2007 TJC H2 MA 0740 PRELIM P2 Qns

# 2007 TJC H2 MA 0740 PRELIM P2 Qns - 2 Section A Pure...

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© TJC/MA9740/P2/PRELIM2007 2 Section A: Pure Mathematics [40 marks] 1 In a chemical reaction, one molecule of P combines with one molecule of Q to form one molecule of R . At the start of the reaction there are n molecules each of P and Q , and no molecule of R . After t hours, x molecules of R have been formed. Write down the number of molecules of P that remains after t hours. [1] The above chemical reaction can be modelled by a differential equation   2 d d x k n x t  where k is a positive constant. State the assumption needed to formulate this differential equation. [1] Solve the differential equation, expressing x in terms of n , k and t . [4] 2 (i) Write down, in the form e i r , the roots of the equation w 7 = 1. [2] (ii) Prove that e1 tan 2 i i i . [2] (iii) Find the roots of the equation   7 7 (1 ) 1 iz iz , giving your answers in trigonometric form. [5] 3 A curve has parametric equations 2 1 1 , , 0 1 x t y t t t   . Find the equations of the tangent and normal to the curve at the point 2 1 1, t t    . [4] The tangent to the curve at point 3 ,2 2 P cuts the y -axis at T while the normal to the curve at the same point cuts the x -axis at N . Find the exact coordinates of T and N and deduce the area of triangle PTN. [7] [Turn over

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© TJC/MA9740/P2/PRELIM2007 3 4 A plane 1 has equations r .   2 3 2 ij . The points A and B have position vectors 4 i j + p k and 2 i + 5 j + q k respectively where p , q . (i) Find, in terms of p , the position vector of the foot of perpendicular X , from A to 1 . [3] Deduce the exact value of p if the acute angle between line OX and the z -axis is 45 o , where O is the origin. [3] (ii) A plane 2 , which is parallel to vector AB  , has equation 10 x y z     . Using the value of p found in part (i) , find the exact value of q . [2] (iii) A plane 3 is parallel to 1 and passes through the origin. Find the perpendicular distance between 1 and 3 . [2] (iv) Another plane 4 has equation r = ( i + 3 j + 2 k ) + ( 2 i +3 j ) where , .
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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 TJC H2 MA 0740 PRELIM P2 Qns - 2 Section A Pure...

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