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(b) ……………………………………..........
(Total 4 marks) 15. rr
The vectors i , j are unit vectors along the xaxis and yaxis respectively.
r
rr
r
r
The vectors u = – i + j and v = 3 i + 5 j are given.
(a) r
r
r
r
Find u + 2 v in terms of i and j .
r
r
r
A vector w has the same direction as u + 2 v , and has a magnitude of 26. 19 (b) r
r
r
Find w in terms of i and j . Working: Answers:
(a) …………………………………………..
(b) ……………………………………..........
(Total 4 marks) 16. Two functions f and g are defined as follows:
f(x) = cos x, 0 = ≤ x ≤ 2π; g(x) = 2x + 1, x∈ . Solve the equation (g o f) (x) = 0.
Working: Answers:
....……………………………………..........
(Total 4 marks) 20 17. The function f is given by
F( x ) = 2 x + 1 , x ∈
x−3
(a) (i) , x ≠ 3. Show that y = 2 is an asymptote of the graph of y = f(x).
(2) (ii) Find the vertical asymptote of the graph.
(1) (iii) Write down the coordinates of the point P at which the asymptotes intersect.
(1) (b) Find the points of intersection of the graph and the axes.
(4) (c) Hence sketch the graph of y = f(x), showing the asymptotes by dotted lines.
(4) (d) Show that f′(x) = −7
and hence find the equation of the tangent at
( x − 3) 2 the point S where x = 4.
(6) (e) The tangent at the point T on the graph is parallel to the tangent at S.
Find the coordinates of T.
(5) (f) Show that P is the midpoint of [ST].
(l)
(Total 24 marks) 18. One thousand candidates sit an examination. The distribution of marks is shown in the
following grouped frequency table.
Marks
1–10
Number of
15
candidates 11–20 21–30 31–40 41–50 51–60 61–70 71–80 81–90 91–100
50 100 170 260 220 90 45 30 20 21 (a) Copy and complete the following table, which presents the above data as a cumulative
frequency distribution.
(3) Mark ≤10 ≤20 Number of
candidates 15 65 (b) ≤30 ≤40 ≤50 ≤60 ≤70 ≤80 ≤90 ≤100 905 Draw a cumulative frequency graph of the distribution, using a scale of 1 cm for 100
candidates on the vertical axis and 1 cm for 10 marks on the horizontal axis.
(5) (c) Use your graph to answer parts (i)–(iii) below,
(i) Find an estimate for the median score.
(2) (ii) Candidates who scored less than 35 were required to retake the examination.
How many candidates had to retake?
(3) (iii) The highestscoring 15% of candidates were awarded a distinction.
Find the mark above which a distinction was awarded.
(3)
(Total 16 marks) 22 19. 6
The circle shown has centre O and radius 6. OA is the vector , OB is the vector 0 − 6 and
0 5
OC is the vector, 11 . y C
B (a) O A x Verify that A, B and C lie on the circle.
(3) (b) Find the vector AC .
(2) (c) Using an appropriate scalar product, or otherwise, find the cosine of angle OAC.
(3) (d) Find the area of triangle ABC, giving your answer in the form a 11 , where a ∈ . (4)
(Total 12 marks) 20. a 2 Let M = 2 − 1 , where a ∈ (a) . Fi...
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This note was uploaded on 06/24/2013 for the course HISTORY exam taught by Professor Apple during the Fall '13 term at Berlin Brothersvalley Shs.
 Fall '13
 Apple
 The Land

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