Engineering Mathematics

If f(x) = x^3 – 3x^2 – 4x + 12, what is f'(2)?

a) 16
b) 10
c) 4
d) 0

Answer: c) 4

Explanation: We can use the power rule to find the derivative of f(x): f'(x) = 3x^2 – 6x – 4. Substituting x = 2, we get f'(2) = 3(2)^2 – 6(2) – 4 = 4.

Which of the following is NOT a complex number?

a) 3 + i
b) -2 – 4i
c) 2/3
d) i/2

Answer: c) 2/3

Explanation: A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit. Option c) is not of this form and is therefore not a complex number.

What is the inverse Laplace transform of F(s) = (s+1)/(s^2 + 4s + 5)?

a) e^(-t) sin(t)
b) e^(-t) cos(t)
c) e^(-t/2) sin(t/2)
d) e^(-t/2) cos(t/2)

Answer: d) e^(-t/2) cos(t/2)

Explanation: We can use partial fraction decomposition and the inverse Laplace transform tables to find the solution. After partial fraction decomposition, we get F(s) = (s+1)/(s^2 + 4s + 5) = (s+1)/[(s+2)^2 + 1]. Using the inverse Laplace transform table, we find that the inverse Laplace transform of (s+1)/[(s+2)^2 + 1] is e^(-at) cos(bt), where a = -2 and b = 1. Therefore, the answer is d) e^(-t/2) cos(t/2).

What is the Fourier series of the periodic function f(t) = t^2 on the interval [-π, π]?

a) π^2/3 + 4π^2/π^2 cos(t) – 2π/π^2 cos(2t) + 4π^2/9π^2 cos(3t) – 2π/π^2 cos(4t) + …
b) π^2/3 + 4π^2/π^2 cos(t) – 2π/π^2 cos(2t) + 4π^2/9π^2 cos(3t) – 2π/9π^2 cos(4t) + …
c) π^2/3 + 4π^2/π^2 sin(t) – 2π/π^2 sin(2t) + 4π^2/9π^2 sin(3t) – 2π/π^2 sin(4t) + …
d) π^2/3 + 4π^2/π^2 sin(t) – 2π/π^2 sin(2t) + 4π^2/9π^2 sin(3t) – 2π/9π^2 sin(4t) + …

Answer: b) π^2/3 + 4π^2/π^2 cos

What is the value of lim x->0 (sin x / x)?

a) 1
b) 0
c) -1
d) Does not exist

Answer: a) 1

Explanation: We can use L’Hopital’s rule to find the limit. Taking the derivative of the numerator and denominator with respect to x, we get lim x->0 (cos x / 1) = 1.

What is the Laplace transform of f(t) = e^(-2t)sin(3t)?

a) (s+2)/(s^2 + 5s + 13)
b) (s+2)/(s^2 – 5s + 13)
c) (s-2)/(s^2 + 5s + 13)
d) (s-2)/(s^2 – 5s + 13)

Answer: d) (s-2)/(s^2 – 5s + 13)

Explanation: We can use the Laplace transform properties and tables to find the solution. Using the time shift property and the Laplace transform of sin(t), we get L{e^(-2t)sin(3t)} = (s+2)/[(s+2)^2 + 9]. Using partial fraction decomposition, we can simplify this to (s-2)/(s^2 – 5s + 13).

What is the inverse Fourier transform of F(w) = δ(w – 3)?

a) e^(3it)
b) e^(-3it)
c) sin(3t)/πt
d) cos(3t)/πt

Answer: b) e^(-3it)

Explanation: The inverse Fourier transform of a delta function is a complex exponential with frequency equal to the argument of the delta function. Therefore, the answer is e^(-3it).

What is the derivative of ln(cos(x))?

a) -tan(x)
b) -cot(x)
c) sec(x)
d) -csc(x)

Answer: a) -tan(x)

Explanation: Using the chain rule, we can find the derivative of ln(cos(x)) as -tan(x) / cos(x) = -tan(x).

Which of the following is a solution to the differential equation y” + 4y = 0?

a) y = e^(2x)
b) y = e^(-2x)
c) y = sin(2x)
d) y = cos(2x)

Answer: c) y = sin(2x) and d) y = cos(2x)

Explanation: The characteristic equation of the differential equation is r^2 + 4 = 0, which has roots r = ±2i. Therefore, the general solution is y = c1sin(2x) + c2cos(2x), where c1 and c2 are constants. Therefore, both c) and d) are solutions.

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