RVHS 2023 JC2 H2 MA Test 3 Solutions (Student Copy)

RVHS 2023 JC2 H2 Maths Lecture Test 3 1 Name : Index Number : Class : Date : 12 May 2023 Duration : 50 min Max. No. of Marks : 30 Formulae from MF26 Vectors The point dividing AB in the ratio : has position vector   + + ab Vector product: 1 1 2 3 3 2 2 2 3 1 1 3 3 3 1 2 2 1 a b a b a b a b a b a b a b a b a b −            =  = −           −      ab [Answer all the questions on writing papers. Up to 1 mark will be deducted for poor presentation.] 1 The line l has equation ( )2 = + + + − +r i j k i k where  is a parameter. The plane π1 has equation 2 2 2x y z+ + =− and the point A is at ( )1,1, 2 . (i) Find the acute angle between the plane π1 and the line l. [2] (ii) The plane π2 contains the line l and is perpendicular to plane π1. Show that the equation of plane π2 is 2 3 2 3x y z− + = . [2] (iii) Find a vector equation of the line of intersection between π1 and π2. [2] (iv) Find the shortest distance from the point A to the plane π1. [3] 2 (a) The complex number 1iv=+ is a root of the equation 22 1 i 0z cz− + − = where c is a complex number. (i) Without the use of a calculator, find the value of c. [4] (ii) Without calculating the other root, explain why v* is not the other root of the equation 22 1 i 0z cz− + + = . [1] (b) The complex number w is given by ππ2 cos isin55w =−  . Showing your working clearly, find the modulus and argument of 2i 2i 2 3 w − . [6] River Valley High School 2023 JC2 Mathematics (9758) Lecture Test 3 (Term 2)

RVHS 2023 JC2 H2 Maths Lecture Test 3 2 End of Paper 3 A disc with sectors labelled 1, 2 and 3 with a free spinning arrow and a fair coin are used in a game booth at a bazaar. When the arrow on the disc is spun, it comes to a stop pointing at the sector labelled 1 with probability of p while it has a probability of 2 3 p pointing at the sector labelled 2. In a game, the arrow is spun, the n the coin is tossed 1, 2 or 3 times according to the sector where the arrow points on the disc. The random variable X denotes the score of the game obtained by adding the total number of heads from the resulting coin toss(es). (i) Show that ( ) 35P1 8 24Xp= = + . Construct a probability distribution table for X with the expressions of the probabilities in terms of p. [5] (ii) Given that ( ) 5E 6X = , find the value of p. [2] (iii) It is given that ( ) 5Var 9X = and that X1 and X2 are two independent observations of X. Find 12 1Var 32 XX − . [3]

RVHS 2023 JC2 H2 Maths Lecture Test 3 3 RVHS 2023 JC2 H2 Maths Lecture Test 3 Solution Question 1 [9] Vectors (Planes) (i) Let the acute angle be  . 1 1 222 1 11 20 21 11 20 21 12 1 32 13.6 sin sin 2 (1 d.p. 2 i ) sn  − − − −          