MARKS Page six 14. The circle with equation x2 + y2 − 12x − 10y + k = 0 meets the coordinate axes at exactly three points. What is the value of k? 15. The rate of change of the temperature, T ºC of a mug of coffee is given by 1 25 = – dT t k dt ,  0 ≤ t ≤ 50 • t is the elapsed time, in minutes, after the coffee is poured into the mug • k is a constant • initially, the temperature of the coffee is 100 ºC • 10 minutes later the temperature has fallen to 82 ºC. Express T in terms of t. [END OF QUESTION PAPER] 2 6 [BLANK PAGE] DO NOT WRITE ON THIS PAGE Page seven [BLANK PAGE] DO NOT WRITE ON THIS PAGE Page eight H © National Qualications 2015 Total marks — 70 Attempt ALL questions. You may use a calculator Full credit will be given only to solutions which contain appropriate working. State the units for your answer where appropriate. Answers obtained by readings from scale drawings will not receive any credit. Write your answers clearly in the spaces in the answer booklet provided. Additional space for answers is provided at the end of the answer booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the examination room you must give your answer booklet to the Invigilator; if you do not, you may lose all the marks for this paper. X747/76/12 Mathematics Paper 2 WEDNESDAY, 20 MAY 10:30 AM – 12:00 NOON HTP Page two FORMULAE LIST Circle: The equation x2 + y2 + 2gx + 2fy + c = 0 represents a circle centre (−g, −f ) and radius The equation (x − a)2 + (y − b)2 = r2 represents a circle centre (a, b) and radius r. Scalar Product: a.b = |a||b| cos θ, where θ is the angle between a and b or a.b = a1b1 + a2b2 + a3b3 where a = 1 1 2 2 3 3 = and a b a b a b b ⎛ ⎞ ⎛ ⎞ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ⎝ ⎠ ⎝ ⎠ . Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B ± sin A sin B sin 2A = 2 sin A cos A cos 2A = cos2 A − sin2 A = 2 cos2 A − 1 = 1 − 2 sin2 A Table of standard derivatives: Table of standard integrals: f (x) f ′(x) sin ax cos ax a cos ax –a sin ax f (x) ∫ f (x)dx sin ax cos ax cos ax + c sin ax + c 1 – a 1 a 2 2 + – . g f c