sure that your answer agrees with Question 8.

Hint: Recall that sec2(u) = 1+tan2(u) and∫

sec(u)du = log | tan(u)+sec(u)|+C.

10. Use the substitution x = cosh(u) to evaluate the integral

2∫

1

1√x2 − 1

dx.

Hint: Remember that cosh2(u)− 1 = sinh2(u).

11. Evaluate the integral1

2∫

0

x2

√1− x2

dx.

12. Find the length of the curve y = x3

2 over the interval [0, 1].

13. Find the length of the graph of the curve f(x) = log(1−x2) over the interval [0, 12].

14. Find the length of the graph of the function f(x) = cosh(x) over the interval [0, 1].

15. Find the arc length of the curve y = x4 + 132x2 from x = 1

2to x = 1.

16. Find the arc length of the graph of the function

y = x2 − 1

8log(x)

over the interval [1, 2].

17. Find the length of the graph of the function

f(x) = log(cos(x))

over the interval [π6, π4].

Hint: You might find the hint from Question 9 useful at some point during yourcalculation.

18. Find the length of the graph of the function

f(x) = − log(cosec(x))

over the interval [π4, π3].

MAS111 24 Spring 2016-17

19. Show that the arc length of the graph of the function

f(x) =1

x

over the positive real axis is infinite.

Hint: It is not necessary to evaluate this integral; you only need to show that it isbigger or equal to something that is infinite.

20. Find the length of the curve y = x2 over the interval [0, 1].

21. Find the area of the surface obtained by rotating the graph of f(x) = x3 about thex-axis over the interval [0, 1].

22. Find the area of the surface obtained by rotating the graph of f(x) = 2√x about

the x-axis over the interval [3, 8].

23. Find the area of the surface obtained by rotating the graph of f(x) = cosh(x) aboutthe x-axis over the interval [0, 1].

24. Find the area of the surface obtained by rotating the graph of f(x) = cos(x) aboutthe x-axis over the interval [0, π

4].

25. Find the area of the surface obtained by rotating the graph of f(x) = x3

12+ 1

xabout

the x-axis over the interval [1, 2].

26. Find the area of the surface obtained by rotating the graph of

f(x) = ex

about the x-axis over the interval [0, 1].

27. Show that the area of the surface obtained by rotating the graph of

f(x) =1

x

about the interval [1,∞) is infinite.

Hint: It is not necessary to evaluate this integral; you only need to bound theintegral below by something that is not finite.

28. Find the surface area of the doughnut obtained by rotating the circle centred at(0, 2) of radius 1 about the x-axis.

29. Show that the surface obtained by rotating the curve determined by the equationy = e−x over the interval [0,∞) is π(

√2 + log(1 +

√2)).

Hint: Show that arcsinh(1) = log(1 +√2).

30. Find the volume of the solid obtained by rotating the region bounded below thegraph of f(x) = x2 about the x-axis above the interval [0, 4].

31. Find the volume of the solid obtained by rotating the region bounded below thegraph of f(x) = ex about the x-axis above the interval [0, 1].

MAS111 25 Spring 2016-17

32. Find the volume of the solid obtained by rotating the region bounded below thegraph of f(x) = cosh(x) about the x-axis above the interval [−1, 1].

33. By rotating the graph of a certain function of the form f(x) = mx about the x-axisover the interval [0, h] find the volume of a cone over a disc of radius r and heighth.

Hint: What value of m makes the base disc have radius r?

34. Find the volume of the solid obtained by rotating the region bounded below thegraph of

f(x) =1

x

about the interval [1,∞).

35. Find the volume of the solid obtained by rotating the region bounded between theloci of the equations y = x and y = x2 about the x-axis.

36. Find the volume of the solid obtained by rotating the region bounded between theloci of the equations y = x2 and y = 1 about the x-axis.

37. Find the volume of the doughnut bounded by rotating the circle centred at (0, 2) ofradius 1 about the x-axis.

38. The surface obtained by rotating the graph of

f(x) =1

x

is called “Gabriel’s horn”. From Question 34, Gabriel’s horn bounds a region withvolume equal to π, but has infinite surface area (see Question 27). Is the followingstatement a paradox: Gabriel’s horn can be filled with a finite amount of paint, butcannot be painted using a finite amount of paint.

Chapter 11

1. Evaluatey=1∫

y=0

x=1∫

x=0

xy dx dy.

2. Sketch the region D bounded by the lines x = 0, x = 1, y = x, y = 1 + x.

3. Let A be the region bounded by y = x2, y = x, y = 1 depicted below.

x

y

−1 1

1

A

MAS111 26 Spring 2016-17

(a) Write∫∫

Af(x, y) dA as an iterated integral integrating first with respect to x.

(b) Let A be the region in the plane described above. Write∫∫

Af(x, y) dA as an

iterated integral integrating first with respect to y.

4. For each f(x, y) and D described in (a)-(c), express∫∫

Df(x, y)dA as an iterated

integral two ways; integrating first w.r.t x and then w.r.t y, AND integrating firstw.r.t. y and then w.r.t. x.

(a) f(x, y) = x2 − y2, and D = {(x, y) : 1 ≤ x ≤ 3,−2 ≤ y ≤ 2}(b) f(x, y) = x+ y, and D = triangle with vertices (0, 1), (1,−1), (−1,−1).

(c) f(x, y) = xy, and D = {(x, y) : x2 + y2 ≤ 1, x ≥ 0, y ≥ 0}.

5. Evaluate each of the six iterated integrals from Question 4.

6. Find the area bounded by the line y = 1, and the curves described by y = x1/2, andx = − sin(πy).

7. Let S be the square {(x, y) : 1 ≤ x ≤ 2, 0 ≤ y ≤ 1}.

(a) Evaluate∫∫

S

(x2 − 3y2) dx dy.

(b) Evaluate∫∫

S

1

(x+ y2)2dx dy.

8. Let D be the region of the plane {(x, y) : x ≥ 0, y ≥ 0, x + y ≤ 1}. Sketch theregion D in the plane. Evaluate the following integral either first with respect to xand then with respect to y, or else the other way round:

∫∫

D

(x2 − 3y2) dx dy.

(Check with someone who did the integration the other way that the answers arethe same.)

9. Let R be the triangular region of the (x, y)-plane with the vertices (0, 1), (1, 2) and(2, 1). Evaluate

∫∫

R

2x+ y + 2

x+ y + 1dx dy.

10. The integral1

∫

0

√y

∫

0

e(3x−x3) dx dy

at first sight is impossible. Express it as a double integral over a region D. SketchD. Evaluate the double integral by calculating the iterated integral in which they-integral is done first.

MAS111 27 Spring 2016-17

11. Let a > 0. By changing the order of integration in the repeated integral prove that

a∫

0

a∫

x

x√

(x2 + y2)dy

dx =1

2(√2− 1)a2.

12. By changing the order of integration in the repeated integral, evaluate

1∫

0

1∫

y2

y9√

(1− x6) dx dy.

13. Let D be the region in the first quadrant bounded by the curve x2 − y2 = 1/4 andthe lines x = 1, y = 0. Evaluate

∫∫

D

y√

(4x3 − 3x+ 1)dx dy.

14. Evaluate∫∫

Dex

2

dA where D is the triangle with vertices (0, 0), (1, 0), and (1, 1).

15. Compute the following double integrals:

(a)∫∫

Dex

2+y2dA where D is the region in the upper right quadrant of the xy-planebounded by x2 + y2 = 1, x2 + y2 = 4, the y-axis and the line y = x.

Hint: Switch to polar coordinates.

(b)∫∫

Dsin(x

2

9+ y2

4)dA where D = {(x, y) : x2

9+ y2

4≤ π

2}.

Hint: First consider the change of variables x(u, v) = 3u, y(u, v) = 2v, andthen make another sensible change of variables. . . .

16. Let D be the region in the (x, y)-plane for which x2 + y2 < 1, y ≥ 0. By changingto polar coordinates show that

∫∫

D

(x+ y)2

1 + (x2 + y2)2dx dy =

π

4ln 2.

17. The formula for the area of a surface z = f(x, y) over a region D is

∫∫

D

√

1 +(∂z

∂x

)2

+(∂z

∂y

)2

dx dy.

By considering that part of the sphere x2 + y2 + z2 = R2 which lies above the setD = {(x, y) : x ≥ 0, y ≥ 0, x2 + y2 ≤ R2} use the above formula to show that thesurface area of a sphere of radius R is

8

∫∫

D

R√

R2 − x2 − y2dx dy.

By changing to polar coordinates confirm that this surface area is 4πR2.

MAS111 28 Spring 2016-17

18. Let D be the region {(x, y) : x ≥ 0, 0 ≤ y ≤ ex and 1 ≤ x2 + y2 ≤ 2}. Letu = x2 + y2, v = ye−x. Evaluate the Jacobian ∂(u, v)/∂(x, y) and use it to find thedouble integral

∫∫

D

(x+ y2)e−x

x2 + y2 + 1dx dy.

19. Let D be the region in the first quadrant of the (x, y)-plane bounded by the curves

y = x2, 2y = x2, x = y2, 2x = y2.

Sketch D. By using the substitution u = x2/y, v = y2/x prove that

∫∫

D

ln(xy) dx dy =2

3(2 ln 2− 1).

20. The triangle D is bounded by the lines y = 1/2, x = y, 3y = x + 2. Sketch D. Byusing the substitution u = x− y, v = x+ y, evaluate

∫∫

D

(x− y)(2x− 2y + 1)ex2−y2 dx dy.

21. Let D be the region in the first quadrant bounded by the curves xy = 1, xy = 2,x2 + y2 = 6, x2 + y2 = 8 for which x > y. Evaluate

∫∫

D

x2 − y2

(x2 + xy + y2)2dx dy

by making the substitution u = xy, v = x2 + y2.

22. Find the area of the region bounded by the curve described by 9x2 + 4y2 = 1.

Hint: start with the change of variables x = u3and y = v

2.

MAS111 29 Spring 2016-17

chapter1 · mas111: mathematics coreii problems booklet – chapters 1–4 chapter1 1. find the...

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