IB MAI HL Functions Topic 2.1 Linear Equations & Graphs

1#

The diagram below shows ai+byx;1ybav k/5i kj, straight line

L_{1}

which passes through A(0,−2) and B(8,0).

1.Write down the coordinates of the midpoint of line segment [AB]. (a,b) a= b=
Another line,

L_{2}

, intersects the

y

-axis at C(0,3) and is parallel to

L_{1}

.
2.Find the gradient of

L_{2}

.

L_{2}

=
3.Find the equation of

L_{2}

, giving your answer in the form y=mx+c. m = c =

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2#

填空题 ( 1.0 分) 切至整卷模式搜藏此题

A small town is planning the construction of a new .uh:0v5 tsolh m-.ozg road. The new road will pass through the point A(0,5) and will be perpendicular to the road connecting the points B(3,0) and C(6,6). This information is shown i.hz5 -0:l.vosuo tghm n the following diagram.

1.Find the gradient of the line through points B and C.

m_{B C}

=
2.Hence, state the gradient of the line through points A and D.

m_{A D}

=
3.Find the equation of the line through A and D. Give your answer in the form y=mx+c. m =    c =
4.Point D lies on the x-axis. Find the coordinates of point D. (a,b) a=   b=

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3#

填空题 ( 1.0 分) 切至整卷模式搜藏此题

The equation of a lie+p s,n6wqtw b0wij25ne

L_{1}

is y=0.5x+p. The point A(2,−1) lies on

L_{1}

1.Find the value of p. p =
A second line,

L_{2}

, is perpendicular to

L_{1}

and intersects

L_{1}

at point A.
2.Find the gradient of

L_{2}

.

L_{2}

=
3.Find the equation of

L_{2}

. Give your answer in the form y=mx+c. m = c =
4.Write your answer to part (c) in the form ax+by+d=0, where a,b,d∈Z. a = b = c =

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4#

填空题 ( 1.0 分) 切至整卷模式搜藏此题

The equation of a lin(w 7b rg5 i0vyapu9jt8e

L_{1}

is y−3x+5=0.
1.For the line

L_{1}

, find:
(1)the

x

-intercept;x =
(2)the gradient. y = ax+b a = b =
A second line,

L_{2}

, intersects the

y

-axis at P(0,2)(0,2) and is parallel to

L_{1}

.
2.Find the equation of

L_{2}

. Give your answer in the form y=mx+c. m = c =
A third line,

L_{3}

, passes through the point Q(3,1) and is perpendicular to

L_{1}

.
3.Find the gradient of the line

L_{3}

.

m_{L_{3}}

=
4.Find the equation of

L_{3}

, giving your answer in the form ax+by+d=0,
where a,b,d∈Z. a = b = d =

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5#

填空题 ( 1.0 分) 切至整卷模式搜藏此题

The diagram shows th-+ lyhsz /s(jge straight line

L_{1}

, which intersects the

x

-axis at A(−8,0) and the

y

-axis at B(0,4).

1.Write down the coordinates of M, the midpoint of line segment [AB]. (a,b) a= b=
2.Calculate the gradient of

L_{1}

.
The line

L_{2}

is perpendicular to

L_{1}

and passes through the point P(1,2).

L_{1}

=
3.Find the equation of

L_{2}

. Give your answer in the form y=mx+c. m = c =

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6#

填空题 ( 1.0 分) 切至整卷模式搜藏此题

The straight line,

L_{1}

, has the equation y=

\frac{1}{3}

x−5.
1.Write down the

y

-intercept of

L_{1}

. (a,b) a= b=
2.Write down the gradient of

L_{1}

.

m L_{1}

=
The line

L_{2}

is perpendicular to

L_{1}

and passes through the point A(2,4).
3.Find the gradient of

L_{2}

.

m L_{2}

=
4.Find the equation of

L_{2}

, giving your answer in the form ax+by+d=0,
where a,b,d∈Z. a = b = d =

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7#

填空题 ( 1.0 分) 切至整卷模式搜藏此题

The coordinates of point *6mh6lq 6rckin g ,wqv)xgjp k99i9;o P are(−4,6) and the coordinates of point Q are(5,1). M is ti66gqgxk)lq, i v6r9jh 99* p mw;nckohe midpoint of [PQ].
1.Find the coordinates of M.

L_{1}

is the line which passes through P and Q. (a,b) a= b=
2.Find the gradient of

L_{1}

.
A new line,

L_{2}

, is perpendicular to

L_{1}

and passes through M.

m_{L_{1}}

=
3.(1)Write down the gradient of

L_{2}

.

m_{L_{2}}

=
(2)Write down the equation of

L_{2}

in the form y=mx+c. m = c =

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8#

填空题 ( 1.0 分) 切至整卷模式搜藏此题

The coordinates of point K are (−2,−3);c-k bf3( u6tf horuji7 .yi-zh+e)yo and the coordinates of point N are (8,6). M is the midpoint of [KN-;tj coz6k(y3oeuif yu-. b)hr7 i+hf].
1.Find the coordinates of M.

L_{1}

is the line which passes through K and N. (a,b) a= b=
2.Find the gradient of

L_{1}

.
A new line,

L_{2}

, is perpendicular to

L_{1}

and passes through M.

L_{1}

=
3.(1)Write down the gradient of

L_{2}

.

L_{2}

=
(2)Write down the equation of

L_{2}

in the form y=mx+c. m = c =

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9#

填空题 ( 1.0 分) 切至整卷模式搜藏此题

The equation of a lib8n.t:u10oht : k7qu crk9so( dwl)bune

L_{1}

is 2x+3y=−5.
1.Find the gradient of

L_{1}

.
A second line,

L_{2}

, is perpendicular to

L_{1}

.

m L_{1}

=
2.Find the gradient of

L_{2}

.
The point P(4,0)(4,0) lies on

L_{2}

.

m L_{1}

=
3.Find the equation of

L_{2}

, giving your answer in the form ax+by+d=0, where a,b,d∈Z. a = b = d =
The point Q is the intersection of

L_{1}

and

L_{2}

.
4.Find the coordinates of (a,b) a= b=

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10#

填空题 ( 1.0 分) 切至整卷模式搜藏此题

The coordinates of point A are (6,−3) and the coordinates of point B arfxf y( ;q4: anxv3byq7r:*yo de (−2,−1).4d; ff:yy3qn q:v(7b* a yxorx

L_{1}

is the line which passes through A and B.
1.Find the equation of

L_{1}

.
Point M is the midpoint of [AB]. The line

L_{2}

is perpendicular to

L_{1}

and passes through M. y = ax+b a = b =
2.Find the equation of

L_{2}

. y = ax+b a = b =

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11#

填空题 ( 1.0 分) 切至整卷模式搜藏此题

The coordinates of point A are (−1,−7) and the coordinates of point B are (5,wvt/ gzgz 259s4tk6uk2).2gt4st6kzz wkg/5vu 9

L_{1}

is the line which passes through A and B.
1.Find the equation of

L_{1}

.
Point M is the midpoint of [AB]. The line

L_{2}

is perpendicular to

L_{1}

and passes through M. y = ax+b a= b =
2.Find the equation of

L_{2}

. y = ax+b a= b =

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12#

填空题 ( 1.0 分) 切至整卷模式搜藏此题

The diagram below shows the straight linek oj/o is )p445+sanbss

L_{1}

1 and

L_{2}

.

1.Find:
(1)the gradient of

L_{1}

;

m L_{1}

=
(2)the equation of

L_{1}

, giving your answer in the form y=mx+c. m = c =
The equation of

L_{2}

is x−2y=0. y =
2.Find the area of the shaded triangle. A =

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13#

填空题 ( 1.0 分) 切至整卷模式搜藏此题

Bruce goes into a car dea.av 3q(oybt2 nlership to purchase a new vehicle. The one he wants to buy costs 16000 dollars , however he doesn't have that much money in his bank. The salesman offers him a financing option of a 30 % deposit followed by 12 mo2t(3nabo v y.qnthly payments of 1150 dollars.
1.Find the amount of the deposit. PV =
2.Calculate the total cost of the loan under this financing option.
Bruce's father generously offers him an interest free loan of 16000 dollars to buy the car to avoid the expensive loan repayments. They agree that Bruce will repay the loan by paying his father

x

in the first month and

y

every following month until the 16000 dollars is repaid.
The total amount Bruce's father receives after 12 months is 5200 dollars. This can be expressed by the equation x+11y=5200. The total amount that Bruce's father receives after 24 months is 10600dollars.FV =
3.Write down a second equation involving x and y. ax+by=c a =    b =    c =
4.Determine the value of x and the value of y. x =    y =
5.Calculate the number of months it will take Bruce's father to receive
the 16000 dollars. n =
Bruce decides to buy a cheaper car for 12000 dollars and invest the remaining 4000 dollars. He is considering two investment options over four years.
Option A: Compound interest at an annual rate of 6.5 %.
Option B: Compound interest at a nominal annual rate of 6 %, compounded monthly.
Express each answer in part (f) to the nearest dollar.
6.Calculate the value of each investment option after four years.
(1)Option A. ≈
(2)Option B. ≈

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14#

填空题 ( 1.0 分) 切至整卷模式搜藏此题

The diagram below shows the triangle ABC. The point C has coordinates (9,eifcv: 2:7 n/q cf),z+ vhrlsx5) and the eqexvc:f sri+v nlh),f:cz/q 27uation of the line (AB) is x+4y=12.

1.Find the coordinates of:
(1)A; y =
(2)B. x =
2.Show that the length of [AB] is 12.412.4 correct to three significant figures. AB ≈
The point D lies on the line (AB). The line (CD) is perpendicular to the line (AB).
3.Find:
(1)the gradient of (CD);

m_{C D}

=
(2)the equation of (CD). y =ax+b a =    b =
4.Find the coordinates of D. (a,b) a=   b=
5.Calculate the length of [CD] correct to three significant figures. CD ≈
6.Calculate the area of triangle ABC. A ≈

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15#

填空题 ( 1.0 分) 切至整卷模式搜藏此题

The equation of the lpy2ju 9*e)h wyine

L_{1}

is 3y−x−5=0. The line

L_{1}

is shown on the diagram below.

The point K has coordinates(4,3). The point T has coordinates (2,−3). The point M is the midpoint of [TK].
1.lculate the coordinates of the point M. (a,b) a= b=
2.Show that the point K lies on the line

L_{1}

.
3.lculate the length of [TK]. TK ≈
The line

L_{2}

passes through the point M and is perpendicular to

L_{1}

.
4.Find the gradient of the line

L_{2}

. gradients is
5.Find the equation of

L_{2}

. Give your answer in the form ax+by+d=0, where a,b and d are integers.
The point N is the intersection of

L_{1}

and

L_{2}

. ax+by+c=0 a = b = c =
6.lculate the coordinates of N. (a,b) a= b=

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16#

填空题 ( 1.0 分) 切至整卷模式搜藏此题

The line

L_{1}

has equation 2y−x−10=0 and is shown on the diagram.

The point A has coordinates (2,6)(2,6).
1.Show that A lies on

L_{1}

.
The point C has coordinates (8,18)(8,18). M is the midpoint of [AC].
2.Find the coordinates of M. (a,b) a= b=
3.Find the length of [AC]. AC ≈
The straight line,

L_{2}

, is perpendicular to [AC] and passes through M.
4.Show that the equation of

L_{2}

is 2y+x−29=0.
The point D is the intersection of

L_{1}

and

L_{2}

.
5.Find the coordinates of D.
The length of [MD] is

\frac{5 \sqrt{5}}{4}

. (a,b) a=   b=
6.Write down the length of [MD] correct to three significant figures.
The point B is such that ABCD is a rhombus. MD ≈
7.Find the area of ABCD. the area ≈

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