Algebra

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Q1. In the expansion of (x + 3)^6, what is the coefficient of x^4?

Solution:

Using the binomial theorem, the coefficient of x^4 in (x + 3)^6 is given by 6C4 * (3)^2 = 15 * 9 = 135.

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Q2. What is the value of (−2)²? (2022)

Solution:

(−2)² = 4.

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Q3. If z1 = 2 + 2i and z2 = 1 - i, find z1 + z2.

Solution:

z1 + z2 = (2 + 2i) + (1 - i) = 3 + i.

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Q4. What is the value of (−7) + (−3)? (2019)

Solution:

−7 + (−3) = −10.

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Q5. If z1 = 2 + 2i and z2 = 2 - 2i, what is the value of z1 * z2? (2019)

Solution:

z1 * z2 = (2 + 2i)(2 - 2i) = 4 - 4i^2 = 4 + 4 = 8.

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Q6. What is the value of (1 + 2)^5 using the binomial theorem?

Solution:

Using the binomial theorem, (1 + 2)^5 = 3^5 = 243.

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Q7. What is the value of the discriminant for the quadratic equation 3x^2 + 12x + 9 = 0? (2019)

Solution:

The discriminant D = b^2 - 4ac = 12^2 - 4*3*9 = 0, indicating equal roots.

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Q8. In the expansion of (2x + 5)^3, what is the coefficient of x^2?

Solution:

The coefficient of x^2 in (2x + 5)^3 is given by 3C2 * (2x)^2 * 5^1 = 3 * 4 * 5 = 60.

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Q9. If x = 3 and y = 4, what is the value of x² + y²?

Solution:

x² + y² = 3² + 4² = 9 + 16 = 25.

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Q10. What is the value of the coefficient of x^4 in the expansion of (x + 5)^6?

Solution:

The coefficient of x^4 in (x + 5)^6 is given by 6C4 * 5^2 = 15 * 25 = 375.