Advanced Higher Grade

Transcription

1 Prelim Eamination / 5 (Assessing Units & ) MATHEMATICS Advanced Higher Grade Time allowed - hours Read Carefully. Full credit will be given only where the solution contains appropriate woring.. Calculators may be used in this paper.. Answers obtained by readings from scale drawings will not receive any credit.. This eamination paper contains questions graded at all levels. Pegasys

2 All questions should be attempted cos. (a) Given that f ( ), obtain and simplify ( ) (b) Differentiate ( sin ) f. (). (). Use Gaussian elimination to solve the following system of equations y y y z z 5z 6 (5). Given that z i is a root of the equation z z 8 z find the value of. () Hence find all the roots of this equation (). Find the term independent of in the epansion of () 5. The parametric equations of a curve are cos t t sin t and y sin t t cos t. Show that dy d tant () Find the equation of the tangent to the curve at the point where π t. () 6. Epress 8 in partial fractions. () Hence evaluate 8 d, giving your answer correct to decimal places. () Pegasys

3 7. Use the substitution u to obtain d 6. () 8. The function f is defined by ( ) f where R,. (a) Find the equations of the asymptotes of the graph of f ( ). () (b) Prove that the graph of f ( ) has no turning points or points of inflection. () (c) Setch the graph of f ( ) indicating all the important features. () 9. (a) Find the equation of the line through the points (, ) and (6, 5). () Hence find the volume of the solid shape generated by rotating the line segment, shown in the diagram, about the -ais. (, ) y (6, 5) () (b) Prove that the volume of the solid shape generated with height h and radii r and R, where r < R, is given by the formula y (h, R) ( R R r r ) π h (, r) (5) Pegasys

4 . (a) The sum of the first n terms of a series is given by S n n 5n Find an epression for the n th term, u n. () Prove that the series is arithmetic and write down the first four terms. () (b) Find the value of θ, π < θ <, such that 6 cos θ cos θ cos θ... (5) [ END OF QUESTION PAPER ] Pegasys

5 Maring Scheme - Advanced Higher Prelim / 5 (a) (b) Give one mar for each Illustrations for awarding each mar sin mars ans: ( ) differentiates cosine correctly nows how to use quotient rule differentiates quotient correctly simplifies answer ( sin ) ans: 9 nows how to use chain rule nows how to differentiate sin - mars sin ( ) ( ) ( ) sin ( ) ( ) sin ( ) ( ) sin simplifies answer ( sin ) 9. ans:, y, z 5 mars using augmented matri 6 5 first modified system second modified system 8 finding z finding and y z y, Pegasys

6 Maring Scheme - Advanced Higher Prelim Mathematics &. Give one mar for each Illustrations for awarding each mar ans: i, i, 5 mars nows to substitute i in for z finds correctly states conjugate root finds quadratic factor divides cubic by quadratic factor and finds third root. ans: mars nows to find general term finds general term correctly nows to put power of equal to zero finds correct coefficient ( i ) ( i) 8( i) i z z z r ( ) r 6r r 6 r r 9 9 r 5. ans: π y 6 mars nows to use product rule d dy finds and correctly dt dt dy nows how to find and proves result d nows how to find - and y- coordinates nows how to find gradient finds correct equation of line d sint sin t t cost t cost dt dy cost cost t sint t sint dt dy dy dt t sin t tant d d t cost dt π π ; y π m tan π y Pegasys

7 Maring Scheme - Advanced Higher Prelim Mathematics & Give one mar for each Illustrations for awarding each mar 6. ans: 68 6 mars 7. now how to find partial fractions finds A and B substitutes epression into integral integrates terms correctly deals with logs correctly substitutes limits correctly π ans: represent integral in terms of u integrates correctly changes limits evaluates limits mars A B 8 A ; B 6 6 d 8 6 ln ln 6 6 ln 6 ln du u tan u u ; u tan tan tan ( ) 6( ) d π u π 8(a) ans:, y mars finds equation of vertical asymptote restates function states equation of horizontal asymptote 8(b) ans: proofs mars differentiates f and shows there is no solution sets ( ) finds second derivative shows there is no solution to f ( ) f ( ) y f ( ) ( ) no solution 8 f ( ) ( ) 8 no solution Pegasys

8 Maring Scheme - Advanced Higher Prelim Mathematics & Give one mar for each Illustrations for awarding each mar 8(c) ans: see graph mars finds intercept setch of graph 9(a) ans: 78 π 5 mars finds gradient of line finds equation of line uses correct formula for finding volume of revolution integrates correctly evaluates limits correctly y ; y (, ) correct behaviour at asymptotes 5 m 6 y 6 Volume π y d 6 π 78 π 6 d π 6 9(b) ans: h ( R R r r ) π 5 mars finds gradient finds equation of line uses correct formula for finding volume of revolution integrates correctly evaluates limits correctly R r m h R r y r h h R r Volume π r d h ( ) R r r R r π r h h π h ( R R r r ) as required h (a) ans: u n n, d Sequence is 6, 8,,, 6 mars nows how to find the n th term substitutes n and n into formula for S n finds correct epression for u n nows to find common difference by substituting n and n into formula for u n finds common difference, hence proving series is arithmetic finds first four terms u n Sn Sn [ ] n 5n ( n ) 5( n ) u n n u n u u n [ ( n ) ] n u n u n n ( ) a S 5 6, Sequence is 6, 8,, Pegasys

9 Maring Scheme - Advanced Higher Prelim Mathematics & (b) Give one mar for each Illustrations for awarding each mar π ans: θ 5 mars recognises series is geometric finds common ratio finds epression for sum to infinity puts sum to infinity equal to and attempts to solve finds θ cos θ cos θ r cos θ cos θ a S n r cos θ sin θ cos θ sin θ π π sinθ ± θ as < θ < Graph for Question 8(c) 67 Mars y y Pegasys

10 Higher Still - / 5 MATHEMATICS Advanced Higher Grade Mini Prelim (Unit Units / Revision) Time allowed - 5 minutes Read Carefully. Full credit will be given only where the solution contains appropriate woring.. Calculators may be used.. Answers obtained by readings from scale drawings will not receive any credit.. This Unit Test contains questions graded at all levels. Pegasys

11 All questions should be attempted n. Prove by induction that n for all n N. (5) d y dy. Find the particular solution to the differential equation y, d d dy given that y and when. () d. Find the first five terms of the Maclaurin series for Use this series to find an approimate value for e. () e d. () Use integration by parts to find the eact value of the above integral. (). If A show that A 6A 5I, where I is the identity matri. () 5. Find the Cartesian equation of the plane with vector equation ( t ) i ( t u ) j ( t u ) r and show that the point (,, ) lies on the plane. () 6. (a) Epress the comple number z i in polar form. () (b) Find 8 z, epressing your answer in the form a bi. () Pegasys

12 7. (a) A parallelogram has vertices A(, ), B(6, ), C(7, 7) and D(, 6). 8 Show that the image of this parallelogram under the transformation is also a parallelogram. () (b) A rectangle has vertices P(, ), Q(, -), R(6, -) and S(, ). 8 Show that the image of PQRS under the transformation is not a rectangle, and state what shape the image actually is. (5) 8. Water runs into a tan at a rate of cm per second. The tan is in the shape of a cone as shown in the diagram and has a height of cm and a base radius of cm. How fast is the water level rising when the water is 75 cm deep? [ Note: The volume of a cone is given by V π r h ] cm cm () [ END OF QUESTION PAPER ] Pegasys

13 Pegasys Maring Scheme Mini Prelim Give one mar for each Illustrations for awarding each mar. ans: proof 5 mars show true for n state inductive hypothesis consider the case for n carry out manipulation state conclusion ; RHS LHS Assume Consider ( ) So, if the formula is true for n, it is valid for n. Since it is valid for n, it is therefore true for all n.. ans: e y sin cos mars creating and solving the auiliary equation stating the complementary function nowing how to find the particular solution stating the particular solution i m m m ± ( ) B A e y sin cos B when d dy A when y e y sin cos

14 Maring Scheme Mini Prelim (cont.). Give one mar for each Illustrations for awarding each mar ans: e mars finds f ( ) and f ( ) iv finds f ( ), f ( ) and f ( ) substitutes coefficients correctly into Maclaurin s epansion substitutes series correctly into integral integrates correctly substitutes limits in correctly nows the formula for integration by parts applies formula correctly handles the second integral evaluates limits. ans: proof mars finds A nows identity matri finds 6A 5I and hence proves result f ( ) ; f ( ) iv f ( ) ; f ( ) 8; f ( ) 6 y 5 d [ ( ) ]. e d e d d e e e or equivalent e 7 6 A 8 9 I A 5I ans: 5 y z 5 mars method substituting t in terms of into y substituting u in terms of and y into z proving point lies on plane. t ; y t u ; z t u t y u u y 5 y z 5 5 ( ) 5 as required Pegasys

15 Maring Scheme Mini Prelim (cont.) 6(a) Give one mar for each Illustrations for awarding each mar π π ans: z cos isin mars finds modulus finds argument writes comple number in polar form z ( ) π θ π π z cos isin 6(b) ans: 6 ( ) 6 i mars nows to use De Moivre s Theorem applies De Moivre s correctly to modulus applies De Moivre s correctly to argument writes in the form a bi 7(a) ans: proof mars nows how to find images of vertices finds images of vertices correctly nows to chec that opposite sides are parallel checs gradients for both pairs of sides ( ) 8 6 π 8 6π 6 cos6π isin 6π 6 i ( ) ( ) (or individually) A 7,5, B 6,6, C 77,, D, 5 m A B mc D ; mb C ma D 7 ( ) ( ) ( ) ( ) 7(b) ans: parallelogram 5 mars finds images of vertices correctly nows to find gradients of adjacent sides shows that gradients do not multiply to give - finds slopes of other two sides concludes that image is a parallelogram P ( 9.5), Q (,5), R (,), S ( 8,) 5 m P Q ; mq R 9 5 image is not a rectangle 9 5 m R S ; mp S 9 image is a parallelogram Pegasys

16 Maring Scheme Mini Prelim (cont.) Give one mar for each Illustrations for awarding each mar 8. ans: cm/s mars nows to find epression for r in terms of h finds correct epression epresses formula for volume in terms of h nows to differentiate differentiates V with respect to h uses chain rule dh finds epression for dt dv nows from question that dt dh subs information into epression for dt finds rate of change of water level at 75 cm r r h h h V π h π h 7 dv π h dh 9 dv dv dh dt dh dt dh 9 dv dt π h dt dv dt dh 9 dt π 75 cm/s 5 Mars Pegasys