CXC/CSEC Mathematics WORKSHEET #2: Algebra Functions, Relations and Graph
Question 1:(Paper 2-Question # 2- May/June 2009)
(a) Factorise completely: (i) 2ax + 3ay - 2bx - 3by [2 3by [2 marks] (ii) 5.x2 – 20 20 [2 marks] (iii) 3.x2 + 4x- 15 [2 marks] (b) One packet of biscuits costs $x and one cup of ice cream costs $y. One packet of biscuits and two cups cup s of ice cream cost $8.00, while three packets of biscuits and one cup of ice cream cost $9.00. (i) Write a pair of simultaneous equations in x and y to represent the given information above. [2 marks] (ii) Solve the equations obtained in (b) (i) above to find the cost of one packet of biscuits and the cost of one cup of ice cream. [4 marks]
(i) the minimum value of 2_x2 - 3x + 1 [1 mark] (ii) the value of x of x for which the minimum occurs. [1 mark] (d) Sketch the graph of y of y = 2_x2- 3x + 1, clearly showing the coordinates of the minimum point the value of they-intercept the values of x of x where the graph cuts the x-axis. [4 marks] (e) Sketch on your graph of y of y = 2_x2- 3x + 1, the line which intersects the curve at the values of x of x and y calculated in (a) above. [2 marks] Question 3:(Paper 2-Question # 2 - May/June 2008)
(a) Find the value of EACH of the following when a = 2 , b = -1, c = 3 (i) a(b (i) a(b +c) [1 mark]
Question 2:(Paper 2-Question # 9 - May/June 2009)
(ii)
(a) Solve the pair of simultaneous equations y=4-2x y = 2_x2 - 3x + 1. [4 marks] (b) Express 2_x2 - 3x + 1 in the form a(x + h)2 + k, where a, hand k are real numbers. [3 marks] (c) Using your answer from (b) above, or otherwise, calculate 1|P
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[2 marks]
(b) Change the following statements into algebraic expressions: (i) Four times the sum of x of x and 5 [1 mark] (ii) 16 larger than the product of a and b [2 marks] (c) Solve the equation 15- 4x = 2(3 x + 1) [2 marks]
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(d)
Factorise completely b [2 marks] (i) 6 (ii) 2 +9m – 5 [2 marks]
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Question 4: (Paper 2-Question # 9 - May/June 2008)
(a)
Simplify (i)
× ÷
[1 mark]
(ii) ×
√
[2 marks] (b)If f ( x) = 2 x -3, find the value of (i) f (2) [1 mark] (ii) − (0) [2 marks] − (iii) [2 marks] f (2) (c) The temperature, K , of a liquid t minutes after heating is given in the table below.
Evaluate (i) 4 * 8 (ii) 2 * (4 * 8) [4 marks] (b) Simplify, expressing your answer in its simplest form
5 +
[2 marks]
(c) A stadium has two sections, A and B. Tickets for Section A cost $a each. Tickets for Section B cost $b each. Johanna paid $105 for 5 Section A tickets and 3 Section B tickets. Raiyah paid $63 for 4 Section A tickets and 1 Section B ticket. (i) Write two equations in a and b to represent the information above. (ii) Calculate the values of a and b. [5 marks] Question 6:(Paper 2-Question # 9b - May/June 2007)
(b) The length of the rectangle below is (2x1) cm and its width is (x + 3) cm. (i) Using a scale of 2 cm to represent 10 minutes on the horizontal axis and a scale of 2 cm to represent 10 degrees on the vertical axis, construct a temperature-time graph to show how the liquid cools in the 60 minute interval. Draw a smooth curve through all the plotted points. [4 marks] (ii)Use your graph to estimate a) the temperature of the liquid after 15 minutes b) the rate of cooling of the liquid at t = 30 minutes. [3 marks]
(i) Write an expression in the form a + bx + c for the area of the rectangle. (ii) Given that the area of the rectangle is 294 c , determine the value of x. (iii) Hence, state the dimensions of the rectangle, in centimetres. [8 marks]
Question 5:(Paper 2-Question # 2 - May/June 2007)
(a) Given that a * b = ab - .
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(iii) the equation of the line which passes through A and B. [2 marks]
Question 7: (Paper 2-Question # 2 - May/June 2012)
(a)
(b)
(c)
Factorise completely: (i) 2 x3 y + 6 x2 y2 [2 marks] (ii) 9 x2 – 4 [1 mark] (iii) 4 x2 + 8 xy – xy – 2 y2 [2 marks]
Question 9: (Paper 2-Question # 5 - May/June 2012)
(a)
Solve for x:
− 5− = 3
[3 marks]
Solve the simultaneous equations: 3 x – 2 y = 10 2 x + 5 y = 13 [4 marks]
Question 8:(Paper 2-Question # 6 - May/June 2011)
(a) The functions f and g are defined by
(ii) Calculate the value of g
.
[2 marks] (ii) Write an expression for g f ( x) in its simplest form. [2 marks] (iii) Find the inverse function − ( x). [2 marks] (b) The diagram below shows the line segment which passes through the points A and B.
(i) Using a ruler, a pencil and a pair of compasses, construct triangle PQR with PQ = 8 cm,
(a) Given that
( ) =
x + 4.
+ 5 and f ( x) =
(i) Calculate the value of g (−2) . (ii) Write an expression for gf ( x) in its simplest form. (iii) Find the inverse function − ( x). [7 marks]
Question 11: (Paper 2-Question # 5 May/June 2009)
Determine (i) the coordinates of A and B [2 marks] (ii) the gradient of the line segment AB [2 marks] 3|P
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(a) Given that f(x) = 2x- 5 calculate the value of (i) f (-2) (ii) g f (1) (iii) − (3). (b) Given that y = *2 2x – 3
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and g(x) = x2- 31, [1 mark] [2 marks] [2 marks]
(i) Copy and complete the table below. [2 marks]
(ii) Using a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the y-axis, draw the graph of y= [5marks]
2 3 -4 ≤ ≤ 2.
Question 12: (Paper 2-Question # 5 May/June 2013)
(a) The incomplete table below shows one pair of values for A and R where A is directly proportional to the square of R.
(i) Express A in terms of R and a constant k . [1 mark] (ii) Calculate the value of the constant k . [2 marks] (iii) Copy and complete the table. [2 marks] (b) Given that
() =
determine the values of: (i) fg (2)
−(3)
(ii)
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+ and g( x) = 4 x + 5, [3 marks] [3 marks]
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