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 hamster