Complex numbers and quadratic equations form the cornerstone of many mathematical concepts that students will encounter later in their academic journey. These topics not only provide a rich exploration into the abstract world of numbers but also lay the groundwork for understanding higher-level mathematical constructs. To test and bolster your knowledge, we’ve compiled 20 insightful MCQs on complex numbers and quadratic equations, complete with answers and explanations.

## 1. The square root of -1 is denoted by:

### Answer:

### Explanation:

By definition, the square root of -1 is represented by the imaginary unit 'i'.

## 2. The complex number 3 + 4i is an example of:

### Answer:

### Explanation:

Complex numbers are in the form a + bi, where 'a' is the real part and 'bi' is the imaginary part.

## 3. If z = 3 + 4i, the magnitude or modulus of z is:

### Answer:

### Explanation:

The modulus of z = √(3^2 + 4^2) = 5.

## 4. The conjugate of the complex number 5 – 3i is:

### Answer:

### Explanation:

The conjugate of a complex number changes the sign of its imaginary part.

## 5. The solutions to the quadratic equation x^2 + 4 = 0 are:

### Answer:

### Explanation:

The solutions are imaginary, given by ±√(-4) = ±2i.

## 6. The quadratic equation with roots 3 + 2i and 3 – 2i is:

### Answer:

### Explanation:

Using Vieta's formulas, the sum of the roots is 6 and the product is 13.

## 7. The value of i^45 is:

### Answer:

### Explanation:

Since i^2 = -1, i^45 can be simplified as i^44 * i = (i^2)^22 * i = (-1)^22 * i = -i.

## 8. The discriminant of the quadratic equation 3x^2 – 4x + 2 = 0 is:

### Answer:

### Explanation:

The discriminant is given by b^2 – 4ac = 16 – 24 = -8.

## 9. A quadratic equation with real coefficients can have:

### Answer:

### Explanation:

Depending on the discriminant, a quadratic equation can have either real roots (if discriminant is non-negative) or imaginary roots (if discriminant is negative).

## 10. The complex number which when added to its conjugate gives a real number is:

### Answer:

### Explanation:

When a complex number is added to its conjugate, the imaginary parts cancel out, resulting in a real number.

## 11. If z1 and z2 are complex numbers, then the product z1*z2 is:

### Answer:

### Explanation:

If z1 and z2 are purely imaginary, their product results in a real number. For example, if z1 = ai and z2 = bi, then z1*z2 = -ab, which is a real number.

## 12. For the quadratic equation ax^2 + bx + c = 0, if b^2 < 4ac, then:

### Answer:

### Explanation:

The nature of the roots depends on the discriminant (b^2 – 4ac). If it's negative, the equation has imaginary roots.

## 13. The argument of the complex number -1 – i is:

### Answer:

### Explanation:

The argument of a complex number is the angle it makes with the positive real axis. In this case, it's in the second quadrant.

## 14. The quadratic equation x^2 + x + 1 = 0 has:

### Answer:

### Explanation:

The discriminant of the equation is 1 – 4(1)(1) = -3, which is negative, hence the roots are imaginary.

## 15. If i^x = 1, then x can be:

### Answer:

### Explanation:

Any number raised to the power of 0 is 1. Therefore, i^0 = 1.

## 16. The sum of a complex number z and its conjugate is always:

### Answer:

### Explanation:

The imaginary parts of z and its conjugate will cancel each other out, leaving only the real component.

## 17. The quadratic equation whose one root is the conjugate of the other is:

### Answer:

### Explanation:

The sum and product of the roots are both real for such equations, making the coefficients real.

## 18. If z is a complex number, the square of the modulus of z is:

### Answer:

### Explanation:

|z|^2 = z * conjugate(z).

## 19. If the roots of the quadratic equation ax^2 + bx + c = 0 are imaginary, then:

### Answer:

### Explanation:

It's the discriminant (b^2 – 4ac) that determines the nature of the roots, not the individual coefficients.

## 20. The multiplicative inverse of the complex number a + bi is:

### Answer:

### Explanation:

The multiplicative inverse is found by dividing the conjugate of the number by the square of its modulus.