• We need your support!

    We are currently struggling to cover the operational costs of Xtremepapers, as a result we might have to shut this website down. Please donate if we have helped you and help make a difference in other students' lives!
    Click here to Donate Now (View Announcement)

Mathematics: Post your doubts here!

Messages
41
Reaction score
26
Points
28
Take the co-ordinates of both the object and image and place them into two matrices, and let x be the required matrix which will take the form
(a b)
(c d)
Solve the matrices to get the required matrix :) Hope it helps.
Cheers
Can you do one example please, so I can fully get it and able to do the rest :D.
 
Messages
20
Reaction score
15
Points
13
Triangle B Co-ordinates:
B= {2 5 5}
{1 1 3}
Triangle C Co-ordinates:
C= { 1 1 3}
{-2 -5 -5}

Let the required transformation matrix be:
T= {a b}
{c d}

Now lets apply the transformation:

TA = C
{a b} * {2 5 5} = { 1 1 3}
{c d} {1 1 3} {-2 -5 -5}

{a*2+b*1 a*5+b*1 a*5+b*3} = { 1 1 3}
{c*2+d*1 c*5+d*1 c*5+d*3} {-2 -5 -5}

{2a+b 5a+b 5a+3b} = { 1 1 3}
{2c+d 5c+d 5c+3d} {-2 -5 -5}

Now solve first two simultaneous equations:

2a+b =1 (1)
5a+b =1 (2)
- - -
_________
-3a=0
a=0

Substitute the value of a in equation (1)
2a+b=1
2(o)+b=1
b = 1

Now solve two other simultaneous equations:

2c+d = -2 (3)
5c+d = -5 (4)
- - +
_________
-3c=3
c=-1

Substitute the value of c in equation (3)

2c+d = -2
2(-1)+d = -2
-2+d= -2
d = 0

So the required transformation matrix is:
{a b}
{c d}

= {0 1}
{-1 0}
 
Last edited:
Messages
41
Reaction score
26
Points
28
Triangle B Co-ordinates:
B= {2 5 5}
{1 1 3}
Triangle C Co-ordinates:
C= { 1 1 3}
{-2 -5 -5}

Let the required transformation matrix be:
T= {a b}
{c d}

Now let apply transformation:

TA = C
{a b} * {2 5 5} = { 1 1 3}
{c d} {1 1 3} {-2 -5 -5}

{a*2+b*1 a*5+b*1 a*5+b*3} = { 1 1 3}
{c*2+d*1 c*5+d*1 c*5+d*3} {-2 -5 -5}

{2a+b 5a+b 5a+3b} = { 1 1 3}
{2c+d 5c+d 5c+3d} {-2 -5 -5}

Now solve first two simultaneous equations:

2a+b =1 (1)
5a+b =1 (2)
- - -
_________
-3a=0
a=0

Substitute the value of a in equation (1)
2a+b=1
2(o)+b=1
b = 1

Now solve two other simultaneous equations:

2c+d = -2 (3)
5c+d = -5 (4)
- - +
_________
-3c=3
c=-1

Substitute the value of c in equation (3)

2c+d = -2
2(-1)+d = -2
-2+d= -2
d = 0

So the required transformation matrix is:
{a b}
{c d}

= {0 1}
{-1 0}
Thanks a lot for your time
 
Messages
20
Reaction score
15
Points
13
I don't know whats happening but the brackets format starts to change when I paste the solution over here so i am attaching the solution file, download and check from there.
 

Attachments

  • sol pro.txt
    959 bytes · Views: 15
Last edited:
Messages
40
Reaction score
50
Points
28
I am really stuck in matrics, I don't know how to find the position vectors like [1 0 0 -1], is there a easy way?
12191880_774230529347498_8461933746463586607_n.jpg

if we look at B to C, the answer is [0 -1, 1 0] but why?







Special thanks to mr.Stefan Salvatore for clearing up the doubts Explicitly..


And Yeah one thing more which book is this..?
 
Messages
20
Reaction score
15
Points
13
f(x) = x^2 + 4x - 6 (1)

The question gives you the hint that the function of x can be converted into algebraic identity form by completing the square method and then by comparison you can deduce the values
of m and n.

By looking closely you can easily see that it can be converted into algebraic identity (a + b)^2

Take the co-efficient of x i.e 4 If you remember a(x)^2 + bx +c​
f(x) = x^2 + 4x - 6 Divide it by 2 ----> 2
= x^2 + 4x - 6 Square The Number ----> 4
= x^2 + 4x - 6 + 4 - 4 Then add and subtract it at the same time to make the equation balance
= x^2 + 4x + 4 - 6 - 4
= (x)^2 + (2 * x * 2 ) + (2)^2 - 6 - 4 Rearranging the equation in the form (a)^2 + (2*a*b) + (b)^2 gives:
= (x + 2)^2 - 6 - 4
= (x + 2)^2 - 10 (2)

Finally by comparison of 1 from 2 the values of m and n can be deduced:
m = 2 and n = -10

Cheers :)
 
Last edited:
Messages
20
Reaction score
15
Points
13
The Rate Of Fuel for (x+20) kilometers is given by the same formula just replacing x with x + 20 as the distance has changed:
(i) This gives Rate Of Fuel = 600/(x+20)

(ii) In part two as clearly stated that the rate of fuel has been decreased by 1.5 liters per kilometer

This means in the second part of journey the rate of fuel was 1.5 less than the rate of fuel for first part of journey. This helps us in constructing the equation:

This means:
(600/x) - (600/x+20) = 1.5
Multiply each term by x(x+20)
600x + 12000 - 600x = 1.5 (x^2 + 20x)
12000 = 1.5(x)^2 +30x
1.5(x)^2 + 30x - 12000 = 0
Divide each term by 1.5
(x)^2 + 20x - 8000 = 0
Hence proved.

(iii) Now what you gotta do is solve the equation by using quadratic formula.
(x)^2 + 20x - 8000 = 0 a = 1 , b = 20 , c = -8000

x = -20 (+ or -) √ (-20)^2 - 4 * (1) * (-8000) / 2

This gives the following roots: 80 , -100

(iv) As we need to find the rate of fuel used by Alan’s car for the complete journey we need to add both the distances for each part of journey:

Also before computing the distances we need to choose one value of x. As the value of x (i.e the distance) can never be negative so we choose the positive root (i.e 80) and substitute it into each of the distance equations giving:

First Distance = x
First Distance = 80

Second Distance = x+ 20
Second Distance = 100

Total Distance = 80 + 100 = 180

So rate of fuel for whole journey is calculated by dividing 600 with the total distance giving answer in liters per 100 kilometer :

Rate Of Fuel For Complete Journey = 600/Total Distance
= 600/180
= 10/3
=3.33 liters per 100 km

Cant add the radical sign in the quadratic formula part (iii). Hope you get it.
 
Messages
37
Reaction score
517
Points
83
The Rate Of Fuel for (x+20) kilometers is given by the same formula just replacing x with x + 20 as the distance has changed:
(i) This gives Rate Of Fuel = 600/(x+20)

(ii) In part two as clearly stated that the rate of fuel has been decreased by 1.5 liters per kilometer

This means in the second part of journey the rate of fuel was 1.5 less than the rate of fuel for first part of journey. This helps us in constructing the equation:

This means:
(600/x) - (600/x+20) = 1.5
Multiply each term by x(x+20)
600x + 12000 - 600x = 1.5 (x^2 + 20x)
12000 = 1.5(x)^2 +30x
1.5(x)^2 + 30x - 12000 = 0
Divide each term by 1.5
(x)^2 + 20x - 8000 = 0
Hence proved.

(iii) Now what you gotta do is solve the equation by using quadratic formula.
(x)^2 + 20x - 8000 = 0 a = 1 , b = 20 , c = -8000

x = -20 (+ or -) √ (-20)^2 - 4 * (1) * (-8000) / 2

This gives the following roots: 80 , -100

(iv) As we need to find the rate of fuel used by Alan’s car for the complete journey we need to add both the distances for each part of journey:

Also before computing the distances we need to choose one value of x. As the value of x (i.e the distance) can never be negative so we choose the positive root (i.e 80) and substitute it into each of the distance equations giving:

First Distance = x
First Distance = 80

Second Distance = x+ 20
Second Distance = 100

Total Distance = 80 + 100 = 180

So rate of fuel for whole journey is calculated by dividing 600 with the total distance giving answer in liters per 100 kilometer :

Rate Of Fuel For Complete Journey = 600/Total Distance
= 600/180
= 10/3
=3.33 liters per 100 km

Cant add the radical sign in the quadratic formula part (iii). Hope you get it.
THNX MAN BUT a(ii) is still very confusing btw thnx for ur time
 
Messages
37
Reaction score
517
Points
83
Question: 25% of a magazine is used to sell 2nd hand cars . 62.5% of the remaining pages were used to show other things and 36 pages of them were used for advertisement ..HOW MANY PAGES ARE THERE TOTALLY??????? TIP : ONLY 17% IN THIS WHOLE WORLD GAVE THE ANSWER WRITE AS MY TEACHER SAID .IF U WANT TO BE THE 18 TH % TTTRRRRYYYYY THHHIISSSSS
 
Messages
37
Reaction score
517
Points
83
Question: 25% of a magazine is used to sell 2nd hand cars . 62.5% of the remaining pages were used to show other things and 36 pages of them were used for advertisement ..HOW MANY PAGES ARE THERE TOTALLY??????? TIP : ONLY 17% IN THIS WHOLE WORLD GAVE THE ANSWER WRITE AS MY TEACHER SAID .IF U WANT TO BE THE 18 TH % TTTRRRRYYYYY THHHIISSSSS
OTHER good tip

Answer is 128 find it how
 
Messages
1,171
Reaction score
4,151
Points
273
Question: 25% of a magazine is used to sell 2nd hand cars . 62.5% of the remaining pages were used to show other things and 36 pages of them were used for advertisement ..HOW MANY PAGES ARE THERE TOTALLY??????? TIP : ONLY 17% IN THIS WHOLE WORLD GAVE THE ANSWER WRITE AS MY TEACHER SAID .IF U WANT TO BE THE 18 TH % TTTRRRRYYYYY THHHIISSSSS
Btw don't you think you have exagerrated this one like only 17% of the world solved this LoL !!
Solution:

Let the total pages= y
Percentage of pages used for selling cars= 25% of y = 0.25y
Remaining percentage= 100 - 25 = 75%
The percentage of remaining pages = 75% of y = 0.75y
62.5% of remaining pages = 0.75y x (62.5/100) = 0.46875y (Do not round this off if you want an accurate answer)
Pages for ads= 36

Therefore the equation becomes like this

0.25y + 0.46875y + 36 = y
0.71875y + 36 = y
Y - 0.71875y = 36
0.28125y=36
Y= (36/0.28125) = 128

Cheers ! I hope you'd get it !
 
Top