evaluation, induction, angles, systems, progressions, etc.

[real_name_x]

im stuck on these 5 questions. if anyone can help me, i would be eternally greatfull. thank you

#1. a) evaluate (1 + i)^2, where i = Squareroot(-1)
b) prove, by mathematical induction, that (1 + i)^(4n) = - 4^n, where n € N
c) hence or otherwise, find (1+ i)^32

#2. a farmer owns a triangular field ABC. The side [AC] is 104m, the side [AB] is 65m and the angle between these two sides is 60 degrees.
a) calculate the length of the third side of the field
b) find the area of the field in the form p *squareroot*(3), where p is an integer

(same prob) Let D be a point on [BC] such that [AD] bisects the 60 degree angle. The farmer divides the field into two parts by constructing a straight fence [AD] of length x meters.
c) (I) show that the area of the smaller part is given by (65x) / 4 and find an expression for the area of the larger part
(II) hence, find the value of x in the form q *squareroot* (3), where q is an integer
d) prove that BD / DC = 5 / 8

#3. the variables x,y,z satisfy the simultaneous equations....

x + 2y + z = k
2x + y + 4z = 6
x - 4y + 5z = 9

where k is a constant.
a) (I) show that these equations do NOT have a unique solution.
(II) find the value of k for which the equations are consistent (that is, they can be solved).
b) for this value of k, find the general solutions of these equations

#4. the ratio of the fifth term to the twelfth term of a sequence in an arithmetic progression is 6 / 13. If each term of this sequence is positive, and the product of the first terms and the third term is 32, find the sum of the first 100 terms of this sequence.

#5. solve the equation l e^(2x) - ( 1/ (x+2) ) l = 2

 

[soroban]

Re: MATH EMERGENCY!! please help me

Hello, real_name_x!

This is a wide variety of problems.
. . Where are they coming from?

#4. the ratio of the \(\displaystyle 5^{th}\) term to the \(\displaystyle 12^{th}\) term of a sequence in an arithmetic progression is \(\displaystyle \frac{6}{13}\).
If each term of this sequence is positive, and the product of the \(\displaystyle 1^{st}\) terms and the \(\displaystyle 3^{rd}\) term is 32,
find the sum of the first 100 terms of this sequence.

We're expected to know these formulas for an Arithmetic Progression.

Given that: \(\displaystyle a\) = first term, \(\displaystyle \,d\) = common difference

. . the \(\displaystyle n^{th}\) term is: \(\displaystyle \:\bf{{\color{blue}a_n\:=\:a\,+\,(n-1)d}}\)

. . the sum of the first \(\displaystyle n\) terms is: \(\displaystyle \:\bf{{\color{blue}S_n \:=\:\frac{n}{2}\left[2a + (n-1)d\right]}}\)


The \(\displaystyle 5^{th}\) term is: \(\displaystyle \,a_5\:=\:a\,+\,4d\)
The \(\displaystyle 12^{th}\) term is: \(\displaystyle \,a_{12} \:=\:a\,+\,11d\)

We are told that: \(\displaystyle \:\frac{a\,+\,4d}{a\,+\,11d} \:=\:\frac{6}{13}\)

. . Hence: \(\displaystyle \:13a\,+\,52d \:=\:6a\,+\,66d\;\;\Rightarrow\;\;7a \:=\:14d\;\;\Rightarrow\;\;d\:=\:\frac{a}{2}\;\) [1]


The \(\displaystyle 1^{st}\) term is: \(\displaystyle \,a_1\:=\:a\)
The \(\displaystyle 3^{rd}\) term is: \(\displaystyle \,a_3\;=\:a\,+\,2d\)

We are told that: \(\displaystyle \:a(a\,+\,2d) \:=\:32\)

. . Hence: \(\displaystyle \:a^2\,+\,2ad\:=\:32\;\) [2]


Substitute [1] into [2]: \(\displaystyle \:a^2\,+\,2a\left(\frac{a}{2}\right)\:=\:32\;\;\Rightarrow\;\;2a^2\:=\:32\;\;\Rightarrow\;\;a^2\:=\:16\)

. . Hence: \(\displaystyle \:a \:=\:\pm4\;\;\;\)Since the terms are positive, \(\displaystyle \,\fbox{a\:=\:4}\)

Substitute into [1]: \(\displaystyle \:d \:=\:\frac{4}{2} \;\;\Rightarrow\;\;\fbox{d\:=\:2}\)


Then: \(\displaystyle \:S_{100} \:=\:\frac{100}{2}\left[2(4)\,+\,99(2)] \:=\;\)\(\displaystyle \L\bf{{\color{blue}10,300}}\)

 

[stapel]

This is a wide variety of problems. Where are they coming from?

Many appear to be from International Baccalaureate programs, assigned as summer homework, due (generally) at or near the end of August.

Perhaps we could refrain from posting complete worked solutions, especially when no effort has been shown...?

im stuck on these 5 questions.

It is unfortunate that you appear to have displayed little effort other than posting the same questions multiple times in multiple locations. Kindly please refrain from sending any more private-message requests for help; the volunteers here are under no obligation to send the solutions to your e-mail address.

Thank you.

Eliz.