RI Yr 5 TP H2 Math Topical Revision (Qn) - Updated

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 _____________________ 2025 Y5 Timed Practice Topical Revision Page 1 of 23 Year 5 H2 Math Timed Practice Topical Revision: Questions Please attempt the questions only after you have completed your revision for the topic(s). You should practice under time constraint based on the guideline of not more than 1.8 minute for each mark. In other words, a 10 mark question should not take you more than 18 minutes to solve. The solutions will be released in Ivy on 30 May (Friday). C2: Vectors 1 ACJC Promo 9758/2022/Q7 modified (a) Referred to the origin O, points A and B have position vectors a and b respectively, where a and b are non-zero and non parallel vectors . Point C lies on OA, between O and A, such that OC : CA = 2 : 1. Point D lies on OB produced such that 5BD OB . Find, in terms of a and b, the position vector of the point E where the lines AB and CD meet. [3] (b) The position vectors of the points D and E relative to the origin O are d and e respectively, where d and e are non-zero and non parallel vectors. It is given that the length OE is 2 units and 3 d e . The point F, with position vector f, is the reflection of the point D in the line OE. (i) Express f in terms of vectors d and e. [3] (ii) Show that the area of triangle ODF can be expressed as k d e , where k is a constant to be determined. [2] [(a) 5 3+8 8a b (bi) 3 2 e d (bii) 3 4 ] 2 CJC Promo 9758/2022/Q9 Relative to the origin O, the position vectors of points A, B and C are a , b and c respectively, where b is a unit vector. It is given that b and b a are perpendicular and C lies on AB such that : 3:1AC CB  . Show that seca , where  is the angle between a and b . [3] By expressing c in terms of a and b , find the value of c b and state the geometrical interpretation of c b , [4] (iii) find the value of   b c a b . [2] [(ii) 1 (iii) 1 4 ]

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________ ______________________ 2025 Y5 Timed Practice Topical Revision Page 2 of 23 3 EJC Promo 9758/2022/Q7 Referred to the origin O, the points A and B have position vectors a and b respectively, where a and b are non-parallel vectors. The lines AB and OC intersect at point X such that : 3 :1AX XB  and : 3 : 5OX XC  (see diagram). (a) Express OC   in terms of a and b. [2] (b) Given that OC is perpendicular to AB and : 3 : 2OA OB  , show that 23. 8a b b . [3] (c) Hence, by considering 2 OC   , find OC : OB. [3] [(a)  2 33 a b (c) 6 :1] 4 NYJC Promo 9758/2022/Q9 The line l has equation 1 , 1 2 zx y    , where  is a constant. (i) Given that the point of intersection between the line l and the yz-plane is  0,1,5 , show that 3  . [2] Referred to the origin, the points A and B have position vectors 3 j k and 12 5 i j k respectively. (ii) Find the cartesian equation of the plane 1 which contains the point A and the line l. [3] (iii) Find the acute angle between the line AB and 1 . [3] The plane 2 has cartesian equation 2 5x z  . (iv) Find the cartesian equations of the planes such that the perpendicular distance from each plane to 2 is 2 5 . [4] [(ii) 2 4x y z   (iii) 24.1 (iv) 2 15 or 2 5x z x z    ] A X O B C

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________ ______________________ 2025 Y5 Timed Practice Topical Revision Page 3 of 23 5 HCI Promo 9758/2022/Q5 Relative to the origin O, the points M, S and T have coordinates (3, ,0.03)p , (1,1,0.02) and ( ,0,0.01)q respectively, where p and q are real constants. The line l passes through the point M and is parallel to the vector  i j . (i) Given that l is perpendicular to the line passing through S and T, show that 2q . [2] (ii) Hence find the area of triangle OST , leaving your answer correct to 6 decimal places. [2] (iii) It is further given that l and the line passing through S and T do not intersect. Determine the set of possible values of p. [3] [(ii) 21.000125 units (iii)  | 5p p  ] 6 EJC Promo 9758/2022/Q12 A drone flies in a straight line from point A to point B with coordinates (40, 40, 5). At B, it makes a sharp 90 turn and thereafter flies in a straight line in the direction of 2 2 i j k . Eventually, it crashes into a cliff face which is a part of the plane 8 4 595,x y z   at point C (see diagram). (a) Find the coordinates of C. [3] Determine the acute angle, α, that the path of the drone makes with the cliff face at C. [2] (c) Given that the coordinates of A are (q, 120, 1), find the value of q. [3] Max is trying to track the drone using his radar. From point D with coordinates (36, 36, 0), his radar emits constant signals in the directions of cos sin  i j k for all values of θ. Assuming that signals travel indefinitely along straight lines, show that none of the signals emitted succeed in hitting the drone as it flies from B to C. [5] [(a) (50, 45, 15) (b) 54.6 (c) 4 ] A B (40, 40, 5) C Cliff face Flight path of drone

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________ ______________________ 2025 Y5 Timed Practice Topical Revision Page 4 of 23 7 TJC Promo 9758/2022/Q12 As part of a structure for a marble race, a plane p is placed at an angle  to the horizontal ground. The ground is taken to be the x-y plane and the line of intersection between p and the ground is the y-axis (see diagram). (i) Given that a point A with coordinates (4, 1, 2) lies on p, find a vector equation of p in the scalar product form. Hence find the value of , leaving your answer in degrees correct to one decimal place. [4] (ii) A straight rod is attached to p from A to a point B on the y-axis, as shown in the diagram. Given that AB = 6 units, find the position vector of B. [4] A marble is released from rest at A and it moves along the rod AB towards B. It is observed that at time t seconds after the marble is released, the height of the marble above the ground is 22 kt units where k is a positive constant less than 1. (iii) Find in terms of k, (a) the time taken for the marble to reach B. [1] (b) the coordinates of the marble when t = 0.5. [4] [(i) 1 r 0 0 2         , 26.6 (ii) 0 5 0         (iiia) 2 k s (iiib) 2 2 4(4 ,1 , 2 )k k k   ] Ground   y x z A B p O

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________ ______________________ 2025 Y5 Timed Practice Topical Revision Page 5 of 23 8 YIJC Promo 9758/2022/Q11 The diagram above shows part of the structure of a rock -climbing wall, with a horizontal base OABC, a vertical rectangular wall OADE and an inclined rectangular wall DEFG. With respect to the origin O, the vectors i, j and k each of length 1 m, are parallel to OA, OC and OE respectively. It is given that OE = 10 m. The plane OADE has equation 0 1 , 0 p         r and the plane DEFG has equation 0 3 0 0 2 1 , 10 4 q                                  r where p and q are constants, and   are parameters. (a) Give a reason why 0p . [1] (b) Write down a vector equation of the line