Straight LineHigher Mathematics PSfrag replacements O x y 18. (a) The line l1 passes through the point (1,10) and is perpendicular to the linewith equation x +2y = 1. Find the equation

Higher Mathematics

PSfrag replacementsOxy

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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PSfrag replacementsOxy Page 1

Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes

Higher Mathematics


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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Higher Mathematics


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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Higher Mathematics


[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]


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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Higher Mathematics


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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Higher Mathematics


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

PSfrag replacements

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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Higher Mathematics


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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Higher Mathematics


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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Higher Mathematics


Paper 21.[SQA] Find the equation of the perpendicular bisector of the line joining A(2,−1) and

B(8, 3) . 4

2.[SQA]


3.[SQA]


4.[SQA]


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Higher Mathematics


5.[SQA]


6.[SQA]


7.[SQA]


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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Higher Mathematics


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]


13.[SQA]


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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Higher Mathematics


15.[SQA]


16.[SQA]


17.[SQA]


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Higher Mathematics


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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Higher Mathematics


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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Higher Mathematics


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


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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Straight LineHigher Mathematics PSfrag replacements O x y 18. (a) The line l1 passes through the point (1,10) and is perpendicular to the linewith equation x +2y = 1. Find the equation