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. [tex] \frac{\sqrt{-9}}{3-2i+(1+5i)} [/tex]
first simplify the denominator which is √-9 = √9 *√-1 = 3*i = 3i
now simplify the denominator which is 3-2i+(1+5i) = 4+3i
[tex] = \frac{{3i}}{(4+3i)} [/tex]
now multiply and divide it by (4-3i)(which is the conjugate of 4+3i)
[tex] = \frac{{3i}}{(4+3i)} * \frac{(4-3i)}{(4-3i)} [/tex]
[tex] = \frac{{12i-9i^{2}}}{(16-9i^{2})} [/tex]
⇒ i² = -1 so
[tex] = \frac{{12i+9}}{(16+9)} [/tex]
[tex] = \frac{{12i+9}}{25} [/tex]
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 [tex] ()^{\frac{1}{3}} [/tex].
= [tex] 3*(2*2^{3})^{\frac{1}{3}} [/tex]
= [tex] 3* (2)^{\frac{1}{3}} [/tex] * [tex] (2^{3})^{\frac{1}{3}} [/tex]
= 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.
