Algebra

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Q1. What is the value of (2 + 3i) + (4 - 2i)?

Solution:

To add the complex numbers, we combine the real parts and the imaginary parts: (2 + 4) + (3 - 2)i = 6 + 1i = 6 + i.

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Q2. From a group of 8 people, how many ways can a team of 3 be selected? (2022)

Solution:

The number of ways to choose 3 people from 8 is given by 8C3 = 56.

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Q3. What is the value of (−2)²? (2022)

Solution:

(−2)² = 4.

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Q4. What is the value of k if the equation x^2 - kx + 9 = 0 has roots 3 and 3?

Solution:

The sum of the roots is 3 + 3 = 6, so k = 6.

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Q5. If the roots of the equation x^2 + 3x + k = 0 are real and distinct, what is the condition on k? (2022)

Solution:

The discriminant must be positive: 3^2 - 4*1*k > 0 leads to 9 - 4k > 0, thus k < 9.

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Q6. For the polynomial x^3 - 3x^2 + 3x - 1, which of the following is true about its roots?

Solution:

The polynomial can be factored as (x - 1)^3, indicating that all roots are real and equal.

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Q7. For the quadratic equation x^2 + 6x + 9 = 0, what type of roots does it have? (2019)

Solution:

The discriminant D = 6^2 - 4*1*9 = 0, indicating real and equal roots.

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Q8. What is the real part of the complex number z = 4 - 3i? (2023)

Solution:

The real part of z is 4.

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Q9. In the expansion of (a + b)^6, what is the coefficient of a^2b^4?

Solution:

The coefficient of a^2b^4 in (a + b)^6 is given by 6C2 = 15.

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Q10. The roots of the quadratic equation x^2 - 4x + k = 0 are 2 and 2. What is the value of k? (2022)

Solution:

If the roots are both 2, then k = 2^2 - 4*2 = 4 - 8 = -4. Thus, k = 4.