Do NOT Answer all – 4 – …dbhs.wvusd.k12.ca.us/ourpages/auto/2010/8/2... · HL MATH –...

HL MATH – Classwork: Tuesday, 5/5/15 (WITH CALCULATOR)

M12/5/MATHL/HP2/ENG/TZ1/XX– 4 –

3. [Maximum mark: 7]

� $�WHDP�RI���SOD\HUV�LV�WR�EH�VHOHFWHG�IURP����YROOH\EDOO�SOD\HUV��RI�ZKRP���DUH�ER\V�and 2 are girls.

(a) ,Q�KRZ�PDQ\�ZD\V�FDQ�WKH�WHDP�EH�VHOHFWHG" [2 marks]

(b) ,Q�KRZ�PDQ\�RI�WKHVH�VHOHFWLRQV�LV�H[DFWO\�RQH�JLUO�LQ�WKH�WHDP" [3 marks]

(c) ,I�WKH�VHOHFWLRQ�RI�WKH�WHDP�LV�PDGH�DW�UDQGRP��¿QG�WKH�SUREDELOLW\�WKDW�H[DFWO\�RQH�JLUO�LV�LQ�WKH�WHDP� [2 marks]

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0416

M12/5/MATHL/HP2/ENG/TZ1/XX– 5 –

turn over

4. [Maximum mark: 5]

The planes 2 3 5x y z+ − = and x y z k− + =2 intersect in the line 5 1 9 5 5x y z+ = − = − .)LQG�WKH�YDOXH�RI��k .

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0516

M12/5/MATHL/HP2/ENG/TZ1/XX– 12 –

Do NOT write solutions on this page.

Section B

Answer all questions on the answer sheets provided. Please start each question on a new page.

11. [Maximum mark: 14]

� 7KH�IXQFWLRQ� f x x x( ) sin cos= +3 4 �LV�GH¿QHG�IRU�0 2< <x π .

(a) :ULWH�GRZQ�WKH�FRRUGLQDWHV�RI�WKH�PLQLPXP�SRLQW�RQ�WKH�JUDSK�RI��f . [1 mark]

(b) The points P( , )p 3 and Q( , )q 3 , q p> , lie on the graph of y f x= ( ) . Find p and q . [2 marks]

(c) Find the coordinates of the point, on y f x= ( ) , where the gradient of the graph is 3. [4 marks]

� �G�� )LQG�WKH�FRRUGLQDWHV�RI�WKH�SRLQW�RI�LQWHUVHFWLRQ�RI�WKH�QRUPDOV�WR�WKH�JUDSK�DW�WKH�points P and Q. [7 marks]

12. [Maximum mark: 22]

� $�VNL�UHVRUW�¿QGV�WKDW�WKH�PHDQ�QXPEHU�RI�DFFLGHQWV�RQ�DQ\�JLYHQ�ZHHNGD\��0RQGD\�WR�)ULGD\��LV��������7KH�QXPEHU�RI�DFFLGHQWV�FDQ�EH�PRGHOOHG�E\�D�3RLVVRQ�GLVWULEXWLRQ��

(a) Find the probability that in a certain week (Monday to Friday only)

(i) there are fewer than 12 accidents;

(ii) WKHUH�DUH�PRUH�WKDQ���DFFLGHQWV��JLYHQ�WKDW�WKHUH�DUH�IHZHU�WKDQ����DFFLGHQWV� [6 marks]

� 'XH�WR�WKH�LQFUHDVHG�XVDJH��LW�LV�IRXQG�WKDW�WKH�SUREDELOLW\�RI�PRUH�WKDQ���DFFLGHQWV�LQ�D�GD\�DW�WKH�ZHHNHQG��6DWXUGD\�DQG�6XQGD\��LV�������

(b) $VVXPLQJ�D�3RLVVRQ�PRGHO�

(i) FDOFXODWH�WKH�PHDQ�QXPEHU�RI�DFFLGHQWV�SHU�GD\�DW�WKH�ZHHNHQG��6DWXUGD\�DQG�6XQGD\��

(ii) FDOFXODWH�WKH�SUREDELOLW\�WKDW��LQ�WKH�IRXU�ZHHNHQGV�LQ�)HEUXDU\��WKHUH�ZLOO�EH�PRUH�WKDQ���DFFLGHQWV�GXULQJ�DW�OHDVW�WZR�RI�WKH�ZHHNHQGV� [10 marks]

� ,W�LV�IRXQG�WKDW������RI�VNLHUV�KDYLQJ�DFFLGHQWV�DUH�DW�OHDVW����\HDUV�RI�DJH�DQG������DUH�XQGHU����\HDUV�RI�DJH�

(c) $VVXPLQJ� WKDW� WKH� DJHV� RI� VNLHUV� KDYLQJ� DFFLGHQWV� DUH� QRUPDOO\� GLVWULEXWHG��¿QG�WKH�PHDQ�DJH�RI�VNLHUV�KDYLQJ�DFFLGHQWV� [6 marks]

1216

HL MATH – Classwork: Tuesday, 5/5/15 (WITH CALCULATOR) SOLUTIONS

– 7 – M12/5/MATHL/HP2/ENG/TZ1/XX/M

SECTION A

1. 2d 3 12dy x x kx � � M1A1

For use of discriminant 2 4 0b ac� or completing the square � �23 2 12x k� � � (M1) 144 12 0k� (A1)

Note: Accept trial and error, sketches of parabolas with vertex (2,0) or use of second derivative.

12k A1

[5 marks]

2. 1

2

11 2

1

12 d 1 ( 0.61556 )2 d

x

x

k x kx

� }³³

(M1)(A1)

12

1E( ) 2 d 2.39....xX k x x k ³ or 1.47 M1A1

Note: Condone missing dx in any part of the question. [4 marks]

3. (a) 10

2106

§ · ¨ ¸

© ¹ (M1)A1

[2 marks]

(b) 8

2 1125§ ·

u ¨ ¸© ¹

(M1)A1A1

Note: Accept 210 – 28 – 70 = 112

[3 marks]

(c) 112 8 0.533210 15

§ · ¨ ¸© ¹

(M1)A1

[2 marks]

Total [7 marks]

HL MATH – Classwork: Tuesday, 5/5/15 (WITH CALCULATOR)

– 8 – M12/5/MATHL/HP2/ENG/TZ1/XX/M

4. point on line is 1 5

5x O� � ,

9 55

y O� , z O or similar M1A1

Note: Accept use of point on the line or elimination of one of the variables using the equations of the planes

1 5 9 5 25 5

kO O O� � �� � M1A1

Note: Award M1A1 if coordinates of point and equation of a plane is used to obtain linear equation in k or equations of the line are used in combination with equation obtained by elimination to get linear equation in k.

2k � A1 [5 marks] 5. (a) 50 A1 [1 mark] (b) Lower quartile is 4 so at least 26 obtained a 4 R1 Lower bound is 26 A1

Minimum is 2 but the rest could be 4 R1

So upper bound is 49 A1

Note: Do not allow follow through for A marks.

Note: If answers are incorrect award R0A0; if argument is correct but no clear lower/upper bound is stated award R1A0; award R0A1 for correct answer without explanation or incorrect explanation. [4 marks]

Total [5 marks]

6. ( ) ( 3) 2 ln( 3) 2h x f x x � � � � (M1)(A1) ( ) ( ) 2 ln( 3)g x h x x � � � M1

Note: Award M1 only if it is clear the effect of the reflection in the x-axis: the expression is correct OR there is a change of signs of the previous expression OR there’s  a  graph  or  an  explanation  making it explicit

2lne ln( 3)x � � M1

2eln3x

§ · ¨ ¸�© ¹

A1

[5 marks]

HL MATH – Classwork: Tuesday, 5/5/15 (WITH CALCULATOR)

– 12 – M12/5/MATHL/HP2/ENG/TZ1/XX/M

12. (a) (i) ~ Po(11)X (M1) P( 11) 0.579X d (M1)A1

(ii) P( 8 12)X X! � (M1)

P(8 12) P( 11) P( 8) 0.3472... or or P( 12) P( 11) 0.5792...

X X XX X

§ ·� � d � d ¨ ¸� d© ¹

A1

0.600 A1 N2

[6 marks] (b) (i) ~ Po( )Y m P( 3) 0.24Y ! (M1) P( 3) 0.76Y d (M1)

2 31 1e 1 0.762 6

m m m m� § ·� � � ¨ ¸© ¹

(A1)

Note: At most two of the above lines can be implied.

Attempt to solve equation with GDC (M1) 2.49m A1

(ii) ~ Po(4.98)A P( 5) 1 P( 5) 0.380A A! � d } M1A1 ~ B(4, 0.380 )W } (M1) P( 2) 1 P( 1) 0.490W Wt � d M1A1

[10 marks] (c) P( 25) 0.8A� , P( 18) 0.4A�

25 0.8416PV�

} (M1)(A1)

18 0.2533 (or 0.2534 from tables)PV�

� } � (M1)(A1)

solving these equations (M1) 19.6P A1

Note: Accept just 19.6, 19 or 20; award A0 to any other final answer. [6 marks]

Total [22 marks]

HL MATH – Classwork: Tuesday, 5/5/15 (WITH CALCULATOR)