Limit, Continuity and Differentiability for Competitive Exams

Limit, Continuity and Differentiability

Q. If f(x) = 3x^2 - 12x + 9, what is f(2)?

Solution

Substitute x = 2 into the function: f(2) = 3(2)^2 - 12(2) + 9 = 12 - 24 + 9 = -3.

Correct Answer: A — 0

Q. What is the solution to the equation 5(x - 1) = 3(x + 2)?

Solution

Distributing gives 5x - 5 = 3x + 6. Rearranging gives 2x = 11, so x = 5.5.

Correct Answer: C — x = 3

Q. What is the value of k if the equation 2x^2 + kx + 3 = 0 has one solution?

Solution

For one solution, the discriminant must be zero: k^2 - 4(2)(3) = 0. Thus, k^2 = 24, so k = ±√24 = ±2√6. The integer value is -4.

Correct Answer: B — -4

Q. What is the value of k if the equation x^2 + kx + 16 = 0 has one real solution?

Solution

For one real solution, the discriminant must be zero: k^2 - 4(1)(16) = 0. Thus, k^2 = 64, so k = ±8.

Correct Answer: B — k = -8

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