Editorial - Matlan - TO TKA 2

Problem 1

A=[[5,3],[-2,1]]

B=[[1,-3],[-4,2]]

C=[[-3,9],[5,8]]

D=[[1,-3],[2,7]]

A*B+C-2*D=[[5,3],[-2,1]]*[[1,-3],[-4,2]]+[[-3,9],[5,7]]-[[2,-6],[4,14]]

[[-7,-9],[-6,8]]+[[-5,15],[1,-7]]=[[-12,6],[-5,1]]

det(A*B+C-2*D)=-12+30=18

(B)

Problem 2

A=[[3,x],[2,4]]

B=[[y,3],[-1,-x]]

C=[[26,10],[12,-4]]

D=[[x,-y],[2*y,3*x]]

2*A*B=C

2⋅[[3,x],[2,4]]*[[y,3],[-1,-x]]=2*[[3*y-x,9-x2],[2*y-4,6-4*x]]=[[26,10],[12,-4]]

6-4*x=-2

4*x=8

∴x=2

2*y-4=6

2*y=10

∴y=5

Subproblem 2.1

A+D=[[5,-3],[12,10]]

A+D=[[3,2],[2,4]]+[[2,-5],[10,6]]=[[5,-3],[12,10]]

The statement is true.

Subproblem 2.2

2*B-D=[[-8,-11],[12,10]]

2*B-D=2*[[5,3],[-1,-2]]-[[2,-5],[10,6]]=[[10,6],[-2,-4]]-[[2,-5],[10,6]]=[[8,11],[-12,-10]]

The statement is false.

Subproblem 2.3

A*D=[[26,-3],[44,14]]

A*D=[[3,2],[2,4]]*[[2,-5],[10,6]]=[[26,-3],[44,14]]

The statement is true.

Problem 3

5*x+3*y=27500

4*x+2*y=21000

[[5,3],[4,2]]*[[x],[y]]=[[27500],[21000]]

This proves the first statement is true and the second statement is false.

[[x],[y]]=[[5,3],[4,2]](-1)*[[27500],[21000]]

[[x],[y]]=1/(-2)*[[2,-3],[-4,5]]*[[27500],[21000]]=1/2*[[-2,3],[4,-5]]*[[27500],[21000]]

This proves the fourth statement is true and statements three and five are false.

Problem 4

(2+3)+(-2+5)=8

(D)

Problem 5

x2-x-1

with remainder -2*x+5

(A)

Problem 6

Subproblem 6.1

V→x3+x2+6*x-36=0

x=t

p=30-2*t

l=20-2*t

V=t⋅(20-2*t)*(20-2*t)=(600-60*t-40*t+4*t2)*t=600*t-100*t2+4*t3=600

4*t3-100*t2+600*t-600=0

t3-25*t2+150*t-150=0

The statement is false.

Subproblem 6.2

p=30-2*t

Yeah, correct.

Subproblem 6.3

V→x3-25*x2+150*x-150=0

Yeah, correct.

Subproblem 6.4

l=20-x

False.

Subproblem 6.5

t=x

True.

Problem 7

Subproblem 7.1

t=15

l=20

t=15→p=15+17=32

l=20→p=20+5=25

Inconsistent. The statement is false.

Subproblem 7.2

t=3

l=20

t=3→p=3+17=20

l=20→p=20+5=25

Inconsistent. The statement is false.

Subproblem 7.3

t=3

p=20

t=3→p=3+17=20

p=20→l=20-5=15

V=20⋅15⋅3=900

True.

Subproblem 7.4

A=810

l=15

l=15→p=15+5=20

p=20→t=20-17=3

V=20⋅3⋅15=900

A=2*(20⋅3+20⋅15+15⋅3)=810

True.

Subproblem 7.5

A=810

p=20

True. Same as before.

Problem 8

|x-5|≤0.2

x-5≤0.2 and x-5≥-0.2

Statements 1 and 3

Problem 9

2*x2-8*x+6>0

x2-4*x+3>0

(x-1)*(x-3)>0

x>3 or x< 1

ezpz (C)

Problem 10

a*b0=a=3

Proves the first statement is true.

a*b2=12

3*b2=12

b=2

Proves the third statement is false.

3⋅21=6

Proves the second statement is true.

HIHIHIHA

Problem 11

ƒ(x)={[x2-1,x<0],[2*x+3,0≤x≤2],[(log_2)(x),x>2])

It's so cool that corca has cases system.

1. ƒ(-2)+ƒ(1)=4-1+2+3=8 False

2. ƒ(0)⋅ƒ(4)=3⋅2=6 True

3. ƒ(ƒ(1))=ƒ(5)=(log_2)(5) False

4. True because it's linear

Problem 12

ƒ(0)=1=a*cos2(0)-b

∴a-b=0

That's free.

Either statement is equivalent, so it really doesn't matter which one we get. We just have to prove we can find the equation using 1, which proves we can use the other interchangably.

ƒ(30)=1/2=a*cos2(30)-b

1/2=(3*a)/4-b

2=3*a-4*b

That's a system of equations that's easily solvable, so the answer is (D)

Problem 13

Well, we need to find the amplitude, so we obviously need the first statement. Can it be done, though?

Well, you only need to know the first statement, as we already know where peaks for ƒ(x) and g(x) are located, which means we can find the shift. Finding the frequency is easy with the coordinates we have on the graph, which means we can find the amplitude easily.

I'm not going to write the full process down.

Problem 14

CE=2*u-v

DF=-2*v+u

CE+DF=3*u-3*v

(B)

Problem 15

[[2+1],[3+5]]*[[4-3,2+4]]=3+48=51

(C)

Problem 16

First statement is true if using the standard coordinate system.

Second statement is obviously false.

Third statement is obviously false.

Fourth statement is true if you just add up the vectors.

Fifth statement is true because it's a space diagonal. l=√(,36+36⋅2)=6√(,3)

Problem 17

First statement is obviously false.

Second statement is true.

FH=[[-5],[12],[0]]

GD=[[-5],[0],[-8]]

cos(θ)=25/(√(,25+144)√(,25+64))=25/1157√(,89)

Woah, wrong.

Problem 18

|u|=√(,1+4+9)=√(,14)

The third statement is false.

|u+v|2=|u|2+|v|2+2*(u⋅v)

11=14+|v|2+2*(u+v)

|v|2+2*(u+v)=-3

|u-v|2=|u|2+|v|2-2*(u⋅v)

35=14+|v|2-2*(u⋅v)

|v|2-2*(u⋅v)=21

2*|v|2=18

|v|=3

The first statement is true.

9+2*(u⋅v)=-3

2*(u⋅v)=-12

u⋅v=-6

The second statement is false.

Let's prove the fourth statement.

(u⋅v)/|v|=-6/3=-2

The fourth statement is true.

Next is the fifth statement.

-2⋅1/3*v

How do I prove this without knowing v? By contradiction.

-2/3⋅(a*i+b*j+c*k)=-4/3*i-2/3*j-4/3*k

v=2*i+j+2k

Let's assume it's true, then all of the other statements we have proven have to be true, right?

|v|=√(,4+1+4)=3

So far, so good.

u⋅v=-2+2+6=6

This is a contradiction. We found that u⋅v=-6 earlier.

So, the fifth statement is false.

Problem 19

Via vectors.

Let L be the origin, as it's used in both vectors LT and LP

Let the side length of the cube be s

LT=[[-s],[-s/2],[s]]

LP=[[0],[-s],[s]]

|LT|=√(,s2+(s2)/4+s2)=3/2*s

|LP|=√(,0+s2+s2)=s√(,2)

LT⋅LP=(s2)/2+s2=3/2*s2

cos(θ)=(LT⋅LP)/(|LT|⋅|LP|)=(3/2*s2)/(3/2*s⋅s√(,2))=√(,2)/2

∴θ=45

Yay!!!

Problem 20

First statement is obviously true. It's a 90 angle.

|AX|=√(,36+36)/2=3√(,2)

Second statement is false.

|AY|=[[0],[0],[3/2]]

Third statement is obviously true.

Problem 21

First statement is obviously true.

(r_1)+(r_2)=√(,(10-2)2+(3-3)2)=8

Second statement is proven to be true.

Third statement is true, karena bersinggungan di luar.

Problem 22

(x-3)2-9+(y+4)2-16+9=0

(x-3)2+(y+4)2=16

Centered around (3,-4) and radius 4

First statement is true.

(x-1)2-1+(y+2)2-4+1=0

(x-1)2+(y+2)2=4

Centered around (1,-2) and radius 2

Second statement is true.

d=√(,22+22)=√(,8)

Third statement is true.

If the 2 circles intersect internally, then d=|(r_1)-(r_2)|

√(,8)=4-2=2

The fourth statement is false.

Problem 23

Radius is 3 just by distance from center to tangent 4-3=1

This means the first statement is true.

The second statement is also true.

The third statement is obviously false.

The fourth statement is true.

The fifth statement is obviously false.

Problem 24

Either statement is enough, but the second option leaves us with 2 options for the equation of the tangent. I'm guessing it doesn't matter. (D)

Problem 25

y=a*(x+1)2+3

Plug in (0,1)

1=a+3

a=-2

y=-2*(x+1)2+3=1-4*x-2*x2

Double rotation. 35+55=90

[[(x^′)],[(y^′)]]=[[cos(90),-sin(90)],[sin(90),cos(90)]]*[[x],[y]]=[[0,-1],[1,0]]*[[x],[y]]=[[-y],[x]]

x=(y^′)

y=-(x^′)

Substitute.

-x=1-4*y-2*y2

x=2*y2+4*y-1

(D)

Problem 26

Subproblem 26.1

y-1=2*(x-3)2-5*(x-3)+3

y=37-17*x+2*x2

First statement is false.

Subproblem 26.2

y+2=2*(x-4)2-5*(x-4)+3

y=53-21*x+2*x2

Second statement is true.

Subproblem 26.3

T=[[7],[-1]]

y+1=2*(x-7)2-5*(x-7)+3

y=135-33*x+2*x2

Third statement is true.

Subproblem 26.4

-b/(2*a)=33/4

Fourth statement is false.

Subproblem 26.5

Obviously true, as the coefficient of x2 is positive.

Problem 27