2024 JPJC J1 EYE Revision Package Qn

JC1 H2 Math Year-End Exam Revision Package \ Pg 1 2024 JC1 H2 Mathematics (9758) Year-End Examination Revision Package Name:______________________________ Class:__________ Contents Topical Revision Page Topic 1: Vectors 3 Topic 2: Graphing Techniques I & II 6 Topic 3: Equations & Inequalities 9 Topic 4: Functions 11 Topic 5: Differentiation Techniques & Applications 13 Topic 6: Integration Techniques 16 Paper Revision Revision Paper A 18 Revision Paper B 23 Revision Paper C 29 Revision Paper D 33 Answer Key 39 Worked solutions will be uploaded on JPJC Math Google Site on 30/8 (Topical 1-3), 9/9 (Topical 4-6), 18/9 (Revision A-D)

JC1 H2 Math Year-End Exam Revision Package \ Pg 2 INFORMATION FOR STUDENTS JC1 H2 Mathematics Year-End Examination: 27 Sept 2024 (Fri) SCHEME OF EXAMINATION PAPERS For the year-end examination in H2 Mathematics, it will be a 3 hour paper consisting of about 10 to 13 questions of different lengths and marks based on the following Pure Mathematics topics: Topic 1: Vectors Topic 2: Graphing Techniques I & II Topic 3: Equations and Inequalities Topic 4: Functions Topic 5: Differentiation Techniques & Applications Topic 6: Integration Techniques (Chp 1 to 8 of Lecture Notes) The total number of marks is 100. Candidates will be expected to answer all questions. USE OF GRAPHIC CALCULATOR (GC) The use of GC without computer algebra system will be expected. The examination papers will be set with the assumption that candidates will have access to GC. As a general rule, unsupported answers obtained from GC are allowed unless the question states otherwise. Where unsupported answers from GC are not allowed, candidates are required to present the mathematical steps using mathematical notations and not calculator commands. For questions where graphs are used to find a solution, candidates should sketch these graphs as part of their answers. Incorrect answers without working will receive no marks. However, if there is written evidence of using GC correctly, method marks may be awarded. Students should be aware that there are limitations inherent in GC. For example, answers obtained by tracing along a graph to find roots of an equation may not produce the required accuracy.

JC1 H2 Math Year-End Exam Revision Package \ Pg 3 H2 Mathematics Year-End Exam Revision Package JC1- 2024 Topic 1: Vectors 1. [CJC/2016/Promo/3] The position vectors of the points A , B and C with respect to the origin O are a , b and c respectively. Point C is on AB produced such that 5 3 .AB AC Given that 2a ci , a is a unit vector and the angle AOB is π 3 , show that 16 5b . [5] 2. [HCI/2016/Promo/7] Relative to the origin O, three distinct fixed points A, B and C have position vectors a , b and c respectively. It is known that b is a unit vector, 3a , 2c and angle AOC = 60 . (i) State the geometrical interpretation of b ci . [1] It is further given that 2  a b b c . (ii) Find the ratio of the area of triangle AOB to the area of triangle BOC. [2] (iii) Show that 2 k a c b where , 0k k  . By considering    2 2 a c a ci , find the exact values of k. [6] 3. [HCI/2016/P1/3] Referred to the origin O, the points A and B are such that OA a  and OB b  with a not parallel to b . The point P is on AB produced with : 3:1AP AB  and the position vector of point Q is 2a . (a) Find the position vector of the point of intersection of lines OB and PQ, giving your answer in terms of b. [4] (b) It is given that 2 4 2   a b i j k and the point  0, 3, 4C does not lie on the plane OAB. Find the coordinates of the foot of the perpendicular from C to the plane OAB. [4] 4. [MJC/2016/PROMO/9] The points A and B have coordinates  0, 2,0 and  1,0,2 respectively, and the plane 1 has equation 2 2 4 1        r i . (i) Find a vector equation for the line AB. [2] (ii) Show that the line AB lies in plane 1 . [1] (iii) The plane 2 contains the line AB and is perpendicular to plane 1 . Show that the equation of plane 2 is 2 1 2 2        r i . [3]

JC1 H2 Math Year-End Exam Revision Package \ Pg 4 (iv) The point  ,0,0P p is equidistant from the planes 1 and 2 . Calculate the value of p. [5] (v) Hence write down an equation of a line that is equidistant from planes 1 and 2 which does not intersect both planes 1 and 2 . [2] 5. [DHS/2016/P1/11] The diagram below shows a tetrahedron ABCD. The equation of the plane ABD is 4 2 16.x y z   (i) Given that A is on the x-axis, find the coordinates of A. [1] The equation of the plane CBD is 7 11 5 23.x y z   (ii) Find a vector equation of the line that passes through B and D. [2] (iii) Given that B is on the xy-plane, find the coordinates of B. [2] The cartesian equation of the line that passes through A and D is 4 .2 2 3 x y z   (iv) Find the coordinates of D. [3] The coordinates of C are (−1, 1, 1). (v) By considering the area of triangle ABC, find the exact volume of the tetrahedron ABCD. [5] [Volume of tetrahedron = 1 3 area of base perpendicular height  ] 6. [MI/2016/PROMO/13] The lines 1l and 2l have equations r 4 1 1 1 , 10 6                        and r 1 4 2 3 , 4 4                        respectively. (i) Find the acute angle between 1l and 2l . [2] (ii) Find a vector equation of the line 3l which passes through the point A with coordinates (1, −2, 4) and is perpendicular to both 1l and 2l . [2] (iii) The plane p1 contains both 1l and 2l . Find the equation of p1 in scalar product form. [2] (iv) Show that the point B with coordinates (−5, 10, 1) lies on 3l . By deducing the position vector of the point of intersection of 3l and p1, find the exact perpendicular distance from B to p1. [3] C A B D

JC1 H2 Math Year-End Exam Revision Package \ Pg 5 (v) The plane p2 has equation a b k c        r i , where a, b, c, and k are constants. What can be said about the values of a, b, c and k when (a) plane p2 and line 2l are perpendicular to each other, [2] (b) planes p1 and p2 do not intersect? [2] 7. [MJC/2011/Promo/12] In the figure below, the pyramids ABCDE and ABCDF are symmetrical about the parallelogram ABCD. Referred to an origin O, the position vectors of the points A, B and E are 2 5 i j k , 7 2 i j k and 4 7 i j k respectively. The line l, which lies on the plane ABCD has vector equation 2 2 1 1 , 5 2                      r  . (i) Find a vector equation of the plane ABCD in the form r . n = d. [3] (ii) Find the acute angle between the lines AE and AB. [2] (iii) Find the position vector of the foot of the perpendicular from the point E to the plane ABCD. Hence find the shortest distance from point E to the plane ABCD. [4] (iv) Find a vector equation of the line AF. [2] 8. The line l has equation 2 133 3    zyx and the plane p has equation 023  zyx . (i) Show that l is perpendicular to p. [2] (ii) Find the coordinates of the point of intersection of l and p. [3] (iii) Show that the point C with coordinates  9,1, 7  lies on l. Find the coordinates of the point C’ which is the mirror image of C in p. [3] E A D C F B

JC1 H2 Math Year-End Exam Revision Package