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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) Find the equation of the line / with gradient 3 and passing through (-2, 8). Hence, write down a possible equation of a line that is perpendicular to 1.(b) Find the equation of the line / with gradient 2 and passing through (7, -3). Hence, write down a possible equation of a line that is
Find the equation of the perpendicular bisector of the line segment joining A(6, 3) and (2,15).
Find the equation of the line passing through the point(a) (2,7) and is perpendicular to y=x+5,(b) (2, 3) and is perpendicular to 2y+x-2,(c) (4,9) and is parallel to 3y + 2x+5-0(d) (-5,-2) and is parallel to 6x-2y-7.
Find the equation of the line I with gradient 2 and passing through (1, 4).Hence, write down a possible equation of a line that is perpendicular to 1.
Find the equation of the line / that passes through the points A(1, 2) and B(3, 8). Hence, write down a possible equation of a line that is perpendicular to L
Use two methods to determine whether the four points (5, 8), (7.5), (3, 5) and (5, 2) are the vertices of a rhombus, showing your working clearly.
Use two methods to determine whether the four points (2, 1), (-1,-5), (1, 5) and (-2,-1) are the vertices of a parallelogram, showing your working clearly.
14. The coordinates of three points are O(0, 0), P(a,b) and Q(c, d). Find an example of the values ofa, b, c and d if OP is perpendicular to OQ.
13. The coordinates of three points are A(-1,-3), B(2, 3) and ((6, k). If AB is perpendicular to BC, find(i) the value of k.(ii) the gradient of AC,(iii) the acute angle that AC makes with the x-axis.
12. Give two reasons why 3x + y = 8 is not the perpendicular bisector of the line segment joining (4, 1) and (8,7).
11. The line x + 3y = 1 intersects the curve 5y = 20 - 3x - x ^ 2 at the points Pand Q. Given that the equations of AB and CD are y - 3x = 4 and 3y + x = 4 what can you say about the lines(a) AB and PQ?(b) CD and PQ
10. Given that A is the point (0, 4) and 8 is (6, 6), find(a) the coordinates of point Con the x-axis such that ABBC,(b) the coordinates of point D on the y-axis such that angle ABD = 90 deg .
9. Show that P-1, 3), (Q(6, 8) and R(11, 1) are the vertices of an isosceles triangle. Is APQRa right-angled triangle? Give reasons for your answer.
8. Explain whether the two lines with equations 3y - 2x = 4 and 4x6y-8 are parallel to each other.
7. A point P is equidistant from R(-2, 4) and S(6,-4) and its x-coordinate is twice its y-coordinate.(i) Find the coordinates of P.(ii) Hence, explain why P. Rand S are not collinear.
The coordinates of three points are A(m, n), B(4,6) and C15, 3). Find an example of the values of m and if AB is perpendicular to BC.
Explain whether the following pairs of lines are parallel, perpendicular or neither.(a) and y = 7x + 16 y = 7x + 14(b) y = 4x - 7 y=-x+5 4(c) y = - 2x + 2 and | 2y = x + 6(d) x + 2y = 6 and y-3x-8
The coordinates of three points are A(-1,-6), B(3,-12) and C(k, 6). Find the value of k if(i) AB is perpendicular to AC,(ii) A, B and Care collinear.
The vertices of AABC are at A(-1,-3), B(2, 3) and C(k + 5,k) Find the value of k if AB is perpendicular to BC
The coordinates of three points are A(1, 1), 8(-1,4) and C(6, k), Find the value of k if AB is perpendicular to BC
The coordinates of four points are O(0, 0), A(2, 3k), B(4k, 6) and C(10k, 7). Find the value(s) of k if(i) OA is parallel to BC,(ii) the points O, A and B are collinear.
The coordinates of three points are O(0, 0), A(5, 1) and B(p, q). Find a possible set of values of p and q if OA is perpendicular to OB.
The coordinates of three points are O(0, 0), P(m, n) and Q(6, 3). Find a possible set of values of m and n if OP is perpendicular to OQ.
The vertices of AABC are at A(0, -5), B(-2, 1) and C(10, 5). Is AABC a right-angled triangle? Give reasons for your answer.
The coordinates of four points are A(0, 9), B(k + 1, k+4), C(2k, k+3) and D(2k+2, k+6).Find the value(s) of k if(i) AB is parallel to CD,(ii) the points A, B and Care collinear.
Given four points O(0, 0), A(2, k), B(2k, 9) and C(3k, 2k +7), find the value(s) of k if(i) OA is parallel to BC,(ii) the points O, A and B are collinear.
10. In APQR, the midpoints of the sides PQ, QR and PR are A(-2, 3), B(5, -1) and C(-4, -7) respectively. Find the coordinates of P, Q and R.
9. The diagram shows a triangle ABC with vertices at A(-2, 2), B(8, 6) and C(10, 1). P and Q are the midpoints of AB and AC respectively. PQ is parallel to BC.(i) Given that PQ produced meets the x-axis at R, find the coordinates of R (ii) Is Q the midpoint of PR? Explain your answer. B(8,6) A(-2,
8. ABCD is a rectangle.(a) How will the midpoint of AC change when the rectangle is moved 4 units in the positive direction of the x-axis?(b) The coordinates of A and Care (h, k) and (m, n) respectively. Given that the rectangle is moved k units in the negative direction of the y-axis and then k
7. Given that the line x + 2y = 5 meets the curve 5x+4y²29-12x at the points A and B,(i) find the coordinates of the midpoint of AB,(ii) calculate the length of AB.
6. The line y = x + 2 intersects the curve y = x² + 5x-3 at the points P and Q. Find the coordinates of the midpoint of PQ.
5. Three of the vertices of a rhombus PRQS are P(1,2), R(5, 0) and Q(7, 4). Find the fourth vertex S.
(b) Use two different methods to explain why F(-6, 1) is not the midpoint of the line segment joining the points P(-7, -1) and Q(-5,5).
4. (a) Explain why E(k, 4) is not the midpoint of the line segment joining the points P(k, 5.5) and Q(k, 7). What can you say about the points E, P and Q?
3. Three of the vertices of a parallelogram ABCD are A(-3, 1), B(4, 9) and C(11, -3). Find(i) the midpoint of the diagonal AC,(ii) the fourth vertex D.
2. In each of the following, M is the midpoint of PQ.Find the coordinates of Q.(a) M(5,0) and P(-5, 0)(b) M(3,7) and P(3, -10)(c) M(6, 2) and P(1, 1)(d) M(0,3) and P(4,-1)
1. Find the coordinates of the midpoint of the line segment joining each of the following pairs of points.(a) (1, 1) and (7,3) (b) (3,-2) and (-2, 7)(c) (4, 4) and (8,4)(d) (0,-2) and (0, 6)(e) (2a+b, 3ba) and (ba, 2ab)(f) (ah², 2ah) and (ak², 4ak)
2. The line 5x + y = 17 intersects the curve 5x² + y² 49 at the points P and Q.(i) Find the coordinates of the midpoint of PQ.(ii) Calculate the length of PQ.
1. ABCD is a parallelogram. The coordinates of A, B and Care (-3, 5), (2, 7) and (4,6) respectively. Find(i) the coordinates of D,(ii) the length of the diagonal BD.
If A(2,-4), B(7, 1) and C(-1, 5) are three vertices of a parallelogram ABCD, find the midpoint of AC. Hence, find(i) the coordinates of D,(ii) the length of AD.
(a) Find the coordinates of the midpoint of the line segment joining the points(i) A(5, 3) and B(-1,7),(ii) C(2, 3) and D(6, 3),(iii) E(1, 4) and F(1,-2).(b) If (2, 0) is the midpoint of the line segment joining A(8, -3) and B(p, q), find the value of pand of q.
(a) Find the coordinates of the midpoint of the line segment joining the points(i) A(2, 5) and B(-6, -1),(ii) C(-3, 4) and D(5, 4),(iii) D(5, 4) and E(5, 1).(b) If the coordinates of the midpoint of the line segment joining P(-5, 3) and Q(a,b) are (3, 4), find the value of a and of b.
17. The coordinates of the points P and Q are (2, 3) and (9, 5) respectively.(i) Find the coordinates of the point where the line passing through P and Q intersects the x-axis.(ii) Given that y = 5 is the line of symmetry of APQR, find the coordinates of R.(iii) Calculate the length of PQ.(iv)
16. The equation of the line l is 3x + 4y = 24. It crosses the x-axis at the point A and the y-axis at the point B. Find(i) the coordinates of A and of B,(ii) the length of the line segment AB,(iii) the equation of the line OC, where O is the origin and the point C is equidistant from the
15. If the line mxny + 2 has the same gradient as the x-axis, find the value of m. State the condition for the line to be parallel to the y-axis instead.
14. A straight line I passes through the points A(0, 3) and B(3, 12).(a) Find(i) the gradient of the line I,(ii) the equation of the line l.(b) The line x-3 is the line of symmetry of ∆ABC Find the coordinates of C.
13. The line I has equation 5x+6y+300. Given that P is the point (3,-1), find(i) the coordinates of the point where I crosses the x-axis,(ii) the coordinates of the point of intersection of/with the line x-2.(iii) the equation of the line passing through P and having the same gradient as I,(iv) the
12. (1) Find the equation of the straight line which passes through the point (3, 1) and with gradient 3.(ii) Hence, find the coordinates of the point of intersection of the line in part (i) with the line
11. Find the equation of the straight line passing through the point (3,-2) and having the same gradient as the line 2y-5x+7.
10. (i) Find the equation of the straight line which passes through the point (-3, 5) and with gradient 2(ii) Given that the line in part (i) also passes through the point (p. 3), find the value of p.
9. Given the line+ -1,(i) make y the subject of the formula+-1,(ii) find the gradient of the line,(iii) find the coordinates of the point at which the line cuts the x-axis.
8. The lines 2x-5-ky and (k+1)x6y-3 have the same gradient. Find the possible values of k.
7. The diagram shows AABC with vertices A(1, 1), B(1, 3) and C(5,5).Find (i) the area of AABC, (ii) the gradient of the line passing through B and C (iii) the equation of the line passing through A and C. 5 4 3 2+ 1 G 0 2 B 5
6. In each of the following diagrams, find the gradient and the y-intercept of the line where possible. State the equation of each line. (a) (c) 1 (b) x 0 0 x (d) x 1.5 0 2 x
5. Write down the equation of the straight line which passes through the origin and with gradient 2.
Find the equation of each of the following straight lines, given the gradient and the coordinates of a point that lies on it. (a) 13. (0,0 0) (c) -3, (2,-5) (e) 0, (5, 4) (b) 3, (1, 1) (d), (5,7) (f) a, (0, a)
3. Find the equation of the straight line passing through each of the following pairs of points.(a) A(0, 0) and 8(1,-1)(b) C(1,3) and D(2, 5)(c) E(2, 4) and F(-2, 3)(d) G(-6,-5) and H(4, 4)(e) 1(-2,-4) and J(1,-7)(f) K(-7,-5) and L(-1,-1)(g) M(8,0) and N(-9,0)(h) O(0,0) and P(0, 7)
The point (-3, 3) lies on the line y4x+k. Find the value of k
Given that the line yx+cpasses through the point (1, 2), find the value of c.
Find the equation of the straight line passing through each of the following pairs of points.(a) A(1, 2) and B(3, 7)(b) C(2, 3) and D(7, 3)(c) E(5, 1) and F(5, 6)
Find the equation of the straight line passing through each of the following pairs of points.(a) A(-2, 1) and B(5, 3)(b) C(6, 4) and D(-4, 4)(c) E(-3, 5) and F(-3, 8)
9. The coordinates of the vertices of a square ABCD are A(0, 6), B(2, 1), C(7, 3) and D(5,8).(i) Find the gradient of all 4 sides of ABCD.(ii) What do you observe about the gradients of the opposite sides of a square?
8. The line joining the points A(2, 1) and B(7, 2²+7) has a gradient of 2. Find the possible values of t.
7. The points P(2,-3), Q(3,-2) and R(8, z) are collinear, ie. they lie on a straight line. Find the value of z.
6. The points P, Q and R have coordinates (6, -11), (k,-9) and (2k, -3) respectively. If the gradient of PQ is equal to the gradient of PR, find the value of k
5. The gradient of the line joining the points (9,a) and (2a, 1) is, where a = 0. Find the possible values of a.
4. The coordinates of A and B are (3k, 8) and (k, -3) respectively. Given that the gradient of the line segment AB is 3, find the value of k.
3. If the gradient of the line joining the points (-3,-7) 3 and (4, p) is find the value of p.
2. The points A(0, 1), B(7, 1), C(6, 0), D(0, 5) and E(6, 4) are shown in the diagram.Find the gradient of each of the line segments AB, AE, DC and DE. D(0,5) 5 E(6,4) 4+ 3- 2+ 1 A(0, 1) 0 + B(7,1) C(6,0) 1 2 3 4 5 6 7 x
1. Find the gradient of the line passing through each of the following pairs of points.(a) A(0,0) and B(-2, 1)(b) C(2,-3) and D(1, 7)(c) E(-2, 4) and F(-5, 8)(d) G(-4,7) and H(1,-8)(e) 1(-2,-5) and J(2, 6)(f) K(-7, 9) and L(6, 9)
If the gradient of the line joining the points (k, 5) and (2, k) is -2, find the value of k.
Find the gradient of the line passing through each of the following pairs of points.(a) C(3, 1) and D(6, 3)(b) H(5,-7) and K(0, -2)(c) M(-4, 1) and N(16, 1)
Find the gradient of the line passing through each of the following pairs of points.(a) A(2, 3) and B(7, 5)(b) P(-2, 8) and Q(1,-1)
5. (a) When y_{2} - y_{1} > 0 and x_{2} - x_{1} < 0 what do you notice about the sign of the gradient?(b) When y_{2} - y_{1} < 0 and x_{2} - x_{1} > 0 what do you notice about the sign of the gradient?(c) When the signs of y, y, and x,x, are the same, what do you notice about the sign of the
Using your answer in Question 2, find the gradient of the line passing through each of the following pairs of points.(a) (-1, 4) and (3,7)(b) (-4,-3) and (2,-11)(c) (6,3) and (-4, 3)(d) (2,-1) and (2, 8)Compare your answers with those obtained by your classmates.
2. Given any two points A(x, y) and B(x, y), how would you find the gradient of the line passing through A and B? B(x, y) A(x, y) x
1. In Fig. 5.5(a) and (b), A and B are two points on each line.For each of the two lines shown in Fig. 5.5:(i) Find the vertical change from point A to point B.(ii) Find the horizontal change from point A to point B.(iii) Find the gradient of the line segment AB.(iv) Choose two other points C and D
11. The vertices of APQR are P(1, 3), (Q(5, 4) and R(5, 15). Find the length of the perpendicular from Q to PR
10. By showing that the points P(3, 4), Q(3, 1) and R(8, 4) are the vertices of a right-angled triangle, find the length of the perpendicular from P to QR.
9. (i) Show that the points A(-1, 2), B(5, 2) and C(2, 5) are the vertices of an isosceles triangle.(ii) Find the area of ∆ABC
8. The distance between the points (1, 2f) and (1-1,1) is √11-9t units. Find the possible values of t.
7. The diagram shows triangle ABC with vertices A(- 1, 1) B(- 1, - 2) and C(3,- 1)(i) Find the lengths of AB, BC and AC.(ii) Find the area of ∆ABC.The coordinates of a point E are (3, k) and the area of ∆BCE is 14 units2.(iii) Find the possible values of k. 2 B A 2 ++ 1+ 3 C
6. The diagram shows triangle ABC with vertices A(- 2, 1) B(1, 1) and C(3,4)(i) Find the area of triangle ABC (ii) Find the length of AC, giving your answer correct to 2 decimal places.(iii) Given that ABCD is a parallelogram, find the coordinates of D.(iv) Given that K is the point (t, 4) and the
5. The vertices of AABC are A(-4, -2), B(8,-2) and C(2,6). 6- 4+ 2 C(2,6) 0 2 2 6 8. 2 A(-4,-2) B(8, 2) (i) Find the perimeter and the area of AABC. (ii) Hence, find the length of the perpendicular from A to BC.
4. A line segment has two endpoints M(3, 7) and N(11,6). Find the coordinates of the point W that lies on the y-axis such that W is equidistant from M and from N.Hint: The term 'equidistant' means 'same distance.
3. Given that the coordinates of the points P and Q are (-2, 6) and (9, 3) respectively, find(a) the coordinates of the point R that lies on the y-axis such that PR - QR,(b) the coordinates of the point S that lies on the x-axis such that PSQS.
2. If the distance between the points A(p, 0) and B(0, p) is 10 units, find the possible values of p.
1. Find the length of the line segment joining each of the following pairs of points.(a) A(2, 3) and B(9, 7)(b) C(3, 6) and D(-5, 9)(c) E(-1, 4) and F(8,-3)(d) G(-10, 2) and H(-4, -7)
13. A triangle has vertices A(0, -5), B(-2, 1) and C(10, 5). Show that AABC is a right-angled triangle and identify the right angle.
12. Given that the coordinates of the points C and D are (4,-1) and (-2, 7) respectively, find(a) the coordinates of the point F that lies on the y-axis such that CE = DE,(b) the coordinates of the point F that lies on the x-axis such that CF-DF.Hence, find the area of AOFF, where O is the origin.
11. Given that the coordinates of the points A and B are (-3, 2) and (1,-6) respectively, find the coordinates of the point C that lies on the y-axis such that AC BC. Hence, find the area of ΔACO, where O is the origin.
10. Find the length of the line segment joining each of the following pairs of points.(a) C(6, 2) and D(3,-2)(b) M(-1,5) and N(6, -4)(c) P(2,7) and Q(8,7)
9. Given that the coordinates of the points A and B are (-4, 1) and (6,5) respectively, find the length of the line segment AB.
The table shows the approximate population of the world in the past centuries.Find (i) the increase in population from 1549 to 1649 (ii) the number of times that the population in 18-49 is as large as that in 16-49, (iii) the number of times that the population of China in 2020 is as large as that
14. On a journey from Planet P to Venus, a rocket is travelling at a constant speed. During this journey the rocket travels pust Planet Qan 4 days. The distance from Planet P to Planet is 4.8 x 10 km.(i) Find the distance travelled by the rocket in 12 days. Give your answer in standard form.(ii)
13. Light travels at a speed of 300 000 000 m/s(1) Express this speed in standard form(4) Given that the mean dutance from the Sun to Jupiter is 778.5 million kilometres, find the time taken, in minutes and seconds, for light to travel from the Sun to fupiter
12. Given that R = M/EI, find the value of R when M = 6 * 104, E = 45x108 and 1 = 4 * 102 Give your answer in standard form.
11. Given that M = 3.2 x 106 and N = 5.0 * 10.7 find the value of each of the following, giving your answer in standard form.(a) MN(b) M/N
10. Given that x = 2 * 10 ^ 10 andy-7x10", evaluate By, giving your answer in standard form.
Given that P = 7.3 * 10 ^ 3 and 25 * 10 ^ 2, x! each of the following in standard form.(a) 2P x 4Q(b) Q - P
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