MCQ Mathematics Chapter 4: COMPLEX NUMBERS AND QUADRATIC EQUATIONS, HS 1st year

 

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MCQ 1:

Which of the following numbers is a complex number?
A) $3 + 4i$
B) $-5$
C) $7$
D) $\sqrt{5}$

Answer: A) $3 + 4i$

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MCQ 2:

What is the real part of the complex number $z = 2 - 3i$?
A) $2$
B) $-3$
C) $0$
D) $5$

Answer: A) $2$

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MCQ 3:

If $z_1 = 3 + 4i$ and $z_2 = 1 - 2i$, what is $z_1 + z_2$?
A) $4 + 2i$
B) $2 + 6i$
C) $3 + 2i$
D) $4 + 6i$

Answer: A) $4 + 2i$

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MCQ 4:

What is the multiplicative identity in the system of complex numbers?
A) $0 + 0i$
B) $1 + 0i$
C) $i$
D) $1 - i$

Answer: B) $1 + 0i$

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MCQ 5:

Which of the following is the modulus of the complex number $z = 3 + 4i$?
A) $5$
B) $7$
C) $25$
D) $3$

Answer: A) $5$
Explanation: The modulus is calculated as $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$.

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MCQ 6:

The conjugate of the complex number $z = 5 - 2i$ is:
A) $5 + 2i$
B) $5 - 2i$
C) $-5 + 2i$
D) $-5 - 2i$

Answer: A) $5 + 2i$

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MCQ 7:

What is the square of $i$ in the system of complex numbers?
A) $-1$
B) $1$
C) $i$
D) $0$

Answer: A) $-1$

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MCQ 8:

If $z_1 = 2 + i$ and $z_2 = 1 - i$, what is $z_1 \cdot z_2$?
A) $3 - i$
B) $2 + 2i$
C) $1 - 2i$
D) $3 + i$

Answer: D) $3 + i$
Explanation: $z_1 \cdot z_2 = (2 + i)(1 - i) = 2 - 2i + i - i^2 = 3 + i$.

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MCQ 9:

The equation $x^2 + 1 = 0$ has solutions in the form of which numbers?
A) Real numbers
B) Rational numbers
C) Complex numbers
D) Natural numbers

Answer: C) Complex numbers

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MCQ 10:

If $z = 4 + 3i$, then $\bar{z} \cdot z = ?$
A) $25$
B) $13$
C) $7$
D) $16$

Answer: A) $25$
Explanation: $\bar{z} \cdot z = (4 - 3i)(4 + 3i) = 4^2 + 3^2 = 16 + 9 = 25$.

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MCQ 11:

What is the imaginary part of the complex number $z = -5 + 7i$?
A) $-5$
B) $7$
C) $0$
D) $-7$

Answer: B) $7$

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MCQ 12:

The addition of two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ results in:
A) $(a + b) + i(c + d)$
B) $(a + c) + i(b + d)$
C) $(a - c) + i(b - d)$
D) $(a + d) + i(b + c)$

Answer: B) $(a + c) + i(b + d)$

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MCQ 13:

If $z = 3 + 4i$, what is its modulus squared, $|z|^2$?
A) $7$
B) $5$
C) $25$
D) $15$

Answer: C) $25$
Explanation: $|z|^2 = 3^2 + 4^2 = 9 + 16 = 25.$

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MCQ 14:

Which of the following properties is true for multiplication of complex numbers?
A) Closure law
B) Commutative law
C) Associative law
D) All of the above

Answer: D) All of the above

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MCQ 15:

The complex conjugate of $z = -3 - 4i$ is:
A) $-3 + 4i$
B) $-3 - 4i$
C) $3 + 4i$
D) $3 - 4i$

Answer: A) $-3 + 4i$

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MCQ 16:

If $i^4 = 1$, then $i^{5}$ is equal to:
A) $i$
B) $-1$
C) $-i$
D) $1$

Answer: A) $i$

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MCQ 17:

In the Argand plane, the x-axis corresponds to:
A) Imaginary axis
B) Real axis
C) Positive axis
D) Complex axis

Answer: B) Real axis

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MCQ 18:

For a complex number $z = 5 - 2i$, what is its modulus?
A) $\sqrt{23}$
B) $23$
C) $7$
D) $5$

Answer: A) $\sqrt{23}$
Explanation: $|z| = \sqrt{5^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{23}.$

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MCQ 19:

The equation $z^2 + 4 = 0$ has solutions:
A) $\pm 2i$
B) $\pm 2$
C) $2i, -2$
D) No solution

Answer: A) $\pm 2i$

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MCQ 20:

The product of $z_1 = 2 + 3i$ and $z_2 = 1 - i$ is:
A) $5 + i$
B) $5 - i$
C) $2 - 3i$
D) $5$

Answer: B) $5 - i$
Explanation: $z_1 z_2 = (2 + 3i)(1 - i) = 2 - 2i + 3i - 3i^2 = 5 - i.$

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MCQ 21:

For any integer $k$, $i^{4k + 3}$ is equal to:
A) $i$
B) $-1$
C) $-i$
D) $1$

Answer: C) $-i$

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MCQ 22:

The multiplicative inverse of a non-zero complex number $z = a + bi$ is:
A) $\frac{1}{a + bi}$
B) $\frac{a}{a^2 + b^2} + \frac{-b}{a^2 + b^2}i$
C) $a + bi$
D) $a - bi$

Answer: B) $\frac{a}{a^2 + b^2} + \frac{-b}{a^2 + b^2}i$

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MCQ 23:

If $z_1 = 2 + i$ and $z_2 = 3 - 2i$, the division $\frac{z_1}{z_2}$ results in:
A) $1 + i$
B) $\frac{4 + i}{13}$
C) $-1 + i$
D) $3$

Answer: B) $\frac{4 + i}{13}$

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MCQ 24:

The conjugate of $z_1 z_2$ is equal to:
A) $\bar{z_1} + \bar{z_2}$
B) $\bar{z_1} \bar{z_2}$
C) $z_1 + z_2$
D) None of the above

Answer: B) $\bar{z_1} \bar{z_2}$

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MCQ 25:

If $z = x + yi$, then $|z|^2 = ?$
A) $x^2 + y^2$
B) $x^2 - y^2$
C) $|x| + |y|$
D) $(x + y)^2$

Answer: A) $x^2 + y^2$

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MCQ 26:

If $z = 4 - 3i$, then its conjugate $\bar{z}$ is:
A) $4 + 3i$
B) $4 - 3i$
C) $-4 + 3i$
D) $-4 - 3i$

Answer: A) $4 + 3i$

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MCQ 27:

The solution of the equation $x^2 + 9 = 0$ is:
A) $x = \pm 3$
B) $x = \pm i3$
C) $x = \pm i9$
D) No solution

Answer: B) $x = \pm i3$

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MCQ 28:

What is $(1 + i)^2$?
A) $1 + 2i$
B) $2i$
C) $2i - 1$
D) $1 - 2i$

Answer: C) $2i - 1$
Explanation: $(1 + i)^2 = (1 + i)(1 + i) = 1 + 2i + i^2 = 1 + 2i - 1 = 2i.$

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MCQ 29:

What is the polar representation of $z = 1 + i$?
A) $\sqrt{2} e^{i\pi/4}$
B) $\sqrt{2} e^{-i\pi/4}$
C) $\sqrt{2} e^{i\pi/2}$
D) $2e^{i\pi/4}$

Answer: A) $\sqrt{2} e^{i\pi/4}$

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MCQ 30:

Which of the following is not true for complex numbers?
A) $z + \bar{z} = 2\text{Re}(z)$
B) $z \cdot \bar{z} = |z|^2$
C) $\bar{z_1 + z_2} = \bar{z_1} + \bar{z_2}$
D) $|z_1 + z_2| = |z_1| + |z_2|$

Answer: D) $|z_1 + z_2| = |z_1| + |z_2|$
Explanation: The modulus of a sum is not always the sum of the moduli.

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MCQ 31:

If $z = a + bi$, then $(\bar{z})^2$ is:
A) $a^2 + b^2$
B) $a^2 - b^2 - 2abi$
C) $a^2 - b^2 + 2abi$
D) $(a - bi)^2$

Answer: D) $(a - bi)^2$

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MCQ 32:

The cube of $i$, $i^3$, is equal to:
A) $i$
B) $-1$
C) $-i$
D) $1$

Answer: C) $-i$
Explanation: $i^3 = i^2 \cdot i = -1 \cdot i = -i.$

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MCQ 33:

What is the multiplicative identity for complex numbers?
A) $0$
B) $1 + i$
C) $1$
D) $i$

Answer: C) $1$

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MCQ 34:

Which equation represents a circle in the Argand plane?
A) $|z - 1| = 2$
B) $|z| = 2$
C) $|z + 2| = 0$
D) $|z| = -2$

Answer: A) $|z - 1| = 2$

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MCQ 35:

If $z = 3 + 4i$, then its modulus is:
A) $5$
B) $7$
C) $3$
D) $4$

Answer: A) $5$

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MCQ 36:

What is the result of dividing $z_1 = 2 + 3i$ by $z_2 = 1 - i$?
A) $\frac{1 + 5i}{2}$
B) $\frac{7 - i}{2}$
C) $\frac{5 + i}{2}$
D) $\frac{1 - i}{5}$

Answer: C) $\frac{5 + i}{2}$

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MCQ 37:

Which of the following statements about $z = x + iy$ is true?
A) $z \cdot \bar{z} = x^2 + y^2$
B) $z + \bar{z} = 2y$
C) $\bar{z} = -x - yi$
D) $z \cdot \bar{z} = 2x^2 + 2y^2$

Answer: A) $z \cdot \bar{z} = x^2 + y^2$

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MCQ 38:

What is the square of the complex number $z = 1 + i$?
A) $2 + i$
B) $2 - i$
C) $1 + 2i$
D) $0$

Answer: A) $2 + i$

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MCQ 39:

For $z = x + yi$, the conjugate $\bar{z}$ lies where in the Argand plane?
A) Above the real axis
B) Below the real axis
C) On the imaginary axis
D) Same as $z$

Answer: B) Below the real axis

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MCQ 40:

If $z = 3 + 4i$, then $|z|^2 = ?$:
A) $5$
B) $25$
C) $7$
D) $0$

Answer: B) $25$

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MCQ 41:

In the polar form, the argument of $z = 1 - i$ is:
A) $\pi/4$
B) $-\pi/4$
C) $\pi/2$
D) $-\pi/2$

Answer: B) $-\pi/4$

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MCQ 42:

Which of the following represents the standard form of a complex number?
A) $a + b$
B) $a + b\sqrt{-1}$
C) $a + bi$
D) $ab + i$

Answer: C) $a + bi$

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MCQ 43:

For $z = 2 + 3i$, the argument is:
A) $\tan^{-1}(3/2)$
B) $\sin^{-1}(3/2)$
C) $\cos^{-1}(2/3)$
D) $\tan^{-1}(2/3)$

Answer: A) $\tan^{-1}(3/2)$

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MCQ 44:

If $z = x + yi$, the modulus of $z$ is the distance from the origin to:
A) $x + iy$
B) $x - iy$
C) $z + 1$
D) $z - 1$

Answer: A) $x + iy$

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MCQ 45:

The power $i^{10}$ equals:
A) $1$
B) $-1$
C) $i$
D) $-i$

Answer: B) $-1$

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MCQ 46:

Which of the following is true for the conjugate of $z = a + bi$?
A) $\bar{z} = a + bi$
B) $\bar{z} = -a - bi$
C) $\bar{z} = a - bi$
D) $\bar{z} = -a + bi$

Answer: C) $\bar{z} = a - bi$

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MCQ 47:

For any complex number $z$, $|\bar{z}|$ is equal to:
A) $-|z|$
B) $|z|$
C) $z$
D) $-z$

Answer: B) $|z|$

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MCQ 48:

The square of $z = i$ is:
A) $-1$
B) $i$
C) $1$
D) $-i$

Answer: A) $-1$

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MCQ 49:

Which of the following equations represents a straight line in the Argand plane?
A) $|z| = 5$
B) $|z - 3| = 2$
C) $|z - i| + |z + i| = 6$
D) $\text{Re}(z) = 3$

Answer: D) $\text{Re}(z) = 3$

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MCQ 50:

What is $(2 + 3i)(2 - 3i)$?
A) $-13$
B) $13$
C) $12 - 9i$
D) $4 + 9i$

Answer: B) $13$
Explanation: $(2 + 3i)(2 - 3i) = 4 - (3i)^2 = 4 + 9 = 13.$

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MCQ 51:

The magnitude of $z = 0 + 4i$ is:
A) $0$
B) $4$
C) $2$
D) $16$

Answer: B) $4$

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MCQ 52:

What is the solution of $x^2 + 16 = 0$?
A) $\pm 4$
B) $\pm i4$
C) $\pm 16i$
D) No solution

Answer: B) $\pm i4$

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MCQ 53:

The argument of $z = -1 + i$ is:
A) $3\pi/4$
B) $\pi/4$
C) $-\pi/4$
D) $5\pi/4$

Answer: A) $3\pi/4$

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MCQ 54:

For $z = 2 + 2i$, the modulus is:
A) $\sqrt{8}$
B) $4$
C) $2$
D) $\sqrt{2}$

Answer: A) $\sqrt{8}$

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MCQ 55:

The equation $z \cdot \bar{z} = |z|^2$ represents:
A) Conjugate property
B) Modulus property
C) Multiplication property
D) Square root property

Answer: B) Modulus property

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MCQ 56:

If $z = 1 - i$, then $z^2$ is:
A) $2 - 2i$
B) $0$
C) $-2i$
D) $2i$

Answer: A) $2 - 2i$

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MCQ 57:

The polar form of $z = -1$ is:
A) $e^{i\pi}$
B) $e^{-i\pi}$
C) $e^{i\pi/2}$
D) $e^{-i\pi/2}$

Answer: A) $e^{i\pi}$

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MCQ 58:

What is $i^{7}$?
A) $-i$
B) $1$
C) $i$
D) $-1$

Answer: A) $-i$
Explanation: $i^7 = i^{4 \cdot 1 + 3} = i^3 = -i.$

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MCQ 59:

For $z_1 = 3 + 2i$ and $z_2 = 1 + 4i$, the sum $z_1 + z_2$ is:
A) $4 + 6i$
B) $2 - 2i$
C) $4 - 6i$
D) $2 + 2i$

Answer: A) $4 + 6i$

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MCQ 60:

If $z_1 = 2 + i$ and $z_2 = 1 - i$, what is the product $z_1 \cdot z_2$?
A) $3 + i$
B) $3 - i$
C) $1 + i$
D) $1 - i$

Answer: B) $3 - i$

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MCQ 61:

For $z = a + bi$, $|z|^2 = z \cdot \bar{z}$ evaluates to:
A) $a^2 + b^2$
B) $a^2 - b^2$
C) $|a + b|$
D) $a^2 + 2ab + b^2$

Answer: A) $a^2 + b^2$

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MCQ 62:

The argument of $z = i$ is:
A) $\pi/4$
B) $\pi/2$
C) $-\pi/2$
D) $0$

Answer: B) $\pi/2$

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MCQ 63:

The power $i^{100}$ is:
A) $1$
B) $-1$
C) $i$
D) $-i$

Answer: A) $1$

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MCQ 64:

Which equation represents a circle of radius 3 centered at the origin in the Argand plane?
A) $|z| = 3$
B) $|z - 3| = 0$
C) $|z + 3| = 3$
D) $|z| = 0$

Answer: A) $|z| = 3$

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MCQ 65:

If $z = 2 + 5i$, then $\bar{z} = ?$:
A) $2 + 5i$
B) $2 - 5i$
C) $-2 + 5i$
D) $-2 - 5i$

Answer: B) $2 - 5i$

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MCQ 66:

The product of $z = 1 + i$ with its conjugate is:
A) $1$
B) $2$
C) $0$
D) $-1$

Answer: B) $2$
Explanation: $z \cdot \bar{z} = (1 + i)(1 - i) = 1^2 + 1^2 = 2.$

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MCQ 67:

If $z = x + yi$, then $\bar{z} \cdot z = ?$:
A) $x^2 + y^2$
B) $x^2 - y^2$
C) $x^2 + y^2i$
D) $x^2 - y^2i$

Answer: A) $x^2 + y^2$