RI_H2_MATHS_P2_Qns

RI H2 Mathematics 2017 Prelim Exam Paper 2 Question 1 Referred to the origin O, the points A, B and C have position vectors a, b and c respectively such that 23aij k , 523 bij k and 42 ci j k . (i) Given that M is the mid-point of AC, use a vector product to find the exact area of triangle ABM. [4] (ii) Find the position vector of the point N on the line AB such that MN  is perpendicular to AB  . [4] 2 (a) (i) Show that   121 2 .11 1 1rr rr r r    [1] (ii) Hence find  3 4 .11 n r rr r  (There is no need to express your answer as a single algebraic fraction). [4] (b) Amy and her brother Ben are saving money togeth er for their family trip. In the first week of 2017, Amy saves $25 and Ben saves $2. In each subsequent week, Amy saves $4 more than the amount she saved in the previous week, and Ben saves 22% more than the amount he saved in the previous week. (i) Which is the first week in which Ben saves more than Amy in that week? [2] (ii) They need a combined total of $2400 for the trip. How many complete weeks do Amy and Ben need to save before they can achieve their targeted amount? [2] 3 The function f is defined as follows. f: 3 s i n c o s , , 0 .xx x x x    (i) Write  f x as sin( )Rx  , where R and  are constants with exact values to be found. [2] (ii) Sketch the graph of =f ,y x stating the axial intercepts, and find the range of f. [3] (iii) Hence, solve  f1 x  exactly. [2] The function g is defined as follows: g: 2c o s , , . 66x xx x b      (iv) Write down the largest exact value of b, for 1g to exist. [1] (v) Taking the value of b found in part (iv), show that the composite function 1gf exists and solve  1gf x x  exactly. [3]

4 The line 1l has equation 1 31 2 4 x yz  and the line 2l has equation 11 434 x zy . (i) Show that 1l and 2l are skew lines. [3] (ii) Find a cartesian equation of the plane p which is parallel to 1l and contains 2l . [3] (iii) The point (0, ,1)A a is equidistant from p and 1l . Calculate the possible values of a exactly. [ 6 ] 5 For events X and Y, it is given that  1P| 2XY  ,  2P| 3YX  and  5P 6XY . Find (i) P, X [3] (ii) P' .XY [2] 6 The power consumption of a randomly chosen Effixion laptop has a normal distribution. The salesman at Elf Superstore claims that the av erage power consumption of an Effixion laptop is 100 watts. The power consumption, w watts, is measured for a random sample of 50 Effixion laptops. The results are summarised as follows. ( 100) 26w