2007 TPJC Prelim Paper 2

2007 TPJC Prelim Paper 2 - 9740 H2 Mathematics Preliminary...

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1 TAMPINES JUNIOR COLLEGE Preliminary Examination 2007 Mathematics Higher 2 9740/02 18 September 2007 8:00 – 11:00 hr Answer All questions Section A: Pure Mathematics [40 marks] 1. Given that two quantities x and y both vary with time t . Express the following statements as differential equations involving x , y and t . i) The rate of change of y with respect to t is inversely proportional to x . [1] ii) The rate of change of x with respect to t is proportional to y . [1] Hence form a differential equation involving x and y only, and prove that the solution is 2 = ln Cx n y where C and n are constants. [3] 2. If 2 1 ) tan 1 ( x y , prove that 2 ) 1 ( 2 ) 1 ( 2 2 2 2 2 dx dy x x dx y d x . Obtain an equation relating dx dy dx y d dx y d and , 2 2 3 3 . Find the Maclaurin‟s series for y , up to and including the term in x 3 . [7] 3. a) Express 13 i in the form of   cos sin ri  . Use De Moivre‟s Theorem or otherwise, show that     88 8 1 3 1 3 2 ii   Hence, find the roots of     4 1 3 1 3 0 z i i i    in the form of i re . [7] b) Indicate on an Argand diagram, with respect to the origin O , the point Q representing the complex number w where 0 arg 4 w  . Also, sketch the locus of P if z w z w on the same diagram. If zw , find possible values of   arg in terms of   arg w . [6]

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2 4. As part of a sculpture, an artist erects a flat triangular sheet ABC in his garden. The vertices are attached to vertical poles DA, EB and FC. The coordinate axes Ox and Oy are horizontal, and Oz is vertical. The coordinates of triangle are A(2, 0, 2), B(–2, 0, 1) and C(0, 4, 3), with units in metres.
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This note was uploaded on 08/29/2009 for the course MA 9740 taught by Professor Moe during the Summer '07 term at Singapore Management.

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2007 TPJC Prelim Paper 2 - 9740 H2 Mathematics Preliminary...

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