Complex Numbers and Quadratic Equations — Class 11 solved problems

53 problems from this chapter, each solved step by step.

Find all values of (1+i)^(1-i) in the form a + bi.
e^(( (1)/(2)ln 2 + (π)/(4) + 2nπ)) [ cos( (π)/(4) + 2nπ - (1)/(2)ln 2) + i sin( (π)/(4) + 2nπ - (1)/(2)ln 2) ] for n Z
Find all real solutions to the equation: √(x + 3 - 4√(x - 1)) + √(x + 8 - 6√(x - 1)) = 1.
The real solutions are x [5, 10].
If the set of all a R, for which the equation 2x²+(a-5)x+15=3a has no real root, is the interval (α, β), and X=x Z: α < x < β, then Σ_x X x² is equal to:
2139
MODIFIED: Given that variable_91, variable_91, variable_91 are roots of x³ - 3x² + 4 = 0, find the value of variable_91⁴ + variable_91⁴ + variable_91⁴.
The value of α^4 + β^4 + ^4 is 33.
If a complex number z lies in the interior or on the boundary of a circle of radius 3 units and centre (-4,0), find the greatest and least values of |z+1|.
The greatest value of |z+1| is 6 and the least value is 0.
Find the value of 2 x^(4)+5 x³+7 x²-x+41, when x=-2-√3 i
6
Express (-√3+√(-2))(2 √3-i) in the form of a+i b
(-√3+√(-2))(2 √3-i) = (-6+√2) + i(√3+2√6)
Find the value of P such that the difference of the roots of the equation x²-P x+8=0 is 2.
The value of P is ± 6.
Let A = θ [0, 2π]: 1 + 10 Re ( (2cosθ + isinθ)/(cosθ - 3isinθ)) = 0. Then Σ_θ A θ² is equal to:
(21π²)/(4)
Let z_1 and z_2 be two complex numbers such that z_1+i z_2=0 and (z_1 z_2)=π. Then find (z_1).
(3π)/(4)
If x+i y=(a+i b)/(a-i b), prove that x²+y²=1.
x²+y²=1
The sum, of the squares of all the roots of the equation x² + |2x - 3| - 4 = 0, is
12 - 6√2
Let the set of all values of p R, for which both the roots of the equation x² - (p + 2)x + (2p + 9) = 0 are negative real numbers, be the interval (α, β]. Then
5
If (x+i y)^((1)/(3))=a+i b, where x, y, a, b R, show that (x)/(a)-(y)/(b)=-2(a²+b²)
The identity (x)/(a)-(y)/(b)=-2(a²+b²) is shown to be true.
Express the following in the form of a+b i: (i) (-5 i)((1)/(8) i) (ii) (-i)(2 i)(-(1)/(8) i)³
(i) (5)/(8) + 0i (ii) 0 + (1)/(256) i
Find the value of a such that the sum of the squares of the roots of the equation x²-(a-2) x-(a+1)=0 is least.
The value of a for which the sum of the squares of the roots is least is 1.
Let |z_1 - 8 - 2i| ≤ 1 and |z_2 - 2 + 6i| ≤ 2, z_1, z_2 C. Then the minimum value of |z_1 - z_2| is:
7
The number of real roots of the equation x x - 2 + 3 x - 3 + 1 = 0 is:
The equation has 1 real root.
Find the multiplicative inverse of 2-3 i.
The multiplicative inverse of 2-3i is (2)/(13) + (3)/(13)i.
Let α, β be the roots of the equation x² - ax - b = 0 with Im(α) < Im(β). Let P_n = α^n - β^n. If P_3 = -5√7i, P_4 = -3√7i, P_5 = 11√7i and P_6 = 45√7i, then |α
158
If 4 x+i(3 x-y)=3+i(-6), where x and y are real numbers, then find the values of x and y.
The values are x = (3)/(4) and y = (33)/(4).
If x² + y² + z² = xy + yz + zx, prove that x = y = z and find the value of (x+y+z)²/(x²+y²+z²).
The value of ((x+y+z)²)/(x²+y²+z²) is 3.
Let O be the origin, the point A be z_1 = √3 + 2 √2 i, the point B(z_2) be such that √3 |z_2| = |z_1| and (z_2) = (z_1) + (π)/(2). Then
z_2 = -(2√2)/(√3) + i
If z_1 and z_2 both satisfy z+ z=2|z-1| (z_1-z_2)=(π)/(4), then find Im(z_1+z_2).
Im(z_1+z_2) = 2
Among the statements: (S1): The set z C -i: |z| = 1 and (z - i)/(z + i) is purely real contains exactly two elements, and (S2): The set z C -1: |z| = 1 and (z -
Statement (S1) is incorrect, and statement (S2) is correct.
Express the following in the form a+i b ll (i) (5+√2 i)/(1-√2 i) & (ii) i^(-35)
(i) 1 + 2√2 i (ii) 0 + 1i
If the locus of z C, such that ((z-1)/(2z + i)) + ( z-12 z - i) = 2, is a circle of radius r and center (a,b), then (15ab)/(r²) is equal to:
9
Let z be a complex number such that |z|=1. If 2 + k² zk + z = k z, k R, then the maximum distance of k + i k² from the circle |z - (1 + 2i)| = 1 is:
The maximum distance is √5 + 1.
Let | z - z2z + z | = (1)/(3), z C, be the equation of a circle with center at C. If the area of the triangle, whose vertices are at the points (0, 0), C and (α
The problem statement contains an inconsistency as the given equation does not represent a circle. Therefore, α² cannot be determined.
Let P_n=α^n+β^n, n N. If P_10=123, P_9=76, P_8=47 and P_1=1, then the quadratic equation having roots (1)/(α) and (1)/(β) is:
x² + x - 1 = 0
Express (5-3 i)³ in the form a+i b.
-10 - 198i
Solve the system of equations: x + y + z = 6, x² + y² + z² = 14, x³ + y³ + z³ = 36.
The solutions (x, y, z) are permutations of (1, 2, 3).
Let z_1, z_2, and z_3 be three complex numbers on the circle |z| = 1 with (z_1) = (π)/(4), (z_2) = 0, and (z_3) = (π)/(4). If |z_1 z_2 + z_2 z_3 + z_3 z_1|² = α
13
If z_1, z_2, z_3 C are the vertices of an equilateral triangle whose centroid is z_0, then Σ_k=1³ (z_k - z_0)² is equal to:
0
If z_1, z_2, z_3 are complex numbers such that |z_1|=|z_2|=|z_3|=|(1)/(z_1)+(1)/(z_2)+(1)/(z_3)|=1, then find the value of |z_1+z_2+z_3|.
|z_1+z_2+z_3| = 1
Given that α, β, γ are roots of x³ - 3x² + 4 = 0, find the value of α⁴ + β⁴ + γ⁴.
33
Find the conjugate of ((3-2 i)(2+3 i))/((1+2 i)(2-i)).
The conjugate of the given expression is (63)/(25) + (16)/(25)i.
Locate the points for which 3<|z|<4
The points for which 3<|z|<4 are all points in the complex plane that lie strictly between the circle of radius 3 and the circle of radius 4, both centered at t
If the imaginary part of (2 z+1)/(i z+1) is -2, then show that the locus of the point representing z in the argand plane is a straight line.
The locus of the point representing z is the straight line given by the equation x+2y-2=0.
Let the curve z(1+i) + z(1-i) = 4, z C, divide the region |z-3| ≤ 1 into two parts of areas α and β. Then |α - β| equals:
(π)/(2) + 1
Let α and β be the roots of x² + √3 x - 16 = 0, and and be the roots of x² + 3x - 1 = 0. If P_n = α^n + β^n and Q_n = ^n + ^n, then (P_25 + √3 P_24)/(2P_23) + (
5
Let _θ and _θ be the distinct roots of 2x² + (cos θ) x - 1 = 0, θ (0, 2π). If m and M are the minimum and the maximum values of _θ^4 + _θ^4, then 16(M + m) equa
13
Solve the equation z²= z, where z=x+i y
z = 0, 1, -(1)/(2) + i(√3)/(2), -(1)/(2) - i(√3)/(2)
If α is a root of the equation x² + x + 1 = 0 and Σ_k=1^n (α^k + α^(-k))² = 20, then n is equal to:
n = 11
The product of all the rational roots of the equation (x² - 9x + 11)² - (x - 4)(x - 5) = 3, is equal to
14
Two numbers k_1 and k_2 are randomly chosen from the set of natural numbers. Then, the probability that the value of i^(k_1) + i^(k_2), (i = √(-1)) is non-zero,
The probability that the value of i^(k_1) + i^(k_2) is non-zero is (3)/(4).
Let α be a solution of x² + x + 1 = 0, and for some a and b in R, [4 a b] 1 & 16 & 13 -1 & -1 & 2 -2 & -14 & -8 = [0 0 0]. If (4)/(α^4) + (m)/(α) + (n)/(α²) = 3
-10
Let z_1 and z_2 be two complex numbers such that |z_1+z_2|=|z_1|+|z_2|. Then show that (z_1)- (z_2)=0.
We have shown that (z_1)- (z_2)=0 (modulo 2π).
The number of real solution(s) of the equation x² + 3x + 2 = min |x - 3|, |x + 2| is:
2
The sum of the squares of the roots of |x + 2|² + |x - 2| - 2 = 0 and the squares of the roots of x² - 2|x - 3| - 5 = 0, is:
26
If the set of all a R 1, for which the roots of the equation (1 - a)x² + 2(a - 3)x + 9 = 0 are positive is (-∞, -α] [β,), then 2α + β + is equal to:
7
If |z²-1|=|z|²+1, then show that z lies on imaginary axis.
The complex number z lies on the imaginary axis.
The number of complex numbers z, satisfying |z| = 1 and |(z)/(2) + (2)/(z)| = 1, is:
0

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