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1. Mention the principles of programmed instruction. (2)
2. Explain the concept of TPCK. (2)
3. 'Mathematics knowledge is a priori'. Comment. (2)
4. ‘Spiral approach is most recommended in organizing school Mathematics curriculum’. Why? (2)
5. What are the major developments in Mathematics in 20ᵗʰ century? (2)
6. Bringout the importance of Link Practice in the training of teaching skills. (2)
7. What are the process abilities in Mathematics that are expected to be developed among students by learning Mathematics ? (2)
8. Citing an example from Mathematics establish Bruner's statement 'The foundations of any subject may be taught to anybody at any age in some form'. (2)
9. Does there exist a linear transformation T: ℝ²⁰²¹ → ℝ²⁰²¹ such that the range and null space of T are identical? Justify your claim. (3)
10. Suppose that xₙ ≥ 0 and lim ((−1)ⁿxₙ) exists. Show that (xₙ) converges. (3)
11. If f is continuous on [0, 1] and ∫₀¹ f(x)xⁿ dx = 0 for n = 0, 1, 2, …, show that f(x) = 0 for every x ∈ [0, 1]. (3)
12. Let X be a topological space. Show that every subset of X is open if and only if each subset containing a single point is open. (3)
13. Show by an example that a non-zero prime ideal of a commutative ring with unity need not be maximal. (3)
14. Is 10 = (3 + i) × (3 − i) = 2 × 5 an example of non-unique factorization in ℤ[i]? Justify your claim. (3)
15. Suppose f(z) and g(z) are entire functions, g(z) is never zero and |f(z)| ≤ |g(z)| for all z. Show that there is a constant c such that f(z) = cg(z). (3)
16. Consider the real vector space V of polynomials of degree less than or equal to n. For P ∈ V define ‖P‖ₖ = max{|P(0)|, |P⁽¹⁾(0)|, …, |P⁽ᴷ⁾(0)|}, where P⁽ⁱ⁾ denotes the iᵗʰ derivative of P. Show that ‖·‖ₖ denotes a norm iff K ≥ n. (3)
17. Show that y′ = 4t³y/(1 + t⁴), 0 ≤ t ≤ 1, y(0) = 1 has a unique solution, and find it. (3)
18. Solve the congruence 25x ≡ 15 (mod 120). (3)
19. Explain the knowledge dimensions in Revised Bloom Taxonomy with suitable examples from Mathematics. (3)
20. Elucidate the need of diagnostic testing and remedial teaching in Mathematics. (3)
21. Differentiate between Holistic rubric and Analytic rubric. (3)
22. Let V be the space of all real-valued continuous functions on ℝ and define T : V → V by (Tf)(x) = ∫₀ˣ f(t) dt. Show that T has no eigenvalues. (5)
23. Let μ be Lebesgue measure, f ≥ 0, and ∫ₑ f dμ = 0. Show that f = 0 almost everywhere on E. (5)
24. Show that if X is a compact metric space, then X is separable. (5)
25. Prove or disprove. The symmetric group Sₙ has exactly 3 normal subgroups for n ≥ 5. (5)
26. Let f(z) be a non constant analytic function defined in a domain D = {z: |z| < r} and continuous on ∂D such that |f(z)| > m on ∂D. If |f(0)| < m, show that there exists at least one zero in |z| < r. (5)
27. Suppose X and Y are F-spaces, Λ: X → Y is linear and G = {(x, Λ(x)): x ∈ X} is closed in X × Y. Then show that Λ is continuous. (5)
28. Solve the PDE (y² + z²)∂z/∂x − xy∂z/∂y + xz = 0. (5)
29. If a prime p does not divide a then prove that aᵖ⁻¹ ≡ 1 (mod p). (5)
30. Explain briefly the cognitive development model. (5)
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