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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
A solid is made up of two identical cones, each with a base diameter of 14 cm and a slant height of 15 cm. Find(1) the total surface area,(ii) the volume, of the solid. 15 cm 15 cm -14 cm-
18. For the following cone, draw two planes of symmetry and one axis of rotational symmetry.
17. A circular cone has a height of 17 mm and a slant height of 21 mm. Find(i) the volume,(ii) the total surface area, of the cone.
16. A cone has a circular base of radius 8 cm and a slant height of 20 cm. Find the volume of the cone.
15. A cone has a circular base of radius 5 cm and a height of 12 cm. Find the curved surface area of the cone.
14. The semicircle shown is folded to form a right circular cone so that the arc PQ becomes the circumference of the base. Given that the diameter of the semicircle, PQ, is 10 cm, find(i) the diameter of the base, (ii) the curved surface area, of the cone. P 0 +10 cm- Q
13. An open cone has a circular base of radius 10 cm and a slant height of 20 cm. Draw the net of the cone and label its dimensions.
12. A conical block of silver has a height of 16 cm and a base radius of 12 cm. The silver is melted to form coins 1 6 cm thick and 1½ cm in diameter. Find the number of coins that can be made.
11. A conical funnel of diameter 23.2 cm and depth 42 cm contains water filled to the brim. The water is poured into a cylindrical tin of diameter 16.2 cm. If the tin must contain all the water, find its least possible height, giving your answer to the nearest integer.
10. For a right circular cone, state(i) the number of planes of symmetry, (ii) the number of axes of rotational symmetry.
9. Find the order of rotational symmetry of the following cone. X
8. An open cone has a slant height of 5 m and a curved surface area of 251 m². Find the radius of the circular base.
A cone has a total surface area of 1000 cm³. Find Pe a possible slant height and radius of the circular base. (Taken to be 3.142.)
A cone has a circular base of radius 6 mm. Given that the curved surface area of the cone is 84 mm², find its slant height.
5. Find the total surface area of each of the following cones (a) 7 cm- 4 cm (b) 28 mm -30 mm- (c) 25 cm Circumference of base - - 132 cm
A cone has a height of 14 cm and a volume of 132 cm³. Find the radius of the circular base. (Take to be)
3. The volume of a cone is 320 cm³ and its base area OPE is ar³ cm³, where r is a prime number. Suggest a value of r and find the corresponding height of the cone.
Find the volume of each of the following cones.2. A cone has a base area of 20 m² and a volume of 160 m². Find the height of the cone. (a) 6 cm 14 cm (c) 7 cm (b) 5 cm (d) Area of base 154 cm 28 mm 14 cm-> Circumference of base - 132 mm
19. The length of each edge of a regular tetrahedron, whose faces are identical equilateral triangles, is 8 cm. Find its(i) slant height,(ii) volume. 23 h 11 h
18 VWXYZ is a rectangular pyramid where WX is longer than XY. Is the slant height VA longer or shorter than the slant height VB? Explain your answer. Z W A X B Y
17. A solid pyramid has a rectangular base of sides 15 cm by 10 cm and a height of 20 cm. It is placed inside an open cubical tank of sides 30 cm. 'The tank is then completely filled with water. If the pyramid is removed, what will be the depth of the remaining water in the tank?
16. Copy and draw the following figures. For each figure, draw two planes of symmetry and one axis of rotational symmetry. (a) (b) A right pyramid with a rectagular base A regular tetrahedron
15. A pyramid has a square base of length 4.8 m and a total surface area of 85 m². Find the volume of the pyramid.
14. A pyramid has a rectangular base of sides 16 m by 14 m. Given that the volume of the pyramid is 700 m³, find its(i) height,(ii) total surface area.
13. VPQRS is a pyramid with a rectangular base of sides 10 cm by 8 cm. Given that the volume of the pyramid is 180 cm³, find its(i) height,(ii) total surface area. S P 10 cm Q R 8 cm
12. VPQRS is a rectangular pyramid with a volume of 100 cm³. Suggest a possible height of the pyramid and the corresponding dimensions of the rectangular base. P S Q R
11. A solid pentagonal pyramid has a mass of 500 g. It is made of a material with a density of 6 g/cm³. Given that the base area of the pyramid is 30 cm³, find its height.
10. VABCD is a pyramid with a rectangular base of sides 15 cm by 9 cm. Given that the slant height VQ of the pyramid is 16 cm, find its(i) height,(ii) volume. D 16 cm C Q9 cm A 15 cm B
9. A glass paperweight is in the shape of a solid pyramid with a square base of length 6 cm and a height of 7 cm. Given that the density of the glass is 3.1 g/cm³, find the mass of four identical paperweights.mass Hint: Density - volume
8. For the following solids, state the number of(i) planes of symmetry,(ii) axes of rotational symmetry. (a) (b) A regular tetrahedron A square pyramid
7. Find the order of rotational symmetry of each of the following solids. X (a) X (b)
6. OWXYZ is a rectangular pyramid where WX66 cm and XY32 cm. Given that the slant heights OA and OB of the pyramid are 56 cm and 63 cm respectively, draw its net and hence, find its total surface area. 63 cm 56 cm B 32 cm W A X 66 cm-
5. The volume of a square pyramid with a height of 12 m is 100 m³. Find the length of its square base.
4. A pyramid with a triangular base has a volume of 50 cm³. If the base and the height of the triangular base are 5 cm and 8 cm respectively, find the height of the pyramid.
3. OXYZ is a pyramid whose base is a right-angled triangle where XY-7 m and YZ-4 m. Given that the height of the pyramid is 5 m, find its volume. X 5 m 7 m 0 Z 4 m Y
OABCDEF is a hexagonal pyramid with a base area of 23 cm³ and a height of 6 cm. Find the volume of the pyramid. A F E 6 cm D B C
OABC is a triangular pyramid with a base area of 15 cm³ and a height of 4 cm. Find the volume of the triangular pyramid. A 4 cm B C
The graph shows the speed-time graph of a car. Speed (m/s) 10 Time (s) 0 23 2 3 4 56 (i) Find the acceleration of the car in the first 2 seconds. (ii) What is the total distance travelled by the car for the whole journey?
3. The figure shows the distance-time graph of a car.(i) Find the duration during which the car is not moving.(ii) Find the speed of the car in the first hour of the journey.(iii) Find the speed of the car for the last part of the journey.(iv) Sketch the speed-time graph of the car for the whole
2. Imran starts a 30-km journey at 0900 hours. He maintains a constant speed of 20 km/h for the first 45 minutes and then stops for a rest. He then continues his journey at a constant speed of 30 km/h, finally arriving at his destination at 1120 hours.(i) Find the distance that he travelled in the
A cyclist set out at 9 a.m. for a destination 40 km away. He cycled at a constant speed of 15 km/h until 10.30 a.m. He then rested for half an hour before completing his journey at a constant speed. He then reached his destination at 11.53 a.m.(i) Draw the distance-time graph to represent his
18. The variables x and y are connected by the equation x+1 y=x²+4x The table below shows some values of x and the corresponding values of y, correct to 1 decimal place. x -8 y -0.2 -6 -5.5 -5 -4.5 4.2 -3.8 -3.5 -0.5 -1.6 3.7 -3 -2.5 -2 -1 -0.8 -0.5 -0.2 0.2 y 0.7 0.3 -0.1 x 0.5 0.8 13 5 y 0.7 0.4
17. A radioactive substance decays at a rate of 15% per hour.(i) Write down an equation for the percentage of substance, S, left after x hours.(ii) By selecting an appropriate scale for each axis, draw the graph that depicts the percentage of substance left after x hours.(iii) Given that the
16. Albert invested PKR 1500 in a financial product that earns 5% interest compounded yearly.(i) Show that the amount of money in the bank after x years is given by 1500(1.05).(ii) By drawing a graph, determine the number of years needed for his investment to double in value.
15. (i) Using a suitable scale, draw the graph of y = 1 + 1/x for 0.5
14. Using a scale of 4 cm to represent 1 unit on the x-axis and 1 cm to represent 2 units on the y-axis, draw the graph of y = 2 ^ x + 1/(x ^ 2) for - 2
13. (i) Using a scale of 2 cm to represent 1 unit on the x-axis and 2 cm to represent 5 units on the y-axis, draw the graph of y = 12 + 10x - 3x ^ 2 for - 2
12. The table below shows some values of x and the corresponding values of y, where y = (x + 2)(4 - x)(a) Copy and complete the table.(b) Using a scale of 2 cm to represent 1 unit on both axes, draw the graph y = (x + 2)(4 - x) for - 2 (c) By drawing a tangent, find the gradient of the curve at the
11. The table below shows some values of x and the corresponding values of y, correct to 2 decimal(i) Copy and complete the table, leaving your answers to 2 decimal places where necessary.(ii) Find the asymptote(s) of the function f(x) = (2x - 5)/(x ^ 2 - 6x + 9)(iii) Using a scale of 1 cm to
10. The sketch represents the graph of yka, where a > 0. 3. Write down the value of k. x
9. On the same axes, sketch the graphs of y = 3 ^ x y = 6 ^ x + 1 and y = - 2 ^ x + 3.
8. Using a scale of 4 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 = 3 ^ pi for - 2
7. The table below shows some values of x and the corresponding values of y, correct to 1 decimal place, where y = 2 + 2 ^ x(i) Find the value of a and of b.(ii) Using a scale of 4 cm to represent 1 unit on the x-axis and 2 cm to represent 1 unit on the y-axis, draw the graph of y = 2 + 2 ^ x for -
6. The table below shows some values of x and the corresponding values of y, correct to 2 decimal places, where y = 2 / x-7. X 4 2 0 2 4 4.5 5 5.5 6 6.8 y-0.18 -1.33 x 7.2 8 8.5 9 9.5 10 12 14 16 18 1.33 0.67 0.22 (i) Copy and complete the table, leaving your answers to 2 decimal places where
5. The table below shows some values of x and the corresponding values of y, correct to 2 decimal places, where y = (3x - 7)/(2x - 5)(iv) Use your graph to find (a) the value of y when x = 3.3, (b) the value of x when y = -2.2, x -5 -3 -1 0 1 1.5 2 y 1.43 1.25 x 2.2 2.8 3 3.5 4 6 8 y 0.67 1.75 (i)
4. (i) Find the asymptote(s) of the function f(x) = (x ^ 2 - 1)/(2x ^ 2 + 3x + 2)(ii) Using a scale of 1 cm to represent 1 unit on the x-axis and 2 cm to represent 1 unit on the y-axis, draw the graph of y = (x ^ 2 - 1)/(2x ^ 2 + 3x + 2) and the asymptote(s) in part (i), for - 8
3. Sketch the graph of the following.(a) y = 5 ^ x - 125(b) y = - 4 ^ x + 16(c) y = 2.5 ^ x + 2(d) y = - 2 ^ x - 1.5
2. The variables x and y are connected by the equation y = 3(2 ^ x) The table below shows some values of x and the corresponding values of y, correct to 1 decimal place where necessary.(i) Copy and complete the table.(ii) Using a scale of 4 cm to represent 1 unit on the x-axis and 1 cm to represent
1. The table below shows some values of x and the corresponding values of y, where y = 4 ^ x(i) Copy and complete the table.(ii) Using a scale of 4 cm to represent 1 unit, draw a horizontal x-axis for - 1 (iii) Use your graph to find(a) the value of y when x = 1.8(b) the value of x when y = 0.4 -1
4. For each of the graphs in Question 3, answer the following questions.(a) The degree of a polynomial is the highest power of its individual terms with non-zero coefficients. For example in Question 3(a), the degree of the polynomial 2x - 3 is 1, and that of 3x ^ 2 + x - 2 is 2. Let the degree of
3. Using a graphing software, draw each of the following graphs.(a) y = (2x - 3)/(3x ^ 2 + x - 2)(b) y = (x ^ 2 - 2x + 1)/(4x ^ 3 + x)(c) y = (2x)/(2x ^ 3 - x ^ 2 + 3x - 2)(d) y = (2x + 3)/(4x)(e) y = (x ^ 2 + 4x - 3)/(3x ^ 2 + 5)(f) y = (4x ^ 3 - 3x + 1)/(4x ^ 3 + 2x)(g) y = (x ^ 2 - 5)/(2x +
2. For each of the graphs in Question 1, answer the following questions.(a) Find the value(s) of x for which the denominator of the function is equal to zero.(b) Is it always true that the value(s) of x in part (a) correspond(s) to the vertical asymptote(s) of the graph?
1. Using a graphing software, draw each of the following graphs.(a) y = (x - 1)/((x - 2)(x + 3))(b) y = ((2x + 1)(x - 1))/(x + 2)(c) y = (x ^ 2)/((4x - 1)(x + 3))(d) y = ((x + 1)(x - 3))/(x + 1)
6. How does the value of k affect the shape of the graph of y =ka^ * ?
5. For each of the graphs in Question 4, answer the following questions.(a) Write down the coordinates of the point where the graph intersects the y-axis.(b) As x increases, what happens to the value of y?(c) Does the graph intersect the x-axis?
3. (a) How does the value ofa, where a is a positive integer and a ne1, affect the shape of the graph of y= a ^ 2 ?(b) What will the graph of y = a ^ x look like if a =1?
2. For each of the graphs in Question 1, answer the following questions.(a) Write down the coordinates of the point where the graph intersects the y-axis.(b) As x increases, what happens to the value of y?(c) Does the graph intersect the x-axis?
1. Using a graphing software, draw each of the following graphs.(a) y = 2 ^ x(b) y = 3 ^ x(c) y = 4 ^ x(d) y = 5 ^ x
Using a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to represent i unit on the y-axis, draw the graph y = 1/4 * x ^ 2 + 8/x - 9 for 0.5 (i) Use your graph to find the minimum value of y in the given range.(ii) By drawing suitable straight lines on the same axes, solve each of the
The variables x and y are connected by the equation 1-1.The table below shows some values of x and the corresponding values of y, correct to decimal place(ii) Using a scale of 1 cm to represent I unit on both ares, draw the graph of y = x + 1/(2x) - 1 for 0.1x4.(iii) Use your graph to find the
14. Using a suitable scale, draw the graph of y = - (x ^ 3)/2 + 3x + 2 for - 4
Using a suitable scale, draw the graph of y = x ^ 3 - 2x - 1 for - 3
12. Using a scale of 2 cm to represent 1 unit on both axes, draw the graph of y = x + 2/(x ^ 2) for 1 < x
11. The table below shows some values of x and the corresponding values of y, where y = x - 3/x The values of y are correct to 1 decimal place, where necessary. x 0.5 1 2 345 6 y -5.5 -2 0.5 h 3.3 4.4 k (i) Find the value of h and of k. (ii) Using a scale of 2 cm to represent 1 unit on 3 both axes,
10. Using a scale of 4 cm to represent 1 unit on both axes, draw the graph of y---1 for 1 2 ≤x≤4.Use your graph to find(i) the value of y when x = 2.5(ii) the value of x when y = - 1.6
9. Sketch the graphs of the following.(a) y = x ^ 3 - x ^ 2 - 2x(b) y = - x ^ 3 - 2x ^ 2 + 15x(c) y = - 1/3 * x ^ 3 + 4/3 * x ^ 2 - x(d) y = 0.5x ^ 3 - x ^ 2 - 4x
8. Using a suitable scale, draw the graph of y = x ^ 3 - 6x ^ 2 + 13x for 0
7. Using 1 cm to represent I unit on the x-axis and 1 cm to represent 2 units on the y-axis, draw the graph for 1.1
6. Using 1 cm to represent 2 units on the x-axis and 1 cm to represent 2 units on the y-axis, draw the graph of y = 6sqrt(x) for 0 < x
5. The table below shows some values of x and the corresponding values of y, where y = 10/(x ^ 2) The values of y are correct to 1 decimal place, where necessary. x 1 2 3 4 5 y 10 2.5 a 0.6 b (i) Find the value of a and of b. (ii) Using a scale of 2 cm to represent 1 unit on both axes, draw the
4. Sketch the graphs of the following.(a) y = - 2/x(b) y = 3/x + 4(c) y = 2/x - 5(d) y = - 1/x - 6
3. The table below shows some values of x and the corresponding values of y, where y = 4/x(i) Copy and complete the table, leaving your answers to 1 decimal place where necessary.(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
2. Sketch the graphs of the following.(a) y = 64 - x ^ 3(b) y = 8 + x ^ 3(c) y = 27x ^ 3 - 1(d) y = - 0.5x ^ 3 - 62.5
2. Sketch the graphs of the following.(a) y = 64 - x ^ 3(b) y = 8 + x ^ 3(c) y = 27x ^ 3 - 1(d) y = - 0.5x ^ 3 - 62.5(iii) Use your graph to find (a) the value of y when x = 1.2 (b) the value(s) of x when y = 0
1. The table below shows some values of x and the corresponding values of y, where y = x ^ 3 - 3x ^ 2 + 1(i) Copy and complete the table above, leaving your answers to I decimal place where necessary.(ii) Using a scale of 2 cm to represent I unit, draw a horizontal x-axis for - 1 (iii) Use your
Using 1 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 = 2.5sqrt(x) for 0
The variables x and y are connected by the equation y = - x ^ 3 - 3 Some corresponding values of x and y are given in the table below.(a) Find the value of p.(b) Using a scale of 2 cm to represent I unit, draw a horizontal x-axis for - 3 (c) The equation - x ^ 3 = 10 has only one solution. Explain
14. The coordinates of the points P and Q are P(-1, 10) and Q(2, 4).(i) Find the equation of the perpendicular bisector of the line joining P and Q.(ii) If the perpendicular bisector of PQ cuts the x- and y-axes at points H and K respectively, calculate the length of HK, giving your answer correct
18. The coordinates of the vertices of a triangle ABC are A(1, 2), B(6, 7) and C(7, 2). Find the equations of the perpendicular bisectors of(a) AB,(b) BC.Hence, find the coordinates of the point equidistant from A, B and C.
12. In the diagram, the coordinates of three points A, B and Care A(-3, 1), B(6, 3) and C(1, 8).(i) Find the gradient of BC and of AB.(ii) If H is the point on the y-axis such that B, C and H are collinear, find the coordinates of H.(iii) Find the coordinates of the point D such that ABCD is a
11. The vertices of AABC are at A(1, 5), B(4, 4) and C(8, 6). Given that Pis the foot of the perpendicular from A to CB produced, find(i) the equation of AP,(ii) the coordinates of P.(iii) the lengths of AP, BC and AC,(iv) the area of AABC,(v) the length of the perpendicular from B to AC. C(-2,8) y
10. The coordinates of three points are A(4, 0), B(-3, 1) and C(-2,8).(i) Find the equation of the line segment joining the points A(4, 0) and C(-2, 8), (ii) Explain why ABBC and that ZABC-90°, showing your working clearly.(iii) If A, Band Care the three vertices of a square, find the coordinates
9. The diagram shows a quadrilateral ABCD where A is (6, 1), B lies on the x-axis and Cis (1, 3). The diagonal BD bisects AC at right angles at M. Find(i) the equation of BD,(ii) the coordinates of B.(iii) the coordinates of D such that ABCD is a parallelogram. C(1,3) M D B A(6, 1) x
A rhombus ABCD is such that the coordinates of A and Care (0, 2) and (12, 8) respectively. Find the equation of the perpendicular bisector of AC and of BD
7. Given that the x-intercept of a line is twice its y-intercept and that the line passes through the point of intersection of the lines 3y+x-3 and 4y3x5, find the equation of this line.
6. Find the equation of the line that is parallel to 2y-x-7 and bisects the line segment joining the points (3, 1) and (1,-5).
Find the equation of the perpendicular bisector of the line segment joining the points(a) (0, 2) and (2.0),(b) (1,8) and (2, 3),(c) (0,7) and (6,7),(d) (5, 4) and (5,9).
Find the equation of the line segment joining the points whose x coordinates on the curvey-2-3 are-1 and 1.
Find the equation of the line passing through the point(a) (-2,5) and is parallel to the line 3y+7-29,(b) (-1,-6) and is perpendicular to the line 42-7y-5,(c) (4,8) and is parallel to the line 3x + y = 17,(d) (2,-3) and is perpendicular to the line y+2x-13.
Find the equation of the line passing through(a) A(2,-1) and 8(5, 5),(b) C(1,4) and D(0, 6),(c) (7.3) and F(-3, 3),(d) G(-9,-2) and H(-9,8).
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