O Level Elementary Mathematics Algebra Functions Quiz

O-Level Elementary Mathematics Quiz – Algebra Functions

ANSWER KEY AND MARKING SCHEME

Total Marks: 50

Section A: Short Answer (Questions 1–8)

1. ( a = -3, b = 5 )
( 2a^2 - 3b + 1 = 2(-3)^2 - 3(5) + 1 )
( = 2(9) - 15 + 1 )
( = 18 - 15 + 1 )
( = 4 )
[M1] for correct substitution; [A1] for correct answer.
Answer: 4

2. ( 4x - 3(2 - x) + 5 )
( = 4x - 6 + 3x + 5 )
( = 7x - 1 )
[M1] for correct expansion of brackets; [A1] for correct simplified answer.
Answer: ( 7x - 1 )

3. ( 6p^2q - 9pq^2 )
( = 3pq(2p - 3q) )
[M1] for identifying common factor ( 3pq ); [A1] for correct factorisation.
Answer: ( 3pq(2p - 3q) )

4. ( (2x - 5)(x + 3) )
( = 2x^2 + 6x - 5x - 15 )
( = 2x^2 + x - 15 )
[M1] for correct expansion; [A1] for correct simplified answer.
Answer: ( 2x^2 + x - 15 )

5. ( x^2 - 7x + 12 )
( = (x - 3)(x - 4) )
[M1] for identifying factors with product 12 and sum –7; [A1] for correct factorisation.
Answer: ( (x - 3)(x - 4) )

6. ( V = \frac{4}{3}\pi r^3 )
( 3V = 4\pi r^3 )
( r^3 = \frac{3V}{4\pi} )
( r = \sqrt[3]{\frac{3V}{4\pi}} )
[M1] for multiplying both sides by 3 and dividing by ( 4\pi ); [A1] for correct expression with cube root.
Answer: ( r = \sqrt[3]{\frac{3V}{4\pi}} )

7. ( \frac{3x^2 - 12}{x^2 - x - 6} )
( = \frac{3(x^2 - 4)}{(x - 3)(x + 2)} )
( = \frac{3(x - 2)(x + 2)}{(x - 3)(x + 2)} )
( = \frac{3(x - 2)}{x - 3} )
[M1] for factorising numerator and denominator; [M1] for cancelling common factor ( (x + 2) ); [A1] for correct simplified answer.
Answer: ( \frac{3(x - 2)}{x - 3} )

8. ( \frac{2}{x} + \frac{3}{x+1} )
( = \frac{2(x+1) + 3x}{x(x+1)} )
( = \frac{2x + 2 + 3x}{x(x+1)} )
( = \frac{5x + 2}{x(x+1)} )
[M1] for correct common denominator ( x(x+1) ); [M1] for correct addition of numerators; [A1] for correct simplified answer.
Answer: ( \frac{5x + 2}{x(x+1)} )

Section B: Structured Questions (Questions 9–15)

9. ( T_n = n^2 - 3n + 2 )

(a) ( T_6 = 6^2 - 3(6) + 2 = 36 - 18 + 2 = 20 )
[A1] for correct answer.
Answer: 20

(b) ( n^2 - 3n + 2 = 42 )
( n^2 - 3n - 40 = 0 )
( (n - 8)(n + 5) = 0 )
( n = 8 ) or ( n = -5 ) (reject, as ( n ) must be positive)
[M1] for setting up equation; [M1] for solving quadratic; [A1] for correct term number.
Answer: 8th term

10. ( 3(2x - 1) - 2(x + 4) = 7 )
( 6x - 3 - 2x - 8 = 7 )
( 4x - 11 = 7 )
( 4x = 18 )
( x = 4.5 )
[M1] for expanding brackets; [M1] for collecting like terms; [A1] for correct answer.
Answer: ( x = 4.5 )

11. ( 2x^2 + 7x - 15 )
( = 2x^2 + 10x - 3x - 15 )
( = 2x(x + 5) - 3(x + 5) )
( = (2x - 3)(x + 5) )
[M1] for splitting middle term; [M1] for factorising by grouping; [A1] for correct factorisation.
Answer: ( (2x - 3)(x + 5) )

12. ( x^2 - 5x - 14 = 0 )
( (x - 7)(x + 2) = 0 )
( x = 7 ) or ( x = -2 )
[M1] for factorising; [M1] for setting each factor to zero; [A1] for both correct solutions.
Answer: ( x = 7 ) or ( x = -2 )

13. ( C = 40 + 25t )

(a) ( C = 40 + 25(6) = 40 + 150 = 190 )
[A1] for correct answer.
Answer: $190

(b) ( 190 = 40 + 25t )
( 150 = 25t )
( t = 6 )
[M1] for setting up equation; [A1] for correct answer.
Answer: 6 hours

14. ( y \propto x^2 ), so ( y = kx^2 )

(a) ( 75 = k(5^2) )
( 75 = 25k )
( k = 3 )
( y = 3x^2 )
[M1] for substituting to find ( k ); [A1] for correct formula.
Answer: ( y = 3x^2 )

(b) ( y = 3(8^2) = 3(64) = 192 )
[A1] for correct answer.
Answer: 192

15. ( f(x) = 2x^2 - 8x + 5 )

(a) ( f(x) = 2(x^2 - 4x) + 5 )
( = 2[(x - 2)^2 - 4] + 5 )
( = 2(x - 2)^2 - 8 + 5 )
( = 2(x - 2)^2 - 3 )
[M1] for factorising out 2; [M1] for completing the square; [A1] for correct form.
Answer: ( f(x) = 2(x - 2)^2 - 3 )

(b) Minimum value = –3, occurs when ( x = 2 )
[A1] for minimum value; [A1] for value of ( x ).
Answer: Minimum value: –3; Value of ( x ): 2

Section C: Application & Reasoning (Questions 16–20)

16. Let the three consecutive even integers be ( n, n+2, n+4 ).
( n + (n+2) + (n+4) = 78 )
( 3n + 6 = 78 )
( 3n = 72 )
( n = 24 )
The integers are 24, 26, 28.
[M1] for defining variables; [M1] for forming equation; [M1] for solving; [A1] for all three integers.
Answer: 24, 26, 28

17. Let width = ( w ) m, length = ( w + 5 ) m.

(a) ( w(w + 5) = 84 )
( w^2 + 5w - 84 = 0 )
[M1] for expressing area; [A1] for correct quadratic equation.
Answer: ( w^2 + 5w - 84 = 0 )

(b) ( (w + 12)(w - 7) = 0 )
( w = -12 ) (reject) or ( w = 7 )
Length = ( 7 + 5 = 12 )
[M1] for factorising; [M1] for solving and rejecting negative; [A1] for both dimensions.
Answer: Length = 12 m, Width = 7 m

18. ( y = x^2 - 4x - 5 )

(a) At ( x )-axis, ( y = 0 ):
( x^2 - 4x - 5 = 0 )
( (x - 5)(x + 1) = 0 )
( x = 5 ) or ( x = -1 )
Coordinates: ( (5, 0) ) and ( (-1, 0) )
[M1] for setting ( y = 0 ) and factorising; [A1] for both coordinates.
Answer: ( (5, 0) ) and ( (-1, 0) )

(b) ( y = (x - 2)^2 - 4 - 5 = (x - 2)^2 - 9 )
Turning point: ( (2, -9) )
[M1] for completing the square; [A1] for correct coordinates.
Answer: ( (2, -9) )

(c) Sketch should show:

• U-shaped parabola
• ( x )-intercepts at ( (-1, 0) ) and ( (5, 0) )
• ( y )-intercept at ( (0, -5) )
• Turning point at ( (2, -9) )
• All points clearly labelled
  [A1] for correct shape; [A1] for all key points labelled.
  Answer: See sketch.

19. ( \frac{x^2 - 9}{x^2 + 5x + 6} \div \frac{x - 3}{x + 2} )
( = \frac{(x - 3)(x + 3)}{(x + 2)(x + 3)} \times \frac{x + 2}{x - 3} )
( = 1 ) (after cancelling ( (x - 3), (x + 3), (x + 2) ))
[M1] for factorising; [M1] for changing division to multiplication by reciprocal; [M1] for cancelling common factors; [A1] for correct simplified answer.
Answer: 1

20. ( g(x) = \frac{6}{x - 2}, x \neq 2 )

(a) ( g(5) = \frac{6}{5 - 2} = \frac{6}{3} = 2 )
[A1] for correct answer.
Answer: 2

(b) ( \frac{6}{x - 2} = 4 )
( 6 = 4(x - 2) )
( 6 = 4x - 8 )
( 4x = 14 )
( x = 3.5 )
[M1] for setting up equation and cross-multiplying; [A1] for correct answer.
Answer: ( x = 3.5 )

(c) When ( x = 2 ), the denominator ( x - 2 = 0 ), and division by zero is undefined.
[A1] for correct explanation referencing division by zero.
Answer: Division by zero is undefined.

END OF ANSWER KEY