PJC P1

3 1 Without the use of the graphic calculator, solve the inequality 2 2 4 4 1 0 1 xx xx    . Hence solve the inequality 2 2 44 0 1 xx xx    . [5] 2 Given that points A, B, and C have position vectors a , b and c respectively, relative to the origin O, and the point D is such that ABCD is a parallelogram. (i) Write down an expression for the position vector of D, in terms of a , b and c . [1] (ii) Prove that the area of the parallelogram ABCD is given by     a b b c c a . [3] (iii) Show that shortest distance of D to AC is given by       a b b c c a ca . [1] 3 By using the substitution 1e xt  , show that   1 11 01 e11e tan e d tan dxx x t t  . Hence find the exact value of   1 1 0 11e tan e dxx x  . [6] 4 Prove by the method of mathematical induction that 11sin sin 22cos cos 2 ... cos 12sin 2 n n                    for all positive integers n. [6] [Turn over Name : Class :

4 5 The diagram shows the cross sectional view of a cylinder open on both ends inscribed in a hemisphere with fixed radius n cm. If the diameter of the cylinder is cmx , show that the surface area A of the cylinder is 222 x n x  cm2. [2] Given that as x varies, the maximum value of A occurs when the ratio of the diameter of the cylinder to the height of the cylinder is 1 k . Find the value of .k [5] 6 A curve C is defined by the parametric equations cosx t t , 3 sin 2yt , for 0t   . Find (i) the exact values of the turning point(s) to the curve, [3] (ii) the equation of the tangent where 2t  , and [2] (iii) the area of the region bounded by C, the tangent in (ii), and the line 2 42x   . [2] 7 Use the method of difference to find  1 2 2 N r rr  in terms of N. [4] Hence express    3 6 2 42 N r rr    in the form of  faN , where a is a constant and  f N is an expression in terms of N, are to be determined, and deduce the value of   6 2 42r rr    . [4] n x

5 8 A piece of meat is taken from an oven when it reaches a temperature of 20 C and put in a refrigerator where the temperature is set to a constant value of 20 C . The rate of cooling in t minutes is proportional to the difference between the body temperature FC and its immediate surrounding temperature. It was observed that the meat cools to 10 C in 10 minutes. (i) By forming and solving a differential equation, show that 10140 204 t F   . [6] (ii) How many more minutes are required for the meat to cool further to 15 C ? [2] 9 A hamster farm has 500 hamsters to sell. The farmer sells k hamsters at the end of every week , where k is a constant factor of 500 (e.g. 5, 100, etc.). The selling price of a hamster is $10 in the first week and it drops by 5% in each subsequent week. It is assumed that there is neither birth nor death of hamsters in the farm. (i) State the total number of weeks for the farmer to sell all his hamsters in terms of k . [1] (ii) Show that the total proceeds from selling all the hamsters is 500 200 1 0.95 kk   . [3] (iii) Given that the cost of rearing a hamste r is $0.50 a week , find the total cost incurred in rearing the hamsters, when the farmer has sold all his hamsters. [3] (iv) If all other costs are negligible, find the least value of k for the farmer to make a profit. [2] [Turn over

6 10 (a) Solve the simultaneous equations 2 i 2pq , 2 8 2i 0pq    , for the complex numbers and pq in Cartesian form. [3] (b) Express 1i 3i   in both Cartesian and trigonometric forms. Hence show that  6211cos 12 4    . [5] Deduce the exact value of 11tan 12   . [1] 11 The curve C has equation 2 2 2 2 2 0c x b y a   , where a, b and c are constants such that 0abc   . (i) Show that C has no turning points. [3] (ii) Obtain the equations of the asymptotes of C. [2] (iii) State the restriction of x and the axes of symmetry of C. [2] (iv) Sketch C, stating the equations of the asymptotes and the coordinates of the points of intersection with the x-axis. [3] (v) It is given that the equation    22c bk c bk x a   has 2 real roots. By using your sketch in (iv), show that 11 k   . [3]

7 12 The Cartesian equation of the plane 1p is given by 5xy . The position vectors of the points A and B are given by i + 2j- 3k and 4i - j+ 2k respectively. (i) Find the acute angle between the position vector of A and the plane 1p . [2] (ii) Find the position vector of the foot of the perpendicular from A to plane 1p . [3] (iii) Show that B lies in 1p and hence find the vector equation of the reflection of line AB in the plane 1p . [4] (iv) Find an equation for the plane 2p , in the form r  n= p, which contains the points A and B and is perpendicular to 1p . Hence, or otherwise, find the vector equation of l, the line of intersection of 1p and 2p . [5] (v) The plane 3p has a Cartesian equation 23x y az d   , where a and d are constants. It is given that 1p , 2p and 3p have no point in common and the distance of B from 3p is 13 . Find the value of a, and determine the value(s) of d. [3]

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