Simultaneous Equations - IGCSE 10 / Form 4

Dear Students,

1. Maths may appear abstract to many (please see Q1 below). But it need not be once we can relate what seems abstract to real life situation - Q2 below is my attempt to relate Q1 to real life situation to my students.

2. You will also note from Q1 that I made my students revisit Form 3 (PMR) or pre-IGCSE Yr-10 level "solving simultaneous equations involving a pair of linear equations in 2 unknowns" before introducing "solving simultaneous equations involving 1 linear equation and 1 quadratic equation or other non-linear equations" 

3. Maths is a fun subject which you can score nearly 100% if not 100%. In year 2013, I do have a Korean student (in the international school) who has twice scored 100% for A-Maths. After I quit the school in August, she enlisted me to be her tutor since 15 Oct 2013.

Enjoy & try my questions below.

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IGCSE Yr 10 / Form 4: Simultaneous Equations

Tutorial Questions No. 4

1. A relation between 2 quantitative unknowns, say x and y, may be represented mathematically as an equation (e.g. y – x = 20). The same 2 unknown (x and y) may, at the same time, be related to one another by another relation (say, 5x = y) or other equations (say, x^2 = y).
1. What do you call a pair of equations with the same 2 unknowns?
2. Can you find the values of the 2 unknowns if you have 2 different equations with the same 2 unknowns?
3. The non-graphical method of finding the values of the 2 unknowns from 2 equations with the same 2 unknown is known as what?
4. Using 2 of the above 3 equations where x and y > 0, solve the simultaneous equations:

                                                               i.      y – x = 20 and 5x = y (Both linear equations – Form 3 “Flashback”)

1.      By Substitution Method

2.      By Elimination Method

                                                             ii.      y – x = 20 and x^2 = y (1 linear and 1 quadratic – Form 4 now)

                                                            iii.      5x = y and x^2 = y (1 linear and 1 quadratic – Form 4)

2. Real-life representation of Q1: Mei Ling’s mother is y years old and Mei Ling is x years old. Write the equation to show that the mother is 20 yrs older than Mei Ling
1. When Mei Ling was x years old, her mother’s age was 5 times Mei Ling’s. How old was Mei Ling when this happened?
2. When Mei Ling was x years old, her mother’s age was the square of Mei Ling’s age. How old was Mei Ling then?
3. When Mei Ling was x years old: Her mother’s age was 5 times Mei Ling’s age and at the same time also equal to the square of Mei Ling’s age. Use the relevant simultaneous equations to find out Mei Ling’ age and her mother’s age then.
4. Draw the graphs y – x = 20 and 5x = y, for -6 < x < 6, to find out the coordinates of their point of intersection.
5. Draw the graphs of 5x = y and y = x^2, for -6 < x < 6, to find out:

                                                               i.      their points of intersection

                                                             ii.      the point of intersection when x > 0.

3. Solve the following simultaneous equations:

x + 2y = 4

x^2 + xy + y^2 = 7

4. Solve the following simultaneous equations:

x/3 + y/4 = 3/2

3/x + 4/y = 3

5. Solve the following simultaneous equations:

2(x – y) = x + y – 1 = 2x^2 – 11y^2

6. Given that (2h, 3k) are solutions to the simultaneous equations y + 2x = 4 and 2/y – 3/2x = 1, find the values of h and the corresponding values of k.

7. Solve the following simultaneous equations x – y = 1 and x^2 + 3y = 6. Give your answers correct to three decimal places.

8. Amin travels from Kajang Toll to Johor Bahru Toll at an average speed, v km/h. If he travels 10 km/hr faster, he would have arrived 25 minutes earlier. Given that the distance between the two tolls is 300 km, how long would he take to complete the journey with the average speed, v km/hr? (Extracted from opening paragraph of Form 4 Textbook Chapter 4)

9. Aminah spends RM24 and RM15 in buying chicken and duck respectively. The price of 1 kg of duck is RM2 more than chicken. If the total mass of the chicken and duck is 10 kg, determine the price per kg for chicken and duck respectively.

10. The numerator and denominator of a fraction are each increased by 3, the fraction is equivalent to 2/3. If the fraction is multiplied by itself, the result is equivalent to 25/81. Find the fraction.

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Quadratic Equations (for IGCSE 10 / Form 4 Students)

Hello IGCSE Yr 10 or 11 / Forms 4 or 5 Students,

1) Taking Add. Maths for your SPM 2012 / IGCSE 2013 exams and beyond? Yes? Then this blog should interest you. Btw I'm Tan residing in PJ, Selangor, Malaysia and can be reached at: tutortan1@gmail.com or SMS 011-2328 7882.

2) One Form 4 student (who took tuition from me) told me her first encounter with Add Maths was "Quadratic Equations (QE)" (F4 Chapter 2). Thereafter, she learned "Simultaneous Equations" (F4 Chapter 4) and now about to learn "Functions" (F4 Chapter 1). Other schools may start differently...

3) After going through the QE chapter with her, I posed 20 questions to test her. Among which is Q 12 which requires her to derive the "Quadratic Root Formula". It's simple - and I have taught her how to (surprisingly, her F4 textbook omits to show them how to - by "completing the square" method!). Why don't you try Q 12 as well if not all the questions (some drafted by me, others from F4 textbook).

4) Maths is best learnt by understanding rather than by rote learning. I believe if you know how to derive the "Quadratic Root Formula", you need not memorise it. You can rewrite it anytime you wish. Also once you know how to derive it, the Discriminant, "b² – 4ac" will no longer be so alien and distant to you.

5) Another thing is the general name given to that formula - it's known as "Quadratic Formula" instead of "Quadratic Root Formula". The latter name should be used as it is more specific - it is not just any quadratic formula but a quadratic formula for finding the roots of a quadratic equation: Hence, I prefer the term "Quadratic Root Formula" which is more friendly to newbies in Add. Maths.

Happy doing maths...

Form 4 Chapter 2: Quadratic Equations

Tutorial Questions:

1. In your own words, describe a quadratic algebraic expression.

2. Explain why x^2 + 2/x – 3 = 0 is not a quadratic equation.

3. Is x^3 – x^2 = 0 a quadratic equation?

4. By substitution, determine whether x = 1, x = 2 and/or x = -2 are roots of x^2 + x - 2 = 0

5. By inspection, determine whether x = 2, x = -2, x = 3 and/or x = -3 are roots of the equation (x – 2)(x + 3) = 0

6. By inspection, state the roots of the equation (x – p)(x – q) = 0

7. The table below gives the values of the dependent variable y of a quadratic equation: y = x^2 – 2x – 15 in terms of the independent variable x.

x	6	4	2	0	-2	-4
y	9	-7	-15	-15	-7	9

1. By inspection, identify the locations of the roots of the quadratic equation as between which pair(s) of x values?
2. By trial and improvement method, find the roots of the quadratic equation.

8. Find the roots of the following quadratic equations by factorization:
1. x^2 + x -12 = 0
2. 12x^2 = 6 – x
3. x(2x – 5) = 12

9. You can make any quadratic expression in the form of x^2 + bx into a perfect square in what is known as ‘completing the square’. Show me how do you complete x^2 + bx into a perfect square.

10. For the quadratic equation x^2 + 6x – 1 = 0:
1. State whether you can easily factorise it into the form: (x – p)(x – q) = 0;
2. Find its roots by “completing the square” method instead of factorization.

11. Solve the quadratic equation 2x^2 + 4x -3 = 0 by completing the square.

12. Find the roots of the quadratic equation ax^2 + bx + c = 0, where a is not equal to 0, by completing the square.  

13. Use the quadratic roots formula you derive in Q12 above to find the solutions of the equation 10x^2 + 3x = 16. Give your answers correct to 4 significant numbers.

14. Q12 shows that you can find the roots of a quadratic equation by the derived quadratic root formula.
1. What is “b^2 – 4ac” known as?
2. If b^2 – 4ac > 0, what can say about the roots of the quadratic equation?
3. If b^2 – 4ac = 0, what can say about the roots of the quadratic equation?
4. If b^2 – 4ac < 0, what can say about the roots of the quadratic equation?

15.  For the quadratic equation (k +1)x^2 – 4x + 9 = 0, find:
1. The value of k, if the equation has 2 equal roots
2. The range of k, if the equation has 2 different roots
3. The range of k, if the equation has no real roots

16. For the quadratic equation x^2 – 4mx + n^2 = 0, derive the algebraic relations between m and n when the equation has:
1. 2 equal roots
2. 2 different roots
3. no real roots

17. Q6 above shows that the quadratic equation (x – p)(x – q) = 0 has the roots p and q.
1. Expand (x – p)(x – q) = 0
2. Show from Q17a that if the roots of a quadratic equation are known, the quadratic equation can be formed from the given roots as:

x^2 –(sum of roots)x + (product of roots) = 0

18. Form a quadratic equation with the roots 2 and -3.

19. Given that a and b are the roots of the quadratic equation:
1. 2x^2 + 2x – 5 + 0, form a quadratic equation with the roots (a – 2) and (b – 2)
2. 2x^2 + 3x – 9 = 0, form a quadratic equation with the roots

                                                               i.      a/2 and b/2

                                                             ii.      3a and 3b

20. If one of the roots of the equation 2x^2 + px +9 = 0 is twice the other root, find the possible values of p.