DINGWALL ACADEMY NATIONAL QUALIFICATIONS. Mathematics Higher Prelim Examination 2010/2011 Paper 1 Assessing Units 1 & 2.

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1 INGWLL EMY Mathematics Higher Prelim Eaminatin 00/0 Paper ssessing Units & NTIONL QULIFITIONS Time allwed - hur 0 minutes Read carefull alculatrs ma NOT be used in this paper. Sectin - Questins - 0 (0 marks) Instructins fr the cmpletin f Sectin are given n the net page. Fr this sectin f the eaminatin u shuld use an H pencil. Sectin (0 marks). Full credit will be given nl where the slutin cntains apprpriate wrking.. nswers btained b readings frm scale drawings will nt receive an credit. Pegass 00

2 Read carefull heck that the answer sheet prvided is fr Mathematics Higher Prelim 00/0 (Sectin ). Fr this sectin f the eaminatin u must use an H pencil and, where necessar, an eraser. Make sure u write ur name, class and teacher n the answer sheet prvided. The answer t each questin is either,, r. ecide what ur answer is, then, using ur pencil, put a hrizntal line in the space belw ur chsen letter (see the sample questin belw). 5 There is nl ne crrect answer t each questin. 6 Rugh wrking shuld nt be dne n ur answer sheet. 7 Make sure at the end f the eam that u hand in ur answer sheet fr Sectin with the rest f ur written answers. Sample Questin line has equatin. If the pint (k,7) lies n this line, the value f k is 7 5 The crrect answer is. The answer shuld then be clearl marked in pencil with a hrizntal line (see belw). hanging an answer If u decide t change an answer, carefull erase ur first answer and using ur pencil, fill in the answer u want. The answer belw has been changed t. Pegass 00

3 FORMULE LIST ircle: The equatin g f c 0 represents a circle centre ( g, f ) and radius g f c. The equatin ( a) ( b) r represents a circle centre ( a, b ) and radius r. Trignmetric frmulae: sin cs sin cs sin cs cs sin cs cs sin sin sin cs cs sin cs sin Pegass 00

4 SETION LL questins shuld be attempted. sequence is defined b the recurrence relatin U U 6 with U. n n The value f U is 0 0. Tw functins are defined n suitable dmains as f (g()) is 5 0 f ( ) and g ( ) The line abve has as its equatin Pegass 00

5 . Given that is a cnstant f integratin, d equals can be epressed in the frm ( a) b. What is the value f b? 6 6. What is the value f cs + sin The functin f ( ) p 0 is such that f ( ) 0. The value f p is Pegass 00

6 8. The diagram ppsite shws part f the graph f a trignmetrical functin. f() The mst likel functin culd be f ( )... cs cs cs 0 cs 9. If is a factr f the plnmial k 0, then k equals The sketch belw shws part f the curve 5. P The gradient f the tangent t the curve at the pint P(5, 0) is Pegass 00

7 . Part f the graph f the functin f () is shwn belw. (, 5 ) Which f the fllwing graphs represents the related functin f ( )? (8, 5 ) ( 8, 5 ) (, ) 5 (, ) 8. The equatin f the line passing thrugh (, ) and parallel t the line with equatin 5 0 is ( ) ( ) ( ) ( ) Pegass 00

8 . Which f the fllwing epressins is/are equal t (9a)? () () () a 9 7 9a a all three epressins nl epressin () nl epressin () epressins () and (). The equatin f a circle, centre (, 6), with the -ais as a tangent is ( 6) ( ) 6 ( ) ( 6) ( ) ( 6) 6 ( ) ( 6) 6 5. The equatin k 0 has real rts. The range f values f k is k 0 k 0 k k 0 Pegass 00

9 6. functin is defined as f ( ). Which f the fllwing statements is true abut this functin? it is never increasing it is never statinar it is never decreasing it has tw statinar pints 7. If, then 0 equals 8. quadratic functin, f, where f ( ) a b c, is such that a 0 and b ac 0. Which f the fllwing is a pssible sketch f the graph f f()? f() f() f() f() Pegass 00

10 9. If sin 5 and, then tan is equal t 0. The area enclsed between the curves the shaded area in the diagram belw. and 8 is shwn as (0,8) 8 Which f the fllwing gives the area f the shaded sectin? 8 8 d 8 8 d 8 d 8 d [ EN OF SETION ] Pegass 00

11 SETION LL questins shuld be attempted. The diagram belw, which is nt drawn t scale, shws part f the curve with equatin p, where p is a cnstant. is a statinar pint and has as its -crdinate. p (a) (b) (c) cnsidering the derivative f, and using the -crdinate f pint t help u, find the value f p. Establish the crdinates f the ther statinar pint. (all relevant wrking must be shwn) The pint n the curve has as its -crdinate. Find the equatin f the tangent t the curve at.. Slve the equatin cs cs, fr 0. 6 Pegass 00

12 . sequence f numbers is defined b the recurrence relatin U n au n 0, where a is a cnstant. (a) Given that U 6 and U 8, find algebraicall, the value f a. (b) Hence find the limit f this sequence. (c) The sequence is such that L ku 0, where L is the limit f the sequence, U 0 is the the initial value and k is a number. Find the value f k.. In the diagram belw, which is nt drawn t scale, triangle is issceles with. is the mid-pint f. units and units as shwn. ngle. (a) Shw clearl that sin. (b) Hence shw that cs = - [ EN OF SETION ] [ EN OF QUESTION PPER ] Pegass 00