1 | (IB/s1/2017/November/Paper2/q10)
[Maximum mark: 17] Note: In this question, distance is in millimetres. Let f(x)=x+asin(x−π2)+a, for x≥0
The graph of f passes through the origin. Let Pt be any point on the graph of f with x-coordinate 2kπ, where k∈N. A straight line L passes through all the points Pk.
Diagram 1 shows a saw. The length of the toothed edge is the distance AB. |
2 | (Continue: 1)
The toothed edge of the saw can be modelled using the graph of f and the line L. Diagram 2 represents this model. The shaded part on the graph is called a tooth. A tooth is represented by the region enclosed by the graph of f and the line L, between Pt and P−z+1
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3 | (IB/s1/2017/May/paper2tz1/q7)
[Maximum mark: 7] A particle P moves along a straight line. Its velocity vpms−1 after t seconds is given by vP=√tsin(π2t), for 0≤t≤8. The following diagram shows the graph of vp.
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4 | (IB/s1/2017/May/paper2tz1/q8)
[Maximum mark: 14] At Grande Anse Beach the height of the water in metres is modelled by the function h(t)=pcos(q×t)+r, where t is the number of hours after 21:00 hours on 10 December 2017 . The following diagram shows the graph of h, for 0≤t≤72. The point A(6.25,0.6) represents the first low tide and B(12.5,1.5) represents the next high tide.
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5 | (IB/sl/2017/May/paper2tz2/q4)
[Maximum mark: 6] The depth of water in a port is modelled by the function d(t)=pcosqt+7.5, for 0≤t≤12, where t is the number of hours after high tide. At high tide, the depth is 9.7 metres. At low tide, which is 7 hours later, the depth is 5.3 metres.
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6 | (IB/s1/2016/November/Paper2/q10)
[Maximum mark: 15] The following diagram shows the graph of f(x)=asinbx+c, for 0≤x≤12. The graph of f has a minimum point at (3,5) and a maximum point at (9,17).
The graph of g is obtained from the graph of f by a translation of (k0). The maximum point on the graph of g has coordinates (11.5,17).
The graph of g changes from concave-up to concave-down when x=w.
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7 | (IB/s1/2016/May/paper1tz1/q3)
[Maximum mark: 7] Let f(x)=3sin(π2x), for 0≤x≤4.
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8 | (IB/s1/2016/May/paper2tz1/q9)
[Maximum mark: 14] A particle P moves along a straight line so that its velocity, v ms−1, after t seconds, is given by r=cos3t−2sint−0.5, for 0≤t≤5. The initial displacement of P from a fixed point O is 4 metres.
The following sketch shows the graph of v.
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9 | (IB/sl/2016/May/paper2tz2/q4)
[Maximum mark: 8] The height, h metres, of a seat on a Ferris wheel after t minutes is given by h(t)=−15cos1.2t+17, for t≥0
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10 | (IB/sl/2019/May/paper1tz2/q7)
[Maximum mark: 7] Consider the graph of the function f(x)=2sinx,0≤x<2π. The graph of f intersects the line y=−1 exactly twice, at point A and point B. This is shown in the following diagram.
Consider the graph of g(x)=2sinpx,0≤x<2π, where p>0.
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11 | (IB/s1/2018/November/Paper1/q7)
[Maximum mark: 6] Given that sinx=13, where 0<x<π2, find the value of cos4x. |
12 | (IB/s1/2018/May/paper2tz1/q10)
[Maximum mark: 15] Let f(x)=12cosx−5sinx,−π≤x≤2π, be a periodic function with f(x)=f(x+2π). The following diagram shows the graph of f. There is a maximum point at A. The minimum value of f is −13.
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13 | (IB/s1/2018/May/paper2tz1/q10b)
A ball on a spring is attached to a fixed point O. The ball is then pulled down and released, so that it moves back and forth vertically. The distance, d centimetres, of the centre of the ball from O at time t seconds, is given by d(t)=f(t)+17,0≤t≤5
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14 | (IB/s1/2018/May/paper2tz2/q6)
[Maximum mark: 8] At an amusement park, a Ferris wheel with diameter 111 metres rotates at a constant speed. The bottom of the wheel is k metres above the ground. A seat starts at the bottom of the wheel. The wheel completes one revolution in 16 minutes.
After t minutes, the height of the seat above ground is given by h(t)=61.5+acos(π8t). for 0≤t≤32,
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15 | (IB/s1/2016/November/Paper1/q2)
[Maximum mark: 5] Let sinθ=√53, where θ is acute.
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16 | (IB/sl/2019/November/Paper2/q6)
[Maximum mark: 6] The diagram below shows a triangular-based pyramid with base ADC. Edge BD is perpendicular to the edges AD and CD. AC=28.4 cm,AB=x cm,BC=x+2 cm,ABC=0.667,BAD=0.611 Calculate AD. |
17 | (IB/s1/2019/November/Paper1/q6)
[Maximum mark: 8] Let f(x)=4cos(x2)+1, for 0≤x≤6π. Find the values of x for which f(x)>2√2+1 |
18 | (IB/sl/2019/May/paper1tz1/q3)
[Maximum mark: 7] The following diagram shows a right triangle ABC. Point D lies on AB such that CD bisects ACB.
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19 | (IB/s1/2019/ May / paper 2tz2/q3)
[Maximum mark: 7] The following diagram shows the quadrilateral ABCD. AB=6.73 cm,BC=4.83 cm,BCD=78.2∘ and CD=3.80 cm
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20 | (IB/s1/2018/November/Paper2/q7)
[Maximum mark: 6] A communication tower, T, produces a signal that can reach cellular phones within a radius of 32 km. A straight road passes through the area covered by the tower's signal. The following diagram shows a line representing the road and a circle representing the area covered by the tower's signal. Point R is on the circumference of the circle and points S and R are on the road. Point $ is 38 km from the tower and RST =43∘,
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21 | (IB/sl/2018/May/paper2tz1/q6)
[Maximum mark: 7] Triangle ABC has a=8.1 cm,b=12.3 cm and area 15 cm2. Find the largest possible perimeter of triangle ABC. |
22 | (IB/s1/2018/May/paper2tz2/q2)
[Maximum mark: 6] The following diagram shows quadrilateral ABCD. AB=11 cm,BC=6 cm,BAD=59∘,ADB=100∘, and CBD=82−
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23 | (IB/s1/2017/November/Paper2/q1)
[Maximum mark: 6] The following diagram shows a triangle ABC. AB=5 cm,CAB=50∘ and ACB=112∘
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24 | (IB/sl/2017/ May / paper 1tz1/q3)
[Maximum mark; 6] The following diagram shows triangle PQR. PQR=30∘,QRP=45∗ and PQ=13 cm Find PR.
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25 | (IB/sl/2017/May/paper2tz2/q9) [Maximum mark: 15] A ship is sailing north from a point A towards point D. Point C is 175 km north of A. Point D is 60 km north of C. There is an island at E. The bearing of E from A is 055 . The bearing of E from C is 134′′. This is shown in the following diagram.
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26 | (IB/s1/2016/May/paper1tz1/q6)
[Maximum mark: 7] The following diagram shows triangle ABC. The point D lies on [BC] so that [AD] bisects BAC AB=2√5 cm, AC=x cm, and DˆAC=θ, where sinθ=23. The area of triangle ABC is 5 cm2. Find the value of x. |
27 | (IB/s1/2016/May/paper2tz1/q3)
[Maximum mark: 7 ] The following diagram shows three towns A,B and C. Town B is 5 km from Town A, on a bearing of 070∘. Town C is 8 km from Town B, on a bearing of 115∘.
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28 | (IB/s1/2016/May/paper2tz2/q2)
[Maximum mark: 6] The following diagram shows a quadrilateral ABCD. AD=7 cm,BC=8 cm,CD=12 cm,DˆAB=1.75 radians, AˆBD=0.82 radians.
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29 | (IB/s1/2015/May/paper1tz1/q5)
[Maximum mark: 7 ] Given that sinx=34, where x is an obtuse angle, find the value of
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30 | (IB/sl/2015/May/paper1tz2/q2)
[Maximum mark: 6] Let f(x)=asinbx, where b>0. The following diagram shows part of the graph of f.
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31 | (IB/sl/2015/May/paper2tz1/q8)
[Maximum mark: 13] The following diagram shows a straight shoreline, with a supply store at S, a town at T, and an island L. A boat delivers supplies to the island. The boat leaves S, and sails to the island. Its path makes an angle of 204 with the shoreline.
It is decided to change the position of the supply store, so that its distance from L is 5 km. The following diagram shows the two possible locations P and Q for the supply store.
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32 | (IB/s1/2015/May/paper2tz2/q1)
[Maximum mark: 6] The following diagram shows triangle ABC. BC=10 cm,ABC=80∘ and BAC=35′
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33 | (IB/sl/2015/November/Paper2/q8)
[Maximum mark: 14] The following diagram shows the quadrilateral ABCD. AD=6 cm,AB=15 cm,ABC=44∘,ACB=83∗ and DAC=θ
The area of triangle ACD is half the area of triangle ABC.
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Answer (Trigo)
1 (a) Show (b)(i) P0=(0,0),P1(2π,2π) (ii) y=x (c) Show
2 (d) 33
3 (a) (i) t=2 (ii) 9.65 (b) k=5.94
4 (a)(i) 6.25 (ii) 0.9 (b) (i) p=0.45 (ii) q=4π25 (iii) r=1.05 (d) 23:00
5 (a) p=2.2 (b) q=π7 (c) 7.01
6 (a)(i) c=11 (ii) b=π6 (iii) a=−6 (b) (i) k=2.5 (ii) g(x)=−6sin(π6(x−2.5))+11 (c)(i) w=8.5 (ii) π
7 (a)(i) 3 (ii) 4 (b) Graph
8 (a) 0.284 (b) t=0.180 (c) 2 (d) 0.744 (e) 3.28
9 (a) h(0)=2 (b) k=1.48 (c) 5.2
10 (a) x=7π6,11π6 (b) p=712
11 1181
12 (a) A(−0.395,13) (b) (i) 13 (ii) 2π (c) f(x)=13cos(x+0.395)
13 (d) 13 (e) 1.02
14 (a) k=6 (b) a=−55.5 (c) t=18.5
15 (a) cosθ=23 (b) cos2θ=−19
16 34.6
17 0≤x<π2,7π2<x<9π2
18 (a) 45 (b) 725 (c) 50
19 (a) 5.50 (b) θ=92.0∘ or 1.61
20 (a) Show (b) 37.5
21 perimeter =40.6
22 (a) DB=9.57 (b) DC=10.6
23 (a) BC=4.13 (b) Area =3.19
24 PR=13√22
25 (a) 235 (b) 146 (c) 193 (d) 27.3
26 x=94
27 (a) AˆBC=135∘ (b) 12.1 (c) AˆCB=17.0∘
28 (a) BD=9.42 (b) BˆDC=1.51
29(a) cosx=−√74 (b) cos2x=−18
30(a)(i) π (ii) 3 (b) (i) a=3 (i) b=2
31(a) 9 (b) SˆQL=142∘,SˆPL=38.0∘ (c) (i) new store is at Q (ii) 4.52
32(a) AC=17.2 (b) area =77.8
33(a) AC=10.5 (b) area =62.9 (c) θ=86.7,93.3 (d) 12.4
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