Question 1

Consider the following inequalities : I. 1+4i>3+2i II. 2+3i III. 4+3i>3+4i where i= -1 How many of the above are valid ?

Accepted Answer

The field of complex numbers cannot be ordered. Inequalities such as z_1 > z_2 or z_1 Since all the given inequalities involve complex numbers with non-zero imaginary parts, they are mathematically meaningless. Therefore, none of the given inequalities are valid. Answer: None

Question 2

If p, q, r are the cube roots of unity, then what is vmatrix p^2+q^2 & r^2 & r^2 \ p^2 & q^2+r^2 & p^2 \ q^2 & q^2 & r^2+p^2 vmatrix equal to ?

Accepted Answer

Let a = p^2, b = q^2, and c = r^2. The given determinant can be written as: = vmatrix a+b & c & c \ a & b+c & a \ b & b & c+a vmatrix Applying the row operation R_1 R_1 - R_2 - R_3, we get: = vmatrix 0 & -2b & -2a \ a & b+c & a \ b & b & c+a vmatrix Expanding the determinant along the first row: = 0 - (-2b) vmatrix a & a \ b & c+a vmatrix + (-2a) vmatrix a & b+c \ b & b vmatrix = 2b(ac + a^2 - ab) - 2a(ab - b^2 - bc) = 2abc + 2a^2b - 2ab^2 - 2a^2b + 2ab^2 + 2abc = 4abc Substituting back a

Question 3

If , , are cube roots of -8, then what is ^2 p^2+ ^2 q^2+ ^2 r^2 ^2 p^2+ ^2 q^2+ ^2 r^2 equal to ?

Accepted Answer

Since , , are the cube roots of -8, they are the roots of the equation z^3 + 8 = 0. Thus, we have ^3 = ^3 = ^3 = -8. The product of the roots of the cubic equation z^3 + 8 = 0 is given by = -8. Using these properties, we can find relations between the roots: ^3 = ^2 = ^3 = ^2 = ^3 = ^2 = Let the given expression be E = ^2 p^2+ ^2 q^2+ ^2 r^2 ^2 p^2+ ^2 q^2+ ^2 r^2 . Consider multiplying the denominator by the fraction : ( ^2 p^2 + ^2 q^2 + ^2 r^2) = p^2 + ^3 q^2 + ^2 r^2 Now, substitute the deri

Question 4

Let Z_1 and Z_2 be complex numbers such that 3Z_1 4Z_2 is purely imaginary. What is | Z_1+Z_2 Z_1-Z_2 | equal to ?

Accepted Answer

Question 5

If is a non-real cube root of unity, then what is a root of the following equation? vmatrix x+1 & & ^2 \ & x+ ^2 & 1 \ ^2 & 1 & x+ vmatrix = 0

Accepted Answer

Applying the column operation C_1 C_1 + C_2 + C_3, the determinant becomes: vmatrix x+1+ + ^2 & & ^2 \ x+1+ + ^2 & x+ ^2 & 1 \ x+1+ + ^2 & 1 & x+ vmatrix = 0 Since 1 + + ^2 = 0 for a non-real cube root of unity, the equation simplifies to: vmatrix x & & ^2 \ x & x+ ^2 & 1 \ x & 1 & x+ vmatrix = 0 Taking x common from the first column: x vmatrix 1 & & ^2 \ 1 & x+ ^2 & 1 \ 1 & 1 & x+ vmatrix = 0 This implies x = 0 is a root of the equation. Answer: x = 0

Question 6

If x + 1 x = 2 , then what is x^3 + 1 x^3 equal to?

Accepted Answer

Given x + 1 x = 2 Using the algebraic identity a^3 + b^3 = (a+b)^3 - 3ab(a+b): x^3 + 1 x^3 = (x + 1 x )^3 - 3 (x + 1 x ) Substituting x + 1 x = 2 : x^3 + 1 x^3 = (2 )^3 - 3(2 ) x^3 + 1 x^3 = 8 ^3 - 6 x^3 + 1 x^3 = 2(4 ^3 - 3 ) Using the trigonometric identity 3 = 4 ^3 - 3 : x^3 + 1 x^3 = 2 3 Answer: 2 3

Question 7

If z 0 is a complex number, then what is amp (z) + amp ( z ) equal to?

Accepted Answer

Let z = r( + i ) where r > 0 and = amp (z). The conjugate of z is given by z = r( - i ). This can be rewritten as z = r( (- ) + i (- )). Thus, the amplitude of z is amp ( z ) = - . Therefore, amp (z) + amp ( z ) = + (- ) = 0. Answer: 0

Question 8

If = -1 + -3 2 , then what is the value of (1 + ^ 19 - ^ 35 )^ 100 - (1 - 3 ^ 25 + ^ 38 )^ 50 ?

Accepted Answer

Given = -1 + -3 2 , which is the complex cube root of unity, . We know that ^3 = 1 and 1 + + ^2 = 0. Substituting = into the given expression, we simplify the powers of : ^ 19 = ^ 19 = ( ^3)^6 = ^ 35 = ^ 35 = ( ^3)^ 11 ^2 = ^2 ^ 25 = ^ 25 = ( ^3)^8 = ^ 38 = ^ 38 = ( ^3)^ 12 ^2 = ^2 The first term becomes: (1 + ^ 19 - ^ 35 )^ 100 = (1 + - ^2)^ 100 Since 1 + = - ^2, this reduces to: (- ^2 - ^2)^ 100 = (-2 ^2)^ 100 = 2^ 100 ^ 200 Since 200 = 3 66 + 2, ^ 200 = ^2. Thus, the first term is 2^ 100 ^2.

Question 9

What is ( 3 +i 3 -i )^3 equal to?

Accepted Answer

Rationalizing the expression inside the bracket: 3 +i 3 -i = ( 3 +i)^2 ( 3 -i)( 3 +i) = 3 + i^2 + 2 3 i 3 - i^2 = 3 - 1 + 2 3 i 3 - (-1) = 2 + 2 3 i 4 = 1 2 + 3 2 i Converting to polar form: 1 2 + 3 2 i = ( 3 ) + i ( 3 ) Using De Moivre's theorem to find the cube: ( ( 3 ) + i ( 3 ) )^3 = (3 3 ) + i (3 3 ) = + i = -1 + 0 = -1 Answer: -1

Question 10

If ( 1-i 1+i )^ 2m ( 1+i 1-i )^ 2n = 1, where i = -1 , then what is the smallest positive value of (m - n)?

Accepted Answer

First, we simplify the expressions inside the parentheses by rationalizing the denominators: 1-i 1+i = (1-i)(1-i) (1+i)(1-i) = 1 - 2i + i^2 1 - i^2 = 1 - 2i - 1 1 - (-1) = -2i 2 = -i 1+i 1-i = (1+i)(1+i) (1-i)(1+i) = 1 + 2i + i^2 1 - i^2 = 1 + 2i - 1 1 - (-1) = 2i 2 = i Substitute these simplified forms back into the given equation: (-i)^ 2m (i)^ 2n = 1 Using the property of exponents, we can rewrite this as: ((-i)^2)^m (i^2)^n = 1 Since i^2 = -1 and (-i)^2 = i^2 = -1, we get: (-1)^m (-1)^n = 1

Question 11

Let 1, , ^2 be three cube roots of unity. If x = a + b, y = a + b ^2, z = a ^2 + b , then what is x^2 + y^2 + z^2 equal to?

Accepted Answer

Given x = a + b, y = a + b ^2, and z = a ^2 + b . Squaring each term, we get: x^2 = (a + b)^2 = a^2 + b^2 + 2ab y^2 = (a + b ^2)^2 = a^2 ^2 + b^2 ^4 + 2ab ^3 z^2 = (a ^2 + b )^2 = a^2 ^4 + b^2 ^2 + 2ab ^3 Using the properties of cube roots of unity, ^3 = 1 and ^4 = . Substituting these into the equations: y^2 = a^2 ^2 + b^2 + 2ab z^2 = a^2 + b^2 ^2 + 2ab Adding x^2, y^2, and z^2: x^2 + y^2 + z^2 = a^2(1 + + ^2) + b^2(1 + + ^2) + 2ab(1 + 1 + 1) Since 1 + + ^2 = 0, the expression simplifies to: x^

Complex Number — Previous Year Questions & Solutions