Topical Revision Paper 3 Unit 3 Equations and Inequalities

Topical Revision Paper 3 Term 1 Unit 3: Equations and Inequalities Question 1: (a) Solve the equation 2 3 2xx . [4] (b) Find the range of values of x for which 2 6xx . Hence, find the range of values of x that satisfy    22 3 6 0x x x    . [6] Solution: (a) Given 2 3 2xx 22 3 2 or 3 2x x x x     Case 1 : 2 32xx or 2 2 3 0xx   ( 3)( 1) 0xx   3 or 1x Case 2 : 2 32xx  or 2 2 3 0xx   ( 3)( 1) 0xx   3 or 1x But 2 3 2 0xx   Hence after verifying, the answer is: x = 3 or 1x (b) 062  xx   3 2 0xx   32 x   For 2(2 3)( 6) 0 (2 3)( 3)( 2) 0 x x x x x x        

From the sketch 33 2x   or 2x . Question 2: Find the possible values of k for which the line 1 kxy is a tangent to the curve 32 xy . Hence ,state the range of values of k for which the line will not meet the curve. [6] Solution:      2 2 2 2 Substitute 1 into 3 40 b 4 0 16 0 4 4 0 4 or 4 y kx y x x kx ac k kk k                  For 4 4 0 44 kk k       Question 3: Solve the equation 2 1 12 7(2 ) 36xx , leaving your answers in two decimal places. [4] Solution:

       2 1 1 2 2 2 2 7(2 ) 36 2 2 7(2 2) 36 2 2 14 2 36 0 Let 2 2 14 36 0 2 4 9 0 2 (rejected) or 9 lg 92 9 3.17 (2 decimal places)lg 2 xx xx xx x x y y y yy yy y                        Question 4: Find the range of values of k for which the graph of 22 13 2 2 2y x kx k k      intersects the x -axis at two distinct points. [3] Solution: 22 13 2 2 0 2x kx k k      has two distinct roots 2 40b ac     2 2 13 4 1 2 2 0 2k k k     229 8 2 8 0k k k    2 2 8 0kk     4 2 0kk   4 or 2kk   Question 5: The diagram above shows part of the graph of 1 33yx with A and B as the x and y intercepts respectively. B O y x A

(i) State the coordinates of A and B. [2] (ii) For each of the cases below, determine the number of intersections of the line y mx c with 1 33yx , justifying your answer. (a) 1 3m and 0c , [2] (b) 1 4m and 3c . [2] (i) ( 9,0) and (0,3)AB (ii) (a) cmxy  is parallel and below 1 3 for all 3y x x . Hence will not intersect at any point or zero intersection (iii) (b) cmxy  will intersect 1 33yx at (0,3) and also intersect 1 3 for 93y x x   once. Hence, two intersections