RVHS 2023 JC2 H2MA Lec Test (Std Soln)

RVHS 2023 JC2 H2 Maths Lecture Test 2 1 Name : Index Number : Class : Date : Duration : 50 mins Max. No. of Marks : 30 List of Formulae [Answer all the questions on writing papers. Up to 1 mark will be deducted for poor presentation.] 1. The lines l1 and l2 have equations: 1 2 2:1 2 51 : 5 0 , 01 yl x z l  += = − −       = − +           r (i) Verify that the point A(1, 0, 2) lies on l1. [1] (ii) Find the position vector of the foot of the perpendicular from A to l2. Hence find the coordinates of the point of reflection of A about l2. [5] (iii) Find the angle between l1 and l2. [2] (iv) Determine whether l1 and l2 are intersecting, parallel or skew lines. [3] 2. A baby’s toy teaches babies music by having them place lettered balls into boxes. The toy comprises five boxes arranged in a row. Seven balls each with a different letter from {A, B, C, D, E, F, G} written on it, are placed into the boxes. Each box must have exactly one ball in it. There will be two balls not used. When the “play” button is pushed, the toy will play the notes corresponding to the letters shown on the balls in order. A baby randomly puts five of the balls into the row of boxes. Find the probability that (i) the notes C, E and G are played consecutively, in that order, [3] (ii) the notes C and G are both played but separated by other notes, [2] (iii) the notes D, F and A are played given that the five notes are played in alphabetical order. [4] River Valley High School Integrated Programme 2023 JC2 H2 Mathematics (9758) Lecture Test 2 (Term 1)

RVHS 2023 JC2 H2 Maths Lecture Test 2 2 3. The points A, B, C, and D are such that ABCD is a parallelogram. It is given that AB= a and AD= b . The point P cuts BC such that BP:BC is  for some 01  . The point Q lies on AP produced such that AP:AQ is also  . (i) Show that 1AQ =+ ab . Hence show that D, C and Q are collinear. [3] (ii) It is given that the area of the parallelogram ABCD is equal to the area of the triangle ADQ. Find the value of  . [4] (iii) Given further that =ab , show that AQ is not perpendicular to DQ . [3] ~ The End ~

RVHS 2023 JC2 H2 Maths Lecture Test 2 3 RVHS 2023 JC2 H2 Lecture Test 2 (Student Solution) 1 Vectors in 3-dimensions [11 marks] (i) Substituting (1, 0, 2) into the Cartesian equation: ( ) ( ) ( )021 2 1 2 += = − (verified) (ii) Let B be (5, -5, 0) which lies on l2 and F be the foot of the perpendicular from A to l2. Then, 11 00 11 4 1 1 1 5 0 02 2 1 1 3 0 3 BF BA  −−       =              − −  −           =                 − = 53 50 03 2 5 3 OF OB BF=+ −       = − +           =− Alternative Let F be the foot of the perpendicular. Since it is a point on l2, 5 1 4 5 0 5 22 AF OF OA   =− −−          