HSC Exam Questions (Linear Functions)
Short Description
A collection of questions relating to the topic Linear Functions from the 1995-2014 HSC Mathematics Advanced (2U) examin...
Description
HSC Exam Questions (Linear Functions) 1995 HSC Q2 The line 𝑙 cuts the 𝑥 axis at 𝐿 (−4, 0) and the 𝑦 axis at 𝑀 (0, 3) as shown. 𝑁 is a point on the line 𝑙 , and 𝑃 is the point (0, 8).
Copy the diagram into your Writing Booklet. (a)
Find the equation of the line 𝑙 .
2
(b)
Show that the point (16, 15) lies on the line 𝑙 .
1
(c)
By considering the lengths of 𝑀𝐿 and 𝑀𝑃, show that Δ𝐿𝑀𝑃 is isosceles.
2
(d)
Calculate the gradient of the line 𝑃𝐿.
1
(e)
𝑀 is the midpoint of the interval 𝐿𝑁. Find the coordinates of the point 𝑁 .
2
(f)
Show that ∠𝑁𝑃𝐿 is a right angle.
1
(g)
Find the equation of the circle that passes through the points 𝑁 , 𝑃 and 𝐿 .
2
1996 HSC Q2 The line 𝑙 is shown in the diagram. It has equation 𝑥 + 2𝑦 + 9 = 0 and cuts the 𝑥 axis at 𝐴. !
The line 𝑘 has equation 𝑦 = − 𝑥 + 6, and is not shown on the diagram. !
Copy or trace the diagram into your Writing Booklet. (a)
Find the coordinates of 𝐴.
1
(b)
Explain why 𝑘 is parallel to 𝑙 .
1
(c)
Draw the graph of 𝑘 on your diagram, indicating where it cuts the axes.
2
(d)
Shade the region 𝑥 + 2𝑦 + 9 ≤ 0 on your diagram.
1
(e)
Write down a pair of inequalities which define the region between 𝑘 and 𝑙 .
2
(f)
Show that 𝑃 (8, 2) lies on 𝑘 .
1
(g)
Find the perpendicular distance from 𝑃 to 𝑙 .
2
(h)
𝑄(4, −6) lies on 𝑙 . Show that 𝑄 is the point on 𝑙 which is closest to 𝑃 .
2
6
1997 HSC Q3(b) Let 𝐴 and 𝐵 be the points (0, 1) and (2, 3) respecrtively.
(i)
Find the coordinates of the midpoint of 𝐴𝐵.
(ii)
Find the slope of the line 𝐴𝐵.
(iii)
Find the equation of the perpendicular bisector of 𝐴𝐵.
(iv)
The point 𝑃 lies on the line 𝑦 = 2𝑥 − 9 and is equidistant from 𝐴 and 𝐵 . Find the coordinates of 𝑃 .
1998 HSC Q3 The diagram shows points 𝐴 1, 0 , 𝐵(4, 1) and 𝐶 (−1, 6) in the Cartesian plane. Angle 𝐴𝐵𝐶 is 𝜃 .
Copy or trace this diagram into your Writing Booklet. (a)
Show that 𝐴 and 𝐶 lie on the line 3𝑥 + 𝑦 = 3.
2
(b)
Show that the gradient of 𝐴𝐵 is .
1
(c)
Show that the length of 𝐴𝐵 is 10 units.
1
(d)
Show that 𝐴𝐵 and 𝐴𝐶 are perpendicular.
1
(e)
Find tan 𝜃.
2
(f)
Find the equation of the circle with centre 𝐴 that passes through 𝐵 .
2
(g)
The point 𝐷 is not shown on the diagram. The point 𝐷 lies on the line 3𝑥 + 𝑦 = 3
! !
between 𝐴 and 𝐶 , and 𝐴𝐷 = 𝐴𝐵. Find the coordinates of 𝐷 . (h)
On your diagram, shade the region satisfying the inequality 3𝑥 + 𝑦 ≤ 3.
2
1
1999 HSC Q1(a) The points 𝐴 and 𝐵 have coordinates (3, −4) and (7, 2) respectively. Find the coordinates of the midpoint of 𝐴𝐵.
2
1999 HSC Q2(b) The diagram shows the points 𝐴(−2, 0), 𝐵 (3, 5) and the point 𝐶 which lies on the 𝑥 axis. The point 𝐷 also lies on the 𝑥 axis such that 𝐵𝐷 is perpendicular to 𝐴𝐶.
8
(i)
Show that the gradient of 𝐴𝐵 is 1.
(ii)
Find the equation of the line 𝐴𝐵.
(iii)
What is the size of ∠𝐵𝐴𝐶?
(iv)
The length of 𝐵𝐶 is 13 units. Find the length of 𝐷𝐶.
(v)
Calculate the area of Δ𝐴𝐵𝐶.
(vi)
Calculate the size of ∠𝐴𝐵𝐶, to the nearest degree.
2000 HSC Q2 The diagram shows the points 𝑃 (0, 2) and 𝑄 4, 0 . The point 𝑀 is the midpoint of 𝑃𝑄. The line 𝑀𝑁 is perpendicular to 𝑃𝑄 and meets the 𝑥 axis at 𝐺 and the 𝑦 axis at 𝑁 .
!
(i)
Show that the gradient of 𝑃𝑄 is − .
1
(ii)
Find the coordinates of 𝑀 .
2
(iii)
Find the equation of the line 𝑀𝑁.
2
(iv)
Show that 𝑁 has coordinates (0, −3).
1
(v)
Find the distance 𝑁𝑄.
1
(vi)
Find the equation of the circle with centre 𝑁 and radius 𝑁𝑄.
2
(vii)
Hence show that the circle in part (f) passes through the point 𝑃 .
1
(viii)
The point 𝑅 lies in the first quadrant, and 𝑃𝑁𝑄𝑅 is a rhombus.
2
!
Find the coordinates of 𝑅 .
2001 HSC Q2(b) The diagram shows the points 𝐴 −2, 5 , 𝐵(4, 3) and 𝑂 (0, 0). The point 𝐶 is the fourth vertex of the parallelogram 𝑂𝐴𝐵𝐶.
(i)
Show that the equation of 𝐴𝐵 is 𝑥 + 3𝑦 − 13 = 0.
2
(ii)
Show that the length of 𝐴𝐵 is 2 10.
1
(iii)
Calculate the perpendicular distance from 𝑂 to the line 𝐴𝐵.
2
(iv)
Calculate the area of parallelogram 𝑂𝐴𝐵𝐶.
2
(v)
Find the perpendicular distance from 𝑂 to the line 𝐵𝐶.
2
2002 HSC Q3(c) The diagram shows two points 𝐴(2, 2) and 𝐵 (1, 5) on the number plane. Copy the diagram into your writing booklet.
(i)
Find the coordinates of 𝑀 , the midpoint of 𝐴𝐵.
1
(ii)
Show that the equation of the perpendicular bisector of 𝐴𝐵 is 𝑥 − 3𝑦 + 9 = 0.
1
(iii)
Find the coordinates of the point 𝐶 that lies on the 𝑦 axis and is equidistant from 𝐴 and 𝐵.
(iv)
1
The point 𝐷 lies on the intersection of the line 𝑦 = 5 and the perpendicular bisector 𝑥 − 3𝑦 + 9 = 0. Find the coordinates of 𝐷 , and mark the position of 𝐷 on your diagram in your writing booklet.
(v)
1
Find the area of triangle 𝐴𝐵𝐷.
2
2003 HSC Q2(b) In the diagram, 𝑂𝐴𝐵𝐶 is a trapezium with 𝑂𝐴 ∥ 𝐶𝐵. The coordinates of 𝑂 , 𝐴 and 𝐵 are (0, 0), (−1, 1) and (4, 6) respectively.
(i)
Calculate the length of 𝑂𝐴.
(ii)
1
Write down the gradient of the line 𝑂𝐴.
1
(iii)
What is the size of ∠𝐴𝑂𝐶?
1
(iv)
Find the equation of the line 𝐵𝐶, and hence find the coordinates of 𝐶 .
2
(v)
Show that the perpendicular distance from 𝑂 to the line 𝐵𝐶 is 5 2.
2
(vi)
Hence, or otherwise, calculate the area of the trapezium 𝑂𝐴𝐵𝐶.
2
2004 HSC Q2(a) The diagram shows the points 𝐴(−1, 3) and 𝐵 (2, 0). The line 𝑙 is drawn perpendicular to the 𝑥 -‐axis through the point 𝐵 .
(i)
Calculate the length of the interval 𝐴𝐵.
1
(ii)
Find the gradient of the line 𝐴𝐵.
1
(iii)
What is the size of the acute angle between the line 𝐴𝐵 and the line 𝑙 ?
1
(iv)
Show that the equation of the line 𝐴𝐵 is 𝑥 + 𝑦 − 2 = 0.
1
(v)
Copy the diagram into your writing booklet and shade the region defined by 𝑥 + 𝑦 − 2 ≤ 0.
1
(vi)
Write down the equation of the line 𝑙 .
1
(vii)
The point 𝐶 is on the line 𝑙 such that 𝐴𝐶 is perpendicular to 𝐴𝐵.
2
Find the coordinates of 𝐶 .
2006 HSC Q3(a) In the diagram, 𝐴, 𝐵 and 𝐶 are the points (1, 4), (5, −4) and (−3, −1) respectively. The line 𝐴𝐵 meets the 𝑦-‐axis at 𝐷 .
(i)
Show that the equation of the line 𝐴𝐵 is 2𝑥 + 𝑦 − 6 = 0.
2
(ii)
Find the coordinates of the point 𝐷 .
1
(iii)
Find the perpendicular distance of the point 𝐶 from the line 𝐴𝐵.
1
(iv)
Hence, or otherwise, find the area of the triangle 𝐴𝐷𝐶.
2
2007 HSC Q1(f) Find the equation of the line that passes through the point (−1, 3) and is perpendicular to 2𝑥 + 𝑦 + 4 = 0.
2
2007 HSC Q3(a) In the diagram, 𝐴, 𝐵 and 𝐶 are the points (10, 5), (12, 16) and (2, 11) respectively.
Copy or trace this diagram into your writing booklet. (i)
Find the distance 𝐴𝐶.
1
(ii)
Find the midpoint of 𝐴𝐶.
1
(iii)
Show that 𝑂𝐵 ⊥ 𝐴𝐶.
2
(iv)
Find the midpoint of 𝑂𝐵 and hence explain why 𝑂𝐴𝐵𝐶 is a rhombus.
2
(v)
Hence, or otherwise, find the area of 𝑂𝐴𝐵𝐶.
1
2
2008 HSC Q2(b) Let 𝑀 be the midpoint of (−1, 4) and (5, 8). !
Find the equation of the line through 𝑀 with gradient − . !
2008 HSC Q3(a) In the diagram, 𝐴𝐵𝐶𝐷 is a quadrilateral. The equation of the line 𝐴𝐷 is 2𝑥 − 𝑦 − 1 = 0.
(i)
Show that 𝐴𝐵𝐶𝐷 is a trapezium by showing that 𝐵𝐶 is parallel to 𝐴𝐷.
2
(ii)
The line 𝐶𝐷 is parallel to the 𝑥 -‐axis. Find the coordinates of 𝐷 .
1
(iii)
Find the length of 𝐵𝐶.
1
(iv)
Show that the perpendicular distance from 𝐵 to 𝐴𝐷 is .
2
(v)
Hence, or otherwise, find the area of the trapezium 𝐴𝐵𝐶𝐷.
2
!
!
2009 HSC Q3(b) The circle in the diagram has centre 𝑁 . The line 𝐿𝑀 is tangent to the circle at 𝑃 .
(i)
Find the equation of 𝐿𝑀 in the form 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0.
2
(ii)
Find the distance 𝑁𝑃.
2
(iii)
Find the equation of the circle.
1
2009 HSC Q5(a) In the diagram, the points 𝐴 and 𝐶 lie on the 𝑦-‐axis and the point 𝐵 lies on the 𝑥 -‐axis. The line 𝐴𝐵 has equation 𝑦 = 3𝑥 − 3. The line 𝐵𝐶 is perpendicular to 𝐴𝐵.
(i)
Find the equation of the line 𝐵𝐶.
2
(ii)
Find the area of the triangle 𝐴𝐵𝐶.
2
2010 HSC Q3(a) In the diagram 𝐴, 𝐵 and 𝐶 are the points (−2, −4), (12, 6) and (6, 8) respectively. The point 𝑁(2, 2) is the midpoint of 𝐴𝐶. The point 𝑀 is the midpoint of 𝐴𝐵.
(i)
Find the coordinates of 𝑀 .
1
(ii)
Find the gradient of 𝐵𝐶.
1
(iii)
Prove that 𝛥𝐴𝐵𝐶 is similar to 𝛥𝐴𝑀𝑁.
2
(iv)
Find the equation of 𝑀𝑁.
2
(v)
Find the exact length of 𝐵𝐶.
1
(vi)
Given that the area of 𝛥𝐴𝐵𝐶 is 44 square units, find the perpendicular distance from 𝐴 to 𝐵𝐶.
1
2011 HSC Q3(c) The diagram shows a line 𝑙! , with equation 3𝑥 + 4𝑦 − 12 = 0, which intersects the 𝑦-‐axis at 𝐵 . A second line 𝑙! , with equation 4𝑥 − 3𝑦 = 0, passes through the origin 𝑂 and intersects 𝑙! at 𝐸 .
(i)
Show that the coordinates of 𝐵 are (0, 3).
1
(ii)
Show that 𝑙! is perpendicular to 𝑙! .
2
(iii)
Show that the perpendicular distance from 𝑂 to 𝑙! is .
!"
1
(iv)
Using Pythagoras’ theorem, or otherwise, find the length of the interval 𝐵𝐸.
1
(v)
Hence, or otherwise, find the area of 𝛥𝐵𝑂𝐸.
1
!
2012 HSC Q5 What is the perpendicular distance of the point (2, −1) from the line 𝑦 = 3𝑥 + 1? (A)
6 10
(B)
6
(C)
5 8
10 (D)
8 5
2012 HSC Q13(a) The diagram shows a triangle 𝐴𝐵𝐶. The line 2𝑥 + 𝑦 = 8 meets the 𝑥 and 𝑦 axes at the points 𝐴 and 𝐵 respectively. The point 𝐶 has coordinates (7, 4).
(i)
Calculate the distance 𝐴𝐵.
(ii)
It is known that 𝐴𝐶 = 5 and 𝐵𝐶 = 65. (Do NOT prove this.) Calculate the size of ∠𝐴𝐵𝐶 to the nearest degree.
(iii)
The diagram shows the line 𝑙 .
What is the slope of the line 𝑙 ? (A)
3
(B)
− 3 1 3
2
2
2013 HSC Q2
(D)
The point 𝑁 lies on 𝐴𝐵 such that 𝐶𝑁 is perpendicular to 𝐴𝐵. Find the coordinates of 𝑁 .
(C)
−
1 3
3
2013 HSC Q12(b) The points 𝐴(−2, −1), 𝐵 (−2, 24), 𝐶 (22,42) and 𝐷 (22, 17) form a parallelogram as shown. The point 𝐸 (18, 39) lies on 𝐵𝐶. The point 𝐹 is the midpoint of 𝐴𝐷.
(i)
Show that the equation of the line through 𝐴 and 𝐷 is 3𝑥 − 4𝑦 + 2 = 0.
(ii)
Show that the perpendicular distance from 𝐵 to the line through 𝐴 and 𝐷 is 20 units. 1
2
(iii)
Find the length of 𝐸𝐶.
1
(iv)
Find the area of the trapezium 𝐸𝐹𝐷𝐶.
2
2014 HSC Q5 Which equation represents the line perpendicular to 2𝑥 − 3𝑦 = 8, passing through the point (2, 0)? (A)
3𝑥 + 2𝑦 = 4
(B)
3𝑥 + 2𝑦 = 6
(C)
3𝑥 − 2𝑦 = −4
(D)
3𝑥 − 2𝑦 = 6
2014 HSC Q12(b) The points 𝐴(0, 4), 𝐵 (3, 0) and 𝐶 (6, 1) form a triangle, as shown in the diagram.
(i)
Show that the equation of 𝐴𝐶 is 𝑥 + 2𝑦 − 8 = 0.
2
(ii)
Find the perpendicular distance from 𝐵 to 𝐴𝐶.
2
(iii)
Hence, or otherwise, find the area of 𝛥𝐴𝐵𝐶.
2
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