BSc Third Year (Sem 5 & 6) Mathematics – Daily Revision MCQ (English Medium)
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What is the dimension of the real vector space of all 2 × 2 real matrices?
Chapter 1: Vector Spaces
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Answer: B. 4
The four matrices with a single 1 in one position and 0 elsewhere form a basis, so the dimension is 2 × 2 = 4.A linear transformation T: ℝ⁵ → ℝ³ has rank 3. What is the nullity of T?
Chapter 2: Linear Transformations
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Answer: D. 2
Rank–nullity theorem: rank + nullity = dim of domain, so nullity = 5 − 3 = 2.If f(z) = u + iv is analytic, which pair of Cauchy–Riemann equations must u and v satisfy?
Chapter 4: Analytic Functions
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Answer: A. u_x = v_y and u_y = −v_x
For an analytic function the Cauchy–Riemann equations are ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x.What is the value of ∮ dz/z taken once anticlockwise around the circle |z| = 1?
Chapter 6: Complex Integration
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Answer: C. 2πi
With z = e^{iθ}, dz/z = i dθ, and θ runs from 0 to 2π, so the integral is 2πi (the pole z = 0 lies inside).Which of the following sets is uncountable?
Chapter 8: Countable Sets and Topology of the Real Line
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Answer: B. The set of real numbers in (0, 1)
ℕ, ℤ and ℚ can be listed one by one (countable), but Cantor's diagonal argument shows the reals in (0, 1) cannot.In a discrete metric space (d(x, y) = 1 if x ≠ y, and 0 if x = y), the open ball B(x, 1/2) is:
Chapter 9: Metric Spaces
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Answer: A. {x}
Only points at distance less than 1/2 from x lie in the ball, and the only such point is x itself (distance 0).In a Boolean algebra, the expression a + a·b simplifies to:
Chapter 13: Lattices and Boolean Algebra
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Answer: D. a
Absorption law: a + a·b = a(1 + b) = a·1 = a.A graph has 7 edges. What is the sum of the degrees of all its vertices?
Chapter 14: Graph Theory
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Answer: C. 14
Handshaking lemma: each edge adds 1 to the degree of each of its two ends, so the sum of degrees = 2 × 7 = 14.For a binomial distribution with n = 10 and p = 0.4, what is the variance?
Chapter 19: Probability Distributions
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Answer: A. 2.4
Variance = npq = 10 × 0.4 × 0.6 = 2.4 (4 is the mean np).If the regression coefficients are b_yx = 0.8 and b_xy = 0.45, what is the correlation coefficient r?
Chapter 20: Correlation and Regression
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Answer: D. 0.6
r = √(b_yx × b_xy) = √0.36 = 0.6, positive because both regression coefficients are positive.
BSc Third Year (Sem 5 & 6): MCQ in other subjects
All questions are written by PadhaiNotes in our own words from the common syllabus of Maharashtra universities (NEP). Your own university's syllabus may differ a little.