Mathematics II (Math.Ed.426) is a critical subject for the BICTE 2nd Semester, covering Calculus, Complex Numbers, and Abstract Algebra. Below are the solved Multiple Choice Questions (MCQs) from the 2082 Examination (New Course).
Use these solutions to verify your answers and understand the step-by-step problem-solving methods.
Group “A”: Attempt All Questions
1. If a = 3 + 2i and b = 3i – 2, what is the value of a + b?
- a. 6i
- b. 1 + 5i
- c. 1 – i
- d. 5 – i
Correct Answer: b. 1 + 5i
Explanation: We add the real parts together and the imaginary parts together. a + b = (3 + 2i) + (-2 + 3i) = (3 – 2) + (2i + 3i) = 1 + 5i
2. If a = 1 and b = 1, what is the value of (a – ib)²?
- a. 2
- b. -2
- c. 2i
- d. -2i
Correct Answer: d. -2i
Explanation: Substitute a=1 and b=1 into the expression: (1 – 1i)² = (1 – i)² Expand the square: 1² – 2(1)(i) + i² Since i² = -1, we get: 1 – 2i – 1 = -2i
3. What is the value of lim (x→∞) (√(x + 2) – √x)?
- a. √2
- b. ∞ – ∞
- c. 0
- d. ∞
Correct Answer: c. 0
Explanation: This is an indeterminate form (∞ – ∞). We rationalize the expression by multiplying and dividing by the conjugate (√(x+2) + √x). Numerator becomes: (x + 2) – x = 2. Denominator becomes: √(x+2) + √x. As x approaches infinity, 2 / ∞ approaches 0.
4. If dy/dx = x³ + 5, which one of the following is the correct value of y?
- a. 3x²
- b. 4x⁴ + 5x
- c. (x⁴ / 4) + 5x
- d. x⁴ + 5x + 6
Correct Answer: c. (x⁴ / 4) + 5x
Explanation: To find y, we integrate the given derivative with respect to x. ∫ (x³ + 5) dx = (x³⁺¹ / 4) + 5x = x⁴/4 + 5x + C. Option (c) matches this form.
5. What is the value of (3 + 4) × 3 (mod 5)?
- a. 1
- b. 2
- c. 3
- d. 4
Correct Answer: a. 1
Explanation: First, calculate the value: (3 + 4) × 3 = 7 × 3 = 21. Now, find 21 mod 5. 21 divided by 5 is 4 with a remainder of 1. So, 21 ≡ 1 (mod 5).
6. What is the x-intercept of the line y = 3x?
- a. 0
- b. 1
- c. 3
- d. 1/3
Correct Answer: a. 0
Explanation: To find the x-intercept, we set y = 0. 0 = 3x x = 0. The line passes through the origin.
7. Which of the following is the equation of directrix of the parabola y² = 8x?
- a. x = -4
- b. x = 4
- c. x = -2
- d. x = 2
Correct Answer: c. x = -2
Explanation: Compare y² = 8x with standard form y² = 4ax. 4a = 8, so a = 2. The equation of the directrix for a right-opening parabola is x = -a. Therefore, x = -2.
8. Given graph G(V,E) with V(G) = {a, b, c} and E(G) = {{a,b}, {a,b}, {a,c}, {a,a}}. What is the degree of the vertex a?
- a. 4
- b. 5
- c. 6
- d. 7
Correct Answer: b. 5
Explanation: The degree is the number of edge ends connected to a vertex.
- {a,b}: 1
- {a,b}: 1 (parallel edge)
- {a,c}: 1
- {a,a}: 2 (a loop contributes 2 to the degree) Total = 1 + 1 + 1 + 2 = 5.
9. Which one of the following is a group under multiplication?
- a. {0}
- b. Z (Integers)
- c. Q (Rational Numbers)
- d. R (Real Numbers)
Correct Answer: c. Q
Explanation: A group requires every element to have an inverse. In integers (Z), numbers like 2 do not have a multiplicative inverse (1/2 is not in Z). Rational numbers (Q), specifically the set of non-zero rationals, form a group under multiplication. In the context of multiple-choice questions comparing Z, Q, and R, Q is the standard answer for the “smallest” field/group set listed after Z.
10. Which one of the following is a field?
- a. Z
- b. 2Z
- c. 3Z
- d. Q
Correct Answer: d. Q
Explanation: A field must be a commutative ring with unity where every non-zero element has a multiplicative inverse.
- Z is an Integral Domain (not a field).
- 2Z and 3Z do not have a multiplicative identity (1).
- Q (Rational numbers) satisfies all conditions to be a field.
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