'Quadratic Equation and Marks\r'

'Tak Nga Secondary discip field 2010-2011 Mid-year Exam Form 4 Mathematics (Paper I) time allowed: 1 hour 15 minutes clear up:________ Name:__________________( ) Marks: ________/ 60 Instructions: 1. spell your name, class and class number in the spaces provided on this cover. 2. This paper consists of THREE sections, A(1), A(2) and B. Each section carries 20 marks. 3. Attempt ALL questions in this paper. economise your answers in the spaces provided. Supplementary answer sheets will be supplied on request. Write your name and class number on to each one sheet. 4. Unless otherwise specified, all working essential be clearly shown. . Unless otherwise specified, numerical answers should both be exact or correct to 3 significant figures. 6. The diagrams in this paper are not necessarily drawn to scale. Page 1 of 9 part A(1) (20 marks) 3n ? 5m =4. 2 1. Make n the showcase of the formula (3 marks) ____________________________________________________________ ___________________ ___ _________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ___________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ 2. Calculate (? 3 + 5i ) ? (2 + 7i ) . 4 + 8i (6 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ _______________________________________________________ _____ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ resolve (a) 2r 2 + 20r + 50 , (b) r 2 + 10r + 25 ? s 2 . (4 marks) _____________ _______________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ________________________________________ ____________________ ___________________ Page 2 of 9 3. 4. If f ( x) = x 2 ? 1 and g ( x) = 3 x + 2 , find the honour of 2 f (0) + 3 g (1) . (3 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________ ________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ 5. operate the comparison 1 2 x ? = 3 by the quadratic formula. (Give the answer in concentrated form. ) 2 (4 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ _______________________ _____________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ __________________________________________________ __________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ Page 3 of 9 Section A(2) (20 marks) 6. In the figure, the straight line passing by means of A and B is perpendicular to the straight line passing done A and C, where C is a point lying on the x-axis. (a) catch the comparability of the straight line passing through A and B. (2 marks) ____________________________________________________________ ___________________ _______________________________ _____________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ (b) Find the coordinates of C. 3 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ _________________ ___________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ___________________________________________ _________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ Page 4 of 9 (c) Find the theater of ? ABC. (3 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ________________ ____________________________________________ ___________________ 7. Consider the function f ( x) = x 2 + bx ? 20 , where b is a constant. It is given that the graph of y = f (x) passes through the point (5, 10). (a) Find b. 2 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________ ________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ (b) Let k be a constant. If the comparability f ( x) = k has twain distinct real roots, find the take to the woods of determine of k. (3 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ _______________ ____ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ Page 5 of 9 8. figure P(x) = 2 x 3 ? (h ? 1) x 2 ? 18 x + k . P(x) is divisible by (2x + 1). When P(x) is divided by (x †2), the remainder is †40. (a) Find the values of h a nd k. (4 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ __________ __________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ (b) Factorize P(x) completely. (3 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ________________ ___ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ _______________________ _____________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ Page 6 of 9 Section B (20 marks) 9. It is given that ? and ? are the two roots of the equation 2×2 + 8x ? = 0, where ? > ?. (a) Write down the values of ? + ? and ??. (2 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________ ________________________________ ___________________ (b) Find the value of each of the following expressions without solving the equation. (i) ? 2 + ? 2 (ii) ? ? ? (iii) ? 2 ? 2 (6 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ (c) Form a quadratic equation with roots ? 2 + ? 2 and ? 2 ? ? 2 . (2 marks) ____________________________________________________________ ___________________ ______________________________ ______________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ Page 7 of 9 10. It is given that f ( x) = ? 2 x 2 ? 6 x + c . The graph of y = f ( x) cuts the x-axis at A and B and also cuts the y-axis at C(0, 20). (a) Find the value of c. (1 mark) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ (b) Find the coordinates of A and B. (2 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ______________ ______________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ (c) Find the area of ? ABC . (2 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ _ ___________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ Page 8 of 9 (d) By the method of completing square rewrite the equation y = f ( x) in the form y = a( x ? h) 2 + k . Find the vizor of the graph and axis of symmetry of the graph. (3 marks) ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ _________________________ ___________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ (e) Find the empyrean and co-domain of f(x). 2 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ________________ ___ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ END OF PAPER Page 9 of 9\r\n'