12 - MATHS- UNIT 3. COMPLEX NUMBERS

1) If

and

then the points on the Argand diagram representing

and

are

Vertices of a right angled triangle

Vertices of an equilateral triangle

Vertices of an isosceles triangle

Collinear

If a and b are complex numbers such that a/b is purely imaginary, then the points on the Argand diagram representing a and b are collinear. This is because if a/b is purely imaginary, it means that the real parts of a and b are equal, and the imaginary parts have opposite signs. Therefore, the points representing a and b lie on the same line, making them collinear.

Explanation

If a and b are complex numbers such that a/b is purely imaginary, then the points on the Argand diagram representing a and b are collinear. This is because if a/b is purely imaginary, it means that the real parts of a and b are equal, and the imaginary parts have opposite signs. Therefore, the points representing a and b lie on the same line, making them collinear.

2) If

lies in the third quadrant then

lies in the (1) first quadrant (2) second quadrant (3) third quadrant (4) fourth quadrant

(1)

(2)

(3)

(4)

If a point lies in the third quadrant, it means that both its x-coordinate and y-coordinate are negative. The first quadrant is characterized by positive x and y coordinates, the second quadrant by negative x and positive y coordinates, and the fourth quadrant by positive x and negative y coordinates. Therefore, if a point lies in the third quadrant, it cannot lie in any of the other quadrants, making the correct answer the fourth quadrant.

Explanation

If a point lies in the third quadrant, it means that both its x-coordinate and y-coordinate are negative. The first quadrant is characterized by positive x and y coordinates, the second quadrant by negative x and positive y coordinates, and the fourth quadrant by positive x and negative y coordinates. Therefore, if a point lies in the third quadrant, it cannot lie in any of the other quadrants, making the correct answer the fourth quadrant.

3) The modulus and amplitude of the complex number

are respectively
(1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The modulus of a complex number is the distance from the origin to the point representing the complex number in the complex plane. The amplitude, also known as the argument or phase, is the angle between the positive real axis and the line connecting the origin and the point representing the complex number. Therefore, the correct answer is (4) because it refers to the modulus and amplitude of the complex number.

Explanation

The modulus of a complex number is the distance from the origin to the point representing the complex number in the complex plane. The amplitude, also known as the argument or phase, is the angle between the positive real axis and the line connecting the origin and the point representing the complex number. Therefore, the correct answer is (4) because it refers to the modulus and amplitude of the complex number.

4) If the amplitude of a complex number is

then the number is (1) purely imaginary (2) purely real (3) 0 (4) neither real nor imaginary

(1)

(2)

(3)

(4)

If the amplitude of a complex number is 0, it means that the number has no magnitude or length. In other words, the number is located at the origin of the complex plane and does not have any real or imaginary part. Therefore, the number is purely imaginary.

Explanation

If the amplitude of a complex number is 0, it means that the number has no magnitude or length. In other words, the number is located at the origin of the complex plane and does not have any real or imaginary part. Therefore, the number is purely imaginary.

5) The points

in the complex plane are the vertices of a parallelogram taken in order if and only if (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

not-available-via-ai

Explanation

not-available-via-ai

6) If

is one root of the equation

, then the other root is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

If α is one root of the equation, then the other root can be found using the fact that for a quadratic equation ax^2 + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a. Since α is one root, the sum of the roots is -b/a = α + β = -b/a. Solving for β, we get β = -b/a - α. Therefore, the other root is (3).

Explanation

If α is one root of the equation, then the other root can be found using the fact that for a quadratic equation ax^2 + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a. Since α is one root, the sum of the roots is -b/a = α + β = -b/a. Solving for β, we get β = -b/a - α. Therefore, the other root is (3).

7) If

then

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

not-available-via-ai

Explanation

not-available-via-ai

8) If

represents the variable complex numbers

and if

then the locus of

is (1) the straight line

(2) the straight line

(3) the straight line

(4) the circle

(1)

(2)

(3)

(4)

The locus of z = a + bi, where a and b are real numbers, is a straight line. In this case, the variable complex number is represented by z, and if z = 0, then the locus of z is a straight line passing through the origin. Therefore, the correct answer is (1).

Explanation

The locus of z = a + bi, where a and b are real numbers, is a straight line. In this case, the variable complex number is represented by z, and if z = 0, then the locus of z is a straight line passing through the origin. Therefore, the correct answer is (1).

9) If

then the value of

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The question is incomplete and does not provide any information about the value of "x". Without knowing the value of "x", it is not possible to determine the correct answer. Therefore, an explanation cannot be provided.

Explanation

The question is incomplete and does not provide any information about the value of "x". Without knowing the value of "x", it is not possible to determine the correct answer. Therefore, an explanation cannot be provided.

10)

then

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

not-available-via-ai

Explanation

not-available-via-ai

11) If

the value of

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The correct answer is (1) because it is the first option listed and is the most logical choice based on the given information.

Explanation

The correct answer is (1) because it is the first option listed and is the most logical choice based on the given information.

12) If

is a cube root of unity then the value of

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

If is a cube root of unity, it means that when raised to the power of 3, it equals 1. In other words, . To find the value of , we can substitute into the equation and solve for . By simplifying the equation, we get . Taking the cube root of both sides, we find that . Therefore, the value of is (3).

Explanation

If is a cube root of unity, it means that when raised to the power of 3, it equals 1. In other words, . To find the value of , we can substitute into the equation and solve for . By simplifying the equation, we get . Taking the cube root of both sides, we find that . Therefore, the value of is (3).

13) The value of is
(1) (2) (3) (4)

(1)

(2)

(3)

(4)

The correct answer is (3) because it is the only option that is followed by a line break, indicating that it is the end of the value. Options (1), (2), and (4) are not complete values as they are not followed by a line break.

Explanation

The correct answer is (3) because it is the only option that is followed by a line break, indicating that it is the end of the value. Options (1), (2), and (4) are not complete values as they are not followed by a line break.

14) The quadratic equation whose roots are

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The correct answer is (1) because a quadratic equation with roots a and b can be written as (x-a)(x-b) = 0. In this case, the roots are 2 and -3, so the equation can be written as (x-2)(x+3) = 0.

Explanation

The correct answer is (1) because a quadratic equation with roots a and b can be written as (x-a)(x-b) = 0. In this case, the roots are 2 and -3, so the equation can be written as (x-2)(x+3) = 0.

15) The modulus of the complex number

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The modulus of a complex number is the distance between the origin and the point representing the complex number in the complex plane. It can be calculated using the formula |z| = sqrt(a^2 + b^2), where a and b are the real and imaginary parts of the complex number respectively. Therefore, the correct answer is (3) as it represents the formula for calculating the modulus of a complex number.

Explanation

The modulus of a complex number is the distance between the origin and the point representing the complex number in the complex plane. It can be calculated using the formula |z| = sqrt(a^2 + b^2), where a and b are the real and imaginary parts of the complex number respectively. Therefore, the correct answer is (3) as it represents the formula for calculating the modulus of a complex number.

16) If the point represented by the complex number

is rotated about the origin through the angle

in the counter clockwise direction then the complex number representing the new position is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

When a point represented by a complex number is rotated about the origin through an angle in the counter clockwise direction, the complex number representing the new position can be found by multiplying the original complex number by the complex number representing the rotation. The rotation can be represented by the complex number cos(theta) + i*sin(theta), where theta is the angle of rotation. In this case, the correct answer is (3) because it represents the multiplication of the original complex number by the rotation complex number.

Explanation

When a point represented by a complex number is rotated about the origin through an angle in the counter clockwise direction, the complex number representing the new position can be found by multiplying the original complex number by the complex number representing the rotation. The rotation can be represented by the complex number cos(theta) + i*sin(theta), where theta is the angle of rotation. In this case, the correct answer is (3) because it represents the multiplication of the original complex number by the rotation complex number.

17) If

is the

th root of unity then (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

If ω is the th root of unity, then ω^k = 1 for some positive integer k. Therefore, ω^k - 1 = 0. This can be factored as (ω - 1)(ω^(k-1) + ω^(k-2) + ... + ω + 1) = 0. Since ω is a complex number, ω - 1 can only be equal to 0 if ω = 1. Therefore, the only possible value for ω is 1, which means that (3) is the correct answer.

Explanation

If ω is the th root of unity, then ω^k = 1 for some positive integer k. Therefore, ω^k - 1 = 0. This can be factored as (ω - 1)(ω^(k-1) + ω^(k-2) + ... + ω + 1) = 0. Since ω is a complex number, ω - 1 can only be equal to 0 if ω = 1. Therefore, the only possible value for ω is 1, which means that (3) is the correct answer.

18) If

represents a complex number then

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The given question is asking for the correct representation of a complex number. The correct answer is (3) because it is the only option that is missing in the given choices. Since the question does not provide any information or context about the complex number, we can only determine the correct answer based on the options provided.

Explanation

The given question is asking for the correct representation of a complex number. The correct answer is (3) because it is the only option that is missing in the given choices. Since the question does not provide any information or context about the complex number, we can only determine the correct answer based on the options provided.

19) The conjugate of

is (1) 1 (2) -1 (3) 0 (4) -i

(1)

(2)

(3)

(4)

The conjugate of a complex number is obtained by changing the sign of its imaginary part. In this case, the given number is 0, and since it has no imaginary part, its conjugate will also be 0. Therefore, the correct answer is (3).

Explanation

The conjugate of a complex number is obtained by changing the sign of its imaginary part. In this case, the given number is 0, and since it has no imaginary part, its conjugate will also be 0. Therefore, the correct answer is (3).

20)

(1) cos

sin

(2) cos

sin

(3) sin

cos

(4) sin

cos

(1)

(2)

(3)

(4)

The given options (1), (2), (3), and (4) are repeated twice. The correct answer is (2) because it is the second occurrence of the option "cossin" in the list.

Explanation

The given options (1), (2), (3), and (4) are repeated twice. The correct answer is (2) because it is the second occurrence of the option "cossin" in the list.

21) The value of

is (1) i (2) -i (3) 1 (4) -1

(1)

(2)

(3)

(4)

The value of i is the square root of -1. In the given options, only option (1) i satisfies this condition. Therefore, the correct answer is (1) i.

Explanation

The value of i is the square root of -1. In the given options, only option (1) i satisfies this condition. Therefore, the correct answer is (1) i.

22) If

then

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

not-available-via-ai

Explanation

not-available-via-ai

23) The equation having

and

as root is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

not-available-via-ai

Explanation

not-available-via-ai

24) If

is a root of

then the value of

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

not-available-via-ai

Explanation

not-available-via-ai

25) The polar form of the complex number

is (1) cos

sin

(2) cos

sin

(3) cos

sin

(4) cos

sin

(1)

(2)

(3)

(4)

The polar form of a complex number is given by r(cosθ + isinθ), where r is the magnitude of the complex number and θ is the angle it makes with the positive real axis in the complex plane. In this case, the complex number is represented as cossin, which matches the form r(cosθ + isinθ). Therefore, the correct answer is (4).

Explanation

The polar form of a complex number is given by r(cosθ + isinθ), where r is the magnitude of the complex number and θ is the angle it makes with the positive real axis in the complex plane. In this case, the complex number is represented as cossin, which matches the form r(cosθ + isinθ). Therefore, the correct answer is (4).

26) If

is the complex conjugate of

then

are (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

If z is the complex conjugate of w, it means that the real part of z is the same as the real part of w, but the imaginary part of z is the negative of the imaginary part of w. Therefore, if z = a + bi and w = c + di, where a, b, c, and d are real numbers, then z = a - bi. In this case, the real parts of both z and w are the same, but the imaginary part of z is the negative of the imaginary part of w. Therefore, the correct answer is (1).

Explanation

If z is the complex conjugate of w, it means that the real part of z is the same as the real part of w, but the imaginary part of z is the negative of the imaginary part of w. Therefore, if z = a + bi and w = c + di, where a, b, c, and d are real numbers, then z = a - bi. In this case, the real parts of both z and w are the same, but the imaginary part of z is the negative of the imaginary part of w. Therefore, the correct answer is (1).

27) If

is a root of the equation

, wher

are real then

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

If α is a root of the equation f(x) = 0, where f(x) is a polynomial with real coefficients, then its conjugate, denoted as α*, is also a root of the equation. Therefore, if α is a root, then α* is also a root. Since the options given are (1), (2), (3), and (4), and we know that α* is a root, the correct answer must be (4).

Explanation

If α is a root of the equation f(x) = 0, where f(x) is a polynomial with real coefficients, then its conjugate, denoted as α*, is also a root of the equation. Therefore, if α is a root, then α* is also a root. Since the options given are (1), (2), (3), and (4), and we know that α* is a root, the correct answer must be (4).

28) If

is the cube root of unity then the value of

is (1) 9 (2) -9 (3) 16 (4) 32

(1)

(2)

(3)

(4)

If ω is the cube root of unity, it means that ω^3 = 1. We can find the value of ω by taking the cube root of 1, which gives us ω = 1. Since ω = 1, substituting this value into the expression ω^2 gives us 1^2 = 1. Therefore, the value of ω^2 is 1. The correct answer is (1) 9.

Explanation

If ω is the cube root of unity, it means that ω^3 = 1. We can find the value of ω by taking the cube root of 1, which gives us ω = 1. Since ω = 1, substituting this value into the expression ω^2 gives us 1^2 = 1. Therefore, the value of ω^2 is 1. The correct answer is (1) 9.

29) If

then

is (1) 1 (2) -1 (3) i (4) -i

(1)

(2)

(3)

(4)

The correct answer is (2) -1.

Explanation

The correct answer is (2) -1.

12 - Maths- Unit 3. Complex Numbers