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Series and quadratic equations
Mathematics class 11 (041)
University of Kerala
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54
Geometric Progression (GP)
A
sequence
of
non-zero
numbers
is
called a geome
progression,
if the
ratio
of
a
term
and
the term
precedin
it is always
constant.
The
constant
ratio,
general
denoted
by
r is
called
the
common
ratio
of the
GP.
In other words,
if a1,a2,4g,..
., ay are
in
GP, then
2= " = 7 = r(say) where r is known as common
(i)
If each term of a
given
AP is
multiplied
or divided
by
a non-zero constant
k,
then the
resulting sequence
is
d
also an AP with common difference kd
or
whered
is the common differenceof the
given
AP.
(1i) A sequence is an AP if its mh term is of the form
An+B, i. a linear expressionin.
an- (iv) In a finite AP the sum of the terms equidistant from
the
beginning
and end is
always
same.
ratio of GP.
11
3-2,...
is
a
GP with first
a t an a2 +an-
tan-2.
eg. The sequence
n Arithmetic Means
between Two Numbers
term and common ratio-
If
a, A, A2, Ag-..,A,,
b are in
AP,
then we
say
that
A1, A2, A3s.,A, are the n arithmetic means (AM)
between two
numbers
a and b. The common difference
(d)
of this AP
is
and mth arithmetic mean is
given
General Term of a GP
The nth term of a GP with first term a and common
ratio r is given by 7, = ar *i or l = ar"7, where l is
the last term.
2-a
n+
by A a+ n-
n+
GP can be written as a, ar,
ar"
,
ar
a, ar,
ar, ar',
ar*,..., ar
accordinglythey are finite or infinite.
By putting m=1, 2, 3,..., 7, we can get the values of
A1, A2., An
The sum of n arithmetic means between two
given
numbers is n times the
single
AM between them i.
A +A2+As t... +A, =n (single AM between a and b)
If there is
only
one
arithmetic mean A' between a and b,
or
ar 7-1|
nth Term from the End of a
Finite GP
The nth term from the end of a
finite GP
consistingo m terms is
ar" ", where a is the first term and r is the
common ratio of the
GP.
then A 4+b
2
EXAMPLE 5. If n arithmetic means are
inserted
between 20 and 80 such that the ratio of first mean to
the last mean is 1:
3,
then find the value
of n.
EXAMPLE 6. If
x,
y,
z are the
pth, qth
and rth
terms
of a
GP, then the
value of
x°-
yf-P zP-
is
equal
to
a. 0
b. 1
a. 12 b. 13 11 d. 14
C.-
d. None of these
C.
Sol. c. Let
A, A2,
A3,., A,
ben arithmetic means between
20 and
80,
and let d be the
common difference of the
AP; 20,
A, Azy., An80.
Sol. b. Let A be the
first
term
and R be the
common
rato
of the GP. We
have,
Tp
=
ARP- =
x
TAR =y
andT, =
AR'
=z
Now, *
=
[ARP*'g-
=A9-
RP- 1) (q-r)
Then,
d80-20 60
using, d
n+1 n+
yP =
[AR--P =A
-P
Rla-1) (r- p)
n+
60
A 20 + d A = 20+
n+
zP
=
[AR'
=P- = AP-
Rr
-1)
(p-g)
9-
y-P zP-
=
[A9-
+r
-p+
p-
Now,
- 20+
An20 + nd An 20 + 20
n+
[RP-1) (q-r)+
(q-1)
(r - p)
- (r
-1) (P-]
A
RO =
and
20(n+ 4)
Selection of
Terms in
GP
n+
Number of terms
Terms
20(4n + 1)
Common ratio
3
, a, ar
n+
n+
1
A
3 ar, ars
4n+1 3
5
4n+ 1 3n + 12 n=
a, ar, ar
MATHEMATICS Sequencesand
Series 55
,
EXAMPLE 7. Find the three numbers in GP, whose
sum is 19 and product is 216.
Sol. d. 767 67 673 7773*.
a. 4,6,
c. 6,5, 8
b. 13, 4, 2
d. None of these
Sol. a. Let three numbers in GP be , a and ar.
Properties of Geometric Progression
r
According to the given condition,
Product of three numbers
(i) If all the terms of a GP be multiplied or divided by
the same non-zero constant, then it remains a GP
with same common ratio.
a ar
216 a
= 216 =
(i)
The reciprocals
of
the terms of
a
given
GP
forms a
GP.
a = 6
(ii)
If each
term of a GP be
raised to
the same
power,
then the
resulting
sequence
also forms
a GP.
(iv)
In a
finite GP the
product
of the
terms
equidistant
from the
beginning
and
the end is
always
same
and
is equal
to
the
product
of
the first and
the last
term.
and sum of three numbers = a+ ar = 19
119
Iput a =6]
6+6r + 6r = 19r
6r2 +6r-19r +6 = 00
6r-13r +6 = 0
(v)
Three
non-zero numbersa, bh
care in GP, iff b
'=ac.
(vi) If the terms of a given GP are chosen at regular
intervals,
then the
new
sequence,
so formed
also
forms a GP.
13-13-4-6-6 -btvb-4ac
(VI1)f 4,
42,
a3.,
ays.
is a GP
of
non-zero,
non-negative terms, then
log
41, lo
42,
log 4..
is an AP and
vice-versa.
2a
13t5 18 832
22 12
n
Geometric
Means
between
Two Numbers
When r, then the numbers are756,6x
3/
i. 4, 6, 9.
If 4, G, G, G3,., G, b are in GP, then
G, G2,
G3,.., G,
are
callled n
geometric
means
between
two
numbers a and
b. The
common ratio
(r)
of
When r , then the numbers are 2/33/
3
i. 9, 6, 4.
1/(n+1)
Hence,
the
numbers
are 4,
6,
9
or 9, 6,
this GP is
and mth geometric mean is given by
Sum of n
Terms
of
a GP
The sun of n
terms
of a
GP
with
first
terms
'a'
and
G
Common ratio r' is given by
S
|-4-
for
|r|<
S, 1-7 1-
By
putting
m=1, 2, 3,..., 7,
we
can
get
the values of
G, Gz,G
The
product
of
n
geometric
means is
equal
to
the nth
power
of
single
geometric
mean
between
the
two
quantities,i:ie. G, G G G4.., = (Wab".
and S,a for |r|>
number of terms
is
infinite,
then
sum
of the
terms is
If there is only one geometric mean 'G° betweena and
b, then
r<1.
G Vab
EXAMPLE 8. What
is
the value
of
76/7/7 7 .. upto?
Note If A and G are the AM and GM between two positive
numbers, then the numbers are A t VA-G
If A
and G
are
respectively
AM and GM
between two positive
numbers
a
and
b,
then
the
quadratic
equation
having
a,
b as
its roots is *-2Ax +G = 0.
a. log
b.
6
C
d. 7
1
QUADRATIC EQUATIONS
AND INEQUALITIES
In
NDA exam,
generally
4- questions
are asked from
this chapter
which
are based
on
nature
of
roots,
finding
roots in different
conditions,
find
equation
when
roots are given
and solving
in
equations etc.
When
we
equate
quadratic
polynomial
of the
form (ax*
bx + c) equal
to
zero
we
get
a quadratic
equation,
where a,
b and
c
are real
numbers
and
a
#0.
POLYNOMIAL
AND
POLYNOMIAL
EQUATION
An expression
of the
form a0*" ta1x
t...+ a-
X+
4,,
where
ao,
aj,
a2.
are
constant
(a
#0)
and
n is
a positive
integer
is
called a
polynomial
in
x of degree
n
If
f(x)
is a
real
or complex
polynomial, then
f(x)=
is
known
as a polynomial
equation.
e.
If x
3x
is
a
real polynomial,
then
x +
3x+2=
is
a polynomial
equation.
QUADRATIC EQUATION
If
fx)
is
a
polynomial of
degree
2,
then
f(x)=
is
called
a quadratic
equation.
The general
form of a
quadratic
equation
is
ax*
bx
+c=0,
where ,
b
and c are
real
numbers
and a # 0.
Here,
x is the
variable
and a, b,
c
are
the
real
coefficients.
Roots
of
a
Quadratic
Equation
The
values
of
the
variable satistying
the
given
quadratic
equation
are called
roots of
that equation.
In
other
words,
x
= o is
a
root
of the equation,
f(x)=0,
if
f(a)
=0.
The set
of
all
roots
of
an equation,
in a
given
domain,
is
called the
solution
set of
the equation.
The
quadratic
equation
axs
bx
+c=0,
where a,
b,
cER
and a
+0 has two roots,
namely
and B=b-VD
2a
.- b+VD
La
where, D
=b
-4ac
is
called the
discriminant.
Note
If one
of
the
root of
the quadratic equation
is a
ib or a
vb,
then the
other root
will be a
ib or a
Vb.
NDA/NA Pathfinder
####### EXAMPLE
If
x
+x
- is a
factor
of
####### polynomial
+Px
+Q,
then
the
values
of P
and
Q are
####### respectively
EXAMPLE 2. If the
equation
a. 8
and
17
b. 7
and
c.
and 4
d.
and 14
(
0)+(27 x3P
- x+4 = 0
has
equal roots, then p
is
equal
to
Sol.
b.
Let
P(x)
= x
+Pax+Q
a. 0
b. 2
d.
None of
these
Since,
- is a
factor
of
Pla).
.Roots of
**
+x-
=
satisfy the
equation Plx)
= 0.
C,
2
Sol. c. The
given equation
will have
equal roots iff
Now, discriminant = 0
(
x 3P
15-
4
x9x4 =
:
D
=b-4ad
+X -6=
x+3x-
2x
- = 0
(27 x 3MP-15 -144 =
(x+3) (x
= 0
(27 x 3MP 152 =
27x 3"P
t
x =- 3, 2
27 x 3P 27
27 x 3MP
=
or
P(-3) 00
-27 +9P +Q=
)
P (2)= 0
3P=
or 3P=
=3?
=
or-
and
8+ 4P+Q=
.ii)
On
solving Eqs.
(i) and
(i), we
get
P = 7
and
Q=
36
But,
cannot be
zero.
So,
p=-
Nature of
the
Roots of a
Quadratic Equation
Relation
between
Roots and
Coefficients
Let the
quadratic
equation
be
ax* + bx + c
=
0, a, b,
ce R
and a # 0. The
nature of the
roots of a
quadratic
equation
is
decided
by discriminant
(i.
D=b-4ac)
Quadratic
equation Consider the
quadratic
equation
ax+
bx+c=0, where
a, h,
ce
Rand a #
(i) Ifb2-
4ac>0,
then the
quadratic
equation has two
If a and
B
are the
roots of
the
equation,
then
real and distinct roots.
Sum of
roots, a
+B=2=*Coefficient of x
Coefficient of x
(ii)
If
b
4ac
=
0,
then the
quadratic equation
has two
equal roots i. a = B=. a
and
product
of
roots, aß =C=_Constant
term
(ii) If b - 4ac< 0, then the quadratic equation has two
distinct complex roots, namely
Coefficient of
Cubic
equation
If
a,ß,y
are the
roots of the cubic
equation
ax +
bx* +
cx+d
=
0,a +0,
then
Sum of
roots,
a+B+ys
-bla
Sum of
product
of
two
roots, aß +By + ya
=
cl
anu
Product of
three
roots,
aßy
=
-d / a
and B=b-itac-b
2a
- b+i4ac - b
O=-
2a
(iv)
If a, b,
ce Qand
Dis a
perfect
square,
then
equation
has rational roots.
(v)
The
roots are
of
the form
p+/g
(p,.qe Q)
iff a,
b,
c
are
rational
and
D is not
a
pertect
square.
EXAMPLE
- If
the
roots of
the
equation
4B
+-2=
are
of the
form-and
then
k+ k+
SOME IMPORTANT POINTS
what is the
value of A
(i)
If
the
roots
of
ar
bx +
c=0 are both positives,
then
the signs of
a
and c
should
be a like
and opposite
to
the sign
as b.
(ii)
If
the
roots of ar
hr
+c=
are
of
opposite
signs,
then
the sign
of
a is opposite
to
the
sign
of c.
(iii)
If
the
roots
of
a
he +
c
=0 are equal
in
magnitude,
but opposite
in sign,
then b=0.
a. 2k
b. 7
C. 2
d. k+
Sol. b.
Letandbe
the
roots
of the equation
4 +AB-2 = 0, then ...)
kk+
k+
(iv)
If
the
roots
of
ax
bx
c=
are
reciprocal
of each
other, then c = a.
and
2
(v
It
roots
are
negative,
then ,
b,
care
of
same
sign.
that
the
roots
of
the equation
k
k+222k=-k -
ka-
(vi) The condition
ax+
bx
+c
= 0 may
be
in
the
ratio
m:n is
mnb
= ac (m
- n).
70
NDA/NA Pathfinder
()
+B =
(o+B)
20B
(ii)
(a-B)
=
(a
+B)
40
(ii)
a
-B'
=
(a-B)(a
+B)=(a
+B)
(iv) y(a+p)-4aß
a
+B
= (a
B)
3aß (a
+B)
(v)
o -B =
(o-B)*
3aß
(a-B)
=
(o-B)(a +oß +ß2)
(vi)
o +B'
= (« +B
-2ap
(vii)
a-B'
=
(a+B)(a-B)(«
+B*)
Use
following
steps
to
solve
it.
Step
I
Equate
the
given
expression
to
y.
Step II Obtain quadratic equation in x by simplifui
ing
the expression in step I.
Step III Put discriminant 20 ot the equation which e
get
in
step
II.
Step IV The values of y obtained by D20 is the
solution set for the given rational expression,
x2-3t ljes between
=
(o+B)(o-B)[(a+
B)
2a]
EXAMPLE 7. The expression
2+3x+
EXAMPLE 6. If a and
B are
the roots of
ax
+2bx+ =0,
then+is equal
to
d
b
c 5
d
5
4b-4ac
b.
-3x+
Sol.
a. Let
y
+3x+
a 45 2ac
C. 2b-2ac 2b-44ac
ac
ac
ac ac
(y -1) + 3ax(y + 1) +4(y - 1) =
Sol. a.
a,B
are the roots of
equation ax
- 2bx + c
= 0
Since, * is real.
D
9 (y
16
(y
20
-7y + 50y -7 207y - 50y + 7 s
a
+B and oß=£
d
(+B)-2ß
aß
(7y -1) (y -7) ss
(4b5/a )-2c/ a_ 4b- 2ac
sys
O cla ac
Maximum and Minimum Value of
INEQUATIONS
ax+ bx +C A
statement
involving
one or
more
variables and
sign
of
inequality >,<,>,
or
S
is
called an
inequation.
a+ bx +
e=a(x 4a
Note For
any
real
number a
.x|
s a
-as xsa
.x
2a
-x
s-a
or
x a
Case I If a >
4ac b
Solution of
Quadratic
####### Inequations
Let
f(x)= ax
bx +6,
where
a, b,
ceRand
a * Then, 0.
fx)20,
flx)>0,
f«)s0,
f(x)<
are called quadratic inequations.
The set of real
values of
x, which
satisfy
the
inequatio
is
called the
solution set.
Then, minimum value of ax + bx+c is
4a
and this value occurs when x=- There is no
2a
maximum value when a > 0.
Case II If a <
4ac- b
Then,
maximum
value of
ax
bx +
c is
4a
Solution of
Linear
Inequations
in Two Variables
There is no
and
this value
occurs when x
=
minimum value when a < 0.
Method
to
Solve
Fractional
Quadratic Polynomial
Consider
the
fractional quadratic
polynomial
be
In
order to
represent the
solution
set of inequation linear
in
two
variables, we
follow the
following
steps
Step
I
Convert
the
given
inequation say
ax+by
into the
equation
graph. ax+by=c and draw
the
a2*
+b2* +
C
71
MATHEMATICS>
Quadratic
Equations
and
Inequalities
2x+3y
S6,
we
first draw
the
line 2x+3y=6.
Let us take a point (0,0).
Step
II
Choose a
point
not
lying
on this
line
ax+by
=c
substitute
its
coordinates
in the
inequation.
If
the
inequation
is satisfied,
then shade
the
portion
of the
plane
which
contains the
chosen
point,
otherwise
shade the
portion
which does
not contain the chosen point.
B(0, 2)
A.
X
Step
II The
shaded
region
obtained
in
step
II
represent
Y
Clearly,
(0,
- satisfy
the
given
inequation, so
the
region
containing
the
origin
is represented
in the figure.
Now,
(1,
1),
(2,0),
(3,0),
(0,1),
(0,2)
are positive
integral
solutions
of 2x+
3y
s6.
the
desired
solution set.
EXAMPLE
The
number of
positive
integral
solutions satisfying
the inequation
2x+3y
6 is
d.
a. 2
b.
c.
Number of
positive
integral
solutions
=
Sol.
c. To represent
the
solution
set of
the
inequation
Series and quadratic equations
Course: Mathematics class 11 (041)
University: University of Kerala
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