RI Equations and Inequalities C3 Add Prac (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ______________________________________________ Additional Practice C3: Equations and Inequalities Page 1 of 3 Additional Practice Questions for Chapter 3: Equations and Inequalities 1 Show that 22 22 21 9 4 2 1 132 32 xx x x xx xx     . [1] Hence, without using a calculator, solve the inequality 2 2 21 9 132 xx xx   . [4] 2 Sketch the graphs of 42yx and  4yx x on a single diagram. [3] Hence solve 42 4 xx x  , giving your answer in exact form. [4] 3 Without using a calculator, solve the inequality  2 25 1 1 xx xx x   . [4] Hence find the exact solution for each of the following: (i)  2e2 e 5 e e1e xx x xx    (ii) 2 2 25 1xx xx x   [6] 4 (i) Solve     2 1 012 3 x xx   . [4] (ii) Hence find the exact solution of     2 1c o s 012 c o s c o s 3     , for 03  . [4] 5 Without the use of a graphing calculator, solve the inequality 2 2 27 6 12 xx xx   . [3] Deduce the range of values of x such that (a) 2 2 2(ln ) 7(ln ) 6 1( l n) ( l n) 2 xx xx   , [2] (b) 2 2 27 6 112 xx xx   , [2] 6 (i) On the same axes, sketch the graphs of 1y xa  and yb x a , where a and b are positive constants. [2] (ii) Hence, or otherwise, solve the inequality 1 bx axa  . [4]

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ______________________________________________ Additional Practice C3: Equations and Inequalities Page 2 of 3 7 (i) Find the exact roots of the equation 223 2 2x xx   [4] (ii) On the same axes, sketch the curves with equations 223 2yxx  and 2.yx Hence solve exactly the inequality 223 2 2 .x xx   [4] 8 Given that   32 2378 2 2 2xxx A x B xx     , where A and B are constants to be determined, solve algebraically the inequality 2 2031 0 1 8 . 1xx x   [4] Hence solve, exactly, the inequality  2 2031 1 0 1 1 8 . 11xx x    [2] 9 A curve C has equation 3 ayb x cx  , where a, b and c are constants. It is given that C passes through the point with coordinates (1, 3) and has a stationary point (5, 21). Find the values of a, b and c. [4] 10 Four friends buy three different kinds of fruit in the market. When they get home they cannot remember the individual prices per kilogram, but three of them can remember the total amount that they each paid. The weights of the fruits and the total amounts paid are shown in the following table. Suresh Fandi Cind y Lee Lian Pineapples (kg) 1.15 1.20 2.15 1.30 Mangoes (kg) 0.60 0.45 0.90 0.25 Lychees (kg) 0.55 0.30 0.65 0.50 Total amount paid in $ 8.28 6.84 13.05 Assuming that, for each variety of fruit, the pri ce per kilogram paid by each of the friends is the same, calculate the total amount that Lee Lian paid. [6] 11 A housewife bought at least one box of laundry deterg ent powder, one tube of toothpaste and one packet of sweets at a supermarket. Given that a box of laundry detergent powder costs $10, a tube of toothpaste costs $3 and a packet of sweets costs $0.50 and the housewife bought a total of 100 such items with $100 at the supermarket, how many of each item did she buy? [6] 12 A corporation wants to lease a fl eet of 12 airplanes with a combined carrying capacity of 220 passengers. The three available types of planes carry 10, 15 and 20 passengers. The monthly cost of leasing each of these types of planes is $8000, $14000 and $16000 respectively. What are the possible combinations of the different types of pl anes that should be leased and which of these combinations would minimise the monthly leasing cost? [6]

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ______________________________________________ Additional Practice C3: Equations and Inequalities Page 3 of 3 13 The usual selling price of a bottle of Vitamin C, Grape Seed Extract and Super Antioxidant is $365 in total. During the year-end sale, two compan ies, Union and Vatson, offered the following discounts to their customers: Company Discounts given for each item Total price after the discount Vitamin C Grape Seed Extract Super Antioxidant Union 10% 15% 15% $314.75 Vatson 8% 25% 15% $301.55 (i) Find the usual selling price of each bottle of Vitamin C, Grape Seed Extract and Super Antioxidant. [3] (ii) The employees from Union are offered a furthe r 5% discount on the usual selling price of each bottle of Vitamin C and Grape Seed Extract . Determine if this additional discount will make it more attractive for the employees to purchase all the 3 items from their own company rather than from Vatson. [2] 14 Mr. Li invested a total of $30000 and divided this su m into three accounts, which paid 2%, 3% and 5% annual interest respectively. At the end of the first year, Mr. Li withdrew a ll the money out from the 2% and 5% accounts and gave the interest earned to his son. The amount in the 3% account, including interest, was re-invested in the same account for another year. At the end of the second year, Mr. Li withdrew all the money out of the 3% account and gave the interest earned to his son. The total interest received by the son from the three accounts was $1423.50. Given that the amount invested in the 2% account was $1000 more than the amount invested in the 5% account, find the amounts invested in each of the three accounts. [4] 15 A Water Show sells tickets at three differe nt prices, depending on the age of the customers. The age categories are under 12 years, between 12 and 62 years, and over 62 years. The cost of the tickets for each group are given in the following table. Age Group Cost per ticket Under 12 years $6 Between 12 and 62 years $9 Over 62 years $5 MI Residence Committee spends $800 to sponsor a group of 100 people from these three age groups to attend the event. By writing down a system of equations, find the range of the possible number of people over the age of 62 sponsored. [4]