Loading [MathJax]/jax/element/mml/optable/GeneralPunctuation.js

User interface language: English | Español

Date November 2013 Marks available 5 Reference code 13N.2.sl.TZ0.9
Level SL only Paper 2 Time zone TZ0
Command term Show that Question number 9 Adapted from N/A

Question

Consider the lines L1 and L2 with equations L1 : r=(1182)+s(431) and L2 : r=(117)+t(2111).

The lines intersect at point P.

Find the coordinates of P.

[6]
a.

Show that the lines are perpendicular.

[5]
b.

The point Q(7,5,3) lies on L1. The point R is the reflection of Q in the line L2.

Find the coordinates of R.

[6]
c.

Markscheme

appropriate approach     (M1)

eg     (1182)+s(431)=(117)+t(2111), L1=L2

any two correct equations     A1A1

eg     11+4s=1+2t, 8+3s=1+t, 2s=7+11t

attempt to solve system of equations     (M1)

eg     10+4s=2(7+3s),{4s2t=103st=7

one correct parameter     A1

eg     s=2, t=1

P(3,2,4)   (accept position vector)     A1     N3

[6 marks]

 

a.

choosing correct direction vectors for L1 and L2     (A1)(A1)

eg     (431),(2111) (or any scalar multiple)

evidence of scalar product (with any vectors)     (M1)

eg     ab(431)(2111)

correct substitution     A1

eg     4(2)+3(1)+(1)(11), 8+311

calculating ab=0     A1

 

Note: Do not award the final A1 without evidence of calculation.

 

vectors are perpendicular     AG     N0

[5 marks]

b.

Note: Candidates may take different approaches, which do not necessarily involve vectors.

In particular, most of the working could be done on a diagram. Award marks in line with the markscheme.

 

METHOD 1

attempt to find QP or PQ     (M1)

correct working (may be seen on diagram)     A1

eg     QP = (431), PQ = (753)(324)

recognizing R is on L1 (seen anywhere)     (R1)

eg     on diagram

Q and R are equidistant from P (seen anywhere)     (R1)

eg     QP=PR, marked on diagram

correct working     (A1)

eg     (324)(753)=(xyz)(324),(431)+(324)

{\text{R}}(–1, –1, 5) (accept position vector)     A1     N3

METHOD 2 

recognizing {\text{R}} is on {L_1} (seen anywhere)     (R1)

eg     on diagram

{\text{Q}} and {\text{R}} are equidistant from {\text{P}} (seen anywhere)     (R1)

eg     {\text{P}} midpoint of {\text{QR}}, marked on diagram

valid approach to find one coordinate of mid-point     (M1)

eg     {x_p} = \frac{{{x_Q} + {x_R}}}{2},{\text{ }}2{y_p} = {y_Q} + {y_R},{\text{ }}\frac{1}{2}\left( {{z_Q} + {z_R}} \right)

one correct substitution     A1

eg     {x_R} = 3 + (3 - 7),{\text{ }}2 = \frac{{5 + {y_R}}}{2},{\text{ }}4 = \frac{1}{2}(z + 3)

correct working for one coordinate     (A1)

eg     {x_R} = 3 - 4,{\text{ }}4 - 5 = {y_R},{\text{ }}8 = (z + 3)

{\text{R}} (-1, -1, 5) (accept position vector)     A1     N3

[6 marks]

c.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.

Syllabus sections

Topic 4 - Vectors » 4.2 » The scalar product of two vectors.
Show 29 related questions

View options