Straight LineHigher Mathematics PSfrag replacements O x y 18. (a) The line l1 passes through the...
Transcript of Straight LineHigher Mathematics PSfrag replacements O x y 18. (a) The line l1 passes through the...
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Higher Mathematics
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Straight Line
Paper 1 Section A
Each correct answer in this section is worth two marks.
1. The line with equation y = ax + 4 isperpendicular to the line withequation 3x + y + 1 = 0.What is the value of a?
A. −3
B. − 13
C. 13D. 3
2. What is the distance, in units,between the points (−1, 2) and(4, 5)?
A.√
8
B.√
16
C.√
34
D.√
58
3. What is the distance, in units,between the points (a, b) and(−b, a)?
A.√
2√
a2 + b2
B.√
2(a + b)
C.√
2(√
a +√
b)
D. 2√
a2 + b2
4. The line through the points (−2, 5)and (7, a) has gradient 3.What is the value of a?
A. 8
B. 22
C. 28
D. 32
5. The equation of a line is 3y = ax + 1where a 6= 0 is a constant.Given that the line has a gradient of75 , what is the value of a?
A. − 215
B. − 75
C. 75
D. 215
6. The line with equation y = − 3a x + 4,where a 6= 0 is a constant, isperpendicular to the line withequation y = 12 x + 1.What is the value of a?
A. −6
B. − 32
C. 32D. 6
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7. The line l passes through (3,−2)and is parallel to the line withequation y = 12 x + 5.What is the equation of l ?
A. x − 2y + 1 = 0
B. x − 2y − 7 = 0
C. x − 2y + 7 = 0
D. x − 2y − 5 = 0
8. Find the equation of the line passingthrough (6,−4) and parallel to theline with equation 2x − 3y − 1 = 0.
A. 2x − 3y − 24 = 0
B. 3x + 2y − 10 = 0
C. 2x − y − 16 = 0
D. 2x − 3y − 18 = 0
9. Given that (1, 0) is the midpoint ofA(−3, a) and B(b, 2) , what are thevalues of a and b?
a bA. −2 4
B. −2 5
C. 2 −5
D. 4 −2
10. Triangle ABC is shown below.PSfrag replacementsOxy
A
B
CD
Here are two statements about theline BD:
I. BD is an altitude of triangleABC
II. BD is the perpendicularbisector of AC
Which of the following is true?
A. neither statement is correct
B. only statement I is correct
C. only statement II is correct
D. both statements are correct
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11. Triangle ABC with verticesA(−4, 1) , B(4, 3) and C(1, 5) isshown below.
PSfrag replacements
O x
y
A
B
C
M
Point M(0, 2) is the midpoint of AB.What is the equation of the medianthrough C?
A. 3x − y + 2 = 0
B. x − 4y + 8 = 0
C. 4x + y − 2 = 0
D. 3x − y − 1 = 0
12. Triangle ABC with vertices A(6, 7) ,B(7, 0) and C(−1,−2) is shownbelow.
PSfrag replacements
Ox
y A
BC
The line through C and B hasgradient 14 . Find the equation of thealtitude through A.
A. 4x + y − 11 = 0
B. x − 4y + 22 = 0
C. 4x + y − 31 = 0
D. 8x − 3y − 27 = 0
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[END OF PAPER 1 SECTION A]
Paper 1 Section B13.[SQA] Three lines have equations 2x + 3y − 4 = 0, 3x − y − 17 = 0 and x − 3y − 10 = 0.
Determine whether or not these lines are concurrent. 4
14.[SQA] The points A and B have coordinates (a, a2) and (2b, 4b2) respectively. Determinethe gradient of AB in its simplest form. 2
15.[SQA] Find the equation of the straight line which is parallel to the line with equation2x + 3y = 5 and which passes through the point (2,−1) . 3
16.[SQA]
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17. The kite ABCD has vertices A(1, 8) , B(0,−2) ,C(−3,−4) and D
(
− 215 , k)
as shown in thediagram.(a) Determine the value of k . 5
(b) Find the area of triangle BCD. 4PSfrag replacements
O x
yA(1, 8)
B(0,−2)
C(−3,−4)
D(
− 215 , k)
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18. (a) The line l1 passes through the point (1, 10) and is perpendicular to the linewith equation x + 2y = 1.Find the equation of l1 . 3
(b) The line l2 passes through the point (6, 5) and makes an angle a with thepositive direction of the x -axis, where tan a = 13 .Find the equation of l2 . 2
(c) Determine the coordinates of the point of intersection of l1 and l2 . 3
19. Triangle ABC has vertices A(1, 2) , B(8, 2) andC(4, 6) .(a) Find the equation of the line through the
collinear points A, C and D. 2
(b) Triangle ABD is right-angled at B. Find theequation of BD. 1
(c) Find the coordinates of D. 2
PSfrag replacements
Ox
y
A(1, 2) B(8, 2)
C(4, 6)
D
20. The equation of a straight line is√
3x + ay + 1 = 0, where a 6= 0 is a constant.Given that this line has a gradient of 1√3 , find the value of a , and hence state thecoordinates of the point where the line cuts the y-axis. 4
21. The line AC is the diameter of a circle with B lying on the circumference as shownbelow.
PSfrag replacementsOxy A(−1, 5)
B(3, 7)
C
A is the point (−1, 5) and B(3, 7) .
Find the equation of the chord BC. 3
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22. The diagram below shows the right-angled triangle OPQ and a circle with centreC(0, 5) and diameter OS.
PSfrag replacements
O x
y
R
S
C
P
Q
3
6
Find the equation of the chord RS. 5
23. Triangle ABC has vertices A(−3, 5) , B(9, 9) andC(9,−3) .(a) Write down the equation of BC. 1
(b) Find the equation of the altitude from A. 2
(c) Find the equation of the perpendicularbisector of AB. 4
(d) Find where the perpendicular bisector of ABand the altitude from A intersect. 2
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O x
y
C
A
B
24. Triangle ABC is shown in the diagram below.
PSfrag replacements
O x
y
A
B(
2√
3, 7)
C
A and C lie on the x -axis, and B is the point(
2√
3, 7)
.
(a) The line AB has a gradient of 12 . Find the equation of AB. 2
(b) BC is part of the line with equation√
3x − y + 1 = 0.Find the length of AC. 2
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25. The line with equation 3x + ay + 1 = 0, where a is a constant, is perpendicular tothe line with equation 2y − x = 2.Find the value of a . 3
26. Triangle ABC has vertices A(4, 7) , B(−2, 1)and C(6,−3) .(a) Find the equation of line p , the median
from A. 3
(b) Find the equation of line q , the altitudefrom C. 3
(c) Find the point of intersection of the linesp and q . 3
PSfrag replacements
O x
y
C
A
B
27. The line with equation 4y + 3x − 24 = 0 intersects the x -axis at P and the y-axisat R.
(a) Write down the coordinates of P and R. 1
(b) The perpendicular bisector of PR meets the line y = −1 at Q.Find the coordinates of Q. 5
(c) Show that P, Q and R could be three vertices of a square. 3
28. Triangle PQR is shown in the diagram.
PSfrag replacements
O x
yP(1, 3)
Q(7,−2)R
The line PR has equation 6x + y − 9 = 0, and QR has equation x − 5y − 17 = 0.(a) Find the coordinates of point R. 2
(b) Hence find the equation of the median through R. 3
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29. The points A(3, 2) , B(2a, 12) and C(a,−1) are collinear.Find the value of the constant a . 4
30. Triangle ABC is shown in the diagram.
PSfrag replacements
O x
yA(−4, 6)
B(4,−3)C
The line AC has equation 8x + 2y = −20, and BC has equation x + 6y = −14.Find the equation of the altitude through C. 5
[END OF PAPER 1 SECTION B]
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Paper 21.[SQA] Find the equation of the perpendicular bisector of the line joining A(2,−1) and
B(8, 3) . 4
2.[SQA]
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3.[SQA]
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4.[SQA]
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5.[SQA]
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6.[SQA]
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7.[SQA]
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8.[SQA] Find the equation of the line through the point (3,−5) which is parallel to the linewith equation 3x + 2y − 5 = 0. 2
9.[SQA]
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10.[SQA] The vertices of a triangle are P(−1, 1) , Q(2, 1) and R(−6, 2) . Find the equation ofthe altitude of triangle PQR, drawn from P. 3
11.[SQA] Find the equation of the median AD of triangle ABC where the coordinates of A,B and C are (−2, 3) , (−3,−4) and (5, 2) respectively. 3
12.[SQA]
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13.[SQA]
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14.[SQA] Triangle ABC has vertices A(−1, 6) ,B(−3,−2) and C(5, 2) .Find(a) the equation of the line p , the
median from C of triangle ABC. 3
(b) the equation of the line q , theperpendicular bisector of BC. 4
(c) the coordinates of the point ofintersection of the lines p and q . 1
PSfrag replacements
Ox
yA(−1, 6)
B(−3,−2)
C(5, 2)
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15.[SQA]
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16.[SQA]
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17.[SQA]
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18.[SQA] P, Q and R have coordinates (1,−2) , (6, 3) and (9, 14) respectively and are threevertices of a kite PQRS.
(a) Find the equations of the diagonals of this kite and the coordinates of the pointwhere they intersect. 7
(b) Find the coordinates of the fourth vertex S. 2
19. Points A(2, 3) and C(8, 11) are the end points of a diagonal of rectangle ABCD.
The diagonal BD is parallel to the x -axis.Find the area of the rectangle. 5
20. The diagram shows a rhombus ABCD.The points B and D have coordinates(3, 5) and (9, 3) respectively.The equation of AD is x − y = 6.Find the size of angle CAD. 6
PSfrag replacements
O x
y
A
B
C
D
21. The diagram shows triangle ABC.A circle passes through all the vertices of thetriangle.AC is a diameter of the circle.The equation of AB is y − 3x = 2.(a) Find the gradient of BC. 3
(b) BC makes an angle of a radians with thepositive direction of the x -axis.Find the value of a . 2
PSfrag replacements
O x
y
A
B
Ca
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22. The line ` passes through points A(−2, 6) and B(5,−1) , as shown below.
PSfrag replacements
O x
y
A(−2, 6)
B(5,−1)C
P
The line through O and P is perpendicular to ` , and C lies on the x -axis.(a) Find the equation of the line through O and P. 3
(b) Hence find the coordinates of P. 3
(c) Calculate the area of the triangle OCP. 3
23. Find the equation of the straight line passing through the point(
2√
3, 7)
, with agradient of 12 . 2
24. The line with equation x + 2y − 6 = 0 makes an angle of φ◦ with the x -axis asshown below.
PSfrag replacements
O x
y
φ◦
x + 2y − 6 = 0
Calculate the value of φ correct to two decimal places. 4
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25. The points R(1, 2) , S(5, 8) and T(11, 4) lie on the circumference of a circle.
PSfrag replacements
O x
y
R
S
T
3x − 2y − 12 = 0
The line with equation 3x − 2y − 12 = 0 is the perpendicular bisector of ST.(a) Find the equation of the perpendicular bisector of RS. 4
(b) The centre of the circle is the point where the perpendicular bisectors of RSand ST intersect.Calculate the coordinates of the centre of the circle. 3
26. Triangle PQR has vertices P(1, 4) , Q(6, 14) andR(7, 6) .(a) Find the equation of the median QS. 3
(b) Find the equation of the altitude RT. 3
(c) The median QS and the altitude RT intersect atA.Find the coordinates of A. 3
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P(1, 4)
Q(6, 14)
R(7, 6)
T
S
27. The vertices of the triangle PQR are P(2, 6) , Q(−4,−4) and R(−3, 7) .
(a) Find the equation of the median through R. 3
(b) Find the equation of the altitude through Q. 3
(c) The median through R and the altitude through Q intersect at point T.Calculate the coordinates of T. 3
[END OF PAPER 2]
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