By Liz Driscoll

This valuable little ebook highlights the true blunders that scholars make within the puppy examination and express tips on how to keep away from them. in line with research of hundreds of thousands of examination scripts, each one unit goals a key challenge zone. transparent reasons and routines support scholars to exploit the language competently. general exams supply scholars an additional chance to examine and consolidate what they've got learnt. point: Pre-intermediate.

Key Features

• in line with genuine errors pointed out in research of over 10,000 examination exhibiting scholars tips on how to steer clear of the most typical mistakes.

• brief, snappy reasons of regular mistakes shop time by means of permitting scholars to target the genuine key challenge areas.

• comprises exam-style workouts to familiarizes scholars with the fashion of workouts they're going to locate within the exam.

• contains commonplace checks so scholars can consolidate studying and construct self belief.

**Read or Download Common Mistakes at PET ... and how to avoid them PDF**

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**Extra info for Common Mistakes at PET ... and how to avoid them**

**Sample text**

1 Trigonometric identities sin x = O H cos x = A H tan x = O A sin x tan x = cos x csc x = H O sec x = H A cot x = A O x tan x = cos sin x H is the Hypotenuse, always being opposite the right angle. Relative to angle x, O is the Opposite and A is the Adjacent. 39 40 CHAPTER 5. 3. 4 cos 2x = (cos2 x) - (sin2 x) cos2 x = 1 + cos 2x 2 2 Hyperbolic functions sinh x = ex - e-x 2 cosh x = ex + e-x 2 tanh x = sinh x cosh x Note: all angles (x) must be expressed in units of radians for these hyperbolic functions.

A capital letter ”A” represents the sum of an arithmetic sequence. For instance, in the same even-counting sequence starting at 2, A4 is equal to the sum of all elements from a1 through a4 , which of course would be 2 + 4 + 6 + 8, or 20. an = an-1 + d an = a1 + d(n-1) Where: d = The "common difference" Example of an arithmetic sequence: -7, -3, 1, 5, 9, 13, 17, 21, 25 . . An = a1 + a2 + . . an An = n (a1 + an) 2 28 CHAPTER 4. 2 Geometric sequences A geometric sequence, on the other hand, is a series of numbers obtained by multiplying (or dividing) by the same value with each step.

2! + 3! 2 1 n! Pi Pi (π) is defined as the ratio of a circle’s circumference to its diameter. 14159 26535 89793 23846 26433 83279 50288 41971 69399 37511 Note: For both Euler’s constant (e) and pi (π), the spaces shown between each set of five digits have no mathematical significance. They are placed there just to make it easier for your eyes to ”piece” the number into five-digit groups when manually copying. 1 Definition of a logarithm If: by = x Then: logb x = y Where, b = "Base" of the logarithm ”log” denotes a common logarithm (base = 10), while ”ln” denotes a natural logarithm (base = e).