G H. Extended Unit Tests A 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 Extended Unit Tests A (more demanding tests covering all levels) Contents Extended Unit Tests Detailed marking schemes Pegasys Educational Publishing Pegasys 005

2 MATHEMATICS Advanced Higher Grade Extended Unit Tests A - UNIT Time allowed - 50 minutes Read Carefully. Full credit will be given only where the solution contains appropriate working.. 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 005

3 All questions should be attempted. Differentiate the following with respect to x : + x y = ln x (). In the expansion of ( + px)( + qx) 5, where p, q R and p q 0, the coefficient of x is zero and the coefficient of x is 70. Find the values of p and q. (). The function f is defined by f(x) = e x sin x, where 0 x. Find the coordinates of the stationary points of f and determine their nature. (5). Use the substitution x = sin t to evaluate the definite integral 0 x 9 x (5) 5. Use Gaussian elimination to solve the following system of equations a a a + b + b + c c 5c = = = (5) 6. A particle is moving along a straight line so that at a time t its displacement x from a fixed point on the line is given by x = cos( t + 6). Prove that v ax is always constant,where v is the velocity and a is the acceleration of the particle at time t. () Pegasys 005

4 7. The function f is defined by f ( x) = x + 9x x, x ± (a) (i) Write down the equations of the vertical asymptotes () (ii) Show that y = f (x) has a non-vertical asymptote and obtain its equation. () (iii) Find the point(s) of intersection with the two axes. () (b) Find the coordinates and nature of the stationary points of f(x). (5) (c) Sketch the graph of y = f (x). (You must show all of the above results in your sketch ) () [ END OF QUESTION PAPER ] Pegasys 005

5 Advanced Higher Grade - Extended Unit Tests A Marking Scheme UNIT. Give mark for each ans: x marks knowing to use chain rule knowing to use quotient rule completing simplification Illustration(s) for awarding each mark x + d + x x x x + x ( x) answer. ans: p = -6, q = marks using Binomial Expansion product of brackets creating a system of equations solving equations + 5( qx) + 0( qx) + 0( qx) qx + 0q x + 0q x px + 5pqx + 0pq x +... q + p = 0 0q + 0pq = 70 answer. ans: π 76 7π,. Max,, 7. 6 Min 5 marks differentiating using product rule equating to zero solving for x evaluating y coordinates 5 justifying nature 5 e x ( sin x+ cos x) sin x + cos x = 0 x = π 7π, y = 7. 6, 7. 6 from f ( x) or nature table Pegasys 005

6 Give mark for each Illustration(s) for awarding each mark. ans: marks dealing with substitution finding and new limits simplifying expression integrating correctly 5 evaluating correctly 9 x = 9cos x π = cos t dt, limits = 0, 6 π 6 tan tdt 0 [ ln(cos )] 5 answer π 6 t 0 5. ans: a =, b=, c= marks using augmented matrix first modified system second modified system finding one value 5 finding other values c = b = - and a = 6. ans: proof marks knowing how to calculate v knowing how to calculate a substituting into statement completing the proof v = = 6sin( t + 6) dt dv a = = 8cos( t + 6) dt v ax = 6sin ( t + 6) + 6cos ( t + 6) 6 Pegasys 005

7 Give mark for each Illustration(s) for awarding each mark 7. (a) i) ans: x = ± stating equations ii) ans: y = x restating the function stating equation mark marks answer 0x y = x+ x y = x iii) ans: (0,0) mark for answer (0,0) (b) ans: (-.565, -6.60) Max (.565, 6.60) Min 5 marks knowing to differentiate using quotient rule knowing to solve f (x)=0 solving f (x) = 0 finding y coordinates 5 justifying nature x x 9 f ( x) = ( x ) x x 9 = 0 x + 80 =, x =± 565. y = ±6. 60 f ( 56. ) < 0 so Max 5 f (. 56) > 0 so Min (c) ans: sketch marks sketch showing all relevant points and turning points showing how curve approaches asymptotes completing curve Total : 8 marks Pegasys 005

8 MATHEMATICS Advanced Higher Grade Extended Unit Tests A - UNIT Time allowed - 50 minutes Read Carefully. Full credit will be given only where the solution contains appropriate working.. 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 005

9 All questions should be attempted. Differentiate the following with respect to x : y = sin ( x) (). A curve has parametric equations x = t and y = t t. Find the equation of the tangent to the curve when t =. (). (a) Verify that z = is a solution of the equation z z 9 z 9 = 0. () (b) Express z z 9 z 9 as a product of a linear factor and a quadratic factor with real coefficients. Hence find all the solutions of z z 9 z 9 = 0 (). The first, fourth and eighth terms of an arithmetic sequence are in geometric progression. Find : (i) the relationship, in its simplest form, between a, the first term, and d, the common difference; () (ii) the value of r the common ratio. () 5. (a) Find partial fractions for x + ( x )( x + ) () (b) Hence show that 5 x + ( x )( x + ) = 5 ln (5) 6. Prove by induction that for all positive integers, n, n r= = r ( r + ) n + (5) Pegasys 005

10 7. (a) Find the stationary point lying between the lines x = and x = of the curve given by the equation x + y = 6xy, x > 0, y > 0. () d y (b) By considering d x, determine the nature of this stationary point. (5) [ END OF QUESTION PAPER ] Pegasys 005

11 Advanced Higher Grade - Extended Unit Tests A Marking Scheme UNIT Give mark for each Illustration(s) for awarding each mark. (a) ans: x( x) marks knowing to use the chain rule d knowing ( sin ) completing the simplification ( x ) x ( x ) x d ( x ). ans: y = 6x + 9 marks finding coordinates of point differentiating w.r.t. x finding gradient of tangent finding the equation of line (-, 7) dy = t, = t t dt dt dy = t t m= 6 answer. (a) ans : Proof mark knowing to sub z = into the polynomial (b) ans : z = ± i (or equiv.) marks writing the expression as a linear and quadratic factor using quadratic formula to solve the quadratic finding the complex roots ( z )( z + z+ ) z = ± 8 z = ±i z = ± i Pegasys 005

12 Give mark for each. (i) ans: a = 9d marks knowing how to find u, u, u8 u u using = in the geometric u u sequence solving equation Illustration(s) for awarding each mark u = a, u = a+ d, u8 = a+ 7d a+ d a d = + 7 a a+ d answer (ii) ans: r = mark answer 5. (a) ans : x x x + marks knowing to express fraction as a sum knowing to find A, B, C calculating A, B, C (b) ans: proof 5 marks knowing to express the integral in PF s integrating x x integrating x + evaluating integral 5 completing proof x + A Bx + C = + x x + ( x )( x + ) x+ = A( x + ) + ( Bx+ C)( x ) A =, B =, C = 0 5 x x x + ln( x ) ln( x + ) ln ln6 ln ln0 5 5 ln 6. ans: proof 5 marks knowing to try for one value of n assume true for n=k attempt to prove true for n=k+ simplifying 5 concluding statement n = LHS =, RHS = true k = r= rr ( + ) k + k + = r = rr ( + ) k + r = rr ( + ) ( k+ )( k+ ) k + 5 By induction, true n Pegasys 005

13 Give mark for each 5 7. (a) ans:, marks differentiating w.r.t. x solving dy = 0 substituting into original equation solving for x and y Illustration(s) for awarding each mark x + y dy = 6y+ 6x dy 6y x x = 0 y = y 6x x ( ) x x x + 6 = x 6 x = 0 x =, y = 5 (b) ans: Maximum 5 marks differentiating w.r.t. x rearranging equation substituting for x and y 5 conclusion 6x 6y dy + + y d y dy + 6x d y dy y dy 6 6x d y = y 6x 6 d y = Answer < 0 Total : 7 marks Pegasys 005

14 MATHEMATICS Advanced Higher Grade Extended Unit Tests A - UNIT Time allowed - 50 minutes Read Carefully. Full credit will be given only where the solution contains appropriate working.. 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 005

15 All questions should be attempted. (a) Use the Euclidean Algorithm to find integers x and y such that x + 7y =. () (b) Express in base 7. (). (a) Find the first four terms in the Maclaurin series for ( 7 x ) ln () (b) Hence show ln( 7x) = ln x x x x (). The n x n matrices A and B satisfy the equation AB = 7 A + I Where I is the n x n identity matrix. If A and B are both invertible, show that A = ( B 7I ) (). The position, s(t) metres, from the origin at a time t seconds, of a particle satisfies the differential equation d s d t + ds dt s = 70sin x If the particle starts from rest at the origin, find s(t). (9) 5. One face of an irregular tetrahedron has two of its edges defined by the following equations x y z x + y 5 z 7 = = and = = (a) Show that these lines intersect and find the point of intersection. (5) (b) Calculate the size of the acute angle between these two edges. () (c) Find the equation of the face defined by these two edges. () [ END OF QUESTION PAPER ] Pegasys 005

16 Advanced Higher Grade - Extended Unit Tests A Marking Scheme UNIT Give mark for each. (a) ans: x = 7, y = 50 marks knowing to find the gcd of 7 & finding the gcd knowing to rearrange the algorithm correctly rearranging the algorithm Illustration(s) for awarding each mark 7 = ( ) + 9 = ( 9) + 9 = ( ) + 7 & = ( 7) = 5() 5 + 5= ( ) + & (7) + 7(-50) = (b) ans: 7 marks converting to base 0 repeated division by 7 recording generated remainders ( ) + ( ) + ( ) + ( ) = = r 7 = r 7= 0r answer. (a) ans : x x x x 8 8 marks i iv & finding f ( 0) substituting above into Maclaurins expansion simplifying expression & 7 f ( 0) = 9 f ( 0) = f ( 0) = 7 06 f ( 0) = x x x 9! 7! 06 8! x answer (b) ans: proof applying rules of logarithms mark ln( 7 ) ln 7 x = x = ln+ ln 7 x Pegasys 005

17 Give mark for each Illustration(s) for awarding each mark. ans: proof marks making I the subject of the formula using AA - =I identifying A - I = AB 7A AB ( 7I) I = I = A ( B 7 I) answer t 5 0 t. ans: S = e e 7sin t cost 9 marks creating and solving auxiliary equation stating the complementary function }, & 5 finding the particular integral 6 stating the general solution 7 finding ds dt 8 evaluating constants using initial conditions 9 stating particular solution m + m = 0, m = & m = S = Ae t + Be t Let S = C sin t + Dcost S = C cost Dsint S = C sin t Dcost 5 C = 7, D = 6 S = Ae 7 S t + Be t t t = Ae Be 7sint cost 7cost + sint A =, B = 9 answer 5. (a) ans: (,, ) 5 marks creating parametric equations equating corresponding coordinates solving two from three equations for parameters showing parameters satisfy third equation 5 finding coordinates x = t +, y = t +, z = t + x = t, y = t + 5, z = t + 7 t + t = t t = t t = 6 t+ t = t 0 t t t = =, = 0 ( ) ( ) = 5 answer Pegasys 005

18 Give mark for each Illustration(s) for awarding each mark o 5. (b) ans : 70.9 marks identifying the direction vectors using dot product calculating angle and cosθ = 6 answer (c) ans: 7x + 5y + z = 8 marks finding the normal to the plane calculating constant stating equation ( i j+ k) ( i+ j+ k) = 7i 5j k 7 0 = 8 answer Total : 5 marks Pegasys 005