G H. Extended Unit Tests B L L. Higher Still Advanced Higher Mathematics. (more demanding tests covering all levels) Contents. 3 Extended Unit Tests

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1 M A T H E M A T I C S H I G H E R Higher Still Advanced Higher Mathematics S T I L L Etended Unit Tests B (more demanding tests covering all levels) Contents 3 Etended Unit Tests Detailed marking schemes Pegasys Educational Publishing Pegasys 005

2 MATHEMATICS Advanced Higher Grade Etended Unit Tests B - UNIT Time allowed - 50 minutes Read Carefully. Full credit will be given only where the solution contains appropriate working.. Calculators may be used. 3. Answers obtained by readings from scale drawings will not receive any credit. 4. This Unit Test contains questions graded at all levels. Pegasys 005

3 All questions should be attempted. Differentiate the following function with respect to, leaving your answer in its simplest form : f ( ) = ln 3 4 ( e ) (3). The first three terms in the epansion of ( + + ) where p, q R. p q are , 5 Find the values of p and q. (4) 3. Find all the possible solutions to the following system of equations : + y + 3z = 4 + 5y z = y 5z = 4 (5) 4. Given y = 4 n d y find a formula for n, the n th derivative of y. (4) 5. Use the substitution = 5sinθ to show that 5 = sin C (5) 6. Find the volume generated, by rotating about the ais, the area enclosed by the 3 curve y = 4 and the ais. (5) Pegasys 005

4 7. Let the function f be given by f ( ) = + 5, (a) (i) Write down the equation of the vertical asymptote. () (ii) Show y = f() has a non-vertical asymptote and obtain its equation. () (b) Find the coordinates and nature of the stationary points of f(). (4) (c) Sketch the graph of y = f ( ). (You must show all of the above results in your sketch ) (4) END OF QUESTION PAPER Pegasys 005

5 Advanced Higher Grade - Etended Unit Tests B Marking Scheme UNIT Give mark for each Illustration(s) for awarding each mark ans: or + 4 knowing to use the chain rule knowing to use product rule completing simplification d ( ) e e ( 3e + 4e ) e answer. ans: p = -4, q = 3 knowing to convert 3 terms to using Binomial epansion creating a system of equations 4 solving equations ( + ( + )) p q. ( ) ( ) + 5 p + q + 0 p + q +.. 5p = 0, 5q+ 0p = 75 4 answer 3. ans: (5 7t, 7t 9, t) 5 marks using augmented matri first modified system second modified system 4 setting z = parameter & finding y in terms of parameter 5 finding in terms of parameter z = t y = 7t 9 = 5 7t Pegasys 005

6 Give mark for each Illustration(s) for awarding each mark 4. ans: n n ( ) 4 n! n+ ( 4 ) finding first & second derivative finding third derivative recognising factorial function 4 recognising pattern dy = 4 ( 4 ) d y 3 = 4 ( 4 ) 3 dy ( ) 3 = = ( ) n n! 4 answer ans: proof 5 marks dealing with substitution finding and simplifying integral using double angle formula 4 integrating correctly 5 replacing θ 5 5sin θ = 5cosθ = 5cosθ dθ.. = 5sin θ dθ 5 5sin θ dθ = ( cosθ) dθ 5 4 θ sin θ 5 answer 6384π 6. (a) ans: cubic units 05 5 marks knowing to find zeros knowing how to find volume integrating 4 evaluating integral 5 answer = 0 = 04, 4 0 ( ) π 4 3 π [ 0] 5 answer Pegasys 005

7 Give mark for each Illustration(s) for awarding each mark 7. (a) i. ans: = stating equation (a) ii. ans: restating the function stating equation mark marks 9 y = + + y = + (b) ans: (-, ) Ma (5, 0) Min knowing to differentiate and solve f ()=0 solving f () = 0 finding y coordinates 4 justifying nature 9 f ( ) = = 0 ( ) 4 5= 0 = 5, = y = 0, f ( ) < 0 so Ma 4 f () 5 > 0 so Min (c) ans: sketch sketch showing all relevant points and turning points showing how curve approaches asymptotes completing curve 4 reflecting negative parts of y = f() in - ais (-,) y 5 (-,-) o 5 Total : 37 marks Pegasys 005

8 MATHEMATICS Advanced Higher Grade Etended Unit Tests B - UNIT Time allowed - 50 minutes Read Carefully. Full credit will be given only where the solution contains appropriate working.. Calculators may be used. 3. Answers obtained by readings from scale drawings will not receive any credit. 4. This Unit Test contains questions graded at all levels. Pegasys 005

9 All questions should be attempted. Find, in its simplest form, dy = when : (a) y (3) (b) y = sin + (4). Find 9 40i and plot both solutions on an Argand diagram. (4) 3. Find the Arithmetic sequence whose sum to n terms is given by the formula S n = 3n 5n. (4) 4. (a) Find partial fractions for 9 a, where a is a constant. (3) (b) Hence show that = ln 9 a 6a 3 a 3+ a + C (3) 5. Use the method of proof by contradiction to show that 5 is irrational. (4) 6. Prove by induction that for all positive integers, n, n rr ( + 4) = nn ( + )( n+ 3 6 r = ) (5) Pegasys 005

10 7. (a) Find the stationary points of the curve given by the equations = 5t y = t 3 6t. (3) d y (b) By considering find the nature of these stationary points., (4) END OF QUESTION PAPER Pegasys 005

11 Advanced Higher Grade - Etended Unit Tests B Marking Scheme UNIT Give mark for each Illustration(s) for awarding each mark. (a) ans: ( ln + ) knowing to take ln of both sides knowing to differentiate implicitly completing the simplification ln y = ln dy = ln + y answer (b) ans : ( + ) knowing to use chain rule d knowing sin knowing to use quotient rule 4 completing the simplification & answer + d +. ans: ± ( 5 4i) using correct method epanding bracket & forming a system of equations solving system of equations 4 diagram Let ( a+ bi) = 9 40 i a + abi b = 9 40i a b = 9, ab= 40 a = ± 5 b= m 4 4 diagram, (5, -4) & (-5, 4) 3. ans : -, 4, 0, 6,.. knowing to find S n knowing method for finding u n finding formula for u n 4 finding sequence S ( n ) ( n = 3 5 n ) un = Sn Sn un = 6n 8 4 answer Pegasys 005

12 Give mark for each Illustration(s) for awarding each mark 4. (a) ans: a( 3 a) a( 3+ a) knowing to epress fraction as a sum knowing to find A, B calculating A, B A B = 9 a 3 a + 3+ a A( 3+ a) + B( 3 a) = A =, B = a a (b) ans: proof knowing to epress the integral in PF s integrating completing proof a( 3 a) a( 3+ a) ln 3 a ln 3+ a + C a 3 3 answer 5. ans: proof stating false assumption knowing to contradict false statement proving statement is false 4 concluding statement a Let 5 = b, ab, Z and have no common factors 5b = a a = 5k, k Z b = 5k b= 5l, l Z 4 contradiction of original statement 6. ans : proof 5 marks knowing to try for one value of n assume true for n = k attempting to prove true for n = k + 4 simplifying 5 concluding statement n= LHS = 5, RHS = 5 true k rr ( + 4) = kk ( + )( k+ 3) 6 r= k + rr ( + 4) r= k = rr ( + 4) + ( k+ )( k+ 5) r= 4 ( k + )( k + )( k + 5) 6 5 By induction, true n Pegasys 005

13 Give mark for each Illustration(s) for awarding each mark 7. (a) ans: (5, -4), (-5, 4) differentiating w.r.t. solving dy = 0 substituting into and y dy dt dy 6t 6 = 6t 6, = 5 = dt 5 6t 6 = 0 t = ± 5 =± 5, y = m4 (b) ans: Ma (-5, 4), Min (5, -4) d y knowing how to find d y finding substituting for t 4 conclusion d y = d dt dy dt d y t = 5 d y d y t =, > 0 and t =, < 0 4 answer Total : 37 marks Pegasys 005

14 MATHEMATICS Advanced Higher Grade Etended Unit Tests B - UNIT 3 Time allowed - 50 minutes Read Carefully. Full credit will be given only where the solution contains appropriate working.. Calculators may be used. 3. Answers obtained by readings from scale drawings will not receive any credit. 4. This Unit Test contains questions graded at all levels. Pegasys 005

15 All questions should be attempted. Use the Euclidean Algorithm to show that 7 and 55 are coprime. Hence find integers and y such that y =. (5). (a) Find the values of k for which the matri A is singular. 3 A= k 3 k 9 k (b) Find the coordinates of the image of the point P( 3, 3) under an anticlockwise rotation of 0 o about the origin. (4) (3) 3. (a) (i) Epress ln (+ ) as a power series up to the term 5. (ii) Epress ln ( 3) as a power series up to the term 5. (4) (b) Hence show that ( ) ln 3 = (3) 4. Find the general solution of the following differential equation dy + ( 3 ) y = 3 (5) 5. (a) Show that these lines intersect and find the point of intersection. y+ 3 z 5 y+ z = = and = = (5) (b) Calculate the size of the acute angle between these two lines. (3) (c) Find the equation of the face defined by these two lines. (3) 6. Find the general solution of the following second order differential equation d y dy 4 y e = 3 (5) END OF QUESTION PAPER Pegasys 005

16 Advanced Higher Grade - Etended Unit Tests B Marking Scheme UNIT 3 Give mark for each Illustration(s) for awarding each mark ). ans: ( 55, 7 = coprime = -6, y = 5 5 marks knowing to find the gcd of 55 & 7 finding the gcd stating coprime 4 knowing to rearrange the algorithm 5 correctly rearranging the algorithm & 55 = 7( 4) = 7( 7) = 8( ) + 8= ( 4) gcd = so coprime 4&5 7(-6) + 55(5) =. (a) ans : k = -5, k =3 knowing to find det A finding det A correctly equating to zero 4 solving equation 4 ( k ) + ( k k) + ( k ) k + k 5 k + k 5= 0 answer (b) ans : ( 3, 3) knowing to find related matri evaluating matri finding image cos0 sin0 sin0 cos0 3 3 answer (a) i) ans : & finding f 4 ( 0) substituting above into Maclaurins epansion & f ( 0) =, f ( 0) =, f ( 0) = 4 5 f ( 0) = 6, f ( 0) = 4 answer ii) ans : mark substituting 3 for above Pegasys 005

17 Give mark for each 3. (b) ans : proof factorising quadratic epressing product as a sum simplifying epression Illustration(s) for awarding each mark ( 3 ) = ( + )( 3) ln ln( ) ln( + ) + ln( 3) answer 4. ans : C e 5 marks rearranging equation to form dy + Py ( ) = Q ( ) knowing to integrate P() and find I.F. correctly evaluating I.F. 4 multiplying equation by I.F. 5 solving final equation dy 3 + y = 3 3 = 3 = ln 3 3 ln 3 ln e e = e e = 3 e y e 3 4 C = answer 5. (a) ans: (-, -7, 6) 5 marks creating parametric equations equating corresponding coordinates solving two from three equations for parameters 4 showing parameters satisfy third equation 5 finding coordinates 4 5 = 3t+, y = 4t 3, z = t+ 5 = t +, y = 3t, z = 3t 3t t = 4t 3t = t + 3t = 5 3 t t = t t 4t 3t = =, = ( ) + 3( ) = 5 answer o (b) ans : 35.9 identifying the direction vectors using dot product calculating angle 3 4 and cosθ = answer Pegasys 005

18 5. (c) ans: Give mark for each 9 8y 5z = 7 finding the normal to the plane calculating constant stating equation Illustration(s) for awarding each mark ( 3i+ 4j k) ( i+ 3j 3k) = 9i+ 8j+ 5k = -7 answer 6. ans : y = Ae + Be + e 7 5 marks creating and solving auiliary equation stating the complementary function &4 finding the particular integral 5 stating general solution m m = 0, m= 4& m= CF = Ae + Be Let y Ce =, y = Ce + 4Ce y = 4Ce + 4Ce + 6Ce y y y = 3e C = 7 5 answer Total : 40 marks Pegasys 005