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Algebra
Complete the columns for grades 5 and 6 in the same way as we have done for grade 4:
Grade 4.
What should learners do?
. Investigate GP&NP.
. Extend GP&NP.
. Identify the relationship between terms in GP&NP.
. Find the rule of GP & NP.
Describe relationship or rule of GP & NP in words.
Grade 5
What should learners do?
Grade 6
What should learners do?
Grade 4.
What should learners do?
. Investigate GP&NP.
. Extend GP&NP.
. Identify the relationship between terms in GP&NP.
. Find the rule of GP & NP.
Describe relationship or rule of GP & NP in words.
Grade 5
What should learners do?
Grade 6
What should learners do?
Algebra
(a) Prove that
(−1 + i
√
3)n + (−1 − i
√
3)n
has either the value 2n+1 or the value −2
n
if n is any integer (positive, negative or zero).
(−1 + i
√
3)n + (−1 − i
√
3)n
has either the value 2n+1 or the value −2
n
if n is any integer (positive, negative or zero).
Algebra
How to factories cubic polynomial????
Ek
Ek
Algebra
4.(a) Prove that
(−1 + i√3)^n + (−1 − i√3)^n
has either the value 2^n+1 or the value −2^n. if n is any integer (positive, negative or zero).
(b) The complex numbers z1 and z2 are connected by the relation
z1 = z2 +1/z2
If the point representing z2 in the Argand diagram describes a circle of radius a and centre at the origin, show that the point representing z1 describes the ellipse
x^2/(1 + a^2)^2 + y^2/(1 − a^2)^2=1/a^2.
(−1 + i√3)^n + (−1 − i√3)^n
has either the value 2^n+1 or the value −2^n. if n is any integer (positive, negative or zero).
(b) The complex numbers z1 and z2 are connected by the relation
z1 = z2 +1/z2
If the point representing z2 in the Argand diagram describes a circle of radius a and centre at the origin, show that the point representing z1 describes the ellipse
x^2/(1 + a^2)^2 + y^2/(1 − a^2)^2=1/a^2.
Algebra
Prove that
(−1 + i√3)^n + (−1−i√3)^n has either the value 2^(n+1) or the value −2^n if n is any integer (positive, negative or zero).
(−1 + i√3)^n + (−1−i√3)^n has either the value 2^(n+1) or the value −2^n if n is any integer (positive, negative or zero).
Algebra
4.(a) Prove that
(−1 + i√3)^n + (−1 − i√3)^n
has either the value 2^n+1 or the value −2^n. if n is any integer (positive, negative or zero).
(b) The complex numbers z1 and z2 are connected by the relation
z1 = z2 +1/z2
If the point representing z2 in the Argand diagram describes a circle of radius a and centre at the origin, show that the point representing z1 describes the ellipse
x^2/(1 + a^2)^2 + y^2/(1 − a^2)^2=1/a^2.
(−1 + i√3)^n + (−1 − i√3)^n
has either the value 2^n+1 or the value −2^n. if n is any integer (positive, negative or zero).
(b) The complex numbers z1 and z2 are connected by the relation
z1 = z2 +1/z2
If the point representing z2 in the Argand diagram describes a circle of radius a and centre at the origin, show that the point representing z1 describes the ellipse
x^2/(1 + a^2)^2 + y^2/(1 − a^2)^2=1/a^2.
Algebra
3. (a) Show that, for any complex number z, zz = |z|^2., z + z = 2Re(z) and Re(z) ≤ |z|. Hence
show that
i. |z1 + z2|^2 = |z1|^2 + |z2|^2 + 2Re(z1z2),
ii. |z1 + z2| ≤ |z1| + |z2|,
where Re(z) is the real part of z and *z the conjugate of z.
(b) If z1 = 1 + 2i, find the set of values of z2 for which
(i) |z1 + z2| = |z1| + |z2|
(ii) |z1 + z2| = |z1| − |z2|.
show that
i. |z1 + z2|^2 = |z1|^2 + |z2|^2 + 2Re(z1z2),
ii. |z1 + z2| ≤ |z1| + |z2|,
where Re(z) is the real part of z and *z the conjugate of z.
(b) If z1 = 1 + 2i, find the set of values of z2 for which
(i) |z1 + z2| = |z1| + |z2|
(ii) |z1 + z2| = |z1| − |z2|.
Algebra
Find all integer solutions to the equation 4x + 3y – 9z = 5 for values of x, y, and z between 0 and 100.
Algebra
If a stock price goes from $10 to $12 from January 1st to January 31, from $12 to $9 from February 1st to February 28th, and from $9 to $15 from March 1st to March 31st is the price change from $10 to $15 a straight line?
It is clear that in each of the three time intervals mentioned there was a complex daily variation of prices as in an electrocardiogram. But what would be a simplified solution for a first naive view of the situation? Would a simple function hold up? What is the simplest function to represent this situation? Does your naive initial and simplified model allow you to predict the behavior of the stock in the next month?
How can I use three “pieces” of lines to describe the price movements from the beginning of January to the end of March? Show the graph for the price movement
It is clear that in each of the three time intervals mentioned there was a complex daily variation of prices as in an electrocardiogram. But what would be a simplified solution for a first naive view of the situation? Would a simple function hold up? What is the simplest function to represent this situation? Does your naive initial and simplified model allow you to predict the behavior of the stock in the next month?
How can I use three “pieces” of lines to describe the price movements from the beginning of January to the end of March? Show the graph for the price movement
Algebra
When it is 11 a.m. in Johannesburg, it is 6 p.m. the same day in Tokyo, Japan. A flight leaves OR Tambo in Johannesburg at 6 a.m. on Tuesday for Tokyo. The flight takes 18 hours in the air, with an additional 1 hour stop-over on land. What time will it arrive in Tokyo?
