2019 H2 Prelim (Vectors) Qns with Ans

(i) Show that the equation of the line BG may be expressed as 4 3 2 1 , 0 t h                         r where t is a parameter. [1] (ii) Find the sine of the angle between the line BG and the rectangular base ABCD in terms of h . [2] It is given that h = 6. (iii) Find the cartesian equation of the plane BCFG . [3] (iv) Find the shortest distance from the point A to the plane BCFG . [2] (v) The line l , which passes through the point A , is parallel to the normal of plane BCFG . Given that, the line l intersects the plane BCFG at a point M , use your answer in part (iv) to find the shortest distance from point M to the rectangular base ABCD . [2] Answers 2 sin 10 h h    2 8 0 x z    16 5 units 5 16 5 CJC JC2 Prelim 9758/2019/01/Q9 Line 1 l has equation 1 2 8 1 3 5                         r where  is a real parameter, and plane p has equation 17 0 a b             r . It is given that 1 l lies completely on p and the point Q has coordinates   1, 2,3   . (i) Show that 1 a   and 2 b  . [3] (ii) Find the foot of perpendicular from point Q to p . Hence find the shortest distance between point Q and p in exact form. [4] Given that the shortest distance between 1 l and the foot of perpendicular of Q onto p is 5 6 3 .

Methane ( 4 CH ) is an example of a chemical compound with a tetrahedral structure. The 4 hydrogen (H) atoms form a regular tetrahedron, and the carbon (C) atom is in the centre. Let the 4 H-atoms be at points P , Q , R , and S with coordinates         9,2,9 , 9,8,3 , 3,2,3 , and 3,8,9 respectively. (i) Find a Cartesian equation of the plane 1  which contains the points P , Q and R . [4] (ii) Find a Cartesian equation of the plane 2  which passes through the midpoint of PQ and is perpendicular to PQ . [2] (iii) Find the coordinates of point F , the foot of the perpendicular from S to 1  . [4] (iv) Let T be the point representing the carbon (C) atom. Given that point T is equidistant from the points P , Q , R and S , find the coordinates of T . [3] Answers (i) 2 x y z     (ii) 1 y z    (iii) (7,4,5) (iv) (6,5,6) EJC JC2 Prelim 9758/2019/02/Q3 The position vectors of points P and Q , with respect to the origin O , are p and q respectively. Point R , with position vector r , is on PQ produced, such that 3 5 PR PQ  . (i) Given that 29  p and 11  p.r , find the length of projection of OQ onto OP . [4] (ii) S is another point such that PS  r . Given that 3 2 4    p i j k and 2 3    r i j k , find the area of the quadrilateral OPSR . [3] Answers (i) 91 5 29 (or 3.38) (ii) 285 HCI JC2 Prelim 9758/2019/01/Q5 Referred to the origin O , points A and B have position vectors a and b respectively. Point C lies on OB produced such that OC OB   where 1.   Point D is such that OCDA is a parallelogram. Point M lies on AD , between A and D , such that : 1: 2 AM MD  . Point N lies on OC , between O and C , such that : 4 : 3 ON NC  .

JPJC JC2 Prelim 9758/2019/02/Q5 The plane p has equation 1 1 2 0 2 3 3 0 4                                      r , and the line l has equation 10 8 0 4 4 1 t                         r , where  ,  and t are parameters. (i) Show that l is perpendicular to p and find the values of  ,  and t which give the coordinates of the point at which l and p intersect. [5] (ii) Find the cartesian equations of the planes such that the perpendicular distance from each plane to p is 2. [5] Answers (i) 1, 2, 1 t       (ii) 8 4 23 x y z    and 8 4 13 x y z     MI PU3 Prelim 9758/2019/01/Q8 (a) Referred to the origin O , the point Q has position vector q such that q 3 1 2 2 2    i j k . (i) Find the acute angle between q and the y -axis. [2] It is given that a vector m is perpendicular to the xy -plane and its magnitude is 1. (ii) With reference to the xy -plane, explain the geometrical meaning of q m and state its value. [2] (b) Referred to the origin O , the point R has position vector r given by    r a b , where  is a positive constant and a and b are non-zero vectors. It is known that c is a non-zero vector that is not parallel to a or b . Given that     c a b c , show that r is parallel to c . [2] It is also given that a is a unit vector that is perpendicular to b and 2  b . By considering r r , show that 2 4 1 k    c , where k is a non-zero constant. [4]

NJC JC2 Prelim 9758/2019/01/Q6 (a) Relative to the origin O , a point A has position vector 6 10 t   i j k , where t is a real and negative constant. Given that the direction cosine of A with respect to the x -axis is 3 35 , find the value of t . Deduce the angle between OA and the yz -plane. [3] (b) The planes 1 p and 2 p have equations 2 2 8 1              r and 2 2 1 k               r respectively, where k is a real constant. It is given that the plane p is equidistant from 1 p and 2 p . (i) Find, in terms of k , a vector equation of p in the scalar product form. [3] (ii) Given 13 k  , find the coordinates of the point on line l with equation 1 2 1 3 3 18 1 x y z      that is equidistant from 1 p and 2 p . [3] Answers (a) 30.5  (b)(ii)   5, 2.5 2.5, NYJC JC2 Prelim 9758/2019/01/Q5 Relative to the origin O , the points A , B , and C , have non-zero position vectors a , b , and 3 a respectively. D lies on AB such that AD AB   , where 0 1    . (i) Write down a vector equation of the line OD . [1] (ii) The point E is the midpoint of BC . Find the value of  if E lies on the line OD . Show that the area of BED  is given by k  a b , where k is a constant to be determined. [5] Answers (i)   : ( 1 ),       OD l r s b a s 3 8 k 

(a) find the exact area of triangle ABC , [1] (b) show that the cartesian equation of the plane ABC is 3 2 7 16 x y z    . [1] (iii) The line 1 l has equation 1 1 1 2 6 1                         r for   . (a) Show that 1 l is parallel to the plane ABC but does not lie on the plane ABC . [3] (b) Find the distance between 1 l and plane ABC . [2] (iv) The line 2 l passes through B and is perpendicular to the xy -plane. Find the acute angle between 2 l and its reflection in the plane ABC , showing your working clearly. [3] Answers (i) 3 2 7           (ii)(a) 62 2 (iii)(b) 62 2 (iv) 54.5 RVHS JC2 Prelim 9758/2019/01/Q8 Two planes have equations given by 1 2 : 5 4 4, : 4. p x y z p x y z       (i) Explain why 1 p and 2 p intersect in a line and determine the cartesian equation of . [3] (ii) The plane 3 p contains the line and the point   5, 3, 6 Q  , find a cartesian equation of 3 p . Describe the geometrical relationship between these 3 planes and . [5] (iii) The line m passes through the points   0,2,0 S and   4,0,0 T . Find the position vector of the foot of perpendicular of S on 2 p . Hence find a vector equation of the line of reflection of m in 2 p . [5] Answers (i) 4 3 2 x z y     (iii) 4 2 4            ; 4 0 0 1 , 0 2                           r

(iv) 31.5  (v) 9 x z   or 1 x z    TJC JC2 Prelim 9758/2019/01/Q2 The position vectors of A , B , C and D are 1 5            , 1 2 1            , 2 3 1            and 1 7            respectively, where α and β are real numbers. Given that BD is a perpendicular bisector of AC , find the values of α and β . [5] Answers 4, 10     TJC JC2 Prelim 9758/2019/01/Q9 (a) Vectors u and v are such that 1   u.v and ( ) u v u   is perpendicular to ( ) u v v   . Show that 1 u v   . [3] Hence find the angle between u and v . [3] (b) The figure shows a regular hexagon ABCDEF with O at the centre of the hexagon. X is the midpoint of BC . Given that a OA  and b OB  , find OF and OX in terms of a and b . [2] Line segments AC and FX intersect at the point Y . Determine the ratio AY : YC . [4] A B C D E F O  X

(ii) Find in terms of a and b . [3] (iii) State the geometrical meaning of . [1] Answers 7 2 a b  3 b a  VJC JC2 Prelim 9758/2019/01/Q10 The point M has position vector relative to the origin O , given by . The line has equation , and the plane  has equation . (i) Show that lies in  . [2] (ii) Find a cartesian equation of the plane containing and M . [3] The point N is the foot of perpendicular from M to . The line is the line passing through M and N . (iii) Find the position vector of N and the area of triangle . [5] (iv) Find the acute angle between and  , giving your answer correct to the nearest . [3] Answers 29 11 2 207 x y z      4 9 4          2 44.2 unit 10.6  YIJC JC2 Prelim 9758/2019/01/Q2 (a) Vectors a and b are such that 0 0   a , b and    a b a b . Show that a and b are perpendicular. [2] PQ OM  PQ OM PQ  6 5 11   i j k 1 l 2 7 3 2 y z x      4 2 30 x y z    1 l 1 l 1 l 2 l OMN 2 l 0.1 

(b) Referred to the origin O , points C and D have position vectors c and d respectively. Point P lies on OC produced such that : 1: 1 OC CP    , where  >1. Point M lies on DP , between D and P , such that : 2:3 DM MP  . Write down the position vector of M in terms of  , c and d . Hence, find the area of triangle OPM in the form k   c d , where k is a constant to be found. [4] (c) Answers (d) (b) 3 10   c d YIJC JC2 Prelim 9758/2019/02/Q3 The plane p contains the point A with coordinates   5, 1, 2  and the line 1 l with equation 3 , 1 2 x y z    . (i) The point B has coordinates   , 2, 2 c . Given that the shortest distance from B to 1 l is 205 5 , find the possible values of c . [3] (ii) Find a cartesian equation of p . [3] The line 2 l has equation 1 4 2 1 1 3                          r , . (iii) Find the coordinates of the point at which 2 l intersects p . [3] The line 3 l has equation 4 2 0 1 a a t t a                        r , , where a is a constant. (iv) Show that p is parallel to 3 l . [1] (v) Given that 3 l and p have no point in common, what can be said about the value of a ? [1] (vi) It is given instead that 1 a  , find the distance between 3 l and p , leaving your answer in exact form. [2] Answers (i) 1 or 13 c c   (ii) 2 4 1 x y z     (iii) 7 7 4 , , 5 5 5         (v) 1 7 a  (vi) 2 21 7