# 4754A A A * * MATHEMATICS (MEI) Applications of Advanced Mathematics (C4) Paper A ADVANCED GCE. Friday 15 January 2010 Afternoon PMT

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2 Section A (36 marks) 1 + x 1 Find the first three terms in the binomial expansion of in ascending powers of x. State the (1 x) set of values of x for which the expansion is valid. [7] Show that cot θ = 1 tan θ tan θ. Hence solve the equation cot θ = 1 + tan θ for 0 < θ < 360. [7] 3 A curve has parametric equations x = e t, y = t 1 + t. (i) Find the gradient of the curve at the point where t = 0. [6] (ii) Find y in terms of x. [] 4 The points A, B and C have coordinates (1, 3, ), ( 1,, 3) and (0, 8, 1) respectively. (i) Find the vectors AB and AC. [] (ii) Show that the vector i j 3k is perpendicular to the plane ABC. Hence find the equation of the plane ABC. [5] (i) Verify that the lines r = ( 3 ) + λ( 0) and r = ( 4) + µ( 4 1 ) meet at the point (1, 3, ). (ii) Find the acute angle between the lines. [4] 1 0 [3] OCR A Jan10

3 3 Section B (36 marks) 6 In Fig. 6, OAB is a thin bent rod, with OA = a metres, AB = b metres and angle OAB = 10. The bent rod lies in a vertical plane. OA makes an angle θ above the horizontal. The vertical height BD of B above O is h metres. The horizontal through A meets BD at C and the vertical through A meets OD at E. a A 10 b B C h O E D Fig. 6 (i) Find angle BAC in terms of θ. Hence show that h = a sin θ + b sin(θ 60 ). [3] 3 (ii) Hence show that h = (a + 1 b) sin θ b cos θ. [3] The rod now rotates about O, so that θ varies. You may assume that the formulae for h in parts (i) and (ii) remain valid. (iii) Show that OB is horizontal when tan θ = In the case when a = 1 and b =, h = sin θ 3 cos θ. 3b a + b. [3] (iv) Express sin θ 3 cos θ in the form R sin(θ α). Hence, for this case, write down the maximum value of h and the corresponding value of θ. [7] [Question 7 is printed overleaf.] OCR A Jan10 Turn over

6 1 A Caesar cipher uses a shift of 11 places. Using lines 36 and 37, write down the shift for the decoding cipher in this case. [1] For Examiner s Use Using lines 143 to 146, write out the first 16 letters of the plaintext message corresponding to the ciphertext message on lines 97 to 101. [1] 3 Table 4 shows an encoding cipher. Complete the table below to show part of the decoding cipher. [] Ciphertext A B C D E Plaintext 4 Line 137 says in string S, the encoded form of the letteraisn. Give two reasons why this is a sensible suggestion. [] 5 Lines 105 and 106 say Taken together, these two shifts suggest that the keyword has length or 4. Explain why this is the case. [] OCR B Jan10

7 6 Lines 107 and 108 say a keyword of length would form a less secure cipher than one of length 4. Explain why this is true. [] 3 For Examiner s Use 7 A long passage is encoded using the Vigenère cipher with keyword ODE. Write down the different ways in which the plaintext wordand could appear in the ciphertext. [] 8 A passage of plaintext is encoded by using the Caesar cipher corresponding to a shift of places followed by the Vigenère cipher with keyword ODE. (i) The first letter in the plaintext passage is F. Show that the first letter in the transmitted text isv. [1] (ii) The first four letters in the transmitted text arevfiu. What are the first four letters in the plaintext passage? [3] (iii) The 800th letter in the transmitted text isw. What is the 800th letter in the plaintext passage? [] OCR B Jan10

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