DINGWALL ACADEMY NATIONAL QUALIFICATIONS. Mathematics Higher Prelim Examination 2009/2010 Paper 1 Assessing Units 1 & 2. Time allowed - 1 hour 30

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1 INGWLL EMY Mathematics Higher Prelim Eaminatin 009/00 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 eaminatinn 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 009

2 Read carefull heck that the answer sheet prvided is fr Mathematics Higher Prelim 009/00 (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 009

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 m sin sin sin cs cs sin cs sin Pegass 009

4 SETION LL questins shuld be attempted. straight line with gradient passes thrugh the pints (,) and ( 6, k). The value f k is 5. Which f the fllwing gives equal rts fr the equatin + k + = 0 k = k = k = 6 k =. The rate f change f the functin f ( ) = when = is. Given that cs = 5, the eact value f sin is 7 5 cannt be fund Pegass 009

5 a 0 5. d is 6a a a 6. The gradient f the tangent t the circle = 0 at the pint (,) n the circumference is 5 7. If f ( ) = and g( ) = 6 then g ( f ()) is equal t Three f the fllwing are equal in value. Which is the eceptin? cs π sin 5 6 π cs π sin 6 π Pegass 009

6 9. Part f the graph f the functin = f () is shwn ppsite. = f () Which f the fllwing graphs represents the related functin = f ( ) +? (, ) (,) (,0) (, ) 0. PQRS is a parallelgram. P, Q and R have crdinates (, ), (,5) and (7,7) respectivel. The crdinates f S are (, ) (,0) (, ) (5,9). recurrence relatinship is defined as U = 0 U b n + 6 n +. If the limit f the recurrence relatinship is 0, the value f b is 50 Pegass 009

7 . The minimum turning value f the functin f ( ) = + is. The diagram belw shws part f the curve with equatin = 0 + k. The curve crsses the -ais at (,0). = k The value f k is If a = b then b equals a a a a Pegass 009

8 5. The equatin f the circle, centre ( 5,6) and having the -ais as a tangent is ( + 5) + ( 6) = 5 ( 5) + ( + 6) = 5 ( + 5) + ( 6) = 6 ( 5) + ( + 6) = 6 6. If cs θ = -, fr 0 θ π, the value f θ is, - and and 7. The range f values f fr which the functin f ( ) = + is decreasing is < > < >. functin is defined as f ( ) = 7. It has a statinar pint at = and = nl = nl = = and = Pegass 009

9 9. The diagram, which is nt drawn t scale, shws parts f the graphs f the curve = and the line =. = = The shaded area abve can be represented b 0 ( ) d ( ) d 0 0 ( ) d ( ) d Given that sin = a and cs = a then tan equals a a + a a a + a + [ EN OF SETION ] Pegass 009

10 SETION LL questins shuld be attempted. The diagram belw, which is nt drawn t scale, shws part f the graph f the curve = and the straight line =. The line intersects the curve at three pints. L = O = (a) Find the crdinates f the pint. (b) The line L is the tangent t the curve at. Establish the equatin f the line L.. functin is given b the equatin, + p 6 = 0, where p is a cnstant. (a) Given that = - is a rt f this functin, find the value f p. (b) Hence, slve the functin full, given that > 0. Pegass 009

11 . n equatin in is given as 5 k = + k, where k is a cnstant and nn-zer. (a) Shw that this equatin can be written as + 5k + k = 0 (b) Hence find the tw values f k fr which the equatin + 5k + k = 0 has equal rts.. PQRS is a rectangle measuring 6 units b units. Pints,, and are pints n the sides f the rectangle such that Q = S = and P = R = as shwn. P Q S R (a) Shw that the area f is given b the functin ( ) = +. (b) Hence find the value f which minimises the area f and calculate this minimum area. 5 [ EN OF SETION ] [ EN OF QUESTION PPER ] Pegass 009