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A number of the form a + ib is called a ...

A number of the form a + ib is called a complex number, where `a,b in R and i=sqrt(-1)` Complex number is usually denoted by Z and the set of complex number is represented by C. Thus `C={a+ib:a,b inB and -=sqrt(-1)}.If Z =a+ib` is a complex number then `bar(z)a=-ib` is called as conjugate. Two complex numbers `z1=a1+ib_(1)&z2=a2+ib2` are equal if and only if their real and imaginary parts are equal respectively. i.e., `z_(1)=z_(2)hArrRe(z_(1))=Re(z_2)and Im(z_1)=Im (z_(2))hArra_(1)=a_(2)and b_(1)=b_(2).` The following algebric operations can be performd on complex numbers.
`{:(1.,"Addition",(a+bi)+(c+bi)=a+bi+c+di=(a+c)+(b+d)i),(2.,"Subtraction",(a+b)-(c+di)=a+bi-c-di=(a-c)+(b-d)i),(3.,"Multipication",(a+b)(c+di)=ac+adl+bci+bdi2=(ac-bd)+(ad+bc)i),(4.,"Multiplication",(a+bi)/(c+di)=(a+di)/(c+di).(c-di)/(c-di)=(ac+bd)/(c^(2)+d^(2))+(bc-ad)/(c^(2)+d^(2))i):}`
Using above comprehension answer the followings :
`((4i^(3)-i)/(2i+1))^(i)` can be expressed in a = ib as

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A number of the form a + ib is called a complex number, where a,b in R and i=sqrt(-1) Complex number is usually denoted by Z and the set of complex number is represented by C. Thus C={a+ib:a,b inB and -=sqrt(-1)}.If Z =a+ib is a complex number then bar(z)a=-ib is called as conjugate. Two complex numbers z1=a1+ib_(1)&z2=a2+ib2 are equal if and only if their real and imaginary parts are equal respectively. i.e., z_(1)=z_(2)hArrRe(z_(1))=Re(z_2)and Im(z_1)=Im (z_(2))hArra_(1)=a_(2)and b_(1)=b_(2). The following algebric operations can be performd on complex numbers. {:(1.,"Addition",(a+bi)+(c+bi)=a+bi+c+di=(a+c)+(b+d)i),(2.,"Subtraction",(a+b)-(c+di)=a+bi-c-di=(a-c)+(b-d)i),(3.,"Multipication",(a+b)(c+di)=ac+adl+bci+bdi2=(ac-bd)+(ad+bc)i),(4.,"Multiplication",(a+bi)/(c+di)=(a+di)/(c+di).(c-di)/(c-di)=(ac+bd)/(c^(2)+d^(2))+(bc-ad)/(c^(2)+d^(2))i):} Using above comprehension answer the followings : If a^(2)+b^(2)=1.Then(1+b+ia)/(1+b-ia)=

A number of the form a + ib is called a complex number, where a,b in R and i=sqrt(-1) Complex number is usually denoted by Z and the set of complex number is represented by C. Thus C={a+ib:a,b inB and -=sqrt(-1)}.If Z =a+ib is a complex number then bar(z)a=-ib is called as conjugate. Two complex numbers z1=a1+ib_(1)&z2=a2+ib2 are equal if and only if their real and imaginary parts are equal respectively. i.e., z_(1)=z_(2)hArrRe(z_(1))=Re(z_2)and Im(z_1)=Im (z_(2))hArra_(1)=a_(2)and b_(1)=b_(2). The following algebric operations can be performd on complex numbers. {:(1.,"Addition",(a+bi)+(c+bi)=a+bi+c+di=(a+c)+(b+d)i),(2.,"Subtraction",(a+b)-(c+di)=a+bi-c-di=(a-c)+(b-d)i),(3.,"Multipication",(a+b)(c+di)=ac+adl+bci+bdi2=(ac-bd)+(ad+bc)i),(4.,"Multiplication",(a+bi)/(c+di)=(a+di)/(c+di).(c-di)/(c-di)=(ac+bd)/(c^(2)+d^(2))+(bc-ad)/(c^(2)+d^(2))i):} Using above comprehension answer the followings : If a^(2)+b^(2)=1.Then(1+b+ia)/(1+b-ia)=

A number of the form a + ib is called a complex number, where a,b in R and i=sqrt(-1) Complex number is usually denoted by Z and the set of complex number is represented by C. Thus C={a+ib:a,b inB and -=sqrt(-1)}.If Z =a+ib is a complex number then bar(z)a=-ib is called as conjugate. Two complex numbers z1=a1+ib_(1)&z2=a2+ib2 are equal if and only if their real and imaginary parts are equal respectively. i.e., z_(1)=z_(2)hArrRe(z_(1))=Re(z_2)and Im(z_1)=Im (z_(2))hArra_(1)=a_(2)and b_(1)=b_(2). The following algebric operations can be performd on complex numbers. {:(1.,"Addition",(a+bi)+(c+bi)=a+bi+c+di=(a+c)+(b+d)i),(2.,"Subtraction",(a+b)-(c+di)=a+bi-c-di=(a-c)+(b-d)i),(3.,"Multipication",(a+b)(c+di)=ac+adl+bci+bdi2=(ac-bd)+(ad+bc)i),(4.,"Multiplication",(a+bi)/(c+di)=(a+di)/(c+di).(c-di)/(c-di)=(ac+bd)/(c^(2)+d^(2))+(bc-ad)/(c^(2)+d^(2))i):} Using above comprehension answer the followings : If (x^(2)+x)+iyand(-x-1)-i(x+2y) are conjugate of each other, then real value of x y are

If z = a+ib is a complex numbers, then

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The modulus of the complex number z =a +ib is

Solve |z|+z= 2+ i , where z is a complex number

Let z be a complex number such that |z|+z=3+I (Where i=sqrt(-1)) Then ,|z| is equal to

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NAGEEN PRAKASHAN ENGLISH-COMPLEX NUMBERS AND QUADRATIC EQUATION -EXERCISE 5B
  1. Write the following in the form of ordered pair : (i) 3-2i (ii)...

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  2. Convert the following in the form of a complex number : (i) (2, -5)...

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  3. Find the values of x and y from the following : (i) (3x -7)+2iy=-5y+...

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  4. If z=1+2i, show that z^(2)-2z+5=0. Hence find the value of z^(3) +7z^...

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  5. Z=-5+4i then Z^4 +9Z^3 +35Z^2 – Z + 4 =

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  6. If z(1)=2-i, z(2)=1+ 2i, then find the value of the following : (i)...

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  7. If x + i y =(a+i b)/(a-i b),prove that x^2+y^2=1.

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  8. (x+iy)^(1/3) =(a+ib) then prove that (x/a+y/b)=4(a^2-b^2)

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  9. If ((a^(2)+1)^(2))/(2a-i) = a + iy , then what is the value of x^...

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  10. Write the least positive integral value of n for which ((1+i)/(1-i))^n...

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  11. The complex number z is purely imaginary , if

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  12. A number of the form a + ib is called a complex number, where a,b in R...

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  13. Find the real values of theta for which the complex number (1+i costhe...

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  14. Find the square root of the following : (i) 3-4i (ii) 4+6isqrt(5) ...

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  15. If x+iy=3/(2+costheta +i sin theta), then show that x^2+y^2=4x-3

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  16. The sum and product of two complex numbers are real if and only if the...

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  17. If (1+x)/(1 -x)=cos2theta +isin2theta, prove that that x=itantheta.

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  18. If x= cos alpha+ i sin alpha, y = cos beta+ i sin beta, then prove th...

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  19. Prove that: x^4=4=(x+1+i)(x+1-i)(x-1+i(x-1-i)dot)

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  20. Evaluate : (4+3sqrt(-20))^(1//2)+(4-3 sqrt(-20))^(1//2)

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