Date | May Specimen paper | Marks available | 7 | Reference code | SPM.1.AHL.TZ0.8 |
Level | Additional Higher Level | Paper | Paper 1 (without calculator) | Time zone | Time zone 0 |
Command term | Find | Question number | 8 | Adapted from | N/A |
Question
The plane П has the Cartesian equation 2x+y+2z=32x+y+2z=3
The line L has the vector equation r =(3−51)+μ(1−2p),μ,p∈R. The acute angle between the line L and the plane П is 30°.
Find the possible values of p.
Markscheme
recognition that the angle between the normal and the line is 60° (seen anywhere) R1
attempt to use the formula for the scalar product M1
cos 60° = |(212)∙(1−2p)|√9×√1+4+p2 A1
12=|2p|3√5+p2 A1
3√5+p2=4|p|
attempt to square both sides M1
9(5+p2)=16p2⇒7p2=45
p=±3√57 (or equivalent) A1A1
[7 marks]
Examiners report
Syllabus sections
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22M.1.AHL.TZ1.11b.i:
Verify that the point P(1, -2, 0) lies on both ∏1 and ∏2.
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17M.1.SL.TZ2.S_9d:
Hence or otherwise, find the distance between P1 and P2 two seconds after they leave A.
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18M.1.SL.TZ1.S_9d:
Point D is also on L and has coordinates (8, 4, −9).
Find the area of triangle OCD.
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17M.1.SL.TZ1.S_8d.ii:
Hence or otherwise, find one point on L2 which is √5 units from C.
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18M.1.AHL.TZ2.H_9a.i:
Explain why ABCD is a parallelogram.
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18M.1.AHL.TZ2.H_9e:
Find the Cartesian equation of Π.
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19M.1.AHL.TZ1.H_11d:
Write down →PA and →PB.
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17M.2.AHL.TZ2.H_9f:
Find a vector perpendicular to the plane Π3.
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19M.1.AHL.TZ1.H_11a.i:
Find how many sets of four points can be selected which can form the vertices of a quadrilateral.
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17M.2.AHL.TZ2.H_9a:
Find the vector equation of the line (BC).
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19N.2.SL.TZ0.S_2b:
Write down a direction vector for L3.
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19M.1.AHL.TZ1.H_11c:
Write down the value of λ corresponding to the point A.
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19N.1.AHL.TZ0.H_8:
A straight line, Lθ, has vector equation r =(500)+λ(5sinθcosθ), λ, θ∈R.
The plane Πp, has equation x=p, p∈R.
Show that the angle between Lθ and Πp is independent of both θ and p.
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19M.1.AHL.TZ1.H_11a.ii:
Find how many sets of three points can be selected which can form the vertices of a triangle.
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17M.1.SL.TZ2.S_9a:
Find the coordinates of A.
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18M.1.AHL.TZ2.H_9d:
Find the vector equation of the straight line passing through M and normal to the plane Π containing ABCD.
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17M.1.SL.TZ1.S_8d.i:
Find a unit vector in the direction of L2.
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17M.2.AHL.TZ2.H_9d:
Show that the line (BC) lies in the plane Π1.
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16N.1.SL.TZ0.S_4a:
Find a vector equation of the line that passes through P and Q.
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18N.2.SL.TZ0.S_8d:
Find the area of triangle ABC.
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17N.1.SL.TZ0.S_9a.i:
Show that →AB=(2−11).
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17M.1.SL.TZ2.S_9c:
Find cosBˆAC.
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16N.1.SL.TZ0.S_4b:
The line through P and Q is perpendicular to the vector 2i + nk. Find the value of n.
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17N.1.SL.TZ0.S_9b:
Find the value of p.
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18M.1.SL.TZ2.S_1a:
Find a vector equation for L1.
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16N.1.AHL.TZ0.H_8a:
find the value of a;
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22M.1.AHL.TZ1.11b.ii:
Find a vector equation of L, the line of intersection of ∏1 and ∏2.
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22M.1.AHL.TZ1.11c:
Find the distance between L and ∏3.
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18N.2.SL.TZ0.S_8b.ii:
Show that →AC=(8−10−1).
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22M.2.AHL.TZ2.11b:
Show that airplane A travels at a greater speed than airplane B.
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18M.1.SL.TZ1.S_9b.i:
Find a vector equation for L.
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17M.1.SL.TZ1.S_8a.ii:
Hence, write down a vector equation for L1.
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EXN.2.AHL.TZ0.11d.ii:
Hence determine the minimum distance, dmin, from D to Π.
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19M.1.SL.TZ1.S_2b:
A second line, L2, is parallel to L1 and passes through (1, 2, 3).
Write down a vector equation for L2.
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18N.1.AHL.TZ0.H_9b:
Find, in terms of b, the equation of the line L which passes through M and is perpendicular to the plane П.
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19M.2.SL.TZ2.S_7:
The vector equation of line L is given by r =(−138)+t(45−1).
Point P is the point on L that is closest to the origin. Find the coordinates of P.
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21M.1.AHL.TZ1.11a.i:
Show that the point (-1, 0, 3) lies on L1.
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21M.1.AHL.TZ1.11b:
Find the possible values of a when the acute angle between L1 and L2 is 45°.
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18N.2.SL.TZ0.S_8a.i:
Find →AB.
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17M.2.AHL.TZ2.H_9c:
Find the Cartesian equation of the plane Π1, which passes through C and is perpendicular to →OA.
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18M.1.AHL.TZ2.H_9a.ii:
Using vector algebra, show that →AD=→BC.
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20N.1.SL.TZ0.S_7b:
Particle P2 also moves in a straight line. The position of P2 is given by r=(-16)+t(4-3).
The speed of P1 is greater than the speed of P2 when t>q.
Find the value of q.
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20N.1.SL.TZ0.S_9b:
Find the value of m.
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18N.2.SL.TZ0.S_8a.ii:
Find |→AB|.
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19N.2.SL.TZ0.S_2c:
L3 passes through the intersection of L1 and L2.
Write down a vector equation for L3.
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18N.2.SL.TZ0.S_8c:
Find the angle between →AB and →AC.
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16N.1.AHL.TZ0.H_8b:
determine the coordinates of the point of intersection P.
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17M.2.AHL.TZ2.H_9b:
Determine whether or not the lines (OA) and (BC) intersect.
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19M.2.SL.TZ1.S_9a:
Find the gradient of L.
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18M.1.AHL.TZ2.H_9b:
Show that p = 1, q = 1 and r = 4.
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19M.1.AHL.TZ1.H_11e:
Let C be the point on l1 with coordinates (1, 0, 1) and D be the point on l2 with parameter μ=−2.
Find the area of the quadrilateral CDBA.
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17M.1.SL.TZ2.S_9b.i:
Find →AB;
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EXN.2.AHL.TZ0.11a:
Find the vectors →AB and →AC.
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17N.1.SL.TZ0.S_9a.ii:
Find a vector equation for L.
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EXN.2.AHL.TZ0.11b:
Use a vector method to show that BˆAC=60°.
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17M.1.SL.TZ1.S_8a.i:
Find →AB.
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21M.1.AHL.TZ1.11a.ii:
Find a vector equation of L1.
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18M.1.SL.TZ1.S_9a:
Show that →AB=(68−5)
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18N.2.SL.TZ0.S_8b.i:
Find the value of y.
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17N.1.AHL.TZ0.H_2b:
Find the coordinates of the point of intersection of the line L with the plane Π.
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19N.2.SL.TZ0.S_2a:
Find the point of intersection of L1 and L2.
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17N.1.SL.TZ0.S_9c:
The point D has coordinates (q2, 0, q). Given that →DC is perpendicular to L, find the possible values of q.
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22M.1.AHL.TZ1.11a:
Show that the three planes do not intersect.
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21N.2.AHL.TZ0.11a.i:
Find the vector →AB and the vector →AC.
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21N.2.AHL.TZ0.11a.ii:
Hence find the equation of Π1, expressing your answer in the form ax+by+cz=d, where a, b, c, d∈ℤ.
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21N.2.AHL.TZ0.11d.i:
Find the reflection of the point B in the plane Π3.
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21N.2.AHL.TZ0.11d.ii:
Hence find the vector equation of the line formed when L is reflected in the plane Π3.
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21N.2.AHL.TZ0.11c.i:
Show that at the point P, λ=34.
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21N.2.AHL.TZ0.11c.ii:
Hence find the coordinates of P.
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18M.1.AHL.TZ2.H_9f.ii:
Find YZ.
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19M.1.AHL.TZ1.H_11b:
Verify that P is the point of intersection of the two lines.
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18M.1.SL.TZ1.S_9c.ii:
Write down the value of angle OBA.
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17M.2.AHL.TZ2.H_9g:
Find the acute angle between the planes Π2 and Π3.
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SPM.2.AHL.TZ0.7:
Two ships, A and B , are observed from an origin O. Relative to O, their position vectors at time t hours after midday are given by
rA = (43)+t(58)
rB = (7−3)+t(012)
where distances are measured in kilometres.
Find the minimum distance between the two ships.
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18M.1.SL.TZ1.S_9b.ii:
Point C (k , 12 , −k) is on L. Show that k = 14.
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18M.1.AHL.TZ2.H_9f.i:
Find the coordinates of X, Y and Z.
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18M.1.AHL.TZ2.H_9c:
Find the area of the parallelogram ABCD.
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18M.1.SL.TZ2.S_1b:
The vector (2p0) is perpendicular to →AB. Find the value of p.
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20N.1.SL.TZ0.S_7a:
Find an expression for the velocity of P1 at time t.
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20N.1.SL.TZ0.S_9a:
Express →AB in terms of m.
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EXN.2.AHL.TZ0.11c:
Show that the Cartesian equation of the plane Π that contains the triangle ABC is -x+y+z=-2.
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EXN.2.AHL.TZ0.11d.i:
Find a vector equation of the line L.
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17N.1.AHL.TZ0.H_2a:
Find a vector equation of the line L passing through the points A and B.
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18N.1.AHL.TZ0.H_9c:
Show that L does not intersect the y-axis for any negative value of b.
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17M.2.AHL.TZ2.H_9e:
Verify that 2j + k is perpendicular to the plane Π2.
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17M.1.SL.TZ1.S_8c:
The lines L1 and L1 intersect at C(9, 13, z). Find z.
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18M.1.SL.TZ1.S_9c.i:
Find →OB∙→AB.
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18N.1.AHL.TZ0.H_9a:
Find, in terms of b, a Cartesian equation of the plane Π containing this triangle.
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20N.1.SL.TZ0.S_9c:
Consider a unit vector u, such that u=pi-23j+13k, where p>0.
Point C is such that →BC=9u.
Find the coordinates of C.
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EXN.1.AHL.TZ0.8a:
Show that l1 and l2 are never perpendicular to each other.
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17M.1.SL.TZ1.S_8b:
A second line L2, has equation r = (113−14)+s(p01).
Given that L1 and L2 are perpendicular, show that p=2.
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19M.1.SL.TZ1.S_2a:
Find c.
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EXN.2.AHL.TZ0.11e:
Find the volume of right-pyramid ABCD.
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17M.1.SL.TZ2.S_9b.ii:
Find |→AB|.
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21M.1.AHL.TZ2.8a:
Show that l1 and l2 do not intersect.
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21M.1.AHL.TZ1.11c:
It is given that the lines L1 and L2 have a unique point of intersection, A, when a≠k.
Find the value of k, and find the coordinates of the point A in terms of a.
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21M.1.AHL.TZ2.8b:
Find the minimum distance between l1 and l2.
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21N.2.AHL.TZ0.11b:
The line L is the intersection of Π1 and Π2. Verify that the vector equation of L can be written as r=(0-20)+λ(11-1).