20 Engineering Mathematics Quiz Questions and Answers

1. Question: What is the limit of \(\lim_{x \to 2} \frac{x^2 – 4}{x – 2}\)?
Options:
A) 0
B) 2
C) 4
D) Undefined
Answer: C) 4
Explanation: Simplifying the expression gives \(\frac{(x-2)(x+2)}{x-2} = x+2\) for \(x \neq 2\), so substituting x=2 yields 4.

2. Question: If \(f(x) = x^3 – 3x^2 + 2x\), what is the derivative \(f'(x)\)?
Options:
A) \(3x^2 – 6x + 2\)
B) \(3x^2 – 3x + 2\)
C) \(x^3 – 6x + 2\)
D) \(3x^2 – 6x\)
Answer: A) \(3x^2 – 6x + 2\)
Explanation: Applying the power rule, the derivative is \(3x^2 – 2 \cdot 3x + 2 = 3x^2 – 6x + 2\).

3. Question: Evaluate the definite integral \(\int_0^1 x^2 \, dx\).
Options:
A) 1/3
B) 1/2
C) 1
D) 2/3
Answer: A) 1/3
Explanation: The antiderivative of \(x^2\) is \(\frac{x^3}{3}\), so evaluating from 0 to 1 gives \(\frac{1^3}{3} – \frac{0^3}{3} = \frac{1}{3}\).

4. Question: Solve the differential equation \(\frac{dy}{dx} = 2y\) with initial condition y(0) = 1.
Options:
A) y = e^{2x}
B) y = 2e^x
C) y = e^x
D) y = 2x
Answer: A) y = e^{2x}
Explanation: This is a separable equation; separating and integrating gives ln|y| = 2x + C, so y = Ce^{2x}. With y(0)=1, C=1, hence y = e^{2x}.

5. Question: For the differential equation \(y” + y = 0\), what is the general solution?
Options:
A) y = A sin(x) + B cos(x)
B) y = Ae^x + Be^{-x}
C) y = A + Bx
D) y = e^x (A + Bx)
Answer: A) y = A sin(x) + B cos(x)
Explanation: The characteristic equation is r^2 + 1 = 0, so r = ±i, leading to the general solution y = A sin(x) + B cos(x).

6. Question: If A = \(\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\), what is the determinant of A?
Options:
A) 5
B) -2
C) 4
D) 0
Answer: B) -2
Explanation: The determinant is (1)(4) – (2)(3) = 4 – 6 = -2.

7. Question: What is the dot product of vectors u = (1, 2, 3) and v = (4, 5, 6)?
Options:
A) 32
B) 20
C) 4
D) 14
Answer: A) 32
Explanation: The dot product is 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.

8. Question: Simplify (2 + 3i) + (4 – i).
Options:
A) 6 + 2i
B) 6 + 4i
C) 2 + 2i
D) 8 + 2i
Answer: A) 6 + 2i
Explanation: Adding real parts: 2 + 4 = 6; adding imaginary parts: 3i – i = 2i, so the result is 6 + 2i.

9. Question: What is the probability of rolling a 6 on a fair six-sided die?
Options:
A) 1/6
B) 1/2
C) 1/3
D) 1
Answer: A) 1/6
Explanation: There is one favorable outcome (rolling a 6) out of six possible outcomes, so the probability is 1/6.

10. Question: For a dataset: 1, 2, 3, 4, 5, what is the variance?
Options:
A) 2
B) 2.5
C) 1
D) 4
Answer: A) 2
Explanation: The mean is 3. The deviations squared are (1-3)^2 + (2-3)^2 + … + (5-3)^2 = 4 + 1 + 0 + 1 + 4 = 10. Variance = 10 / 5 = 2 (for population variance).

11. Question: The Fourier series of a periodic function with period 2π is given by ∑ [a_n cos(nx) + b_n sin(nx)]. What is a_0 for f(x) = x on [-π, π]?
Options:
A) 0
B) π
C) 2π
D) 1
Answer: A) 0
Explanation: a_0 = (1/π) ∫_{-π}^π x dx = (1/π) [x^2/2]_{-π}^π = (1/π) [(π^2/2) – (π^2/2)] = 0.

12. Question: The Laplace transform of e^{-at} is:
Options:
A) 1/(s + a) for s > -a
B) 1/(s – a) for s > a
C) s/(s + a)
D) 1/s
Answer: A) 1/(s + a) for s > -a
Explanation: The standard Laplace transform of e^{-at} is ∫ e^{-st} e^{-at} dt from 0 to ∞, which simplifies to 1/(s + a) for s > -a.

13. Question: Using the bisection method, what is the next midpoint for f(x) = x^2 – 4 = 0 between 1 and 3?
Options:
A) 2
B) 1.5
C) 2.5
D) 3
Answer: A) 2
Explanation: The midpoint of 1 and 3 is (1+3)/2 = 2.

14. Question: For f(x, y) = x^2 + y^2, what is the gradient at (1, 1)?
Options:
A) (2, 2)
B) (1, 1)
C) (2x, 2y)
D) (0, 0)
Answer: A) (2, 2)
Explanation: The gradient is (∂f/∂x, ∂f/∂y) = (2x, 2y), so at (1,1), it is (2*1, 2*1) = (2,2).

15. Question: The partial differential equation \(\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}\) is known as:
Options:
A) Heat equation
B) Wave equation
C) Laplace equation
D) Poisson equation
Answer: A) Heat equation
Explanation: This is the standard form of the one-dimensional heat equation.

16. Question: Does the series ∑ 1/n^2 converge?
Options:
A) Yes
B) No
C) Only for n even
D) Only for n odd
Answer: A) Yes
Explanation: This is a p-series with p=2 > 1, so it converges by the p-series test.

17. Question: In engineering, the identity sin(2θ) = 2 sin(θ) cos(θ) is used for:
Options:
A) Double-angle formula
B) Sum of angles
C) Product of sines
D) Pythagorean identity
Answer: A) Double-angle formula
Explanation: This is the double-angle formula for sine.

18. Question: For Boolean algebra, what is the complement of A?
Options:
A) NOT A
B) A AND 1
C) A OR 0
D) A XOR 1
Answer: A) NOT A
Explanation: The complement of A is the logical negation, denoted as NOT A.

19. Question: To minimize f(x) = x^2 + 2x + 1, what is the critical point?
Options:
A) x = -1
B) x = 1
C) x = 0
D) x = 2
Answer: A) x = -1
Explanation: Set f'(x) = 2x + 2 = 0, so x = -1.

20. Question: Solve for x in the matrix equation AX = B, where A = \(\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\) and B = \(\begin{pmatrix} 5 \\ 11 \end{pmatrix}\).
Options:
A) X = \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\)
B) X = \(\begin{pmatrix} 3 \\ 1 \end{pmatrix}\)
C) X = \(\begin{pmatrix} 2 \\ 1 \end{pmatrix}\)
D) X = \(\begin{pmatrix} 1 \\ 1 \end{pmatrix}\)
Answer: A) X = \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\)
Explanation: Using inverse or Gaussian elimination, solve [1x + 2y = 5; 3x + 4y = 11], yielding x=1, y=2.