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Think! New Syllabus Mathematics 3 8th Edition Dr Yeap Ban Har, Dr Joseph B. W. Yeo, Dr Choy Ban Heng, Teh Keng Seng, Wong Lai Fong, Wong-Ng Slew Hiong - Solutions
Simplify each of the following, leaving your answer in index notation where appropriate. (a) 3' x 8' (b) (56) (c) (-2cd) (h*k)(-5hk) (d) (Shk)
Simplify each of the following, leaving your answer in index notation where appropriate. (a) 24x7 (b) (-2h)5 (c) (xy)*(-3xy)* (3xy*)
Given that x8 x(x3)n / (xn)2 = x10, find the value of n.
1. Simplify each of the following, leaving your answer in index notation where appropriate.
Simplify each of the following, leaving your answer in index notation where appropriate. (a) (5) (b) [(-h)]* (c) (7) x (72)+(73)
We have learnt four methods to solve a quadratic equation:(a) factorisation,(b) completing the square,(c) quadratic formula,(d) graphical method.Write down the advantages and disadvantages of using each method.When solving a quadratic equation, how would you choose which method to use?
Equations that contain one or more algebraic fractions are known as fractional equations. To solve a fractional equation that can be reduced to a quadratic equation: Multiply both sides of the equation by the LCM of the denominators. Reduce the equation to a quadratic equation. Solve the quadratic
11. In a carnival game, participants throw a balloon filled with water from the top of a platform onto a sandpit. Points are allocated based on the horizontal distance from the foot of the platform to where the balloon lands.Cheryl throws a balloon. During the flight, its height above ground level,
10. During a test flight, an aircraft flies from Sandy Land to White City and back to Sandy Land. The distance between Sandy Land and White City is 450 km and the total time taken for the whole journey is 5 hours and 30 minutes. Given that there is a constant wind blowing from Sandy Land to White
9. Two weeks before Nadia went to New York for a holiday, she exchanged S$2000 into US dollars (US$) at XYZ Money Exchange at a rate of US$1S5x.(i) Write down an expression, in terms of x, for the amount of US$ she received from XYZ. Money Exchange.One week before her holiday, she exchanged another
8. A tank, when full, contains 1500 litres of water. Pump A can fill the tank with water at a rate of x litres per minute.(i) Write down an expression, in terms of x, for the number of minutes taken by Pump A to fill the tank completely.Pump B can fill the tank with water at a rate of (x + 50)
7. Sara travels 700 km by coach from Islamabad to Bahawalnagar to visit her grandparents. She returns to Islamabad by car at an average speed which is 30 km/h greater than that of a coach.(i) If the average speed of the car is x km/h and the time taken for the whole journey is 20 hours, formulate
6. Waseem and Yasir started a 10-km hike at 8 a.m. They started walking at the same speed of x km/h. After 2 km, Waseem increased his speed by 1 km/h and walked the remaining distance at a constant speed of (x + 1) km/h. Yasir maintained his speed of x km/h throughout the hike.(i) Write down an
5. A model of a commemorative structure in the shape of a square-based pyramid has a scale of 1: 100. The diagram shows one of the lateral faces of the model, where AP = (x + 5) cm and BC = 2x cm.(i) Write down an expression, in terms of x, for the total surface area of the model.(ii) Given that
4. An object is launched from a platform. The path of the object can be modelled by the equation h+5t+10, where t, in seconds, is the time from the launch and h, in metres, is the height of the object from the ground. Find the difference in height between the object at its highest point and at the
3. Rice from Brand A costs $x per kilogram. A food catering company spent $600 on rice in January.(i) Write down an expression, in terms of x, for the amount of rice that this food catering company ordered in January. In February, the food catering company decides to buy rice from Brand B, which
2. There are 2 printers in a library. Printer A prints 60 pages every x minutes.(i) Write down an expression, in terms of x, for the number of pages printed by Printer A in 1 minute.(ii) Printer B takes 2 minutes longer than Printer A to print 60 pages. Write down an expression, in terms of x, for
The perimeter of a rectangular poster is 112 cm and its breadth is x cm.(i) Write down an expression, in terms of x, for the length of the rectangular poster.(ii) The area of the poster is 597 cm³. Write down an equation to represent this information and show that it simplifies to x56x+597-0.(iii)
Bernard drove 600 km from City P to City Q. The average speed of his return journey was 7 km/h faster and the time taken was 15 minutes less(i) If he drove at an average speed of x km/h on his journey from City P to City Q, write down an equation to represent this information and show that it
Find the values(s) of that satisfy the equations. x(x-3) 3 (x+1) 5
Solve each of the following equations (a) 12 3 + xx-1 x+1 1 (b) x-93-x 3 + 2x+3x-5
Solve each of the following equations (c) 5 (a) --1 x-2 x + I 2x-3 (b) 2 x+1 1- x+5 2 (d) 4K 1 =9 xx-1 (e) (6) + x+2x-2 5 1 7 x+1 x-1 x+3 (g) 4 5 2- x-2 (x-2) (h) x-1(x-1) x 1
Solve each of the following equations (a) x-1 8x x+1 1-x (b) (x-2)(x-3) 2 (x-1)(x+2) 3
Solve each of the following equations (a) x(x+1)=1 (b) 3(x+1)(x-1)=7x (c) (x-1)-2x=0 (d) x(x-5)-7-2x (e) (5x-9)(x-1)-x(x-2)=0 (f) (4x-3)+(4x+3)=25
Solve each of the following equations (a) 8 x =2x+1 x+1 (c) 5-x =x (e) 2x+1=x+1 7 (b) x+--9 2 x (d) 3x-1= 9 3x+5 x-5 (f) 5x =4x+1 x+4
Solve each of the following equations (a) x+5x=21 (c) 8x2-3x+6 (e) 9-5x=-3x (b) 10x-12x-15 (d) 4x+14x (f) 16x-61-
Solve each of the following equations (a) x+4x+1=0 (c) 3x2-5x-17-0 (e) 2+2x2-7x=0 (b) 3x+6x-1-0 (d) -3x-7x+9=0 (f) 10x-5x-2=0
Solve each of the following equations 2 (a) -5 x+6 3-x 5 (b) -3 y-3'y-2 -1
Given the equation y ^ 2 - ay - 6 = 0 where a is a constant, find the solutions for y in terms of a.
7. Solve each of the following equations.(a) a(a + 4) = 3a + 1(b) (b + 1) ^ 2 = 7b(c) (c - 2)(c + 5) = c(d) d(d - 4) = 2(d + 7)
6. (1) Express x ^ 2 + 17x - 30 in the form (x +a) ^ 2 + b(ii) Hence solve the equation x ^ 2 + 17x - 30 = 0 giving your answers correct to one decimal place.
5. Solve each of the following equations by completing the square, giving your answers correct to 2 decimal places.(a) x ^ 2 + 2x - 5 = 0(b) y ^ 2 - 12y + 9 = 0(c) z ^ 2 - 5z - 5 = 0(d) \mathfrak{p} ^ 2 + 1/4 * \mathfrak{p} - 3 = 0(e) q ^ 2 - 6/7 * q + 2/49 = 0(f) r ^ 2 + 0.6r - 1 = 0
4. Express each of the following expressions in the form (x + r) ^ 2 + u(a) x ^ 2 - 6x + 1(b) x ^ 2 + 3x - 2(c) x ^ 2 + 9x - 1.8(d) x ^ 2 - 2/7 * x + 7(e) - x ^ 2 + 10x - 2(f) - x ^ 2 + 13x - 13/2(g) - x ^ 2 - 9x - 20.25(h) - x ^ 2 - 3/4 * x + 3
3. Express each of the following expressions in the form (x + r) ^ 2 + u(a) x ^ 2 + 20x(b) x ^ 2 - 15x(c) x ^ 2 + 1/2 * x x ^ 2 - 2/9 * x(e) x ^ 2 + 0.2x(f) x ^ 2 - 1.4x(g) - x ^ 2 - 10x(h) - x ^ 2 + 11x
2. Solve each of the following equations, giving your answers correct to 2 decimal places where necessary.(a) (x + 1) ^ 2 = 9(b) (2y + 1) ^ 2 = 16(c) (5h - 4) ^ 2 = 81(d) (7 - 3k) ^ 2 = 9/16(e) (m + 3) ^ 2 = 11(f) (2n - 3) ^ 2 = 23(g) (5 - w) ^ 2 = 7(h) (1/2 - t) ^ 2 = 10
Solve each of the following equations.(a) x ^ 2 + 7x - 18 = 0(b) 2x ^ 2 + 5x - 7 = 0(c) 5y ^ 2 - 28y + 15 = 0(d) 4z ^ 2 - 49 = 0
2. Solve the equation (x + 4)(x - 3) = 15
1. Solve each of the following equations, giving your answers correct to 2 decimal places.(a) x ^ 2 + 6x - 4 = 0(b) y ^ 2 + 7y + 5 = 0(c) z ^ 2 - z - 1 = 0
Solve the equation x2 + 4x - 3 - 0, giving your answer correct to 2 decimal places.
2. Express each of the following expressions in the form - (x + r) ^ 2 + u(a) - x ^ 2 + 6x - 2(b) - x ^ 2 + 9x - 3.5(c) - x ^ 2 - 7x + 5(d) - x ^ 2 - 4/9 * x - 1
1. Express each of the following expressions in the form (x + r) ^ 2 + u(a) x ^ 2 + 14x + 5(b) x ^ 2 + 7x - 1.2(c) x ^ 2 - 9x + 3(d) x ^ 2 - 6/5 * x - 4
Express each of the following expressions in the form (x + r) ^ 2 + u(a) x ^ 2 + 10x(b) x²-5x
Express each of the following expressions in the form (x + r) ^ 2 + u(a) x ^ 2 + 12x(b) x ^ 2 - 7x(c) x ^ 2 + 1.6x(d) x ^ 2 - 3/4 * x
Solve each of the following equations.(a) x ^ 2 - 7x - 8 = 0(b) 2y ^ 2 + 3y - 20 = 0
Solve the equation x2 - 5x - 6 = 0.
Why quadratic equations and quadratic functions have useful applications in mathematics and real-world contexts
How to formulate a quadratic equation in one variable to solve problems
How to solve fractional equations that can be reduced to quadratic equations
How to solve quadratic equations in one variable by completing the square for equations of the form x²+bx+c=0 use of formula graphical method
Graphs of quadratic functions of the form y=(xp)+q The coordinates of the turning point are (p, q).The equation of the line of symmetry is x = p. x-pis -the line of x-intercepts symmetry y-intercept- P 0 9 x (p. q) is the minimum point y= (x p)+q y-intercept- (p. q) is the maximum point x-pis the
Graphs of quadratic functions of the form y=(x-h)(x-k)The x-intercepts are h and k.The equation of the line of symmetry is x = (h + k)/2 The x-coordinate of the turning point is h+k 2 line of y-intercept h+k symmetry x k minimum point y=(x-h)(x-k) maximum point h h+k k y-intercept- 2 line of
The general form of the equation of a quadratic function is y = a * x ^ 2 + bx + c wherea, b and c are constants and a≠0. line of symmetry maximum point y-intercept y=ax + bx + c where a > 0 x-intercepts y=ax + bx + c, where a 0 line of symmetry a 0, the graph of y = ax + bx + c opens upwards
If P and Q are factors of an algebraic expression such that PQ = 0 then P = 0 or Q = 0 Show how you can use this principle to help you solve the following quadratic equations:(a) 4x ^ 2 - 9x = 0(b) 4x ^ 2 - 9 = 0(c) 4x ^ 2 - 12x + 9 = 0(d) 4x ^ 2 + 5x - 9 = 0
10. The graph of y = (x - h) ^ 2 + k has a minimum point at (i) State the value of h and of k. (ii) Hence, sketch the graph of y = (x-h)+k, indicating the coordinates of the point of intersection of the graph with the y-axis.
9. (a) (i) Express y = - x ^ 2 + 6x - 6 in the form y = - (x - p) ^ 2 + q(ii) Hence, sketch the graph of y = - x ^ 2 + 6x - 6(b) (i) Express y = x ^ 2 - 8x + 5 in the form y = (x - p) ^ 2 + q(ii) Hence, sketch the graph of y = x ^ 2 - 8x + 5
8. (a) (i) Express y = - x ^ 2 - 10x - 25 in the form y = - (x + p) ^ 2 + q(ii) Hence, sketch the graph of y = - x ^ 2 - 10x - 25(b) (i) Express y = x ^ 2 - 8x + 16 in the form y = (x - p) ^ 2 + q(ii) Hence, sketch the graph of y = x ^ 2 - 8x + 16
7. (a) (i) Express y = - x ^ 2 - 7x - 15 in the form y = - (x + p) ^ 2 + q(ii) Hence, sketch the graph of y = - x ^ 2 - 7x - 15(b)(i) Express y = x ^ 2 - 3x + 4 in the form y = (x - p) ^ 2 + q(ii) Hence, sketch the graph of y = x ^ 2 - 3x + 4
6. Sketch the graph of y = x ^ 2 - 4x + 3
5. (i) Factorise x ^ 2 + x - 6 completely.(ii) Hence, sketch the graph of y = x ^ 2 + x - 6
4. Sketch the graph of y = - (x ^ 2 - x)
3. (i) Factorise x ^ 2 + 3/4 * x(ii) Hence, sketch the graph of y = x ^ 2 + 3/4 * x
Sketch the graph of each of the following functions. Indicate clearly the coordinates of the maximum or the minimum point and the equation of the line of symmetry. (a) y=x-4 (c) y=(x-3)2-2 (e) y=-(x+2)+3 (b) y=-x+6 (d) y=(x+1)2-3 (f) y=-(x-4)+5
Sketch the graph of each of the following functions. (a) y (x+1)(x+3) (c) y=-(x+1)(x-5) (e) y (3-x)(x+2) (b) y (x-2)(x+4) (d) y=-(x-1)(x+6) (f) y (2-x)(4-x)
3. Consider the graph of y = - x ^ 2 - 6x - 11 in Worked Example 14.(i) How many times does this graph cut the x-axis?(ii) Can we express y = - x ^ 2 - 6x - 11 in the form of y=(x-h)(x-k)? Explain.
2. Consider the graph of the quadratic function y = (x - 3)(x - 3)(i) y = (x - 3)(x - 3) in the form of y=(xh)(x-k) or y = plus/minus (x - p) ^ 2 + q or both? Explain.(ii) How many times does the graph of y = (x - 3)(x - 3) cut the x-axis?(iii) Sketch the graph of y = (x - 3)(x - 3)
1. Consider the graph of y = (x - 1) ^ 2 - 4 in Worked Example 13.(i) How many times does this graph cut the x-axis?(ii) Can we express y = (x - 1) ^ 2 - 4 in the form of y=(x-h)(x-k)? If yes, show how it is done.
3. (i) Express y = - x ^ 2 - 6x - 9 in the form y = - (x + p) ^ 2 + q(ii) Hence, sketch the graph of y = - x ^ 2 - 6x - 9
2. (i) Express y = x ^ 2 + x + 1 in the form y = (x + p) ^ 2 + q(ii) Hence, sketch the graph of y = x ^ 2 + x + 1
1. (i) Express y = - x ^ 2 + 4x - 6 in the form y = - (x - p) ^ 2 + q(ii) Hence, sketch the graph y = - x ^ 2 + 4x - 6
(a) What is the relationship between the coordinates of the turning point and the values of p and q?(b) What is the relationship between the equation of the line of symmetry and the x- or y-coordinate of the turning point?(c) When will the curve just touch the x-axis (i.e. there is only one
Use a graphing software to plot the graphs of the form y=(x-p)²+gory=(xp)²+q in Table 2.5. Then copy and complete the table. The first row has been done for you. Unlike the previous Investigation where the values for h and k are interchangeable, the values of p and q are not interchangeable here.
(a) What is the relationship between the x-intercepts and the values of h and k?(b) What is the relationship between the equation of the line of symmetry and the x-intercepts?(c) What is the relationship between the equation of the line of symmetry and the x- or y-coordinate of the turning
Use a graphing software to plot the graphs of the form y=(x-h)(x-k) or y=(xh)(x-k) shown in Table 2.3. Copy and complete the table. The first row has been done for you. Note that the values for h and k are interchangeable, i.e. for the first row, you can also write h = 5 and k = - 1 Function
12. The height, y metres, of an object projected directly upwards from the ground can be modelled by y = 17t - 5t ^ 2 where t is the time in seconds after it leaves the ground.(i) Find the height of the object 2.5 seconds after it leaves the ground.(ii) After how many seconds will the object strike
13. The height, h metres, of a ball projected directly upwards from the ground can be modelled by h56t7t, where t is the time in seconds after it leaves the ground.(i) Find the height of the ball 3.5 seconds after it leaves the ground.(ii) After how many seconds will the ball strike the ground
11. The path of a pirate ship adventure ride at a theme park follows the shape of a parabola. The ship swings back and forth, accelerating to the base and then upwards. The height of a rider above ground level, h m, can be modelled by the equation h = 1.2x ^ 2 - 12x + 30 where x is the horizontal
The value of an asset can be modelled by y=2x²-4x+7, where y is the value of the asset in thousands of dollars and x is the time in years.(i) On a sheet of graph paper, using a scale of 5 cm to represent 1 year on the x-axis and 1 cm to represent $1000 on the y-axis, draw the graph of y=2x-4x+7
The variables x and y are connected by the equation y=10-x-x². Some values of x and the corresponding values of y are given in the table.(a) On a sheet of graph paper, 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
The diagram shows the curve y=-x-x+20. (i) The curve cuts the x-axis at two points A and B, and the y-axis at the point C. Find the coordinates of A, B and C. (ii) The point D(3, h) lies on the curve. Find the value of h. C A D y-x-x+20 B x
The variables x and y are connected by the equation y2-3x-2x². Some values of x and the corresponding values of y are given in the table.(a) Find the value of p and of q.(b) On a sheet of graph paper, using a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to represent I unit on the
The variables x and y are connected by the equation y=x²+2x-8. Some values of x and the corresponding values of y are given in the table.(a) Find the value of a and of b.(b) On a sheet of graph paper, using a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the
In the figure, the curve y = x² + 7x + 6 cuts the x-axis at two points A and B, and the y-axis at the point C. Calculate the coordinates of A, B and C. C B x
2. An athlete throws a javelin across a field. The height, h metres, of the javelin above 1 ground level, can be modelled by h= - 1/85 (x2 - 65x - 204), where x is the horizontal 85 distance from the point it was thrown in metres. (i) What does the value of h at x = 0 mean? (ii) Did the athlete
1. The height, y metres, of a model rocket launched directly upwards from level ground can be modelled by y = 96t-4f, where t is the time in seconds after it leaves the ground. (i) Find the height of the rocket 12 seconds after it leaves the ground. (ii) After how many seconds will the rocket
2. The cross section of a farmland can be modelled by the equation y=1+0.45x0.025x, where x and y are the horizontal distance from the farmhouse and the height of the farmland above sea level respectively, measured in metres.(i) On a sheet of graph paper, using a scale of 1 cm to represent 2 m on
A stone is thrown vertically upwards from the top of a cliff. Its height, h metres, above ground level, can be modelled by h = 28 + 42x - 12x2, where x metres is the horizontal distance travelled by the stone.(a) Assuming that the height of the thrower is negligible, find the height of the
Ken kicks a soccer ball into the air such that the height, Ih metres, of the ball from the ground can be modelled by h27x6x, where x is the horizontal distance travelled by the ball in metres.(a) On a sheet of graph paper, using a scale of 2 cm to represent 1 m on the x-axis and 1 cm to represent 5
The variables x and y are connected by the equation y = x²+2x+2. Some values of x and the corresponding values of y are given in the table.(a) Calculate the value of p.(b) On a sheet of graph paper, using a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to represent 2 units on the
2. Study the graphs and answer each of the following questions.(a) Both graphs pass through a particular point on the coordinate axes. What are the coordinates of this point?(b) State the lowest or highest point of each graph.(c) Both graphs are symmetrical about one of the axes. Name the axis.
1. Using a graphing software, draw each of the following graphs.(a) y = x ^ 2(b) y = - x ^ 2
Nadia walks at an average speed of (x + 2) km/h for (x-3) hours and cycles at an average speed of (3x+5) km/h for x hours. She covers a total distance of 74 km.(i) Form an equation in x and show that it reduces to 4x ^ 2 + 4x - 80 = 0 .(ii) Solve the equation 4x ^ 2 + 4x - 80 = 0(iii) Find the time
Solve the equation 9x ^ 2 * y ^ 2 - 12xy + 4 = 0 expressing y in terms of x.
(i) x = 5 is a solution of the equation x ^ 2 - qx + 10 = 0 find the value of q.(ii) Hence, find the other solution of the equation.
24. Solve the equation x - (2x - 3) ^ 2 = - 6(x ^ 2 + x - 2)
25. When (x + 1) ^ 2 is divided by x2, the quotient is 16 and the remainder is x - 3. Find the possible values of x.
The figure shows an ancient coin which was once used in China. The coin is in the shape of a circle of radius 3 cm with a square of sides x cm removed from its centre. The area of each face of the coin is 7 cm.(i) Form an equation in x and show that it reduces to 2pi - x ^ 2 = 0 (ii) Solve the
19. A piece of wire 44 cm long is cut into two parts. Each part is bent to form a square. Given that the total area of the two squares is 65 cm³, find the perimeter of each square.
18. The length of a side and the corresponding height of a triangle are (x + 3) cm and (2x-5) cm respectively. Given that the area of the triangle is 20 cm², find the value of x.
17. A rectangular field, 70 m long and 50 m wide, is surrounded by a concrete path of uniform width. Given that the area of the path is 1024 m², find the width of the path.
16. The perimeter of a rectangular campsite is 64 m and its area is 207 m². Find the length and the breadth of the campsite.
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