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 ā-9 = ā9 *ā-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]
ā iĀ² = -1 so
[tex] = \frac{{12i+9}}{(16+9)} [/tex]
[tex] = \frac{{12i+9}}{25} [/tex]
C is the correct answer.
3. 4ā24 -4ā8 +ā98
you can write 24 as 6*4, 8 as 4*2, 98 as 49*2.
= 4ā(6*4) -4ā(4*2) +ā(49*2)
= 4*ā6 *ā4 -4*ā4 *ā2 +ā49* ā2
= 4*2*ā6 -4*2 *ā2 + 7* ā2
= 8ā6 -8ā2 +7ā2
= 8ā6 +ā2(-8+7)
= 8ā6 -ā2
D is the correct answer.
4. 3ā16
you can write 16 as 2*2Ā³ and ā 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 * ā2
= 6ā2
B is the correct answer.
5. The average rate of change for a given function from x=-1 to x=2 = Īy/Īx
so,
Īy/Īx = (-1-(-3))/ (0-(-1))
Īy/Īx = (-1+3)/ (0+1)
Īy/Īx = 2/ 1
average rate of change of given function=Īy/Īx = 2
D is the correct answer.
