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1. Integral of sin squared 2x cos raised to 4 2x dx
2. Interal of cot raised to 5 x dx
How many numbers with five decimal digits contain
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 .
show that absolute value of. a linear functional has properties of sublinear functional
P2+ 2 py cot x=y2
Find the number of equivalence relations that can be defined on a set of 6 elements.
a) Obtain the Ferrar graph of the partition 8+7+6+5+5+3+2+1. Find the conjugate
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.
A) Find the generating function of the recurrence
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]
a) Solve the recurrence
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.
a) Let A be an 8×8 Boolean matrix (i.e. every entry is 0 or 1). If the sum of the entries in A is
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?
a) Make a table of the values of the Boolean function
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
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