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Advanced Higher Revision Unit 1 Outcome 1 1. Expand (u 3v) 4 by the binomial theorem. 2. Expand (2 x 3 y)5 by the binomial theorem. 3. D E 2 2 8 5 2 Express x in the form Ax Bx Cx . x x x4 4. 2 Find the y term in the expansion of 3 y 2 y 5. Expand 6. Find the term independent of 4 5 2 (2 x)4 and use your expansion to find x a) (2.1)4 b) (1.9)4 . in the expansions 3 2x 2 x 6 a) 1 4x x 4 b) n 1 n 36 2 1 7. Solve for n: 8. Prove that 9. Express 10. Obtain partial fractions for n n n 1 3 2 3 x in partial fractions. ( x 2)( x 3) a) 10 x2 2 ; ; b) 2 2 x ( x 1) ( x 1)( x 3) b) g ( x) ( x 1)sin 2 x c) x 3 4x 1 . ( x 2)( x 1) Unit 1 Outcome 2 1. Differentiate with respect to x: a) f ( x) exp(sin 2 x) 2. Given that f ( x) x tan 2 x , show that 3. Show that the function equation y c) h( x ) ln 3 x . cos 3 x f ''( x) 4sec2 2x 1 2xtan2x . sin(kx) where x 0 and k is a non-zero constant, satisfies the differential x d 2 y 2 dy k 2 y 0. 2 dx x dx 4. The turning effect, T, of a power boat, is given by the formula T 4 cos x sin x , 0 x 2 where x is 2 the angle (in radians) between the rudder and the central line of the boat. Find the size of x which maximises the turning effect. Unit 1 Outcome 3 1. Find a) x x 3dx 2x 4 b) 2 x 3dx b) x3 dx x 1 c) 3 0 6x dx 2 x2 3 d) sin xcos 4 xdx d) x 3x dx 4 2 2. Integrate by substitution. a) 3. e x x dx a) Show that ln 2 ln 3 2 e x e x 9 dx ln x x e e 5 c) b) Find 3 0 2 1 e2 x dx (e2 x 1)2 12 x3 6 x x 4 x 2 1dx 4. A car is travelling along a straight road. Its acceleration at time t seconds is a(t ) 20 6t 4t 2 measured in metres per second per second. The car started from rest at time t 0 from a point O on the road. Find the speed of the car and its distance from O after t seconds. 5. By using the substitution 1 x sin , evaluate 3 6. If a(t ) 1 t , and when t 2, v 1 & s 4 1 6 0 1 1 2 2 dx . (1 9 x ) 2 find expressions for v (t ) & s (t ). 3 7. A particle P moves in a straight line and passes a fixed point O with a velocity of V m/s. Its acceleration, a m/s2 is given by a 16 4t for 0 t 3 & a t 1 for t 3 , when t is the time in seconds after passing O. Given that the velocity of P when t 3 is 38m/s find the velocity of v (t ) when a) t 2 b) t 4. 8. Find the volume of the solid of revolution which is created when one revolution around the area bounded between the x axis and x axis of the y x 1 between x 1 and x 1. 2 9. Find the volume of the solid of revolution created by the area lying in the first quadrant, bounded by the curve y x 2 , the y axis, and the line y 4 . Unit 1 Outcome 4 1. For each of the following functions, state whether they are even, odd, or neither. a) f ( x) 4 3 x 2 b) f ( x) 3 x 2 x 2. Sketch the function f ( x) x 3. Sketch the graph y c) f ( x) x 1 x d) f ( x) x3 x 1 showing all important points and lines. Use this result to sketch f ( x) . x x2 , marking on any asymptotes, stationary points, and points where axes are x2 1 crossed. 4. Sketch the function f ( x) x 1 , marking on all important features as above. Use this result to x( x 1) f 1 ( x). sketch f ( x 2) and Unit 1 Outcome 5 x 1. Use Gaussian Elimination to solve the equations 4 y x 2y z 5 3 y 2 z 6. 2. For what value of a and b will the following system of equations have 2x x y 3y 3z a) No solution b) Infinitely many solutions c) A unique solution? 5 2z 8 7 x 4 y az b 3. Which of these systems would you consider ill conditioned? a) 2x 3x 7y 5 b) 10 y 6 2x 3x 7y 5 10 y 6 Unit 2 Outcome 1 1. Find the inverse of the following functions a) y x 3 b) y 4 x c) y 2 x 4 d) y 1 . x2 2. Differentiate 1 2 a) sin x b) ex sin 1 2 x 1 c) ln x tan x 2 x2 d) cos 1 (1 x) 3. Find the Cartesian Equation for each of the following parametric forms. x 3t 2 , y 4 t 2 a) 4. Find b) x 1 2 , y t 2 1 t d2y dy and in terms of t when dx dx 2 x t 2, y t2 4 a) b) x t 4t , y 2 1 t. 5. Find the Cartesian Equation for the parametric form x sin 2cos , y 2sin cos . 6. Find dy in terms of x and y if: dx a) 7. Find an expression for 2 y 3 3 y xy x 2 b) 6 x 2 3xy 2 y 4 . dy given that dx a) y 2x b) y x2 x 1 c) y 2xx 8. If r, the radius of a circle increases at the rate of 2cm/s, find an expression in terms of r for the rate at which the area of the circle is increasing. 9. Air is being pumped into a spherical balloon at a rate of 54cm 3/s. Find the rate at which the radius is increasing when the volume of the balloon is 36 cm3. x 1 cos t sin t , y 1 cos t sin t. . 2 2 If v and a are the velocity and acceleration at any instant, prove that v a 4 . 10. A particle is moving in a plane under the equations of motion Unit 2 Outcome 2 1. Express 3 in partial fractions, and hence evaluate 2 2x 5x 2 4 2 3 dx to 3.d.p. 2 x 5x 2 2 2. Find the following integrals. a) 1 25 16 x 2 dx (Hint: let u 4x .) b) 3. Find the following indefinite integrals in terms of a) x( x 2) dx ; u x 2 9 b) 1 2 9 x2 dx c) x 2 1 dx 4 x 29 x using the suggested substitution. 1 1 16 x 2 dx ; u 4x c) 1 2 dx ; x sin u 3 4 9x 2 4. Use integration by parts to evaluate 0 x sin xdx. 5. Find; a) 4 2 ( x 1) ln(2 x)dx b) 4 0 x cos xdx c) e x cos 2 xdx d) 2 0 x3e x dx 8x x3 11x 3 dx dx . 6. Find a) 2 b) ( x 4)5 9 x2 dy 1 2 given that x 1 when y 2. 7. Solve dx x y dy (2 x 2 4 x 1)( y 3) if when x 0 , y 7. 8. Solve ( x 2) dx 9. a) Express 1 in partial fractions. x x2 b) The spread of disease in a large population can be modelled by means of a differential equation. The proportion x of the population infected with the disease after t days satisfies i) Given that x dx 1 1 x x 2 for t 0 . dt 2 2 1 when t 0 , find x in terms of t . 500 ii) Verify that about 6% of the population was infected after seven days. iii) How long will it take for 25% of the population to become infected? t 0 , just one individual with a contagious dn k ( N n)n where n(t ) is disease. Assume that the spread of the disease is governed by the equation dt the number of infected individuals after a time t days and k is a constant. 10. A large population of a) Find N individuals contains, at time n explicitly as a function of t . b) Given that measurements reveal that half the population is infected after 100 days, show that k ln( N 1) /100 N. c) What value does 11. a) Express n(t ) approach as t tends to infinity. 1 in partial fractions. x x2 b) The spread of disease in a large population can be modelled by means of a differential equation. The proportion x of the population infected with the disease after t days satisfies i) Given that x 1 when t 0 , find x in terms of t . 500 ii) Verify that about 6% of the population was infected after seven days. iii) How long will it take for 25% of the population to become infected? dx 1 1 x x 2 for t 0 . dt 2 2 Unit 2 Outcome 3 1. Given that z 2 3i , plot on an Argand diagram the points that represent the complex numbers z , z and 2 z , where z is the complex conjugate of z. 2. Express in the form a ib a) 2 3i 1 i 3. Find the modulus and principle argument of 4. If z 2 cos 3 i sin a) b) 1 i 2i 5 4 3i b) c) 1 6i . 1 6i (3 i ) 2 . Hence express in polar form. 1 i 6 find z . 3 sin 3 3sin 4sin 3 5. Use De Moivre’s Theorem to show that Hence obtain an expression for cos 3 4 cos3 3cos tan 3 in terms of tan . 6. Find the locus of P when a) 7. Given that z 3 b) z 2 4 c) 2 i is a root of the equation z 3 11z 20 0 8. Show that 1 i is a root of the equation Hence find all roots. z 1 3i 5 11. Given 4 . find the remaining roots. x 4 3x 2 6 x 10 0 9. By Using De Moivre’s Theorem or otherwise, find the roots of the equation 10. Given that d) arg z z4 4 0 z 2 i is one root of z 4 z 3 z 2 9 z 30 0 find all the remaining roots. z 2 3 2i find i) z 2 , ii) z10 . 12. Solve the following equations, leaving your answers in polar form. Illustrate the solutions on an Argand diagram. a) z 2 2i 4 b) z 5 3 3 3i . Unit 2 Outcome 4 3 and S 40. 5 2 2. Find the sum of the first seven terms of a geometric series which has u8 and u5 18 . 3 1. Find the first term of a geometric progression which has a common ratio of 3. An arithmetic sequence has a common difference of 7, and the 25th term equals 300. What is the first term? 4. The fourth term of an arithmetic sequence is -3 and the tenth term is -15. a) Identify the sequence b) For what value of n is 5. By writing un 35 ? (k 1)2 k 2 2k 1 show that (k 1) n k 1 n Hence show that k k 1 2 n k 2 2 k n. k 1 n(n 1) . 2 Extend this idea by considering (k 1)3 k 3 to show that n k 2 k 1 n(n 1)(2n 1) . 6 Unit 2 Outcome 5 1. Prove that k (k 2 5) id divisible by 6. Hence prove that, if n is even, then n2 (n2 20) is divisible by 48. 2. Prove by contradiction that given a, b 3. Prove by induction that a) ab is even then at least one of a or b is even. 2n n, n ; b) 2n n2 , n 4, n . n 4. Prove by induction that , if r n r (r 1) n 1 . r 1 n 5. Prove by induction that r 2 (r 1) r 1 n (n 1)(n 2)(3n 1). 12 Unit 3 Outcome 1 2 3 4 1. Find a.(b c) if a 1 , b 1 and c 1 . 4 0 3 2. Find the symmetric form of the equation of a line through the points A(2, 7, 4) and B(1, 2, 4) . 3. Find the equation of the plane passing through the points D(3,1, 4), E (6, 0,1) and F (1, 5, 2). 4. Find the vector and symmetric equation for the line parallel to the vector 3i 4 j k , passing through the point (2,3, 2). 5. The lines and are contained within the plane . Find the equation of the plane. 6. The coordinates of the points L is x 3 y 2 z 2 . 2 2 1 A and B are (0, 2, 5) and (-1, 3, 1) respectively. The equation of the line a) Find the equation of the plane which contains A and is perpendicular to L . b) Verify that B lies in . c) Show that the point C in which L meets is (1, 4, 3), and find the angle between CA and CB . Unit 3 Outcome 2 3 1 4 10 2 , B & C find, where possible; 1 0 3 8 1 T a) A B b) BC c) BA d) AB e) CB f) B 1 g) det B h) B i) the 2x2 matrix D such that BD C . x2 3 3 6 2. Find the possible values x can take given that A , B and AB BA . 2 x 1 3 x 3 2 2 3. If A find the values of m and n if A mA nI . 4 1 1. If A 4. Find the 2x2 matrix which will transform the point (1, 2) to (3, 3) and the point (-1, 1) to (-3, 3). 5. Find the matrices corresponding to the following linear transformations. a) 180o rotation about the origin; b) Enlargement with a scale factor of 3, centre (0, 0); c) Reflection in the line y x ; d) 45o rotation about the origin. 6. Give a geometrical description of the effect of the following transformation matrices; a) 0 1 1 0 b) 5 0 0 5 c) 3 0 0 1 d) 0 3 3 0 e) 1 0 2 1 f) 1 0 . 0 0 1 0 0 2 7. Given that A 0 k 1 , calculate A and find the values of k for which the determinant of the matrix 0 0 k 2 A 2I is zero. ( I is the 3x3 identity matrix.) 1 1 0 1 8. Calculate A where A 2 3 1 2 2 1 Hence solve the system of equations x y 1 2x 3y z 2 2x 2 y z 1 9. For the matrix 2 A , find the values of such that the matrix is singular. 2 3 Write down the matrix A1 when 3. Unit 3 Outcome 3 56. Use Maclaurin’s theorem to find series expansions for each of the following, giving all terms up to and including that in x5 ; a) cos( x) 2. Find the Maclaurin expansion of 3. Prove that the equation b) ex c) e2 x d) (1 x)n . (2 x)2 e x as far as the term in x3 x3 2 x 1 0 has only one root in the interval 1 x 3. Verify that the equation 1 1 can be rewritten as x (2 x 1) 3 . By using the iterative scheme xn1 (2 xn 1) 3 with xo 2 , obtain an approximation to the root which is correct to two decimal places. 1 5 xn 1 xn with x0 2 to calculate x1 , x2 and x3 . 2 xn 4. Apply the recurrence relation Find the fixed points of this recurrence relation. Suggest an initial value to generate a sequence converging to the negative fixed point. Unit 3 Outcome 4 1. An industrial scientist finds that the differential equation t dx 2 x 3t 2 models a production process. dt Find the general solution of the differential equation. Hence find the particular solution given x 1 when t 1. . 2 dy d y 6 x 2 , given that when x 1, 2 and y 3. 2 dx dx 2. Solve the differential equation 3. Find the general solution to each of the following differential equations. d2y dy 3 10 y 0 a) 2 dx dx d2y dy 8 16 y 0 b) 2 dx dx c) d2y dy 2 15 y e4 x 2 dx dx d) 2 e) d2y dy 2 5 y 10 x 1 2 dx dx f) 2 d2y dy 5 3 y 4e5 x 2 dx dx d2y dy 11 12 y 2 x 2 5 x 7 . 2 dx dx Unit 3 Outcome 5 1. Change 523 into a) binary b) octal c) Hexadecimal. 2. Find the gcd of 286 and 142 3. Express the gcd of 132 and 424 in the form 132s 424t where s and t and are to be calculated.