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Gime me ncrt class 11 most important questions from the ch complex numbers and quadratic equation.

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Answered by GPT-4o mini
Sure! Here are some important questions from the chapter "Complex Numbers and Quadratic Equations" that are relevant for Class 11 students following the NCERT curriculum:

### Complex Numbers
1. **Definition**: What are complex numbers? Define a complex number in the standard form \( z = a + bi \).
2. **Imaginary Unit**: What is the value of \( i \) and how is it defined?
3. **Addition and Subtraction**: If \( z_1 = 3 + 4i \) and \( z_2 = 1 - 2i \), calculate \( z_1 + z_2 \) and \( z_1 - z_2 \).
4. **Multiplication**: Multiply \( z_1 = 2 + 3i \) and \( z_2 = 4 + i \). Express the result in standard form.
5. **Modulus**: Find the modulus of the complex number \( z = 3 - 4i \).
6. **Conjugate**: What is the conjugate of a complex number? Find the conjugate of \( z = 5 + 6i \).
7. **Division**: Divide \( z_1 = 4 + 3i \) by \( z_2 = 1 - i \) and express in standard form.
8. **Polar Form**: Convert the complex number \( z = 1 + \sqrt{3}i \) into polar form.
9. **De Moivre's Theorem**: State De Moivre’s theorem and use it to find \( (2(\cos 30^\circ + i\sin30^\circ))^3 \).
10. **Roots of Unity**: Find the cube roots of unity.

### Quadratic Equations
1. **Standard Form**: Write the general form of a quadratic equation. Give examples.
2. **Roots Formula**: State and derive the quadratic formula for the roots of \( ax^2 + bx + c = 0 \).
3. **Nature of Roots**: Using the discriminant \( D \), explain how to determine whether the roots are real and distinct, real and equal, or complex.
4. **Example Calculation**: Find the roots of the quadratic equation \( x^2 - 5x + 6 = 0 \).
5. **Sum and Product**: For the quadratic equation \( ax^2 + bx + c = 0 \), express the sum and product of the roots in terms of \( a, b, \) and \( c \).
6. **Graphical Representation**: Sketch the graph of a quadratic equation and indicate the roots.
7. **Applications**: Solve a word problem that involves formulating and solving a quadratic equation.
8. **Factoring**: Factor the quadratic \( x^2 - 7x + 10 \) and find its roots.
9. **Vertex Form**: Convert the quadratic equation \( y = 2x^2 - 8x + 6 \) into vertex form.
10. **Real-life Applications**: Provide a real-life scenario where quadratic equations can be applied (e.g., projectile motion).

These questions cover a range of fundamental concepts in complex numbers and quadratic equations that are significant for exam preparation. Make sure to practice solving these problems, as well as understanding the underlying theories.
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