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Durham University Mathematical Sciences Department Exam: MATH1561 Solutions, Study notes of Mathematics

The London School of Economics and Political Science (LSE)Mathematics

Solutions to various mathematical problems covered in the MATH1561 exam held at Durham University during the academic year 2018-2019. Topics include trigonometric functions, complex numbers, differentiation, and integration.

Typology: Study notes

2021/2022

Uploaded on 09/12/2022

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DURHAM UNIVERSITY — Department of Mathematical Sciences
COLLECTION 2018-2019
SINGLE MATHEMATICS A
MATH1561
Name: College:
Time allowed: 45 minutes. Answer all questions. Use of electronic calculators is forbidden.
1. (a) Write out the addition formula for cosh(A+B).
(b) Write out the addition formula for sinh(A+B).
(c) Determine the (possibly zero) constants b,cand din
cosh(3A) = bcosh(A) + ccosh2(A) + dcosh3(A)
continued . . .
pf3
pf4
pf5
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Download Durham University Mathematical Sciences Department Exam: MATH1561 Solutions and more Study notes Mathematics in PDF only on Docsity!

DURHAM UNIVERSITY — Department of Mathematical Sciences COLLECTION 2018-

SINGLE MATHEMATICS A MATH

Name: College:

Time allowed: 45 minutes. Answer all questions. Use of electronic calculators is forbidden.

  1. (a) Write out the addition formula for cosh(A + B). (b) Write out the addition formula for sinh(A + B). (c) Determine the (possibly zero) constants b, c and d in cosh(3A) = b cosh(A) + c cosh^2 (A) + d cosh^3 (A)
  1. (a) Differentiate xcos(x). (b) Using z = x + iy, determine the constants a, b and c in | sin(z)|^2 = a sin^2 (x) + b sinh^2 (y) + c
  1. Compute the indefinite integrals I 1 =

∫ (^2) x (^2) − 5 x + 9 (x^2 − 9)(x − 4) dx^ and^ I^2 =

∫ (^) x − 7 x^2 + 9 dx.

  1. Compute the definite integrals I 3 =

1

tanh(x) dx^ and^ I^4 =

∫ (^) π/ 16 −π/ 6

16 sin^2 (x) cos^2 (x) dx.

  1. (a) Compute ∑^20 j=

(j + 2)

(b) Find all solutions to z^4 − 2

3 z^2 + 4 = 0. You can give your answers in polar form. State clearly the number of distinct solutions.

  1. (a) Compute

xlim→ 0 sin(1/x) sinh(x) (b) State the definition of the derivative of a function f (x) as a limit. (c) Use this definition to show that d dx (x^ ln(x)^ −^ x) = ln(x) Note: to compute the limit, you may use wihtout further proof that

ylim→ 0 ln(1 + y y)= 1^. but you are not allowed to use L’Hˆopital’s rule.