Topical Revision Worksheet 2 Equations and Inequlities Solution (LC)

Topical Revision Worksheet 2 Topic: Equations and Inequalities Question 1 [2008 EOY, Q3] marks It is given that , where , is a solution to the pair of simultaneous equations and . (i) Show that . Solution: Sub and into to get since [2] (ii) Find the other solution to the above pair of simultaneous equations. Solution: so hence Hence or 3 so sub to get , i.e. , [4] Question 2 [2009 EOY, Q7] marks (a) Given , where p is real. Find the value(s) of p for which the graph of y = f(x) is a tangent to the x-axis. Solution: [4] (b) Find the range of values of x for which Solution: [3]

Question 3 [2010 EOY, Q17 EITHER i, ii] marks (a) A power generator works in such a way that it takes 10 minutes to warm up before it is able to generate electricity. It is expected to generate 500 units of electricity every 20 minutes after it warms up. (i) Write down an equation in E and t, where E units represent the amount of electricity generated and t minutes (where represents the time after which the generator is turned on. Solution: [1] (ii) Show that it is expected to take at least 5.5 hours to generate at least 8000 units of electricity. Solution: For , So it takes at least 330 minutes or 5.5 hours **Question 4 [2011 EOY, Q10b] marks The roots of the equation are and . Without finding the value of and of , construct a quadratic equation whose roots are and . Solution: M1 Quadratic equation : (Only quadratic expression is written, (not an equation, i.e. no equal sign) answer mark cannot be given.) [5]

Question 5 [2012 EOY, Q9] marks (a) Given that is always positive for all real values of x. Find the range of values of k. Solution: [4] **(b) The roots of the quadratic equation are and , such that . Find the possible values of k. Solution: [5] Question 6 [2013 EOY, Q4] marks (a) Find the range of values of h , , for which the line y = x – h meets the curve . Solution: As the line meets the curve i.e. [3] (b) Find the range of values of k for which is always positive for all real values of x. Solution: As y is always positive and , [3]

Question 7 [2013 EOY, Q11] marks Solve the equation Solution: Given Case 1 : Case 2 : But Hence after verifying, the answer is: x = 3 or Alternative Solution [4] Question 8 [2014 EOY, Q5] marks Find the range of values of m for which is always positive for all real values of x. Solution: is always positive [4]

Question 9 [2014 EOY, Q6] marks It is given that the equation has two real roots . (i) Find the range of values of k. Solution: [3] **(ii) If , determine the value(s) of k. Solution: [4] **Question 10 [2015 EOY, Q1] marks The roots of the quadratic equation are and . (i) Write down the values of and . Solution: and [2] (ii) Find an equation whose roots are given by and . Solution: Hence equation is [3]

Question 11 [2015 EOY, Q2] marks (a) Explain why cannot be greater than 3 for all real values of x. Solution: Since , , so cannot be greater than 3. [3] (b) Find the range of values of p such that is always positive for all real values of x. Solution: Need and i.e. and Hence [5]