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5. (a) Define a compact set. Check whether the
set of integers is compact or not. 2
(b) If a + b + c = — 4 and 4a + 2b + c = 6, then
show that both the roots of the quadratic
equation axe + bx + c = 0 are real.
(c) Using Taylor's Theorem, prove that
, x2 x4
+ V XE R.
2! 4!
6. (a) State the second mean value theorem of
integrability. Verify it for the functions
f and g defined on [1, 2] by fix) = 3x and
g(x) = 5x.
(b) Test the following series for absolute and
conditional convergence : 5
00
(i) E E1)n
3n+1
n=1
E00 sin nx
n3
n=1
4
4
5
MT E-09 4
7. (a) Prove that there is no rational number
whose square is 6. 3
(b) Check whether the following functions are
continuous or not at x = 0. Also, find the
nature of discontinuity at that point, if it
exists. 4
- 2 x
(i) f(x) =
1
1
X
, x 0
, x=0
X
2 + -1 x<_0 ,
(ii) f(x) =
3
- (X3 -F ) , x>0 3
(c) Examine the function f given by
ME) = (x - 8)3 (x + 3), x E R
for extreme values. .
set of integers is compact or not. 2
(b) If a + b + c = — 4 and 4a + 2b + c = 6, then
show that both the roots of the quadratic
equation axe + bx + c = 0 are real.
(c) Using Taylor's Theorem, prove that
, x2 x4
+ V XE R.
2! 4!
6. (a) State the second mean value theorem of
integrability. Verify it for the functions
f and g defined on [1, 2] by fix) = 3x and
g(x) = 5x.
(b) Test the following series for absolute and
conditional convergence : 5
00
(i) E E1)n
3n+1
n=1
E00 sin nx
n3
n=1
4
4
5
MT E-09 4
7. (a) Prove that there is no rational number
whose square is 6. 3
(b) Check whether the following functions are
continuous or not at x = 0. Also, find the
nature of discontinuity at that point, if it
exists. 4
- 2 x
(i) f(x) =
1
1
X
, x 0
, x=0
X
2 + -1 x<_0 ,
(ii) f(x) =
3
- (X3 -F ) , x>0 3
(c) Examine the function f given by
ME) = (x - 8)3 (x + 3), x E R
for extreme values. .
co 7n
(3n + 1) !
n=1
00
1
n=1
(i)
3. (a) Show that the sequence (fn) where
fn(x) = x e [1, .[ is uniformly
1 +2ne
convergent in [1, .4.
(b) Check whether the following sequences (s n)
are Cauchy, where
(i) sn =1+2+3+...+n
4n3 + 3n (ii)
sn 3n + n2
(c) Check whether the function f(xx)) = cos —1 is
x
uniformly continuous on the interval JO, 1[.
Is it continuous on the same interval ?
Justify.
4. (a) Show that the union of two open sets is an
open set.
(b) Verify Inverse Function Theorem for finding
the derivative at a point yo of the domain of
the inverse function of the function
f(x) = cos x, x E [0, 7C] . Hence, find the
derivative at yo.
(c) Test for convergence the following series : 4
3
4
3
3
(3n + 1) !
n=1
00
1
n=1
(i)
3. (a) Show that the sequence (fn) where
fn(x) = x e [1, .[ is uniformly
1 +2ne
convergent in [1, .4.
(b) Check whether the following sequences (s n)
are Cauchy, where
(i) sn =1+2+3+...+n
4n3 + 3n (ii)
sn 3n + n2
(c) Check whether the function f(xx)) = cos —1 is
x
uniformly continuous on the interval JO, 1[.
Is it continuous on the same interval ?
Justify.
4. (a) Show that the union of two open sets is an
open set.
(b) Verify Inverse Function Theorem for finding
the derivative at a point yo of the domain of
the inverse function of the function
f(x) = cos x, x E [0, 7C] . Hence, find the
derivative at yo.
(c) Test for convergence the following series : 4
3
4
3
3
2.
(c)
(d)
(e)
(a)
The greatest integer
R is derivable in
lim cot x does
function
the interval
not exist.
(7c
of Mathematical
is convergent.
[-1
2 4
[x] defined on
Induction,
It It
-> - X --
2 2
The series
nZ
Using the Principle
show that
1 + 3 + 5 + 7 + + (2n - 1) = n2, V n E N.
(b) Show that the function f : [0, 1] -4 R
defined by
11 when x is rational
f(x) =
is not Riemann integrable. 4
(c) Show that the function
f(x) = I x - 5 I + x2 + 3x + 10
is continuous but is not differentiable at the
point x = 5
(c)
(d)
(e)
(a)
The greatest integer
R is derivable in
lim cot x does
function
the interval
not exist.
(7c
of Mathematical
is convergent.
[-1
2 4
[x] defined on
Induction,
It It
-> - X --
2 2
The series
nZ
Using the Principle
show that
1 + 3 + 5 + 7 + + (2n - 1) = n2, V n E N.
(b) Show that the function f : [0, 1] -4 R
defined by
11 when x is rational
f(x) =
is not Riemann integrable. 4
(c) Show that the function
f(x) = I x - 5 I + x2 + 3x + 10
is continuous but is not differentiable at the
point x = 5
(a) Find the sum of the series
VT).
+ + + ,
./ v (n + 3)3 11(n + 6)3 V[n 3 (n — 1)]3
3
(b) By showing that the remainder after n-terms
tends to zero, find Maclaurin's series
expansion of sin 2x. 5
(c) Prove that the function f defined on [0, 1] by
f(x) = (— 1)n+1for 1
< x < ,
n + 1
n = 1, 2, 3, ... is integrable on [0, 1].
VT).
+ + + ,
./ v (n + 3)3 11(n + 6)3 V[n 3 (n — 1)]3
3
(b) By showing that the remainder after n-terms
tends to zero, find Maclaurin's series
expansion of sin 2x. 5
(c) Prove that the function f defined on [0, 1] by
f(x) = (— 1)n+1for 1
< x < ,
n + 1
n = 1, 2, 3, ... is integrable on [0, 1].
1. Which of the following statements are true or
false ? Give reasons for your answer. 10
(a) The function f(x) = 3x2+ 5 I x I is
differentiable at x = — 2.
(b) The singleton set {x} for any x E B is an open
set.
(c) Every bounded sequence is convergent.
(d) Every integrable function is monotonic.
(e) The function fix) = cos x is uniformly
continuous on [0, —E 2l
false ? Give reasons for your answer. 10
(a) The function f(x) = 3x2+ 5 I x I is
differentiable at x = — 2.
(b) The singleton set {x} for any x E B is an open
set.
(c) Every bounded sequence is convergent.
(d) Every integrable function is monotonic.
(e) The function fix) = cos x is uniformly
continuous on [0, —E 2l
Let $a_n\geq \frac{2^n-1}{n}$ and let $\nu\leq \lfloor{\log_2 n}\rfloor$. Show that $a_n-\nu> 0$.
2. Car insurance companies assume that the longer a person has been driving, the less likely they will be in an accident, and therefore charge new drivers higher insurance premiums than experienced drivers. To determine whether driving experience is related to the number of car accidents, you survey a random sample of 12 Torontonians and ask them about the number of years they have been driving, and the number of car accidents they have been involved in during the past year. The data are presented below:
Driver # of years driving (X) # of car accidents (Y)
A 4.5 3
B 2.5 5
C 1.5 3
D 3 3
E 1.5 6
F 5 2
G 5 0
H 2 4
I 3 1
J 4 2
K 1 5
L 3 2
a. Determine whether the assumptions of car insurance companies are valid. Assuming α=0.05, include the hypotheses for a one-tailed test, critical test statistic, conclusion, and all formulas and calculations.
b. Is it appropriate to conclude that lack of driving experience causes accidents? Why or why not?
Driver # of years driving (X) # of car accidents (Y)
A 4.5 3
B 2.5 5
C 1.5 3
D 3 3
E 1.5 6
F 5 2
G 5 0
H 2 4
I 3 1
J 4 2
K 1 5
L 3 2
a. Determine whether the assumptions of car insurance companies are valid. Assuming α=0.05, include the hypotheses for a one-tailed test, critical test statistic, conclusion, and all formulas and calculations.
b. Is it appropriate to conclude that lack of driving experience causes accidents? Why or why not?
Check the continuity of f(x) = {x+1, x is less than 1}
{0, 2 is greater than 1 and x}
{2-x , x is greater than 2}
{0, 2 is greater than 1 and x}
{2-x , x is greater than 2}
A survey was recently done in a certain town to determine readership of newspapers available. 50% of the resident read Daily Nation, 60% read the standard and 20% read both newspapers. Determine the probability that a resident selected does not read any newspaper.
A marketing manager wants to assign four regions to four different salesmen. The salesmen differ in their efficiency and territories also differ in potentiality. An estimated sale (in Ks, 000s) by different sales men in the four territories are given below.
Territory
Determine an optimal assignment schedule for maximum sales. What are the maximum sales ( 9marks)
Territory
Determine an optimal assignment schedule for maximum sales. What are the maximum sales ( 9marks)
