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HOLIDAY ASSIGNMENT 1. Find in dot product form, the equation of the plane r 1 3 2 i 1 4 j k 2a) c) A and B are points with position vectors a and b respectively. D is a point on the line joining A to B such that AD : DB 3 : 4 . Find the position vector of D in terms of a and b . Find the symmetric equation of the straight line passing through the point (1,2,1) and is normal to the plane 2 x 3 y z 2 . Find the point of intersection of the line in (b) above with the plane x y 2 z 9 . 3. Find the shortest distance between two skew lines: b) x 2 y 2 z 1 1 2 1 x 1 y 2 z and 2 3 1 4. A plane passes through the point (1,2,3) and is perpendicular to the vector i 5 j 4k . The plane meets the z plane in P and the y plane in Q . Find: i) the equation of the plane. ii) the distance PQ . 5a) Determine the equation of the plane through the points A(1, 1, 2) , B(2, 1, 3) and C (1, 2, 2) . A line through the point D(13, 1, 2) and parallel to the vector 12i 6 j 3k meets the plane in (a) at point E . Find: i) the coordinates of E . ii) The angle between the line and the plane. b) 6. If i) ii) iii) iv) v) cos ec sin m and sec cos n, prove that m 2 n 2 (m 2 n 2 3) 1. x cos cos 2 , and y sin sin 2 , show that x 2 y 2 cos 2 2 cos 3 cos 4 and that 2 xy sin 2 2 sin 3 sin 4 tan 2 3 tan 2 2 , show that cos 2 3 cos 2 2 x a cos b cos 3 , y a sin b sin 3 , show that x 2 y 2 (a b) 2 4ab sin 2 x tan sin , y tan sin , prove that ( x 2 y 2 ) 2 16 xy k 1 sin A. . Find all the angles for k 1 0 o x 360 o which satisfy the equation 2 tan x tan( 30 o x) 0. 7. Prove that if tan x k tan( A x) , then sin( 2 x A) 8. Prove that if tan tan( ) k , then (k 1) cos( 2 ) (1 k ) cos . Find all the angles for 0 o x 360 o which satisfy the equation tan tan( 3) . © MATHEMATICS DEPARTMENT 2011 Email: [email protected] Page 1 HOLIDAY ASSIGNMENT 9. If cos cos a , sin sin b , prove that cos( ) 12 (a 2 b 2 2) and tan 12 ( ) b a . Hence solve the simultaneous equations cos cos 1 , sin sin 1.5 10. Given that sin sin a , cos cos b , prove that tan 12 ( ) a b and sin( ) 2ab (a 2 b 2 ) . 11. Show that the perpendicular distance from a point P( x1 , y1 ) to a line ax by c 0 is ax1 by1 c 12. 13. 14. 15. . a2 b2 Find the equation of the line through the origin and concurrent with 2 x 5 y 3 and 3x 4 y 2 . Prove that the quadrilateral with vertices (2, 1), (2, 3), (5, 6), (5, 4) is a parallelogram. Find the equations of the straight lines drawn through the point (1, -2), making angles 45o with the x-axis. ABCD is a quadrilateral with A2, 2 , B5, 1 , C 6, 2 and D3, 1 . Show that the quadrilateral is a rhombus. 16. One side of the rhombus is the line y 2 x , and two opposite vertices are the points 0, 0 and 4 12 , 4 12 . Find the equations of the diagonals, the coordinates of the other two vertices and the length of the side. 17. The curve C is given by y ax 2 b x where a and b are constants. Given that the gradient of C at the point 1, 1 is 5 , find a and b . 18. A tangent to the parabola x 2 16 y is perpendicular to the line x 2 y 3 0 . Find the equation of this tangent and the coordinates of its point of contact. 1 1 Show that the tangent to the curve y 4 2 x 2 x 2 at points 1, 4 and , 2 , 2 2 1 1 respectively, pass through the point , 5 . Calculate the area of the curve 2 4 enclosed between the curve and the x axis. 19. 20. Show that the gradient of the curve y x( x 3) 2 is zero at the point P1, 4 , and sketch the curve. The tangent at P cuts the curve again at Q. Calculate the area contained between the chord PQ and the curve. © MATHEMATICS DEPARTMENT 2011 Email: [email protected] Page 2 HOLIDAY ASSIGNMENT 21. If :i) ii) iii) iv) dy d2y and in terms of . dx dx 2 2 dy 3 d y 2 2 and show that (4 x 3 y) 3 0 3x 8 xy 3 y 3 find dx dx 2 2 d2y dy 2 a x ay 2 by c where a, b, c are constants, prove that dx 2 dx 1 1 y A tan 2 x B(2 x tan 2 x), where A and B are constants, prove that x a cos 3 , (1 cos x) y b sin 3 find d2y y dx 2 2 v) vi) vii) d 2 y dy y (5 x 3) , show that y 2 5 dx dx 3 d2y dy 2 show that 4 1 x 2 4x 9y 0 y x 1 x2 2 dx dx 2 d 2 y dy y sec x , prove that y 2 y 4 dx dx 2 22. Express 12 x 8 x 2 5 in the form a( x b) 2 c and deduce the maximum value of the curve. 23. Express 2 3x 4 x 2 in the form a b( x c) 2 and find the turning point stating whether its a maximum or a minimum. 24. The sum of two numbers is 24. Find the two numbers if the sum of their squares is to be minimum. 25. A right circular is circumscribed about a sphere of radius a . If h is the distance form the centre of the sphere to the vertex of the cone, show that the volume of the cone is a 2 (a h) 2 . Find the vertical angle of the cone when the volume is minimum. 3(h a) 26. A right circular cylinder is inscribed in a sphere of given radius a . Show that the volume of the cylinder is h(a 2 14 h 2 ) , where h is the height of the cylinder. Find the ratio of the height to the radius of the cylinder when its volume is greatest. 27. A right circular cone of vertical angle 2 is inscribed in a sphere of radius a . Show that the area of the curved surface of the cone is a 2 (sin 3 sin ) and prove that its greatest area is 8a 2 3 3 . © MATHEMATICS DEPARTMENT 2011 Email: [email protected] Page 3 HOLIDAY ASSIGNMENT 28. A right circular cylinder is inscribed in a sphere of given radius a . Prove that the total area of its surface (including its ends) is 2a 2 (sin 2 cos 2 ) , where a cos is the radius of an end. Hence prove that the maximum value of the total area is a 2 ( 5 1) . TOPICS YOU MUST GO THROUGH BEFORE COMING BACK TO SCHOOL Introducing Pure Mathematics Advanced Level Pure Maths TOPIC SERIES Book 1 and 2 Understanding Pure Mathematics Advanced Level Pure Mathematics By Tranter SUBTOPIC - ARITHMETIC PROGRESSION - GEOMETRIC PROGRESSION - PROOF BY INDUCTION REFERENCE Intrd P M Ch. 9 Under P M Ch. 8 BackHse 1 Ch. 13 Tranter Ch. 3 - PERMUTATIONS - COMBINATIONS BHSE 1 Ch. 12 Tranter Ch. 3 Undersg Pure Mths Ch. 7 BINOMIAL THEOREM - PASCAL’S TRIANGLE - BINOMIAL EXPANSION EXPONENTIAL & LOGARITHMIC FUNCTIONS - EXPONENTIAL FUNCTIONS BHSE 1 Ch. 14 Tranter Ch. 3 BHSE II Ch. 4 Intrdg PM Ch. 10 Understanding PM Ch. 8 BACKHOUSE 2 Ch. 2 Understanding PM Ch .19 Tranter Ch. 13 - LOGARITHMIC FUNCTIONS MACLAURINS’ THEOREM INTEGRATION(II) BACKHOUSE 2 Ch .16 Tranter Ch.13 BACK HOUSE 2 Ch. 1 , 2 & 13 DIFFERENTIAL EQUATIONS Intrdg Pure Mths Ch. 20 BHSE 2 Ch. 19 Understadg Pure Mths Ch. 20 I Please engage the girls into serious research and discussions during this holiday. NOTE: An Assessment will be done when you get back to school. © MATHEMATICS DEPARTMENT 2011 Email: [email protected] Page 4