National Quali cations

Transcription

1 National Quali cations AH08 X747/77/ Mathematics THURSDAY, MAY 9:00 AM :00 NOON Total marks 00 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 answer booklet provided. In the answer booklet you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the eamination room you must give your answer booklet to the Invigilator; if you do not, you may lose all the marks for this paper. *X74777* A/PB

2 FORMULAE LIST Standard derivatives Standard integrals f ( ) f ( ) f ( ) f ( ) d sin cos ( ) sec a tan( a) + c a a sin + c a tan + a + tan + c a a tan sec ln + c cot cosec a e a e a + c sec sec tan cosec ln cosec cot e e Summations Sn n a n d (Arithmetic series) = + ( ) (Geometric series) S n n ( r ), a = r r nn ( + ) nn ( + )( n+ ) n( n+ ) r =, r =, r = 6 4 n n n r= r= r= Binomial theorem n n n n r r a+ b = a b r= 0 r where ( ) n n n! = C = r r r!( n r)! Maclaurin epansion iv 4 f ( 0) f ( 0) f ( 0) f( ) = f( 0) + f ( 0) !! 4! page 0

3 FORMULAE LIST (continued) De Moivre s theorem Vector product [ (cos sin )] n n r θ + i θ = r ( cos nθ + isin nθ ) i j k a a a a a a a b= absin θ nˆ = a a a = i j + k b b b b b b b b b Matri transformation Anti-clockwise rotation through an angle, θ, about the origin, cosθ sin θ sin θ cosθ [Turn over page 0

4 Total marks 00 Attempt ALL questions MARKS. (a) Given f( ) = sin, find f ( ). 5 e (b) Differentiate y = 7 +. (c) For ycos + y = 6, use implicit differentiation to find dy d. 4. Use partial fractions to find 7 d (a) Write down and simplify the general term in the binomial epansion of (b) Hence, or otherwise, find the term independent of. 4. Given that z = + i and z = p 6 i, p, find: (a) zz ; (b) the value of p such that z z is a real number. 5. Use the Euclidean algorithm to find integers a and b such that 06a+ 9b= 7. 4 page 04

5 6. On a suitable domain, a curve is defined parametrically by = t + Find the equation of the tangent to the curve where t =. and y = ln ( t+ ). MARKS 5 7. Matrices C and D are given by: C = 0 0 and D= k+ 0, where k. (a) Obtain C D where C is the transpose of C. (b) (i) Find and simplify an epression for the determinant of D. (ii) State the value of k such that D does not eist. 8. Using the substitution u = sinθ, or otherwise, evaluate π 4 sin θcosθdθ. π Prove directly that: (a) the sum of any three consecutive integers is divisible by ; (b) any odd integer can be epressed as the sum of two consecutive integers. [Turn over page 05

6 0. Given z = + iy, sketch the locus in the comple plane given by z = z + i. MARKS. (a) Obtain the matri, A, associated with an anticlockwise rotation of π radians about the origin. (b) Find the matri, B, associated with a reflection in the -ais. (c) Hence obtain the matri, P, associated with an anticlockwise rotation of π radians about the origin followed by reflection in the -ais, epressing your answer using eact values. (d) Eplain why matri P is not associated with rotation about the origin.. Prove by induction that, for all positive integers n, n = ( ). n r r= 5 page 06

7 . An engineer has designed a lifting device. The handle turns a screw which shortens the horizontal length and increases the vertical height. MARKS The device is modelled by a rhombus, with each side 5 cm. The horizontal length is cm, and the vertical height is h cm as shown. 5 cm h (a) Show that h = 500. (b) The horizontal length decreases at a rate of 0 cm per second as the handle is turned. Find the rate of change of the vertical height when = 0. 5 [Turn over page 07

8 MARKS 4. A geometric sequence has first term 80 and common ratio. (a) For this sequence, calculate: (i) the 7 th term; (ii) the sum to infinity of the associated geometric series. The first term of this geometric sequence is equal to the first term of an arithmetic sequence. The sum of the first five terms of this arithmetic sequence is 40. (b) (i) Find the common difference of this sequence. (ii) Write down and simplify an epression for the nth term. Let S n represent the sum of the first n terms of this arithmetic sequence. (c) Find the values of n for which S n = (a) Use integration by parts to find sin d. (b) Hence find the particular solution of dy y = sin, 0 d given that =π when y = 0. Epress your answer in the form y = f( ). 7 page 08

9 6. Planes π, π and π have equations: MARKS π : y+ z = 4 π : 5y z = π : 7 + y + az = where a. (a) Use Gaussian elimination to find the value of a such that the intersection of the planes π, π and π is a line. (b) Find the equation of the line of intersection of the planes when a takes this value. 4 The plane π 4 has equation 9+ 5y+ 6z = 0. (c) Find the acute angle between π and π 4. (d) Describe the geometrical relationship between π and π 4. Justify your answer. 7. (a) Given f ( ) = e, obtain the Maclaurin epansion for f ( ) the term in. (b) On a suitable domain, let g( ) = tan. (i) Show that the third derivative of g( ) is given by 4 g ( ) = sec + tan sec 4. up to, and including, (ii) Hence obtain the Maclaurin epansion for g( ) up to and including the term in. (c) Hence, or otherwise, obtain the Maclaurin epansion for including, the term in. e tan up to, and (d) Write down the first three non-zero terms in the Maclaurin epansion for e tan + e sec. [END OF QUESTION PAPER] page 09

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