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Total Number of Questions : 30

Time : 2.00 Hours Max. Marks : 100
2. Mention the principles of programmed instruction. (2 Marks)
2. Explain the concept of TPCK. (2 Marks)
3. ‘Mathematics knowledge is a priori’. Comment. (2 Marks)
4. ‘Spiral approach is most recommended in organizing schoo! Mathematics curriculum’. Why ? (2 Marks)
5. What are the major developments in Mathematics in 20" century ? (2 Marks)
6. Bringout the importance of Link Practice in the training of teaching skills. (2 Marks)
7. What are the process abilities in Mathematics that are expected to be developed ‘among students

by learning Mathematics ? (2 Marks)
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 Marks)

9. Does there exist a linear transformation T ۰ R? ‏ب‎ മോ such that the range and null space of T are
identical ? Justify your claim. (3 Marks)
10, Suppose that x, 2 0 and lim (വറ exists. Show that (x,) converges. (3 Marks)

1

11. If fis continuous on [0, 1] and if | f(x)x"dx=0 forn=0, 1, 2,..., then show that f(x) = 0 for every
xe [0, 1]. 7 (3 Marks)

12. Let X bea topological space. Show that every subset of X is open if and only if each subset containing
a single point is open. (3 Marks)

13. Show by an example that a non-zero prime ideal of a commutative ring with unity need not be
maximal. (3 Marks)
34. 15105 (3 +i) x (3 +i) = 2 ×5 an example of non-unique factorization in 21] ? Justify your claim. (3 Marks)

15. Suppose f(z) and g(z) are entire functions, g(z) is never zero and |f(z)|< |g(z)| for all 7. Show that
there is a constant c such that f(z) = ‏.لتاق‎ (3 Marks)

16. Consider the real vector space V of polynomials of degree less than or equal to ൨. For Pe V define

IP Ii ,=max{|P(0)|, |P*0)|,..., |, P*(0) |}, P denotes the i derivative of P. Show that || ||, denotes
anorm iff K =n. (3 Marks)

3,

17. Show that the initial-value problem y’= = , 0 < {< 1, #(0) = 1 has a unique solution, and find the
solution. 1+1 ⋅ (3 Marks)
18. Solve the congruence 25x = 15 (mod 120). (3 Marks)