Quadratic Equation and Marks

Tak Nga Secondary School 2010-2011 Mid-year Exam Form 4 Mathematics (Paper I) Time allowed: 1 hour 15 minutes Class:________ Name:__________________( ) Marks: ________/ 60 Instructions: 1. Write 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. Write your answers in the spaces provided. Supplementary answer sheets will be supplied on request. Write your name and class number on each sheet. 4. Unless otherwise specified, all working must be clearly shown. . Unless otherwise specified, numerical answers should either be exact or correct to 3 significant figures. 6. The diagrams in this paper are not necessarily drawn to scale. Page 1 of 9 Section A(1) (20 marks) 3n ? 5m =4. 2 1. Make n the subject of the formula (3 marks) ____________________________________________________________ ___________________ __________________________________ __________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ___________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ 2. Calculate (? 3 + 5i ) ? (2 + 7i ) . 4 + 8i (6 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ _____ _______________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ Factorize (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 value of 2 f (0) + 3 g (1) . (3 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ _______ ___________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ 5. Solve the equation 1 2 x ? = 3 by the quadratic formula. (Give the answer in surd form. ) 2 (4 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ _________ __________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ________________ ____________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ Page 3 of 9 Section A(2) (20 marks) 6. In the figure, the straight line passing through A and B is perpendicular to the straight line passing through A and C, where C is a point lying on the x-axis. (a) Find the equation 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 area 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 equation f ( x) = k has two distinct real roots, find the range of values of k. (3 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ __ __________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ Page 5 of 9 8. Suppose 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 and 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 2Ãâ€"2 + 8x ? = 0, where ? > ?. (a) Write down the values of ? + ? and . (2 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ (b) Find the value of each of th e 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 vertex of the graph and axis of symmetry of the graph. (3 marks) ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ___ _________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ _____________________________ _______________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ (e) Find the domain and co-domain of f(x). 2 marks) ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ___ _________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ __________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ ____________________________________________________________ ___________________ END OF PAPER Page 9 of 9