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2. Interal of cot raised to 5 x dx

a) the digit ?

b) the digits and , where comes right after and all five digits are distinct?

c) the digits and , where comes somewhere after (not necessarily consecutive) and all five digits are distinct?

Note: A number cannot start with the digit .

partition. Is the partition self conjugate?

b) For the graph in Fig. Fig. 2 on the following page find a minimal colouring.

c) Is Petersen graph bipartite? Give reasons for your answer.

an = 4an−1 −4an−2 +1

with initial conditions a0 = 1, a1 = 1.

B) Express 3x4 +2x3 −2x2 +x in terms of [x]4, [x]3, [x]2 and [x]

6an+3 −13an+2 +9an+1 −2an = 2

−n

,n ≥ 0,if a0 = 1, a1 = 1, a2 = 1

b) How many numbers from 0 to 999(0 and 999 inclusive) are not divisible by 5 or 13?

c) If an apple costs | 15, a papaya costs | 17 and a banana costs | 8, write the generating function

for the number of ways of purchasing these fruits with | n.

51, prove that there is a row i and a column j in A such that the entries in row i and in column j

add up to more than 13. Further, show that there are at least 4 such pairs of rows and

columns.

b) If a planar graph has the degree sequence {2,2,3,3,4,4,4}, how many faces will it have? Draw

a planar graph with this degree sequence and number the faces to check your answer.

c) Give the order and the degree of the recurrence

a

2

n+2 = a

2

n +2an +4

Is the recurrence homogeneous?

f(x1,x2,x3) = x2 ⊕(x1 ∧x3)

Write the function in DNF using the table.

b) Find the general form of the solution to a linear homogeneous recurrence with constant

coefficients for which the characteristic roots are 1 with multiplicity 1, −2 with multiplicity 2

and 2 with multiplicity 3. The relation also has a non-homogeneous part which is a linear

combination of n2n

and (−2)

n