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Date November 2018 Marks available 3 Reference code 18N.1.AHL.TZ0.H_9
Level Additional Higher Level Paper Paper 1 (without calculator) Time zone Time zone 0
Command term Find Question number H_9 Adapted from N/A

Question

Consider a triangle OAB such that O has coordinates (0, 0, 0), A has coordinates (0, 1, 2) and B has coordinates (2b, 0, b − 1) where b < 0.

Let M be the midpoint of the line segment [OB].

Find, in terms of b, a Cartesian equation of the plane Π containing this triangle.

[5]
a.

Find, in terms of b, the equation of the line L which passes through M and is perpendicular to the plane П.

[3]
b.

Show that L does not intersect the y-axis for any negative value of b.

 

[7]
c.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

METHOD 1

n=(012)×(2b0b1)      (M1)

=(b14b2b)      (M1)A1

(0, 0, 0) on Π so (b1)x+4by2bz=0      (M1)A1

 

METHOD 2

using equation of the form px+qy+rz=0      (M1)

(0, 1, 2) on Π ⇒ q+2r=0

(2b, 0, b − 1) on Π ⇒ 2bp+r(b1)=0      (M1)A1

Note: Award (M1)A1 for both equations seen.

solve for pq and r      (M1)

(b1)x+4by2bz=0      A1

 

[5 marks]

a.

M has coordinates (b,0,b12)      (A1)

r(b0b12)+λ(b14b2b)      M1A1

Note: Award M1A0 if r = (or equivalent) is not seen.

Note: Allow equivalent forms such as xbb1=y4b=2zb+14b.

 

[3 marks]

b.

METHOD 1

x=z=0      (M1)

Note: Award M1 for either x=0 or z=0 or both.

b+λ(b1)=0 and b122λb=0      A1

attempt to eliminate λ       M1

bb1=b14b      (A1)

4b2=(b1)2      A1

EITHER

consideration of the signs of LHS and RHS       (M1)

the LHS is negative and the RHS must be positive (or equivalent statement)       R1

OR

4b2=b22b+1

5b22b+1=0

Δ=(2)24×5×1=16(<0)     M1

no real solutions       R1

THEN

so no point of intersection       AG

 

METHOD 2

x=z=0      (M1)

Note: Award M1 for either x=0 or z=0 or both.

b+λ(b1)=0 and b122λb=0      A1

attempt to eliminate b       M1

λ1+λ=114λ      (A1)

4λ2=1(λ2=14)      A1

consideration of the signs of LHS and RHS       (M1)

there are no real solutions (or equivalent statement)       R1

so no point of intersection       AG

 

[7 marks]

c.

Examiners report

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

Syllabus sections

Topic 3— Geometry and trigonometry » AHL 3.14—Vector equation of line
Show 98 related questions
Topic 3— Geometry and trigonometry » AHL 3.17—Vector equations of a plane
Topic 3— Geometry and trigonometry » AHL 3.18—Intersections of lines & planes
Topic 3— Geometry and trigonometry

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