Flashcards

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Keats Mackay English 51 cards

Cards in this deck (51)

1—n, 1/n, upper/lower - (normal - changed)/n
2—Prove that the dot product equals 1 as -1^n if n is odd then its -1 meant det(m) = -det(m) -> this is only true for 0.
Because of this its either infinite or no solutions. For infinite the dot product must be zero which is show in the given maple output, implying that the vectors are perpendicular
3—Circle can only have two points colinear -> therefore there is no circle where this is true
4—Scalar triple product to prove they are on the plane (if it = 0) -> do all the combinations. If this is the case then they all lay within the sphere but within a circle (on the same plane). Therefore there are infinetely many spheres that could contain this circle
5—MTM
6—exp(angle*I). Add 1 for the second one. This si because of vector addition we can add the two
Last we just have to find the area of the parallelogram -> magntitude of cross product of the two vectors either side * the amount of points
7—Area either side cancels out -> an odd function essentially thats been shifted of half the number -> think of it as another function that has been shifted. The integral has also been shifted should be -a a rather than -a + number/half a + number/half
8—Invertible
9—Cant let any the bracket factorisastions tings equal 0 -> this will fuck leading columns otherwise and create parameters
10—Number at the top of vector.
Obious bc if we made another function that made this 0 then we would have a polynomial with more 'roots' than degrees
11—
12—
13—Left + ZX invert
Expand out to get
14—
15—Flipped sign of constant (expand out if need to into one fraction to work out) and multiply by constant in denominator (in brackets)
If it equals that -> can compare 1/x^2 which diverges
Otherwise it would be 1/x $Flipped sign of constant (expand out if need to into one fraction to work out) and multiply by constant in denominator (in brackets) If i...$
16—* ln(x)
17—Maple
Continuous
exp(Answer for the one above)
18—1-1 -> Polynomial so continuous and no turning points
Solve for input
Swap x and y and solve for y and then find the derivative and sub
the coefficient on the RHS multiplied by the answer above
19—-(answer of first) - (answer of second)
Split into two different integrals and integrate by parts on both
Two parts are each function times sin
20—Double the x=
Not differentiable as its not continuous -> show that they approach different numbers
21—Get all the differences between vectors and the parallel ones clearly have the same middle value. You may have to multiple by -1 to make same
Use projection -> draw it out
Both of them calculate the area of the parllelogram
22—Both = 1
This is because the cross products calculate the area of parallelogram so we do not need to multiply
23—Flips over the real axis
Because inverse v = conj v if thats true -> if we add I and conj I it equals a real number
Change the shit in the ^2 to that times its conj instead and then if in maple you get mod v remember its 1
24—
25—
26—A Plane + empty set
Number each vector
X1 + t1(X2-X1) + t2(X3-X1)
27—First and last
Eg (j and n)
28—Because A*lamba = U
A*AT*Lamba = AT*U
Therefore we can find the multiplicative difference
29—Do substitution just remmeber to make the limits equal tan(original), or whatever the given equation is
30—Pi -> sub value in to sin/cos and work around that
infinity -> just make the numerator grow faster
0 -> see what sin and cos equal at that value
31—beta - numerator * p
P + 1
Use the MVT between X and X + 1 remember to sub into C
32—Write that becase I3 is LESS THAN what we got for the previous question: it converges for P>0
33—Do implicit differentiation and should be clear that the constant on the top makes it bigger than 1 and is never undefined as the bottom is never zero due to cosh
34—Split function theorem (differentiability + continuous)
When we are at x=... we know the bottom equals 0, so should the top and so we can solve for that in the top. THen find the derivative of the top at 1 and then solve for the bottom.
35—AG = G - A = scalar * normal
Should be some scalar of the normal
Work this out for each x y and z and plug back into equation to find t
Then sub back fro the worked out x y and z
eg x = 3+4t -> sub that into equation
36—OA - AG = R
Sub R back into original equation and use the answer as the RHS instead
37—The one with the real number eg 5^11
Add on 2*Pi/ to the r (which can be seen as 5) and use the conversion
38—
39—The two that make m thingo 0
The other one is the one that doesnt have a twin
40—Get M5 and divide bottom row by m shit
Then solve each row as each variable is
X Y Z A B C
41—First range - [<1 + val * -1], infinity)
Eg. 20 -> -21
b.i. -> take inside alpha divided constant of bottom
eg. ln(x+9)/5
42—Denominator m + 1 (eg 6)
(6, infinity)
Open interval to infinity
43—1/2 * Pi, -1/2 * Pi, 3/4 * Pi, -3/4*Pi
[0, infinity)
Exponent(<4 or whatever value> * l)
2 * Pi * /
44—Rememebr that r and theta are times next to I in the exponent -> use conversion
Remember first principle
-(a+b*I) for last
45—
46—Paramters for function on each row
Use these as values for mean value theorem
47—ALL
Just infinite
Determinant of A cannot = 0
As Cx has infinite solutions -> det(A) = det(B) * det(C) = 0
48—Increasing as the diff can never be zero -> look at output
Add them = ln(1) which equals 0 -> therefore odd if you
f(-x) + f(x) = 0
f(-x) = -f(x)
49—If FALSE => skey-smmetric & invertible
first is the same
second is ^rows * det(A)
eg (-4)^18 * 39
50—A is the last row of the first matrix
B is always 1
51—Bottom Left of matrix B (could make same row) Plus the two values that make k shit 0
<0, 0, 1> -> try and calculate anyways