Question

98 points :)

1 Which of the following is the conjugate of a complex number with 5 as the real part and −2i as the imaginary part?
5 + 2i
−5 − 2i
−5 + 2i
5 − 2i
2 Simplify the expression The root of negative nine over the quantity of three minus two i plus the quantity of one plus five i..
the quantity of twelve minus nine i all over twenty five.
the quantity of twelve minus nine i all over seven.
the quantity of nine plus twelve i all over twenty five.
the quantity of nine plus twelve i all over seven.

3 Simplify 4 times square root 24 minus 4 square root 8 plus square root of 98.
7 square root 2
23 square root 2
negative square root 2 plus 16 square root 3
negative square root 2 plus 8 square root 6

4 Simplify 3 times the 3rd root of 16.
5 times the 3rd root of 2
6 times the 3rd root of 2
12 times the 3rd root of 2
27 times the 3rd root of 2

5 Use the table below to answer this question:
x y
−1 −3
0 −1
2 3

Find the average rate of change for the given function from x = −1 to x = 2.
−2
−1
1
2

Answer 1

1. Concept: Complex numbers of the form (a+bi) and (a-bi) which have the same real parts and whose imaginary parts differ in sign only, are called conjugate of each other.

Correction: 5 is a real part and -2i is an imaginary number(not imaginary part) (but -2 is known as imaginary part :)

in this question,

z= 5-2i

z* = 5+2i ( conjugate of z with real part as it is and imaginary part(opposite in sign))

A is the correct answer.

2.
`(√(-9))/(3-2i+(1+5i))`

first simplify the denominator which is √-9 = √9 *√-1 = 3*i = 3i

now simplify the denominator which is 3-2i+(1+5i) = 4+3i

`= \frac{{3i}}{(4+3i)}`

now multiply and divide it by (4-3i)(which is the conjugate of 4+3i)

`= \frac{{3i}}{(4+3i)} * ((4-3i))/((4-3i))`

`= \frac{{12i-9i^(2)}}{(16-9i^(2))}`

⇒ i² = -1 so

`= \frac{{12i+9}}{(16+9)}`

`= \frac{{12i+9}}{25}`

C is the correct answer.

3. 4√24 -4√8 +√98

you can write 24 as 6*4, 8 as 4*2, 98 as 49*2.

= 4√(6*4) -4√(4*2) +√(49*2)

= 4*√6 *√4 -4*√4 *√2 +√49* √2

= 4*2*√6 -4*2 *√2 + 7* √2

= 8√6 -8√2 +7√2

= 8√6 +√2(-8+7)

= 8√6 -√2

D is the correct answer.

4. 3∛16

you can write 16 as 2*2³ and ∛ as
`()^{(1)/(3)}`.

=
`3*(2*2^(3))^{(1)/(3)}`

=
`3* (2)^{(1)/(3)}` *
`(2^(3))^{(1)/(3)}`

= 3* 2 * ∛2

= 6∛2

B is the correct answer.

5. The average rate of change for a given function from x=-1 to x=2 = Δy/Δx

so,

Δy/Δx = (-1-(-3))/ (0-(-1))

Δy/Δx = (-1+3)/ (0+1)

Δy/Δx = 2/ 1

average rate of change of given function=Δy/Δx = 2

D is the correct answer.