IIT-JEE Mathematics

Binomial Theorem

BINOMIAL THEOREM FOR POSITIVE INTEGRAL INDEX

Synopsis:1. f x and a are real numbers, then for all ,n N∈

1 1 2 21 2( ) ....n n n o n n n n

ox a C x a C x a C x a− −+ = + + +1 1

1.... ...n n r r n n n o nr n nC x a C x a C x a− −

−+ + + + + i.e

Formula:0

( ) .n

n n n r rr

r

x a C x a−

=

+ =∑

2. MIDDLE TERM IN A BINOMIAL EXPANSION

Synopsis:

If n is an even natural number, then in the binomial expansion of ( ) , 12

n nx a th

+ +

term is the middle term.

Formula: If is odd natural number , then1

2

nth

+

and3

2

nth

+

are the middle terms

In the binomial expansion of (x+a)n.

3. GREATEST TERM

Synopsis:

Let Tr+1 and Tr be(r+1)th and rth terms respectively in the expansion of (x+a)n. Then, 1n n r r

r rT C x a−+ = and

Tr = nCr-1 xn-r+1 ar-1.

∴ 11 1

1

n n r rr r

n n r rr r

T C x a

T C x a

−+

− + −−

= =! ( 1)!( 1)!

.( )! ! !

n r n r a

n r r n x

− − +×−

=1

.n r a

r x

− +

CASE1 When1

1

nx

a

+

+is an integer

Let1

.1

nm

x

a

+ =+

Then, from (i), we have

Tr+1 > Tr for r=1,2,3,…..(m-1) ……(ii)

Tr+1 = Tr for r = m ….(iii)

and, Tr+1 < Tr for r = m +1, ….n …(iv)

∴ T2 > T1, T3 > T2, T4 > T3, ……, Tm>Tm-1 [From (ii)]

Tm+1 = Tm [From (iii)]

and, Tm+2 < Tm+1, Tm+3 < Tm+2, …Tn+1 < Tn [From (iv)]

⇒ T1 < T2 < ….<Tm-1 < Tm=Tm+1 > Tm+2 > …> Tn

This shows that mth and (m+1)th terms are greatest terms.

Case II: When1

1

nx

a

+

+is not an integer.

Let m be the integral part of1

1

nx

a

+

+. Then, from (i), we have

1 1,2.....,r rT T for m+ > = …(v)

and, 1 1, 2,....r rT T for r m m n+ < = + + ….(vi)

∴ 2 1 3 2 1, ,...., m mT T T T T T+> > > [From (v)]

and, 2 1 3 2 1, ,...,m m m m n nT T T T T T+ + + + +< < < [From (vi)]

⇒ 1 2 3 1 2 3... 1.... m m m m nT T T T T T T T+ + + +< < < < < > > >

⇒ (m+1)th term is the greatest term.

4. MULTINOMIAL THEOREM

Using binomial theorem, we have0

) ,n

n n n r rr

r

x a C x a n N−

=

+ = ∈∑

=0

!

( )! !

nn r r

r

nx a

n r r−

= −∑

=!

,! !

s r

r s n

nx a

r s+ =∑ where s=n-r.

This result can be generalized in the following form :

1 2( .... )nkx x x+ + + = 1 2

1 2

1 2.... 1 2

!....

! !.... !k

k

rr rk

r r r n k

nx x x

r r r+ + + =∑

The general term in the above expansion is

31 21 2 3

1 2 3

!....

! ! !..... !kr rr r

kk

nx x x x

r r r r

The number of terms in the above expansion is equal to the number of non-negative integral solution of theequation.

r1+r2+…..+rk = n, because each solution of this equation gives a term in the above

expansion. The number of such solutions is 11

n kkC+ −

− .

5. PARTICULAR CASES

(i)!

( )! ! !

n r s t

r s t n

nx y z x y z

r s t+ + =

+ + = ∑

The above expansion has 3 1 23 1 2

n nC C+ − +− = terms.

(ii)!

( )! ! ! !

n p q r s

p q r s n

nx y z u x y z u

p q r s+ + + =

+ + + = ∑

There are 4 1 34 1 3

n nC C+ − +− = terms in the above expansion.

REMARK The greatest coefficient in the expansion of 1 2( .... )nmx x x+ + + is

!,

( !) [( 1)!]m r r

n

q q− +where q and

r are the quotient and remainder respectively when n is divided by m.

PROPERTIES OF THE BINOMIAL COEFFICEINT

PROPERTY I In the expansion of (1+x)n the coefficients of terms equidistant from the beginning and theend are equal.

PROPERTY II The sum of the binomial coefficients in the expansion of (1+x)n is 2n.

i.e, 1 2 ... 2no nC C C C+ + + + = or,

0

2n

n nr

r

C=

=∑ .

PROPERTY III The sum of the coefficients of the odd terms in the expansion of (1+x)n is equal to the sumof the coefficients of the even terms and each is equal to 2n-1.

i.e, Co+C2+C4+….=C1+C3+C5+……=2n-1.

PROPERTY IV Prove that:

1 21 2

1. . .

1n n n

r r r

n n nC C C

r r r− −

− −−= =−

and so on.

PROPERTY V Co-C1+C2-C3+C4-…+(-1)n Cn=0

1. Find the coefficient of xm in the expression (1+x)n+2(1+x)n-1+3(1+x)n-2+….+(n-m+1) (1+x)m, where0 n≤ .

2. Find the sum of the series :2 3 4

0

1 3 7 15( 1) ....

2 2 2 2

r r rnr n

r r r r rr

C upto m terms=

− + + + +

∑

\

3. Find the sum of the series2 3 4

0

1 3 7 15( 1) ....

2 2 2 2

r r rnr n

r r r r rr

C to=

− + + + + + ∞

∑

4. If k and n be positive integers and sk =1k + …+ nk, then show that 1 1

1

( 1) ( 1)m

m mr r

r

C s n n+ +

=

= + − +∑

5. Prove that : 1 21 22 2 2

0 1 1 2 2k k kn n n n n n

k k k− −− −

− + − −

+…..+ +(-1)k-1 .... ( 1)0

n n k n

k k

− + + − =

, where n

k

nC

k

=

6 In the expansion of the binomial expression (x+a)15, if the eleventh term is the geometric mean of theeighth and twelfth terms, which term in the expansion is the greatest.

7. Prove that the greatest term in the expansion of (1+x)2n has also the greatest coefficient, then1

,1

n nx

n n

+ ∈ + .

8. 1 2 32 3

1 1 2 1 31 ....

1 (1 ) (1 )

x x xC C C

nx nx nx

+ + + − + − + + + +

1( 1) 0

(1 )n

nn

nxC

nx

++ − = +

9. If nCo,nC1,

nC1,nC2, …..,nCn denote the binomial coefficients in the expansion of (1+x)n and p+q=1,

then prove that

(i)0

nn r n r

rr

r C p q np−

=

=∑ (ii) 2 2 2

0

nn r n r

rr

r C p q n p npq−

=

= +∑

10. Evaluate3 2

0

3 2

6 11 6

nn

rr

rC

r r r=

++ + +∑ , Where 0 1, ,.....n n n

nC C C are the binomial coefficients in the

expansion of (1+x)n.

11. When(32)(32)32 is divided by 7, prove that the remainder is 4.

12. A is a set containing n elements. A subset P of A is chosen at random. The set A is reconstructed byreplacing the elements of P. A subset Q is again chosen at random. Find the number of ways selectingP and Q so that.

(i) P and Q are disjoint sets. (ii) P Q∩ contains just one element

(iii) P Q∪ contains just one element

(iv) Q is a subset of P

(v) P Q φ∩ =

(vi) P and Q have equal number of elements

(vii) Q contains just one element more than P

(viii) P Q A∪ = (ix) P = Q

PASSAGE – 1

Numerically greatest Term in the Expansion of (x + a)n

Let Tr and Tr+1 be rth and (r+1)th terms respectively in the expansion of bionomial

(x+a)n. Then Tr = nCr-1 xn-r+1 ar-1 and Tr+1 = nCr xn-r ar

∴ 1 1r

r

T n r a

T r x+ − +=

Now, Tr+1 > , = , < Tr According as1

, , 1,n r a

r x

− + > = <

i.e. according as1

1 , , ,n x

r a

+ − > = < i.e.

according as1

, ,1

nr

x

a

+< = >+

.

So, if1

1

nx

a

+

+is an integer, say p, then Tr+1 > Tr if r < p otherwise Tr+1 ≤ Tr

So, Tp = Tp+1 (numerically) and these are greater than any other term in the expansion.

Next, if1

1

nx

a

+

+is a non-integer, suppose m be its integral part then Tr+1 < Tr if r ≤ m and

Tr+1 < Tr if r > m.

So, Tm+1 is the numerically greatest term among the terms of the expansion.

Again we can also write that kth term is numerically greatest if Tk > Tk+1 and Tk-1.

1. The numerically greatest term in the expansion of (1-2x)8, when x=2 is

a) 8C6 46 b) 8C4 44 c) 21 7 d) None of these

2. Magnitude wise the greatest term in the expansion of (3, -2x)9 when x=1 is

a) 9C2 37 22 b) 9C336 23 c) 9C4 35 24 d) both (b) and (c)

3. If x > 0 and the 4th term in the expansion of10

32

8x +

has maximum value, then

a) 2< x < 3 b) 3 < x <10

3c) 4 < x < 5 d) None of these

4. If n is even positive integer, then the condition that the numerically greatest term in the expansion of(1+x)n may have the greatest coefficient also is

a)2

| |2

n nx

n n

+< <+

b)1

| |1

n nx

n n

+ < <+

c)4

| |4

n nx

n n

+< <+

d) None of these

5. The interval in which x must lies so that the numerically greatest term in the expansion of (1-x)21 hasthe greatest coefficient is, (x > 0).

a)5 6

,6 5

b)5 6

,6 5

c)4 5

,5 4

d)4 5

,5 4

Definite Integrals

THE NEWTON – LEIBNITZ FORMULA OR THE FUNDAMENTAL THEOREM OF INTEGRALCALCULUSIf φ (x) is one the primitives or anti derivatives of a function f(x)defined on [a, b], then the definite integral

of f(x) over [a, b] is given by φ (b) - φ (a) and is denoted by ( )b

a

f x dx∫ .

Thus, if ( ( )) ( ),d

x f xdx

φ = then ( ) ( ) ( )b

a

f x dx b aφ φ= −∫

The numbers a and b are called the limits of integration ‘a’ is called the lower limit and ‘b’ is the upperlimit. The interval [a, b] is called the interval of integration.

INTEGRAL FUNCTION AND ITS PROPERTIESPROPERTY I : The integral function of an integrable function is always continuousPROPERTY II : If φ (x) is the integral function of a continuous function f(x) definedon [a, b], then φ (x) is differentiable on (a, b) and φ 1(x) = f(x) for all x ∈ (a, b).PROPERTY III : The integral function of an odd function is an even function

If f(x) is an odd function, then φ (x) = ( )x

a

f t dt∫ is an even function

PROPERTY IV : If f(t) is an even function, then φ (x) =0

( )x

f t dt∫ an odd function

(1) Prove that/ 2

2 2 2 2 20

sin cos,

( cos sin ) 4 ( )

x x xdx

a x b x ab a b

π π=+ +∫ where a, b > 0.

(2) Prove that2

20

2

, 11 1

2 cos 1, 0 1

1

if aaI dx

a a xif a

a

ππ

π

> −= = − + < < −

∫

(3) Prove that2 2

2 20

2

, 0sin 2

2 cos, 0

2

if a bx adx

a ab x bif b a

b

ππ

π

> >= − + > >

∫

(4) Show that2

0

1, 1.

cos 1dx a

a x a

π π= >− −∫ Also, deduce that

20

1 2

(2 cos ) 3 3dx

x

π π=−∫

(5) If1 12 2220 0

x xx e dx e dxβ − −+ =∫ ∫ , then find the value of β .

(6) For x > 0, let f(x) =1

log.

1

xe t

dtt+∫ Find the function f(x) + f

1

x

and show that1 1

( )2

f e fe

+ =

(7) If in is positive integer, prove that

!0

ax ne x dx n− =∫

2

1 1 ....2! !

na a a

e an

− − + + + +

Also, deduce the value

of0

x ne x dx∞ −∫

PROPERTIES OF DEFINITE INTEGRALS

(12) ( ) ( )b b

a a

f x dx f t dt=∫ ∫ i.e., integration is independent of the change of variable.

(13) ( ) ( ) ( )b c b

a a c

f x dx f x dx f x dx= +∫ ∫ ∫ , where a < c < b.

(14) Property ( ) ( ) .b b

a a

f x dx f a b x dx= + −∫ ∫

(15) Property0

( ) 2 ( ) , ( )a a

a

f x dx f x dx if f x is an even function−

=

∫ ∫

0

2 ( ) , ( )( )

0 , ( )

aa

a

f x dx if f x is an even functionf x dx

if f x is an odd function−

=

∫∫

(16) Prove that2

0

sin 2 sin cos82

2

x x xdx

x

ππ

π π

=

−∫

(17) Evaluate2

2 (1 sin )

1 cos

x xdx

x

π

π−

++∫ (19)

0

( ) { ( ) ( )}a a

a

f x dx f x f x dt−

= + −∫ ∫

(18) Property2

00

2 ( ) , (2 ) ( )( )

0 , (2 ) ( )

aa f x dx if f a x f x

f x dx

if f a x f x

− ==

− = −

∫∫

(19) Property2

0 0

( ) { ( ) (2 )}a a

f x dx f x f a x dx= = −∫ ∫

(20) Property1

( ) ( ) {( ) }b

a a

f x dx b a f b a x a dx= − − +∫ ∫

(21) Property f(x) is a periodic function with period T, then

(i)0 0

( ) ( ) ,nT T

f x dx n f x dx n Z= ∈∫ ∫

(ii) ( ) ( ) , ,0

a nT Tf x dx n f x dx n Z a R

a

+= ∈ ∈∫ ∫

(iii)0

( ) ( ) ( ) , ,TnT

f x dx n m f x dx m n ZmT

= − ∈∫ ∫

(iv)0

( ) ( ) , ,aa nT

f x dx f x dx n Z a RnT

+= ∈ ∈∫ ∫

(v) ( ) ( ) , ,b

a

b nTf x dx f x dx n Z a R

a nT

+= ∈ ∈∫

+∫

INDEFINITE INTEGRALS

(7) Evaluate the following integralsin 2

sin( / 3)sin( / 3)

xdx

x xπ π∫

− +

(11) Evaluate the following integral3

1

sin sin( )dx

x x a+∫

(12) Evaluate the following integral3

2 sin sin 2

( cos )

a x b xdx

b a x

++∫

(13) Evaluate the following integral3/ 2 3/ 2

3 3

sin cos

sin cos sin( )

x xdx

x x x θ+

+∫

(14) Evaluate the following integral 13/ 2 5 / 2 1/ 2(1 )x x+∫ dx

(15) Evaluate4

1

1

x

x

− ∫ +

dx

(16) Evaluate2 3

1

( 2 10)dx

x x∫

+ +(17) For any natural number m, evaluate 3 2 2 1/( )(2 3 6) , 0m m m m mx x x x x dx x∫ + + + + >(18) Evaluate 25/13 27 /13sec cosx ec x dx∫ (19) 3 2sin cos 2x x dx∫

(20)7

2 5(1 )

xdx

x∫

−(21)

3 5

1

sindx

xcs x∫

(22) Evaluate6 6

1

cos sindx

x x∫

+(23) Evaluate ( tan cot )x x dx∫ +

(24) Evaluate3 5cos cos

2

x xdx

+∫ (25) Evaluate

3 2

1

( 1)

xdx

x x x x

−∫

+ + +(28) Prove that

2 2 2 2( )

xdx

x a x b∫

+ +

2 21 2 2

2 22 2

2 2 2 22 2

2 2 2 2 2 2

1tan ,

1log ,

2

x bif a b

a ba b

x b b aif a b

b a x b b a

− + > − − + − − <

− + + −

(29) Evaluate cos 1ecx dx∫ − (30)1 x

dxx

+∫

(31) Evaluate2cos sin

sin cos

x xdx

x x∫

−(32) Evaluate

2

2

cos sin 2

(2cos sin )

x xdx

x x

+∫

−

(33)1

2 3/ 2

sin

(1 )

xdx

x

−

∫−

(34) sin x dx∫

(35) 1 2(sin )x dx−∫ (36)1 1

1 1

sin cos

sin cos

x x

x x

− −

− −

−∫

+

(37) 1 2cot (1 )x x dx−∫ − + (38)2

1 22costan sec

2 sin 2d

θ θ θθ

− ∫ −

(39) 1sinx

dxa x

−∫+

(40)2

2

cos sin 2

(2cos sin )

x xdx

x x

+∫

−

(41)3

sin2

cos sin

cosx x x x

e dxx

−∫

(42 cos 2 log(1 tan )x x dx∫ +

(43) If f(x) is a polynomial function of the degree prove that( ) { ( ) '( ) ''( ) '''( ) .... ( 1) ( )}x x n ne f x dx e f x f x f x f x f x∫ = − + − + + − , where

fn(x) =( )n

n

d f x

dx

(44) Evaluate4 3cos sin cossin cos .

2 2cos

x x x x xx x xex x

− + +∫

dx

(45)1

2 2

tan

(1 )

xdx

x x

−

∫+

(46) 2 2sec log(1 sin )x x dx∫ +

(47)2

2

( 1){ 1 1}

1

xx xe dx

x

+ + +∫

+(48)

2

1

sin (2cos 1)dx

x x∫

−

(49)6 6 2 2 2 2

1

( )dx

x a a x x a∫

− + −(50)

2

1

(1 )x

xdx

x xe

+∫

+

Limits

FORMAL APPROACH TO LIMITBefore we proceed to formulate a definition of the limit of a function at a point on the basis of the discussionmade in the previous article, we intend to discuss some basic concepts which will be used in furtherdiscussions.

NEIGHBOURHOOD (NBD) OF A POINTLet a be a real number and let δ be a positive real number. Then the set of all real numbers lying between a-δ and a+δ is called the neighbourhood of a of radius ‘δ ’ and is denoted by Nδ (a).Thus Nδ (a) = (a-δ , a+δ ) = {x. R|a-δ <x<a+δ }

The set Nδ (a) – {a} is called deleted nbd of a of radius δ . The set (a-δ , a) is called the left nbd of aand the set (a, a+δ ) is known as the right nbd of a.

If δ is very small and x lies in the interval (a-δ , a), then x is said to approach to a from the left andwe write x a→ . If x. (a, a+δ ), then x is said to approach to a from the right which is denoted by x a+→ .

Consider the statement |x-a|<δ . We have|x-a|<δ ⇔ -δ <x-a<δ ⇔ a-δ ⇔ x. Nδ (a).

Thus, |x – a| < δ means that x lies in the nbd of ‘a’ of radius δ as shown in Fig.• • •

Let f(x) be a function with domain D and let ‘a’ be point such that every nbd of a contains infinitely manypoints of D.A real number l is limit of f(x) as x tends to a, if for every nbd of l, there exists a nbd of ‘a’, suchthat images of all points in the deleted nbd of a are in the nbd of l.

In other words, a real number l is lmit of f(x) x tends to a, if for every. > 0 there exists 0aδ > suchthat 0 < |x – a| < δ ⇒ |f(x) – l | < ε ⇔ x. (a - δ , a+δ ), x ≠ a ⇒ f(x). (l+ε )

If l is the limit of f(x) as x tends to a, then we write ( ) 1x aLim f x

→=

THE ALGEBRA OF LIMITSLet f and g be two real functions with domain D. We define four new functions , , /f g fg f g± on

domain D by setting.(f ± g) (x) = f(x) ± g(x), (fg) (x) =f(x) g(x)(f/g) (x) = f(x)/g(x), if g(x) ≠ 0 for any x D∈ .Following are some results concerning the limits of these functions.Let lim

x a→f(x) =l and lim ( ) .

x ag x m

→= If l and m exist, then

(i) lim( )( ) lim ( ) lim ( )x a x a x a

f g x f x g x l m→ → →

± = ± = ±

(ii) lim( )( ) lim ( ).lim ( )x a x a x a

fg x f x g x lm→ → →

= =

(iii)lim ( )

lim ( ) ,lim ( )x a

x ax a

f xf lx

g g x m→

→→

= =

provided 0m ≠

(iv) lim ( ) .lim ( ),x a x a

Kf x K f x→ →

= where K is constant

(v) lim | ( ) lim ( ) | | |x a x a

f x f x l→ →

= =

(vi) ( )lim{ ( )} x m

x af x g l

→=

(vii) If f(x) ≤ g(x) for every x in the deleted nbd of a, then limx a→

f(x) ≤ limx a→

g(x

(viii) If f(x) ≤ g(x) ≤ h (x) for every x in the deleted nbd of a and, lim ( ) lim ( )x a x a

f x l h x→ →

= = then

lim ( ) .x a

g x l→

=

(ix) ( )lim ( ) lim ( ) ( )x a x a

fog x f g x f m→ →

= =

In particular

a δ− a δ+a

(i) limx a→

log f(x) =log ( )lim ( ) logx a

f x l→

=

(ii)lim ( )( ) 1lim .x a

f xf x

x ae e e→

→= =

(iii) If1

lim ( ) , lim 0( )x a x a

f x or thenf x→ →

= +∞ −∞ =

(iv) Evaluate2 2 2

3

[1 (sin ) ] [2 (sin ) ] .... [ 9sin ) ]lim lim

0

x x x

n

x x n x

nx →∞

+ + + →

(v) Evaluate3

33

8lim

8

n

nr

r

r→∞=

−+∏ Here Π stands for the product.

(ix)2

21/

1 cos( )lim

(1 )x

cx bx a

xα α→

− + +−

(x) If f(x+y)=f(x)+f(y) for all x, y ∈ R and f(1) = 1, then evaluate(tan ) (sin )2 2

lim20 (sin )

f x f x

x x f x

−

→

Maxima and Minima

MAXIMUM Let f(x) be a function with domain D ⊂ R. Then, f(x) is said to attaining the maximumvalue at a point a ∈ D if f(x) ≤ f (a0 for all x ∈ D.

MINIMUM Let f(x) be a function with domain D ⊂ R. Then, f(x) is said to attain the minimum value at apoint a ∈ D if f(x) ≥ f(a0 for all x ∈ D.

LOCAL MAXIMUM A function f(x) is said to attain a local maximum at x = 0 a if there exists aneighbourhood (a-δ m a+δ ) of a such thatf(x) < f(a) for all x ∈ (a - δ , a + δ ), x ≠ a (or)

f(x) – f(a) < 0 for all x ∈ (a - δ , a+δ ), x ≠ a. In such a case f(a) is called the local maximum value of f(x)at x = a.

LOCAL MINIMUM A function f(x) is said to attain a local minimum at x = a if there exists aneighbourhood (a - δ , a+δ ) of a such that

f(x) > f(a) for all x ∈ (a - δ , a + δ ), x ≠ a(or) f(x) – f(a0 > 0 for all x ∈ (a - δ , a + δ ), x ≠ aThe value of the function at x=a i.e., f(a0 is called the local minimum value off(x) at x = a.

NECESSARY CONDITION FOR EXTREME VALUES:We have the following theorem which we state without proof.

THEOREM A necessary condition for f (a) to be an extreme value of a function f(x) is that f’(a) = 0, incase it exists.

ILLUSTRATION Let f(x) =3 2 10 , 0

3sin , 0

x x x x

x x

+ + <− ≥

Investigate x = 0 for local maximum/minimum.

PROPERTIES OF MAXIMA AND MINIMA(I) If f(x) is continuous function in its domain, then at least one maxima and one minima must lie

between two equal values of x.(II) Maxima and Minima occur alternately, that is, between two maxima there is one minimum and vice-

versa.(III) If f(x) → ∞ as x →a or b and f’(x) = 0 only for one value of x (say c) between a and b, then f© is

necessarily the minimum and the least value.If f(x) → ∞ as x →a or b, f(c) is necessarily the maximum the greatest value.

(1) The circle x2+y2 =1 cuts the x-axis at P and Q. Another circle with center at Q and variable radiusintersects the first circle at R above the x-axis and the line segment PQ at S. Find the maximum areaof ∆ QSR.

(2) P is a point on the ellipse2 2

2 21

x y

a b+ = whose center is O and N is the foot of the perpendicular from

O upon the tangent at P. Find the maximum area of ∆ OPN and the coordinates of P.(3) Let A(p2, -p), B(q2, q) and C(r2, -r) be the vertices of the triangle ABC. A parallelogram AFDE is

drawn with D, E and F on the lines segments BC, CA and AB respectively. Using calculus show that

the maximum area of such a parallelogram is1

4(p+q) (q+r) (p-r)

(4) (at12, 2ati); i=1, 2, 3 are the vertices of a triangle inscribed in the parabola y2 = 4ax. A parallelogram

AFDE is drawn with D, E, F on the line segments BC, CA and AB respectively. Show that the

maximum area of such a parallelogram is2

1 2 2 3 1 3( ) ( )( ).2

at t t t t t− − −

(8) From a fixed point P on the circumference of a circle of radius a, the perpendicular PM is let fall on

the tangent at point Q. Prove that the maximum area of PQM∆ is23 3

8

a.

(9) Find the values of p for which f(x) = x3 + 6(p-3)x2+3(p2-4)x+10 has positive point of maximum.(10) Find the condition that f(x) = x3 + ax2 + bx + c has

(i) a local minimum at a certain x ∈ R+

(ii) a local maximum at a certain x ∈ R-

(iii) a local maximum at certain x ∈ R- and minimum at certain x ∈ R+.(11) If f(x) = cos3 x + λ cos2x, x ∈ (0, π ). Find the range of λ so that f(x) has exactly one maximum andexactly one minimum.

Monotonic Functions

STRICTLY INCREASING FUNCTION A function f(x) is said to be a strictly increasing function on (a,b) if

1 2 1 2( ) ( )x x f x f x< ⇒ < all 1 2, ( , )x x a b∈

STRICTLY INCREASING FUNCTION A function f(x) is said to be a strictly decreasing function on (a,b) if

1 2 1 2( ) ( )x x f x f x< ⇒ > for all x1, x2 ∈ (a, b)

NECESSARY CONDITION Let f(x) be a differentiable function defined on (a, b). Then f’(x) > 0 or < 0according as f(x) is increasing or decreasing on (a, b).

SUFFICIENT CONDITIONTHEOREM Let f be a differentiable real function defined on an open interval (a, b).COROLLARY Let f(x) be a function defined on (a, b)(a) If f’(x) > 0 for all x ∈ (a, b) except for a finite number of points, where f’(x)=0, then f(x) is

increasing on (a, b).(b) If f’(x) < 0 for all x∈ (a, b0 except for a finite number of points, where f’(x)=0, then f(x) id

decreasing on (a, b).

SOME USEFUL PROPERTIES OF MONOTONIC FUNCTIONS(1) If f (x) is strictly increasing function on an interval [a, b], then f-1 exists and it is also positive.(2) If f(x) is strictly increasing function on an interval [a, b] such that it is continuous, then f-1 I

continuous on [f(a), f(b)].(3) If f(x) is continuous on [a, b] such that f’(c ) ≥ 0 (f’©>0) for each c∈ (a, b), then f(x) is

monotonically (strictly) increasing function on [a, b].(4) If f(x) is continuous on [a, b] such that f’ (c ) ≤ 0 (f’© < 0) for each ∈ (a, b), then f(x) is

monotonically (strictly) decreasing function on [a, b](5) If f(x) and g(x) are monotonically (or strictly) increasing (or decreasing) functions on [a, b], then

gof(x) is a monotonically (or strictly) increasing function on [a, b](6) If one of the two functions f(x) and g(x) is strictly (or monotonically) increasing and other a strictly

(monotonically) decreasing, then g of (x) is strictly (monotonically) decreasing on [a, b].

(1) Let f(x) =2 3

, 0

, 0

axxe x

x ax x x

≤

+ − >, where a is a positive constant.

Find the intervals in which f’ (x) is increasing.(2) If φ (x) =f(x) + f(1-x) and f’’ (x) < 0 for all x ∈ [0, 1]. Prove that φ (x) is increasing in [0, ½] and

decreasing in (1/2, 1].(3) Let g(x) = 2f(x/2) + f(2-x) and f’’(x) < 0 for all x ∈ (0, 2). Find the intervals of increases and

decrease of g(x).

Mean Values Theorems and Some Other Applications of Derivatives

ROLE’S THEOREMSTATEMENT Let f be a real valued function defined on the closed interval [a, b] such that

(i) It is continuous on the closed interval [a, b],(ii) it is differentiable on the open interval (a, b), and(iii) f(a) = f(b).

LAGRANGE’S MEAN VALUE THEOREM

STATEMENT Let f(x) be a function defined on [a, b] such thatit is continuous on [a, b] (ii) it is differentiable on (a, b).

Then there exists a real number c ∈ (a, b) such that( ) ( )

'( )f b f a

f cb a

−=−

(1) Prove that (b-a)sec2a<tanb-tana<(b-a)sec2b, where 0 < a < b <2

π.

(2) If f(x) and g(x) are continuous functions in [a, b] and they are differentiable in (a, b), then prove

that there exists c ∈ (a, b) such that( ) ( ) ( ) '( )

( )( ) ( ) ( ) '( )

f a f b f a f bb a

g a f b g a g c= −

(3) If the function of f: [0, 4] → R is differentiable, then show that(1) (f(4))2 – (f(0))2 = 8f’(a) f(b) for a, b ∈ (0, 4)

(2)4

2 2

0

( ) 2[ ( ) ( ) )]f t dt f fα α β β= +∫ for all 0 < ,α β < 2.

(4) If 2a+3b+6c=0, then prove that the equation ax2+bx+c=0 has at least one realroot in (0, 1)

(5) If aoxn + a1x

n-1 +….+an-1x=0 has a real positive root α , then prove that the equation1 2

0 1 1( 1) ... 0n nnna x n a x a− −

−+ − + + = has a positive real root less than α

Permutations and Combinations

FACTORIALThe continued product of first n natural numbers is called the “n factorial” and is denoted by n or n! i.e.

! 1 2 3 4 ... ( 1)n n n= × × × × × − ×

EXPONENT OF PRIME p in n!Let p be a prime number and n be a positive integer. Then, the last integer amongst 1, 2,

3…., (n-1), n which is divisible by p is ,n

pp

wheren

p

denotes the greatest integer

less than or equal ton

p.

(1) Find the number of zeros at the end of 100!.(2) Find the largest positive integer n for which 35! Is divisible by 3n.

SYMBOL nPr OR, P(n, r) :If n is a natural number and r is a positive integer satisfying 0 ,r n≤ ≤ then the natural number

!

( )!

n

n r−is denoted by the symbol nPr or, P(n, r)

PROPERTIES OF nCr or, C(n, r)We shall now discuss some important properties of nCr.PROPERTY I nCr = nCn-r

for 0 r n≤ ≤PROPERTY II Let n and r be non-negative such that r ≤ n. Then, 1

1n n n

r r rC C C+−+ =

PROPERTY III Let n and r be non-negative integers such that 1 .r n≤ ≤ Then,1

1.n nr r

nC C

r−

−=

PROPERTY IV If 1 ,r n≤ ≤ then 11 1. ( 1) .n n

r rn C n r C−− −= − +

PROPERTY V If nCx = nCy ⇒ x = y or, x+y=n.PROPERTY VI If If n is even, then the greatest value of (0 )n

rC r n≤ ≤ is / 2n

nC

PROPERTY VII If n is odd then the greatest value of nCr (0 )r n≤ ≤ is

1

2

nnC +

or,1

2

nnC −

FUNDAMENTAL PRINCIPLES OF MULTIPLICATION

If there are two jobs such that one of them can be completed in m ways, and when it has been completed inany one of these m ways, second job can be completed in n ways; then the two jobs in succession can becompleted in m × n wasy.

FUNDAMETNAL PRINCIPLE OF ADDTION

If there are two jobs such that they can be performed independently in m and n ways respectively, theneither of the two jobs can be performed in (m+n) ways.

(1) Five persons entered the lift cabin on the ground floor of an 8 floor house. Suppose each of them canleave the cabin independently at any floor beginning with the first. Find the total number of ways inwhich each of the five persons can leave the cabin (i) at any one of the 7 floors (ii) at different floors.

COMBINATIONS Each of the different selections made by taking some or all of anumber of distinct objects or items, irrespective of their arrangements or order in which they are placed, iscalled a combination.

PERMUTATIONS Each of the different arrangements which can be made by taking some or all of a numberof distinct objects is called a permutation.THEOREM 1: Let r and n be positive integers such that 1 .r n≤ ≤ Then prove that the number of allpermutations of n distinct items or objects taken r at a time is.THEOREM 2: The number of all permutations (arrangements) of n distinct objects taken all at a time is n!.THEOREM 3: The number of ways of selecting items or objects from a group of n distinct items or objectsisPERMUTATIONS UNDER CERTAIN CONDITIONSTHEOREM: The number of all permutations (arrangements) of n different objects taken r at a time,

(i) When a particular object is to be always included in each arrangement is 11 !n

rC r−− ×

(ii) When a particular object is never taken in each arrangement is 11 !n

rC r−− ×

(iii) When two specified objects always occur together is 21 ! 2!n

rC r−− × ×

(1) Find the number of ways of arranging m white and n black balls in a row(m > n) so that no two black balls are placed

PERMUTATIONS OF OBJECTS NOT ALL DISTINCT

THEOREM: The number of mutually distinguishable permutations of n things, take all at a time, of whichp are alike of one kind, q alike of second such that p+q=n, is

!

! !

n

p q

(1) Find the number of ways of selecting two integers a and b from theset {1, 2, …, 5n}, n N∈ so that a4 – b4 is divisible by 5.

Progressions

SEQUENCEA sequence is a function whose domain is the set N of natural numbers.REAL SEQUENCEA Sequence whose range is a subset of R is called a real sequence.In other words, a real sequence is a function with domain N and the range a subset ofthe set R of real numbers.

PROGRESSIONS: It is not necessary that the terms of a sequence always follow aCertain pattern or they are described by some explicit formula for the nth term. Thosesequences whose terms follow certain patterns are called progressions.

ILLUSTRATION 1 11, 7, 3, -1, …is an A.P. whose first term is 11 and the commondifference 7-11=-4.

ILLUSTRATION 2 Sow that the sequence <an> defined by an = 2n2 +1 is not an A.P.

PROPERTIES OF AN ARITHMETIC

PROPERTY I: If a is the first term and d the common difference of an A.P., then itsnth terms an is given by an = a+(n-1)d

PROPERTY II: A sequence is an A.P iff its nth term is of the form An+B i.e. a linearexpression in n. The common difference in such a case is A i.e. the coefficient of n.

PROPERTY III: If a constant is added to or subtracted from each term of an A.P.,then the resulting sequence is also A.P. with the same common difference.

PROPERTY IV: If each term of a given A.P. is multiplied or divided by a non-zeroconstant k, then the resulting sequence is also an A.P. with common difference kd ord/k, where d is the common difference of the given A.P.

PROPERTY V: In a finite A.P. the sum of the terms equidistant from the beginningand end is always same and is equal to the sum of first and last term i.e.

ak + an-(k-1) = a1 + an for all k=1, 2, 3, …, n-1.

PROPERTY VI: Three numbers a, b, c are in A.P. iff 2b=a+c.

PROPERTY VII: If the terms of an A.P. are chosen at regulr intervals then they froman A.P.

PROPERTY VIII: If an, an+1 and an+2 are three consecutive terms of an A.P., then2an+1 = an+an+2.

INSERTION OF ARITHMETIC MEANSIf between two given quantities a and b we have to insert n quantities A1, A2, …,An such that a, A1, A2,…..An, b form an A.P., then we say that A1, A2, ….,An are arithmetic means between a and b.

INSERTION OF n ARITHMETIC MEANS BETWEEN a AND b.Let A1, A2, ….,An be n arithmetic means between two quantities a and b. Then,a, A1, A2,…., An, b is an A.P.Let d be the common difference of this A.P. Clearly, it contains (n+2) terms.∴ b = (n+2)th term

⇒ b = a+(n+1) d

⇒

1

b ad

n

−=+

Now, A1 = a + d

⇒ A1 = 1 1

b aA a

n

− = + +

A2 = a + 2d ⇒ 2

2( )

1

b aA a

n

− = + +

An = a + nd ⇒ An =( )

1n

n b aA a

n

− = + +

ILLUSTRATIVE EXAMPLES(1) Between 1 and 31 are inserted m arithmetic means so that the ratio of the 7th and (m-1)th means is 5 :9. Find the value of m.

GEOMETRIC PROGRESSIONA sequence of non-zero numbers is called a geometric progression (abbreviate as G.P.) ifthe ratio of a term and the term preceding to it is always a constant quantity.The constant ratio is called the common ratio of the G.P.In other words a sequence, a1, a2, a3,. …,an…. is called a geometric progression if

1n

n

a

a

+= constant for all n ∈ N.

PROPERTIES OF GEOMETIC PROGRESSIONSIn this section, we shall discuss some properties of geometric progressions and geometric series

PROPERTY I: If all the terms of a G.P. be multiplied or divided by the same non-zeroconstant, then it remains G.P. with the same common ratio.

PROPERTY II: The reciprocals of the terms of a given G.P. form a G.P.

PROPERTY III: If each terms of a G.P. be raised to the same power, the resultingsequence also forms a G.P.

PROPERTY IV: In a finite G.P the product of the terms equidistant from thebeginning and the end is always same and is equal to the product of the first and thelast term.

PROPERTY V Three non-zero numbers a, b, c are in G.P. iff b2 = ac

PROPERTY VI If the terms of a given G.P. are chosen at regular intervals, then thenew sequence so formed also forms a G.P.

PROPERTY VII If a1, a2, a3…., an, … be a G.P. of nonzero non negative terms, thenlog a1, loga2, …, logan,…. is an A.P and vice – versa.

ILLUSTRATIVE EXAMPLES(1) Find all the complex numbers x and y such that x, x+2y, 2x+y are in A.P. and (y+1)2 , xy + 5, (x+1)2

are in G.P. Also, find the progression.

(2) If mth term of a G.P.is m and nth term is n, then prove that its rth term is1/ m nr n

r m

m

n

−−

−

SUM OF n TERMS OF A G.P.THEOREM To prove that the sum of n terms of a G.P. With first term ‘a’ and common ratio ‘r’ is

given by1

1

n

n

rS a

r

−= − or,

1, 1

1

n

n

rS a r

r

−= ≠ − .

ILLUSTRATIVE EXAMPLES(1) Let S denote the sum of the terms of an infinite G.P. and 2σ denote the sum of the squares of the

terms. Show that the sum of the first n terms of this geometric progression is given by2 2

2 21

nS

SS

σσ

−− + (2) Find the geometric progression of real number such that the sum of its first four terms is equal to 30and the sum of the squares of the first four terms is 340.(3) Prove that

91111.....1

timesis not a prime number.

INSERTION OF n GEOMETRIC MEANS BETWEEN TWO GIVEN NUMBERS A AND b.Let G1, G2, … Gn be n geometric means between two given numbers a and b. Then,A, G1, G2, …., Gn, b is a G.P. consisting of (n+2) terms. Let r be the common ratio of this G.P.

Then,B = (n+2)th term = arn+1

⇒ rn+1 =b

a⇒ r=(b/a)1/n+1

∴1/( 1)

1 ,n

bG ar a

a

+ = =

∴2/( 1)

22 ...,

nb

G ar aa

+ = =

( 1)

1

nn b

G ar aa

+ = =

AN IMPORTANT PROPERTY OF GEOMETRIC GEANS THEOREM:If geometric means are inserted between two quantities, then the product of n geometric means is the nthpower of the single geometric mean between the two quantities.

SOME IMPORTANT PROPERTIES OF ARITHMETIC AND GEOMETRICMEANS BETWEEN TWO GIVEN QUANTITIES.

PROPERTY I If A and G are respectively arithmetic and geometric means betweentwo positive number a and b, then A > G.

PROPERTY II If A and G are respectively arithmetic and geometric means betweentwo positive quantities a and b, then the quadratic equation having a, b as its roots isx2 – 2Ax + G2 = 0

PROPERTY III If A and G be the A.M. and G.M. between two positive numbers,

then the numbers are 2 2A A G± −ILLUSTRATIVE EXAMPLES

(1) If one geometric mean G and two arithmetic means A1 and A2 be inserted between two givenquantities, prove that G2 = (2A1 – A2) (2A2-A1).

(2) The A.M. between m and n and the G.M. between a and b are each equal to .ma nb

m n

++

Find m and n

in terms of a and b.

ARITHMETICO-GEOMETRIC SERIES Let a, (a + d)r, (a+2d)r2, (a+3d)r3, … bean arithmetico –geometric sequence. Then, a+(a+d)r+(a+2d)r2+(a+3d)r3+… is anarithmetico geometric series.

ILLUSTRATION Find the nth term of the series2 3

2 3 41 ...

3 3 3+ + + +

SUM OF n TERMS OF AN ARITHMETICO-GEOMETRIC SEQUENCE

THEOREM The sum of n terms of an arthmetico-geometric sequencea, (a + d) r, (a+2d)r2, (a+3d)r3, … is given by

1(1 ) { ( 1) }, 1

1 1 1

[2 ( 1) ], 12

n n

n

a r a n d rdr when r

r r rSn

a n d when r

− − + −+ − ≠ − − −= + − =

EXAMPLE Show that the sum of the series2

2 1 2 11 5 ....

2 1 2 1

n nto

n n

+ + + + + − − n terms is an even or an odd

number according as n is even or odd.

HARMONIC PROGRESSIONDEFINITION A sequence a1, a2,…., an, …. Of non-zero number is called a Harmonic

progression, if the sequence1 2 3

1 1 1 1, , ,..., ,...

na a a ais an Arithmetic progression .

nth TERM OF A HPThe nth term of a H.P is the reciprocal of the nth term of the corresponding A.P. Thus,if a1, a2, a3, …, an, … is a HP and the common difference of the corresponding AP is d

i.e.1

1 1

n n

da a+

= − , then

1

11

( 1)na

n da

=+ −

If a, b, c are in HP, then1 1 1

, ,a b c

are in AP. Therefore2 1 1

b a c= + ⇒

2acb

a c=

+.

(1) If S1, S2 and S3 denote the sum up to n(>1) terms of three non-constant sequence in A.P., whose first

terms are unity and common differences are in H.P., prove that 1 3 1 2 2 3

1 2 3

2

2

S S S S S Sn

S S S

− −=− +

(2) If a, b, c are in HP and a > c, show that1 1 4

b c a b a c+ >

− − −(3) Let a, b, c be positive real numbers. If a, A1, A2, b are in A.P., a, G1, G2, b are in G.P. and a, H1, H2, b

are in H.P., show that 1 2 1 2

1 2 1 2

(2 ) ( 2 )

9

G G A A a b a b

H H H H ab

+ + += =+

PROPERTIES OF ARITHMETIC, GEOMETRIC AND HARMONIC MEANS BETWEEN TWO GIVENNUMBERSLet A, G and H be arithmetic, geometric and harmonic means of two positive numbers a and b. Then.

,2

a bA G ab

+= and2ab

Ha b

=+

. These three means possess the following properties :

PROPERTY12ab

Ha b

=+

A ≥ G ≥ H.

PROPERTY2 A, G, H form a GP i.e, G2 – AH.

PROPERTY3 The equation having a and b as its roots x2 – 2Ax + G2 = 0PROPERTY4 If A, G, H are arithmetic, geometric and harmonic means between three given numbers a, band c, then the equation having a, b, c as its roots is

33 2 33

3 0G

x Ax x GH

− + − =

(1) For what value of n,1 1n n

n n

a b

a b

+ +++

is the harmonic mean of a and b?

(2) If A1, A2; G1, G2; H1, H2 be two A.M.’s, G.M’s and H.M.s between two number a and b, then provethat :(3) If H1, H2,….,Hn be n harmonic means between a and b and n is a root of the equation x2(1-ab)-

x(a2+b2)-(1+ab)=0, then prove that H1 = ab(a-b) 1

1n

H n r

H nr

+=+

METHOD OF DIFFERENCES: Sometimes the nth term of a sequence or a series cannot bedetermined by the methods discussed in the earlier sections. In such cases, we use the following stepsto find the nth term Tn of the given sequence.

STEP 1 Obtain the terms of the sequence and compute the differences between the successive termsof the given sequence. If these differences are in A.P, then take Tn = an2 + bc +c, where a, b, c are constants.Determine a, b, c by putting n=1, 2, 3 and putting the values of T1, T2, T3.

STEP 2 If the successive differences computed in step 1 are in G.P. with common ratio r, then take Tn

= arn-1 + bn+c.

STEP 3 If the differences of the differences computed in step 1 are in A.P., then take Tn = an3 + bn2 +cn + d and find the values of constants a, b, c, d.

STEP 4 If the differences of the differences computed in step 1 are in G.P. with common ratio r, thentake Tn = arn-1 + bn2 + cn + d The following examples will illustrate the above procedure.

(1) Sum the following series to n terms : 5 + 7+ 13 + 31 + 85 +….

(2) If1 1 1

1 ....2 3nH

n= + + + + and,

1'

2n

nH

+= -1 2 3 2

...( 1) ( 1)( 2) ( 2)( 3 2.3

n

n n n n n n

−+ + + + − − − − − show that Hn = Hn

1

(3) Find the sum of first n terms of the series whose nth term is 4 3 21( 2 2 1)

( 1)n n n n

n n+ + + −

+

(4) If1

( 1)( 2)( 3),

12

n

rr

n n n nT

=

+ + +=∑ where Tr denotes the rth term of the series. Find,1

1lim .

n

nr rT→∞ =∑

(5) Sum the series to n terms:1 1 1

....(1 )(1 2 ) (1 2 )(1 3 ) (1 3 )(1 4 )x x x x x x

+ + ++ + + + + +

Scalar or Dot Product of VectorsDEFINITION

Let a�

and b�

be two non-zero vectors inclined at an angleθ . Then, the scalar product of a�

with b�

is

denoted by .a b� �

and is defined as the scalar | | | |cosa b θ� �

(1) Show that the angle between two diagonals of a cube is 1 1cos

3−

.

(2) The length of the sides a, b, c of a triangle ABC are related as a2 +b2= 5c2. Prove, using vectormethods, that the medians drawn to the sides a and b are perpendicular.

(3) Determine the lengths of the diagonals of a parallelogram constructed on the vectors � � �2a α β= − and� �2b α β= −� , where �α and �β are unit vectors forming an angle of 60o.

(4) Two points A and B are given on the curve y=x2 such that OA��

. 1i =� and OB��

. i� =-2. Find

|2 3OA OB−��

|

VECTOR (CROSS) PRODUCT OF VECTORS

DEFINITIONVECTOR (CROSS) PRODUCT Let ,a b

� �

be two non –zero non-parallel vectors. Then

the vector product ,a b×� �

in that order, is defined as a vector whose magnitude is | || |a b� �

sinθ where θ is the angle between a�

and b�

and whose direction is perpendicular to the plane of a�

and b�

in such a way that ,a b� �

and this direction constitute a right handed system.

PROPERTIES OF VECTOR PRODUCT

The vector product has the following properties:(1) Vector product I not commutative i.e. if a

�

and b�

are any two vectors, then

a�

× b�

≠ b�

× a�

, however -b a×� �

.

(2) If a�

and b�

are two vectors and m is a scalar, then

( )ma b m a b a mb× = × = ×� � � ��

(3) If ,a b� �

are two vectors and m, n are scalars, then ( ) ( ) ( )ma nb mn a b m a nb n ma b× = × = × = ×� � � � � � � �

(4) Vector product is distributive over vector addition i.e.( )a b c a b a c× + = × + ×

� � � � � � �

and, ( )b c a b a c a+ × = × + ×� � � � � � �

(5) For any three vectors , ,a b c� � �

, we have ( )a b c a b a c× − = × − ×� � � � � � �

.(6) The vector product of two non-zero vectors is she the null vector if they are collinear or parallel i.e.

0 ||a b a b× = ⇔� � � � �

, where ,a b� �

are non-null vectors

(7) � � � � 0,i i j j k k× = × = × =� � � � � � �, ,i j k j i k× = × = −� � � � � � � � �, , ,j k i k j i k i j i k j× = × = × = × = −� � �

(8) If �1 2 ,a a i a k= +

�� and � �

1 2 3 ,b b i b j b k= + +�

� then

� �

1 2 3

1 2 3

i j k

a b a a a

b b b

× =

�

��

SOME USEFUL RESULTS

RESULT I If a�

and b�

are two non-zero, non-parallel vectors, then unit vectors normal to the plane of a�

and b�

are| |

a b

a b

×±×

� �

� �

RESULT II Vectors of magnitude λ normal to the plane of a�

and b�

are| |

a b

a bλ ×±

×

� �

� ��

RESULT III The area of the parallelogram with adjacent sides a�

and b�

is | |a b×� �

RESULT IV The area of a triangle with adjacent sides a�

and b�

is1

| |2

a b×� �

RESULT V Area of1

| |2

ABC AB AC∆ = ×��

=1

| |2

BC BA×��

=1

| |2

CA CB×��

RESULT VI The area of a plane quadrilateral ABCD is1

| |2

AC BD×��

, where AC and BD are its diagonals.

(1) If , ,a b c� � �

are the position vectors of the vertices A, B, C of a triangle ABC, show that the area of

triangle ABC is1

| |2

a b c c a× + + ×� � � � �

.

(2) Show that the perpendicular distance of the point c�

from the line joining

a�

and b�

is| |

| |

b c c a a b

b a

× + × + ×−

� � � � � �

� �

(3) If A, B, C, D be any four points in space, prove that | |AB CD BC AD CA BD× + × + ×��

= 4 (Area oftriangle ABC).(4) Let , 10 2 ,OA a OB a b= = +

��

and OC b=��

where O is origin. Let p denote the area of the quadrilateralOABC and q denote the area of the parallelogram with OA and OC as adjacent sides. Prove that p = 6q.(5) ABCD is a quadrilateral such that , ,AB b AD d= =

��

.AC mb pd= +��

Show that the area of the

quadrilateral ABCD is1

| || |2

m p b d+ ×� ��

PRODUCT OF THREE VECTORS

SCALAR TRIPLE PRODUCTDEFINITION Let , ,a b c

� � �

be three vectors. Then the scalar ( ).a b c×� � �

is called the scalar product of a�

, b�

and c�

and is denoted by [ ]a b c� � �

.

Thus, [ ]a b c� � �

= ( )a b×� �

. c�

.

PROPERTIES OF SCALAR TRIPEL PRODUCTPROPERTY-I If , ,a b c

� � �

are cyclically permuted the value of scalar triple product remains same,

i.e ( ). ( ).b c a c a b× = ×� � � � � �

(or), [ ] [ ] [ ]a b c b c a c a b= =� � � � � � � � �

PROPERTY-II The change of cyclic order of vectors in scalar triple product changes the sign of the scalartriple product but not the magnitude.i.e.[ ] [ ] [ ] [ ]a b c b a c c b a a cb= − = − = −

� � � � � � � � � � � �

PROPERTY-III In scalar triple product the positions of dot and cross can be interchanged provided thatthe cyclic order of the vectors remains same i.e.

( ). . . ( )a b c a a b c× = ×� � � � � � �

PROPERTY-IV The scalar triple product of three vectors is zero if any two of them are equal.[ ] [ ]a b c a b cλ λ=

� � � � � �

PROPERTY-VI The scalar triple product of three vectors is zero if any two of them are parallel orcollinear.

PROPERTY-VII If ,a b c d� � � ��

are four vectors, then [ ]a b c d+� � � ��

= [ ]a c d� � ��

+ [ ]b c d� � ��

PROPERTY-VIII The necessary and sufficient condition for three non-zero, non-collinear vectors , ,a b c� � �

to be coplanar is that [ ] 0.a b c =� � �

i.e. , ,a b c� � �

are coplanar ⇔ [ ] 0a b c =� � �

PROPERTY-IX For points with position vectors , ,a b c� � �

and d��

will be coplanar if

[ , , ] [ , , ] [ , , ] [ , , ]d b c d c a d a b a b c+ + =��

PROPERTY-X If � �2 3, ,a a i a j a k= + +

��

1 2 3b b i b j b k= + +�

� and � �1 2 3c c i c j c k= + +

�� are three vectors, then

1 2 3

1 2 3

1 2 3

[ ]

a a a

a b c b b b

c c c

=� � �

(1) If the vectors � � �,ai a j ck i kα β= = + = +��

� � and � �ci c j bkγ = + +�

� are coplanar, then prove that c is thegeometric mean of a and b.(2) Let , ,a b c

� � �

three non-zero vectors such that c�

is a unit vector perpendicular to both a�

and b�

. If the

angle between a�

and b�

is / 6π , prove that 2 1[ ]

4a b c =� � �

2 2| | | | .a b� �

(3) Consider the vectors: � �cos( ) cos( )A i j kβ α γ α= + − + −��

�

� �cos( ) cos( )B i j kα β γ β= − + + −��

�

and, � �cos( ) cos( )C i j akα γ β γ= − + − +��

�

where ,α β are γ are different angles in (0, / 2)π . If , ,A B C��

are coplanar vectors, show that a isindependent of ,α β and γ .

(4) If , ,a b c� � �

be three on –coplanar unit vectors equally inclined to one another at an angle θ such that

,a b b c pa qb rc× + × = + +� � � � � � �

find p, q, r in terms of θ . Also, prove that2

2 2 2cos

qp r

θ+ + = .

(5) If , ,a b c� � �

are three non-coplanar vectors, prove that any vector r�

is expressible as

[ [ [.

[ ] [ ] [ ]

r bc r c a r bcr a b c

ab c ab c abc

= + +

� � � � � � � � �

� � � �

� � � � � � � � �

RECIPROCAL SYSTEM OF VECTORS

Let , ,a b c� � �

be three non-coplanar vectors, vectors, so that [ ] 0.a b c ≠� � �

We define another set of three

vectors , ,a b c� � �

as given below . ,[ ] [ ]

b c c aa b

a b c a b c

× ×= =� � � �

� �

� � � � � �

[ ]

a bc

a b c

×=� �

�

� � �

(1) If , ,a b c� � �

are three non-coplanar vectors and , ,a b c� � �

form a reciprocal system of vectors, then provethat

(i) . . . 1a a b b c c= = =� � � � � �

(ii) . . 0; . . 0;a b a c b c b a= = = =� � � � � � � �

. . 0c a c b= =� � � �

(iii)1

[ ][ ]

a b ca b c

=� � �

� � �

(2) If a b c� � �

and ' ', , 'a b c��

��

be the reciprocal system of vectors, prove that

(i) . . . 3a a b b c c+ + =� � � � � �

(ii) 0a a b b c c× + × + × =� � � � � � �

(3) If a�

and b�

are two vectors such that . 0,a b ≠� �

then solve the vector equations

. 0, . 1,r a r b= =� � � �

[ ] 1.r a b =� � �

(4) If ( . ) ,r a r b c d× + =� � � � � ��

then prove that2

(,

( . )| |

a d cr a a

a c aλ

× × = + ×

� ��

� � �

� � � where λ is a scalar

VOLUME OF A TETRAHEDRON

THEOREM (i) If two pairs of opposite edges of a tetrahedron are perpendicular, then the opposite edgesof the third pair are also perpendicular to each other.(ii) The sum of the squares of two opposite edges is the same for each pair of opposite edges(iii) Any two opposite edges in a regular tetrahedronCENTROID OF A TETRAHEDRON

THEOREM The volume V of a tetrahedron whose three coterminous edges in the right-handed system are

,a b c� � �

is given by1

[ ]6

V a b c=� � �

(1) A tetrahedron has three of its vertices of its vertices at A, B and C whre� � � �3 2 , 3 ; 2OA i j OB i j k OC j= + = + − =

�� . Find the unit vector perpendicular to the face ABC. The

fourth vertex D is such that . 0 . .DA AB DA AC= =��

Find the vector equation of AD. If the volume

of the tetrahedron is 3 2 cubic units and D is on the same side as the origin, find the coordinatesof D.

(2) The position vectors of the vertices A, B and C of a tetrahedron ABCD are � � ,i j k i+ +� � and 3i�

respectively. The altitude from vertex D to the opposite face ABC meets the median line throughA of the triangle ABC at a point E. If the length of the side AD is 4 and the volume of the

tetrahedron is2 2

3cubic units, find the position vector of the point E for all its possible

positions.

(3) OABC is a regular tetrahedron. D is the circumcentre of OAB∆ and E is the middle point of theedge AC. Use vector method to find distance DE. .

(4) A pyramid with vertex at the point P whose position vector is � �4 2 2 3i j k+ +� has a regular

hexagonal base ABCDEF. The points A and B have position vectors i� and �2i j+� respectively.

The centre of hexagon has position vector � �3i j k+ +� . Given that the volume of the pyramid is

6 3 and the perpendicular from the vertex meets the diagonal AD, locate the position vectors ofthe foot of this perpendicular.

(5) Let � � � �1 2 3 1 2 3,a a i a j a k b b i b j b k= + + = + +

� �� and � �

1 2 3c c i c j c k= + +�

� be three non-zero vectors such

that c�

is a unit vector perpendicular to both the vectors a�

and b�

. If the angle between a�

and b�

is ,6

πprove that

1 2 3

1 2 3

1 2 3

1

4

a a a

b b b

c c c

= (a12 + a2

2 + a32) (b1

2 + b22 + b3

2)

(6) If , ,a b c� � �

are three on-coplanar vectors and r�

is any vector in space, then prove that

. . .( ) ( ) ( )

[ ] [ ] [ ]

r a r a r cr b c c a a b

a b c a b c a b c= × + × + ×

� � � � � �

� � � � � � �

� � � � � � � � �

(7) If , ,a b c� � �

and d��

are four vectors, then prove that

(i) ( ).( ) ( ).( ) ( ).( ) 0a b c d b c a d c a b d× × + × × + × × =� � � ��

(ii) .[ { )}] [ . ][ ]d a b c d b d a c d× × × =��

Tangents and Normals

ILLUSTRATIVE EXAMPLES:

(i) The curve y=ax3+bx2+cx+5 touches the x-axis at P(-2, 0) and cuts the y-axis at the point Q where itsgradient is 3. Find the equation of the curve completely.

(ii) Find the equation of the normal to the curve y= (1=x)y + sin-1(sin2x)atx=0(iii) Determine the constant c such that the straight line joining the points (0, 3) and 95, -2) is tangent to

the curve1

cy

x=

+(iv) Prove that all normal to the curve x=a cost + at sint, y = a sint - at cost(v) Find the points at which the tangents to the curves y=x3 – x – 1 and

y=3x2 – 4x + 1 are parallel. Also, find the equations of tangents.(vi) Find the equation of the tangent to x3 = ay2 at the point A (at2, at3). Find also the point where this

tangent meets the curve again.(vii) Tangent at point P1 (other than (0, 0) on the curve y=x3 meets the curve again at P2. The tangent at P2

meets the curve at P3 and so on. Show that the abscissae of P1, P2, P3. ….., Pn form a GP. Also, find

the ratio 1 2 3

2 3 4

area PP P

area P P P

∆∆

(viii) For the function F(x) =0

2 | |x

t∫ dt, find the tangent lines which are parallel to the bisector of the angle

in the first quadrant.(ix) If ,α β are the intercepts made on the axes by the tangent at any point of the curve x=a cos3θ ,

y=bsin3θ , prove that2 2

2 21

a b

α β+ = .

(x) If x1 and y1 be the intercepts on the axes of X an Y cut off by the tangent to the curve

1,n n

x y

a b + =

then prove that/ 1 / 1

1 1

1.n n n n

a b

x y

− −

+ =

(xi) Show that the normal to the rectangular hyperbola xy=c2 at the point P 11

,c

ctt

meets the curve

again at the point 22

,c

Q ctt

, if t13 t2 = -1

ANGLE OF INTERSECTION OF TWO CURVESThe angle of intersection of two curves is defined to be the angle between the tangents to the two curves attheir point of intersection.

ORTHOGONAL CURVESIf the angle of intersection of two curves is a right angle, the two curves are said to intersect orthogonallyand the curves are called orthogonal curves.If the curves C1 and C2 are orthogonal, then φ = / 2π

∴ m1 m2 = -1 ⇒

1 2

1C C

dy dy

dx dx = −

EXAMPLES:(i) Find the acute angle between the curves y = |x2 –1| and y=|x2 – 3| at their points of intersection(ii) Show that the curves x3 – 3xy2 = -2 and 3x2y-y3=2 cut orthogonally.

(iii) Find the acute angles between the curves y = |2x2 – 4| and y=|x2 – 5|.

(iv) Show that the curves y2=4ax and ay2=4x3 intersect each other at an angle of tan-1 1

2and also if PG1

and PG2 be the normals to two curves at common point of intersection (other than the origin)meeting the axis of X in G1 and G2, thenG1 G2 = 4a.

LENGTHS OF TANGENT, NORMAL, SUBTANGENT AND SUBNORMALLet the tangent and normal at a point P(x, y) on the curve y=f(x), meet the x-axis at T and N respectively. IfG is the foot of the ordinate at P, then TG and GN are called the Cartesian subtangent and subnormal, whilethe lengths PT and PN are called the lengths of the tangent and normal respectively.

If PT makes angle Ψ with x-axis, then tan Ψ =dy

dx. From Fig we find that

Subtangent = TG = y coty

dy

dx

Ψ =

Subnormal = GN = y tandy

ydx

Ψ =

Length of the tangent = PT = y cosec Ψ

= 21 coty + Ψ

Theory of Equations

SOME DEFINITIONS

REAL POLYNOMIAL Let ao, a1, …., an be real numbers and x is a real variable. Then, f(x)=a0 + a1x+a2x2

+….+anxn is called a real polynomial of real variable x with real coefficients.

For example, 2x3 – 6x2 + 11x-6, x2-4x+3 etc. are real polynomials.

COMPLEX POLYNOMIAL If a0, a1, a2…..an be complex numbers and x is a varying complex number,then f(x) = a0 + a1x+a2x

2 +….+an-1xn-1+anx

n is called a complex polynomial or apolynomial ofcomplex variable with complex coefficients.

POLYNOMIAL EQUATION If f(x) is a polynomial, real or complex, then f(x) = 0 is called a polynomialequation.If f(x) is a polynomial of second degree, then f(x) =0 is called a quadratic equation. The general form of aquadratic equation is ax2+bx+c=0, where a, b, c ∈ C, set of all complex numbers, and a ≠ 0.

ROOTS OF AN EQUATION The values of the variable satisfying the given equation are called its roots.In other words, x = α is a root of the equation f(x)=0, if f(α )=0.The real roots of an equation f(x)=0 are the x-coordinates of the points where the curve y=f(x) crosses x-axis.

SOME RESULTS ON ROOTS OF AN EQUATION

The following are some results on the roots of a polynomial equation with rational coefficients:

I An equation of degree n has n roots, real or imaginaryII Surd and imaginary roots always occur in pairs i.e. if 2-3i is a root of an equation,

then 2+3i is also its root. Similarly, if 2 3+ is a root of a given equation, then 2 3− is also itsroots.

III An odd degree equation has at least one real root, whose sign is opposite to that of its last termprovided that the coefficient of highest degree term is positive.

IV Every equation of an even degree whose constant term is negative and the coefficient of highestdegree term is positive, has at least two reals, one positive and one negative.

POSITION OF ROOTS OF A POLYNOMIAL EQUATIONIf f(x) = 0 is an equation and a, b are two real numbers such that f(a) f(b) < 0, then theequation f(x) = 0 has at least one real root or an odd number of real roots between a and b. In case f(a) andf(b) are of the same sign, then either no real root or an even number of real roots f(x)=0 lie between a and b.DEDUCTIONS

1. Every equation of an odd degree has at least one real root, whose sign is opposite to that of its lastterm, provided that the coefficient of first term is positive.

2. Every equation of an even degree whose last term is negative and the coefficient of first termpositive, has at least two real roots, one positive and one negative.

3. If an equation has only one change of sign, it has one positive root and no more.4. If all the terms of an equation are positive and the equation involves no odd powers of x, then all its

roots are complex.

ILLUSTRATION 1 If a, b, c, d ∈ R such that a < b < c < d, then show that the roots ofthe equation (x-a)(x-c)+2(x-b)(x-d)=0 are real and distinct.

ROOTS OF A QUADRATIC EQUTION WITH REALCOEFFICIENTSAn equation of the form ax2 + bx + c= 0 where a ≠ 0, a, b, c ∈ R is a called a quadratic equation with realcoefficients.

The quantity D=b2 – 4ac is known as the discriminant of the quadratic equation in (i) whose roots are given

by2 4

2

b b ac

aα − + −= and

2 4

2

b b ac

aβ − + −=

The nature of the roots is as given below:1. The roots are real and distinct if D > 0.2. The roots are real and equal if D = 0.3. The roots are complex with non-zero imaginary part if D < 0.4. The roots are rational if a, b, c are rational and D is a perfect square.

5. The roots are of the form P + q ( , )p q Q∈ if a, b, c are rational and D is not a perfect square.

6. If a = 1, b, c ∈ 1 and the roots are rational numbers, then these roots must be integers.

7. If a quadratic equation in x has more than two roots, then it is an identity in x that is a = b = c = 0.COMMON ROOTS

Let a1x2 + b1x+c1=0 and a2x

2 + b2x+c2=0 be two quadratic equation such that a1, a2 ≠ 0 and a1b2 ≠ a2b1. Letα be the common root of these two equations. Then,

21 1 1 0a b cα α+ + =

22 2 2 0a b cα α+ + =

Eliminatingα , we get2

1 2 2 1 1 2 2 1

1 2 2 1 1 2 2 1

b c b c c a c a

a b a b a b a b

− −= − −

SIGN OF A QUADRTIC EXPRESSIONLet f(x) = ax2 + bx+c be a quadratic expression, where a, b, c ∈ R and 0.a ≠ In thissection, we shall determine the sign of f(x) = ax2 + bx + c for real values of x. As thediscriminate of f(x) = ax2 + bx + c can be positive, zero or negative. So, we shall discuss the following threecases.

CASE I : When D = b2 – 4ac < 0If D < 0, then it is evident from Figs 20.12 and 20.13 that f(x) > 0 iff a > 0 and f(x) < 0 iff a < 0.

CASE II When D = b2 – 4ac = 0From Figs, 20.10 and 20.11, we abserve that:When D = 0, we have

f(x) ≥ 0 iff a > 0 and f(x) ≤ 0 iff a < 0.

CASE III When D = b2 – 4ac > 0

From Fig. 20.8 and 20.9, we observe the following if D = b2 – 4ac > 0 and a > 0, then0

( ) 0

0 ,

for x or x

f x for x

for x

α βα β

α β

> < >< < <= =

If D = b2 – 4ac > 0 and a < 0, then

0

( ) 0

0 ,

for x or x

f x for x

for x

α βα β

α β

< < >> < < =

CONDITION FOR RESOLUTION INTO LINEAR FACTORS

THEOREM: The quadratic function ax2 + 2hxy + by2 + 2gx + 2fy+c is resolvable into

linear rational factors if abc+2fgh-af2-bg2-ch2=0 i.e. 0

a h g

h b f

g f c

=

Probability

ELEMENTARY EVENTIf a random experiment is performed, then each of its outcomes is known as an elementary event.

SAMPLE SPACE The set all possible outcomes of a random experiment is called the sample spaceassociated with it and it is generally denoted by S.

ILLUSTRATION Consider the experiment of tossing two coins together or a coin twice.

In this experiment the possible outcomes are.Head on first and Head on secondHead on first and Tail on second,Tail on first and Head on second,Tail on first and Tail on second.

If we defineHH = Getting head on both coins,HT = Getting head on first and tail on secondTH = Getting tail on first and head on second,TT = Getting tail on both coins.

COMPOUND EVENT A subset of the sample space associated to a random experiment is said to define acompound event if it is disjoint union of single element subsets of the sample space.

NEGATION OF AN EVENT Corresponding to every event A associated with a random experiment wedefine an event “not A” which occurs when and only when A does not occur.

PROBABILITYDEFINITION If there are n elementary events associated with a random experiment and m of them arefavourable to an event A, then the probability of happing or occurrence of A is denoted by P(A) and is

defined as ratiom

n.

Thus, P(A) =m

nClearly, 0 m n≤ ≤ . Therefore,

0 1m

n≤ ≤

⇒ 0 ( ) 1p a≤ ≤If p(A) =1, then A is called certain event and A is called an impossible event, if P(A) = 0.The number of elementary events which will ensure the non-occurrence A i.e. Which ensure the

occurrence of A is (n-m). Therefore.

( )n m

P An

−=

= 1m

n−

= 1 – P(A)⇒ P(A) + P ( )A = 1

The odds in favour of occurrence of the event A are defined by m : (n-m) i.e, P(A) : P ( )A and the

odds against the occurrence of A are defined n-m : m i.e, P ( )A : P(A)

1. An unbiased die, with face numbered 1, 2, 3, 4, 5, 6, is thrown n times and the list of n number showingup is noted. What is the probability that, among numbers, 1, 2, 3, 4, 5, 6, only three numbers appear inthis list?

2. Three six faced die are thrown together. Find the probability that the sum of the numbers appearing onthem is (9 14).k k≤ ≤

3. In a bag there are three tickets numbered 1, 2, 3. A ticket is drawn at random and put back, and this isdone four times. Find the probability of getting an even number as the sum.

GEOMETRICAL PROBABILITY1. Two points are taken on a straight line AB of length unity. Prove that the probability that the distance

between them exceeds (0<l<1) is (1-l)2 .2. The outcomes of an experiment are represented by points in the square bounded by x=0, y=0, x=2 and

y=2 in the xy plane. If the probability be distributed uniformly, determine the probability that x2 + y2 >1.

ALGEBRA OF EVENTSIn this section, we shall see how new events can be constructed by combining two orMore events associated to a random experiment.Let A and B be two events associated to a random experiment with sample space S. WeDefine the event “A or B” which is said to occur iff an elementary event favourable toeither A or B or both is an outcome. In other words, the event “A r B” occurs iff eitherA or B or both occur i.e. at least one of A and B occurs. Thus, “A or B” is representedby the subset (( (( ) ( )) ) ( ))P AP A B A B B A B∩ ∪ ∩ ∪ ∩ of the sample space S.Give verbal descriptions of some events and their equivalent set theoretic notations forready reference.

Verbal description of the event Equivalent set theoretic notationNot A AA or B (at least one of A or B) A B∪A and B A ∩ BA but not B A B∩Neither A nor B A B∩At least one of A, B or C A B C∪ ∪Exactly one of A and B ( ) ( )A B A B∩ ∪ ∩All three of A, B and C A B C∩ ∩Exactly two of A, B and C ( ) ( ) ( )A B C A B C A B C∩ ∩ ∪ ∩ ∩ ∪ ∩ ∩

ILLUSTRATION Let A, B and C are three arbitrary events. Find the expression forthe events noted below, in the context of A, B and C.(i) Only A occurs (ii) Both A and B, but not C occur(iii) All the three events (iv) At least one occurs(v) At least two occur (vi) One and no more occurs(vii) Two and no more occursI(viii) None occurs(ix) Not more than two occur.

SOLUTION (i) A B C∩ ∩ (ii) A B C∩ ∩(iii) A B C∩ ∩ I (iv) A B C∪ ∪(v) ( ) ( ) ( )A B B C A C∩ ∪ ∩ ∪ ∩

(vi) ( ) ( ) ( )A B C A B C A B C∩ ∩ ∪ ∩ ∩ ∪ ∩ ∩(vii) ( ) ( ) ( )A B C A B C A B C∩ ∩ ∪ ∩ ∩ ∪ ∩ ∩(viii) (A B C A B C∩ ∩ = ∪ ∪

(ix) ( ) ( ) ( ) 3( )A B B C A C A B C∩ ∪ ∩ ∪ ∩ − ∩ ∩

TYPE OF EVENTSAll events associated to a random experiment are divided into different types on the basis of their nature of

occurrence. In this section, we shall discuss those types.

MUTUALLY EXCLUSIVE EVENTS: Two or more events associated to a random experiment aremutually exclusive if the occurrence of one of them prevents or denies the occurrence all others. .It follows from the above definition that two or more events associated to a random experiment are mutuallyexclusive, if there is no elementary event which is favrourable to all the events.Thus, if two events A and B are mutually exclusive, then ( ) 0P A B∩ =Similarly, if A, B and C are mutually exclusive events, then ( ) 0P A B C∩ ∩ = .

EXHAUSTIVE EVENTS Two or more events associated to a random experiment are exhaustive if theirunion is the sample space. i.e. events A1, A2,…., An associated to a random experiment with sample space Sare exhaustive if 1 2... nA A A S∪ ∪ = .

INDEPENDENT EVENTS Two event A and B associated to a random experiment are independent if theprobability of occurrence or non occurrence of A is not affected by the occurrence or non-occurrence of B.

THEOREM 1 (Addition Theorem for two events) If A and B are two events associated with a randomexperiment, then ( ) ( ) ( ) ( )P A B P A P B P A B∪ = + − ∩COROLLARY If A and B are mutually exclusive events, then there fore

( ) 0, ( ) ( ) ( )P A B P A B P A P B∩ = ∪ = + This is the addition theorem for mutually exclusive events.

THEOREM 2 (addition Theorem for three events). If A, B, C are three events associated with a randomexperiment, then ( ) ( ) ( ) ( ) ( ) ( )P A B C P A P B P C P A B P B C∪ ∪ = + + − ∩ − ∩ ( ) ( )P A C P A B C− ∩ + ∩ ∩COROLLARY If A, B, C are mutually exclusive events, then ( ) ( ) ( )P A B P B C P A C∩ = ∩ = ∩ =

( ) 0P A B C∩ ∩ = .∴ ( ) ( ) ( ) ( )P A B C P A P B P C∪ ∪ = + +

THEOREM 3 Let A and B be two events associated to a random experiment. Then,(i) ( ) ( ) ( )P A B P B P A B∩ = − ∩ (ii) ( ) ( ) ( )P A B P A P A B∩ = − ∩(iii) (( ) ( ))P A B A B∩ ∪ ∩ = P(A)+P(B)-2P(A ∩ B)

THEOREM 4 For any two events A and B, prove that

( ) ( ) ( ) ( ) ( ).P A B P A P A B P A p B∩ ≤ ≤ ∪ ≤ +

THEOREM 4 For any two events A and B, Prove that the probability that exactly one of A, B occurs isgiven by

( ) ( ) 2 ( ) ( ) ( )P A P B P A B P A B P A B+ − ∪ = ∪ − ∩

THEOREM 5 If A, B, C are three events, then prove that(i) P(At least two of A, B, C occur)

= ( ) ( ) ( ) 2 ( )P A B P B C P C A P A B C∩ + ∩ + ∩ − ∩ ∩(ii) P (exactly two of A, B, C occur)

= ( ) ( ) ( ) 3 ( )P A B P B C P A C P A B C∩ + ∩ + ∩ − ∩ ∩(iii) P (Exactly one of A, B, C occurs)

( ) ( ) ( ) 2 ( ) 2 ( )P A P B P C P A B P B C+ + − ∩ − ∩ - 2 ( ) 3 ( )P A C P A B C∩ + ∩ ∩

Example. A die is loaded so that the probability of face I is proportional to i, i =1,2,….6. What is theprobability of an even number occurring when the die is rolled?

CONDITIONAL PROBABILITY

Let A and B be two events associated with a random experiment. Then, the probabilityof occurrence of event A under the condition that B has already occurred and P(B) ≠ 0,is called the conditional probability and it is denoted by P(A/B). Thus, we haveP(A/B)= Probability of occurrence of A given that B has already occurred.Similarly, P(B/A) when P(A) ≠ 0 is defined as the probability of occurrence of event Bwhen A has already occurred.

ILLUSTRATION 1: Let there be a bag containing 5 white and 4 red balls. Two ballsare drawn from the bag one after the other without replacement. Consider the followingeventsA = Drawing a white ball in the first draw,B = Drawing a red ball in the second draw.Now,

P(B/A) = Probability of drawing a red ball in second draw given that a white ball hasalready been drawn in the first draw

= Probability of drawing a red ball from a bag containing 4 white and 4 redballs.

ILLUSTRATION 2: Two integers are selected at random from integers 1 to 11. If the sum is even, find theprobability that both the numbers are odd.

MULTIPLICATION THEOREMS ON PROBABILITY

In this section, we shall discuss some theorems which are helpful in computing the probabilities ofsimultaneous occurrences of two or more events associated with a random experiment.

THEOREM 1 If A and B are two events associated with a random experiment, then( ) ( ) ( / ),P A B P A P B A∩ = if P(A) ≠ 0( ) ( ) ( / ),P A B P B P A B∩ = if P(B) ≠ 0.

THEOREM 2 (Extension of multiplication theorem) if A1, A2,….,An are n events associated with arandom experiment, then

1 2 3( .... )nP A A A A∩ ∩ ∩ = 1 2 1 3 1 2( ) ( / ) ( / )P A P A A P A A A∩

1 2 1...... ( / .... ),n nP A A A A −∩ ∩ ∩ where 1 2 1( / .... )i iP A A A A −∩ ∩ represents the conditional probability of

the occurrence of event Ai, given that the events A1, A2….Ai-1 have already occurred.1. Suppose n persons are asked a question successively in a random order and exactly 3 of the n persons

know the answer.(i) If n > 6, find the probability that the first four of those asked do not know the answer.(ii) Show that the probability that the rth person asked is the fist to know the answer is3( ) ( 1)

( 1)( 2)

n r n r

n n n

− − −− −

where 1 2r n≤ ≤ −

2. A die loaded so that the probability of throwing the number is proportional to i. Find the probability thatthe number 2 has occurred, given that when the die is recalled an even number has turned up.

MORE ON INDEPENDENT EVENTSIn section 40.6, we have defined independent events and we have seen that two eventsand B are independent if P(B/A)=P(B) and P(A/B)=P(A).

Also, ( ) ( ) ( )P A B P A P B∩ = if A and B are independent events.In this section, we shall discuss about pair wise independence and mutual independence

of events.PAIRWISE INDEPENDENT EVENTS: Let A1, A2, …., An be n events associated toa random experiment. These events are said to be pair wise independent if

1( ) ( ) ( )j i jP A A P A P A∩ = for ; , 1, 2...,i j i j n≠ =( ) ( ) ( ) ( ),i j k i j kP A A A P A P A P A∩ ∩ = for ;i j k≠ ≠ j, k = 1,2,….,n

�

1 2 1 2( ..... ) ( ) ( ).... ( )n nP A A A P A P A P A∩ ∩ =

ILLUSTRATION 1 A lot contains 50 defective and 50 non-defective bulbs. Two bulbsare drawn at random, one at a time, with replacement. The events A, B, C are defined asA : “the first bulb is defective”,B : “the second bulb is non-defective”,C : “the second bulb is non-defective”,Determine whether (i) A, B, C are pair wise independent,

(ii) A, B, C are mutually independent.

THEOREM If A and B are independent events associated with a random experiment,then prove that(i) A and B are independent events (ii) A and B are independent events(iii) A and B are also independent events

1. For three independent events A, B and C, the probability to A to occur is a, the probability that A, B andC will not occur is b, and the probability that at last one of thee three events will not occur is c. If pdenotes the probability that c occurs but neither A nor B occurs, prove that p is a root of the equation

ap2 + {ab +(1-a)(1-a-c)} p+b(1-a)(1-c)=0 and deduces that2(1 )

1

a abc

a

− +>−

2. An urn contains five balls alike in every respect except colour. If three of these balls are white and twoare black and we draw two balls at random from this urn without replacing them. If A is the event thatthe first ball drawn is white and B the event that the second ball drawn is black, are A and Bindependent?

THE LAW OF TOTAL PROBABILITYTHEOREM (Law of total probability) Let S be the sample space and let E1, E2,….,En

be n mutually exclusive and exhaustive events associated with a random experiment. IfA is any event which occurs with E1 or E2 or ….or En, then

P(A) = P(E1) P(A/E1) + P(E2) P(A/E2) +…..P(En) P(A/En)

1

( ) ( / ).n

r rr

P E P A E=

=∑

1. Urn contains m white and n black balls. A ball is drawn at random and is put back into the urn alongwith k additional balls of the same colour as that of the ball drawn. A ball is again drawn at random.What is the probability that the ball drawn is now white?

2. An employer sends a letter to his employee but he does not receive the reply (It is certain that theemployee would have replied if he did receive the letter). It is known that one out n letters does notreach its destination. Find the probability that the employee does not receive the letter.

BAYE’S RULETHEOREM (Baye’s Theorem) Let S be the sample space and let E1, E2…, En be n mutually exclusive andexhaustive events associated with a random experiment. If A is any event which occurs with E1 or E2 or….or

En, then k kk n

i ii 1

P(A / E ).P(E )P(E / A)

P(A / E ).P(E )=

=∑

1. A company has two plants to manufacture scooters. Plant 1 manufactures 70% of the scooters and PlantII manufactures 30%. At Plant I, 80% of the scooters are rated as of standard quality and at Plant II,90% of the scooters are rated as of standard quality A scooter is chosen at random and is found to be ofstandard quality. What is the probability that it has come from Plant II?

RANDOM VARIABLE AND ITS PROBABILITY DISTRIBUTION

DEFINITION Let S be the sample space associated with a given random experiment.Then a real valued function X which assigns to each even w S∈ to a unique realnumber X (w) is called a random variable. .

In other words, a random variable is a real valued function having domain as thesample space associated with a random experiment.

MEAN AND VARIANCE OF A RANDOM VARIABLEMAEN If X is discrete random variable which assumes values x1, x2, x3,….,xn withrespective probabilities p1, p2, p3,….,pn, then the mean X of X is defined as

X = p1 x1 + p2x2 +…..+pnxn or,1

n

i ii

X p x=

=∑

VARIANCE If X is a discrete random variable which assumes values x1, x2, x3,…, xn

With the respective probabilities p1, p2, ….,pn, then variance of X is defined asVar (X) = 2 2 2

1 1 2 2( ) ( ) .... ( )n np x X p x X p x X− + − + + − …… 2( )n np x X+ −

2

1

( ) ,n

i ii

p x X=

= −∑ Where1

n

i ii

X p x=

=∑ is the mean of X.

Now,2

2

1

n

i ii

p x X=

= −∑

BINOMIAL DISTRIBUTION A random variable X which takes values 0. 1, 2,.., n is said to followbinomial distribution if its probability distribution function is given byP(X = r) = , 0,1, 2,.... ,n r n r

rC p q r n− = where p, q > 0 such that p + q = 11. An urn contains 25 balls of which 10 balls bear a mark ‘A’ and the remaining 15 balls bear a mark ‘B’.

A ball is drawn at random from the urn, its mark is noted down and it is replaced. If 6 balls are drawn inthis way, find the probability that

(i) all will bear ‘A’ marked(ii) not more than 2 will bear ‘B’ mark(iii) then number of balls with ‘A’ mark and ‘B’ mark will be equal(iv) at least one ball will bear ‘B’ mark

2. Numbers are selected at random one at a time, from the numbers 00, 01, 02,…, 99 with replacement. Anevent A occurs if the product of the two digits of the selected number is 18. If four numbers areselected, find the probability that A occurs at lest 3 time.

Areas of Bounded Regions

THEOREM Let f(x) be a continuous function defined on [a, b]. Then, the area bounded by the curvey=f(x), the x-axis and the ordinates x = a and x = b is given by

( ) ,b

a

bf x dx or y dx

a∫ ∫

The area bounded by the curve x = f(y), the y axis and the abscissae y = c and y = d is given by

( ) ,d

c

df y dy or x dy

c∫ ∫

(1) Let f(x) = maximum {x2, (1-x)2, 2x(1-x)}. Determine the area of the region bounded by the curvey=f(x), x-axis, x = 0 and x = 1.

(2) Let f(x) be defined by f(x) = max (4sinx, 4-2 sinx), 0 ≤ x ≤ 2π . Draw a sketch of y=f(x) and computethe area bounded by the curves y=f(x), the y-axis, the x-axis and the ordinate at x = 2π .

(3) Find the area lying on the same side of the axis of x, as the positive part of the axis of y and which iscontained by y2 = 4ax, x2 + y2 = 2ax and x=y + 2a..

Complex Numbers

COMPLEX NUMBER:If a, b are two real numbers, then a number of the form a+ib is called a complex number.A complex number z is purely real if its imaginary part is zero i.e. Im(z)=0 and purely imaginary if

its real part is zero i.e. Re (z) = 0.

SET OF COMPLEX NUMBERS: The set of all complex numbers is denoted by CI.e. C = {a + ib | a, b ∈ R}.Since a real number ‘a’ can be written as a+0 i, therefore every real number is a complex number.

Hence, ⊂� � , where � is the set of all real numbers.

DEFINITION: Two complex numbers z1 = a1 + ib1 and z2 = a2 + ib2 are equal if a1 = a2 and b1 = b2 i.e.

Re (z1) = Re (z2) and Im(z1) = Im(z2).

POLAR FORM OF z = x + iy FOR DIFFERENT SIGNS OF x and y.

Let | z | = r and be the acute angle given by 1tany

x−

. Let θ be the argument of z.

CASE I Polar form of z = x + iy when x > 0 and y > 0 : In this case, we have θ α= . So, the polar form ofz = x + iy is r(cosα + isinα ).

CASE II Polar form of z = x + iy when x < 0 and y > 0 : In this case, we have θ π α= − . So the polar formof z=x+iy is [cos( ) sin( )r iπ α π α− + − or, ( cos sin )r iα α− +

CASE III Polar form of z = x +iy when x < 0 and y < 0 : In this case, we have ( ).θ π α= − − So, the polarform of z is [cos( ) sin( ( ))]r iπ α π α− + − − or [ cos sin )r iα α− −CASE IV Polar form of z=x+iy when x > 0 and y<0 : In this case, we have θ α= − So, the polar form of zis [cos( ) sin( )]r iα α− + − or, [cos sin ]r iα α−

ILLUSTRATION1 Write the following complex numbers in the polar form:(i) 3 2 3 2i− + (ii) 1 + i (iii) -1 - i (iv) 1 - i

EXAMPLE 1: If z1 z1 and z2 are complex numbers, prove that:

(i) arg ( )z = - arg(z). In general, arg ( )z = 2 arg( )n zπ −(ii) arg (z1 z2) = arg(z1)+arg(z2) (iii) arg 1 2( )z z =arg(z1)-arg(z2)

(iv) arg 11 2

2

arg( ) arg( )z

z zz

= −

EXAMPLE 3:

(i) If z1 and z2 are two computers such both of them satisfy the relation 1

2

z

z⇔ and arg(z1-z2) = , then

4

πfind the imaginary part of (z1 + z2).

(ii) If z is a complex number satisfying |z-1|=1, then prove that2

tan{arg( )}z

i zz

− =

(iii) If z is a complex number such that2

arg ,2 4

za

z

π− = + then prove that |z-2i|= 2 2

(iv) Number such that 1

2 4

z z

z z

π − = + arg, then prove that |z-7-9i|=3 2

THEOREM 1: If z1, z2, z3 are the affixes of the points A, B, and C in the Arg and plane, then

(i) 3 1

2 1

argz z

BACz z

−∠ = − (ii) 3 1 3 1

2 1 2 1

| |

| |

z z z z

z z z z

− −=− −

where BACα = ∠

THEOREM 2: If z1, z2, z3 and z4 are the affixes of the points A, B, C and D respectively in the Arg and

plane. Then, AB is inclined to CD at the angle.

arg 2 1

4 3

z z

z z

− −

(i) Prove that the triangle whose vertices are the points z1, z2, z3 on the Argand plane is an equilateral

triangle if and only if2 3 3 1 1 2

1 1 10

z z z z z z+ + =

− − −(ii) Let z1, z2, z3 be three distinct complex numbers satisfying |z1 -1| = |z2 -1| = |z3 -1|. Let A, B and C

be the points in the Arg and plane representing z1, z2 and z3 respectively. Prove that z1 + z2 + z3 = 3 if and

only if ABC∆ is an equilateral triangle.THEOREM : The equation of a circle described on a line segment joining A(z1) and B(z2) as diameter is

1 2 2 1( )( ) ( )( ) 0z z z z z z z z− − + − − =EXAMPLE 3

(i) A point z moves in the Arg and plane such that arg(z-2-i)=4

π. Find the path traced

by1

.z

(ii) Let z1 = 10 + 6i and z2 = 4+2i be two complex numbers and z be a complex numbers such that arg

1

2

.4

z z

z z

π − = − Find the centre and radius of the locus of complex number z.

De’ MOIVERE’S THEOREMSTATEMENT:(i) If n Z∈ (the set of integers), then (cosθ +isinθ )n=cosnθ +isinnθ(ii) If n Q∈ (the set of rational numbers), then cosnθ +isinnθ is one of the values of (cosθ +isinθ )n.

(1) If z=x+iy is a complex number with rational x and y and |z|=1, show that |z2n-1| is a rational numberfor every n N∈ .

(2) Form an equation whose roots are 2 2 22 3 2sin ,sin ,sin ,...,sin

2 1 2 1 2 1 2 1

n

n n n n

π π π π+ + + +

ROOTS OF A COMPLEX NUMBERlet z = a + ib be a complex number, and let r (cos+isin) be the polar form of z. Then by De moivere’s

theorem 1/ cos sinnr in n

θ θ +

is one of the values of z1/n. Here, we shall show that z1/n has n distinct

values. We know that cos sin cos(2 ) sin(2 ),i m i mθ θ π θ π θ+ = + + +m=0, 1, 2, .... So, 1/ 1/ 1/[cos(2 ) sin(2 ]n n nz r m i mπ θ π θ= + + + =

1/ 2 2cos sinn m m

r in n

π θ π θ+ + +

Now by giving m the values 0, 1, 2, ..., (n - 1); we shall obtain distinct values of z1/n.

For the values m = n, n+1, ...., the valuesof z1/n will repeat for example, if m=n,then

1/ 1/ 2 2cos sinn n n n

z r in n

π θ π θ+ + = +

1/ cos sinnr in n

θ θ = +

which is same as the value obtained by taking m=0. On taking m=n+1, the value comes out to be

identical with corresponding to m=1, Hence, z1/n has n distinct values.

(i)3 5 1

cos cos cos7 7 7 8

π π π = − (ii)3 5 7

cos cos cos4 14 14 8

π π π =

ROOTS OF UNITYIn this section, we shall obtain various order roots of unity by using De’ Moivere’s theorem. We

shall also learn about some poperties of these roots.

nth ROOTS OF UNITY : Let z = 11/n. Then, z = (cos0o + isin0o01/n

⇒1/(cos 2 sin 2 ) ,nz r i r Zπ π= + ∈

2 2cos sin , 0,1,2,..., ( 1)

r rz i r n

n n

π π= + = −

[Using De’ Moivere’s2 11, , ,...., ,nα α α − where 2 / 2 2

cos sini ne in n

π π πα = = +

PROPERTIES OF nTH ROOTS OF UNITY(i) nth roots of unity form a G.P. With common ratio 2 /i ne π

(ii) Sum of nth roots of unity is always zero(iii) Sum of pth power of nth roots of unity is zero, if P is not a multiple of n.(iv) Sum of pth powers of nth roots of unity is n, if p is a multiple of n.

(v) Product of nth roots of unity is (-1)n-1 (vi) nth roots of unity lie on the unit circle | z | = 1 and divide its circumference into n equal parts.

EXAMPLE 1(i) If 1

1 21, , ,...., nα α α − are the nth roots of unity then prove that

(i) 1 2 1(1 )(1 )....(1 ) 1,nα α α −+ + + = if n is an odd natural number

(ii) 1 2 1(1 )(1 )....(1 ) 0,nα α α −+ + + = if n is an even natural number.

(ii) If 1 2 3 41, , , ,α α α α are the roots of the equation x5-1=0, then prove that

1 22 2

1 2

, .ω α ω αω α ω α

− −− −

3 42 2

3 4

,ω α ω α ωω α ω α

− − =− −

Continuity and Differentiability

CONTINUITY AT A POINTA function f(x) is said to be continuous at a point x = a of its domain, if lim ( ) ( )f x f a

x a=

→

CONTINUITY ON AN OPEN INTERVALA function f(x) is said to be continuous on an open interval (a, b) if it is continuous at every point on theinterval (a, b).

CONTINUITY ON A CLOSED INTERVAL:A function f(x) is said to be continuous on a closed interval [a, b] if(i) f is continuous on the open interval (a, b)(ii) lim ( ) ( )f x f a

x a+=

→(and) (iii) lim ( ) ( )f x f b

x b−=

→In other words, f(x) is continuous on [a, b] if it is continuous on (a, b) and it is continuous at a from

the right and at b from the left.

PROPERTIES OF CONTINUOUS FUNCTIONS(I) If f, g are two continuous functions at a point a of their common domain D, then f ± g, fg are

continuous at a and if g(a) ≠ 0, then f/g is also continuous at a.(II) If f is continuous at a and f(a) ≠ 0, then there exists an open interval

(a-δ , a+δ ) such that for all ( , ), ( )x a a f xδ δ∈ − + has the same sign as f (a)(III) If a function f is continuous on a closed interval [a, b], then it is bounded on

[a, b] i.e there exist real numbers k and K such that ( )k f x K≤ ≤ for all [ , ].x a b∈(IV) If f is a continuous function defined on [a, b] such that f (a) and f(b) are of opposite signs, then there

exists at least one solution of the equation f(x) = 0 in the open interval (a, b)(V) If f is continuous on [a, b], then f assumes atleast once, every value between its minimum and

maximum values i.e. if K is any real number between minimum and maximum values of f(x), thenthere exists at least one solution of the equation f(x) = K in the open interval (a, b).

(VI) If f is continuous on [a, b] and maps [a, b] into [a, b], then for some [ , ]x a b∈ we have f(x) = x.(VII) If g is continuous at a and f is continuous at g(a), the fog is continuous at a.

(1) A function is defined as follows1

sin , 0( )

0, 0

mx xf x x

x

≠ = =

What condition should be imposed on m so

that f(x) may be continuous at x = 0 ?(2) If f(x) = min {|x|, |x-2, 2-|x-1|}, draw the graph of f(x) and discuss its continuity and differentiability.

(3) If f(x) = x2 – 2 |x| and{ ( ) : 2 , 2 0}

( ){ ( ) : 0 , 0 3}

Min f t t x xg x

Max f t t x x

− ≤ ≤ − < ≤ ≤ ≤ ≤

(i) Draw the graph of f(x) and discuss its continuity and differentiability.(ii) Find and draw the graph of g(x). Also, discuss the continuity and

differentiability of g(x).

(4) If 13

cos3cos ,

cos

xy

x−= then show that

3

cos cos3

dy

dx x x=

Differential Equations

DIFFERENTIAL EQUATION: An equation containing an independent variable, dependent variable anddifferential coefficients of dependent variable with respect to independent variable is called a differentialequation.

GENERAL SOLUTION: The solution which contains as many as arbitary constants as the order of thedifferential equation is called the general solution of the differential equation.

PARTICULAR SOLUTION: Solution obtained by giving particular values to the arbitrary constants in thegeneral solution of a differential equation is called a particular r solution.ILLUSTRATIVE EXAMPLES(1) Obtain the differential equation of all circles of radius r.(2) find the differential equation of all the circles in the first quadrant which touch the coordinate axes.

(3) Find the differential equation of all conics whose center lies at the origin.

Solve the following differential equations by inspection method

(4) 3( ) cos ( )x

ydx xdy xy xdy ydxy

− = +

(5)2 2 2

3

cos ( )xdx ydy x x y

ydx xdy y

+ +=−

(6) xdy-ydx = (x2 + y2) dx

LINEAR DIFFERENTIAL EQUATIONS OF THE FORMdx

Rx Sdy

+ =

Sometimes a linear differential equation can be put in the form

,dx

Rx Sdy

+ =

Where R and S are functions of y or constants.Note that y is independent variable and x is a dependent variableThe following algorithm is used to solve these types of equations

ALGORITHM

STEP I Write the differential equation in the formdx

Rx Sdy

+ = and obtain R and S.

STEP II Find I.F by using I.F. =e∫ R dy

STEP III Multiply both sides of the differential equation in step I by I. F.STEP IV Integrate both sides of the equation obtained is step III w.r.t y to obtain the solution given byx(I.F.) = ∫ S(I.F) dy + C

Where C is the constant of integration. Following examples illustrate the procedure.

(1) Solve y dx – (x + 2y2)dy = 0

(2) If y1 and y2 are the solutions of the differential equationdy

Py Qdx

+ = , where P and Q are functions

of x alone and y2 = y1 z, then prove that z = 1 + C1

Qe dx

y

−∫ , where C is an arbitrary constant.

(3) Let u(x) and v(x) satisfy the differential equation ( ).da

P xdx

+ u=f(x) anddv

dx+p(x). v=g(x)

respectively where p(x), f(x) and g(x) are continuous functions. If u(x1) > v(x1) for some x1 and f(x)> g(x) for all x > x1, prove that any point (x, y), wherex > x1, does not satisfy the equation y=u(x) and y = v(x).

EQUATIONS REDUCIBLE TO LINEA FORM BERNOULLI’S DIFFERENTIAL EQUATIONS

The equations of the form ndyPy Qy

dx+ = Where P and Q are constants or functions of x alone and n

is a non-zero constant other than unity, are known as Bernoulli’s equations.

(1) Solvedy

dx+xsin2y = x3 cos2 y (2) Solve 2

2log (log )

dy y yy y

dx x x+ =

Solve each of the following differential equations:

(3) 3dy yy

dx x+ (4) 2

dyy

dx− secx = y3 tanx

(5) 2xdy yxe y

dx x+ = (6)

32 1/ 2( ) 0xxy e dx x y dy− − =

EQUATIONS SOLVABLE FOR YIf the given differential equation is expressible in the form ( , )y f x p= then we say that it is solvable for y.

Differentiating (i) with respect to x, we get , , ,dy dp dp

f x p or p f x pdx dx dx

= =

This equation contain two variables x and p. Solving this equation, we obtain( , , ) 0x p cφ =

The solution of differential equation (i) is obtained by eliminating pbetween (i) and (iii).

Following examples will illustrate the above procedure.

(1) Solve the differential equation y=(1+p)x+ap2, where P=dy

dx.

(2) Solve the differential equation x2p2 + xyp – 6y2 = 0(3) A right circular cone with radius R and height H contains a liquid which evaporates at a rateproportional to its surface area in contact with air (proportionally constant = k > 0).

Functions

INTERVALS:Let a and b two given real numbers such that a < b. Then the set of all real numbers x such that a x b≤ ≤ iscalled a closed interval and is denoted by [a, b].i.e. [a, b] = {x ∈ r |a ≤ x ≤ b}For example, [1, 2] = {x ∈ R| 1 ≤ x ≤ 2} i.e., the set of all real numbers lying between 1 and 2, includingthe end points.

OPEN INTERVAL: Let a and b be two given real numbers such that a < b. Then the set of all real numbersx such that a ≤ x b is called a closed interval and is denoted by (a, b).i.e (a, b) = {x ≤ R | a ≤ x ≤ b}

REAL FUNCTION:If the domain and co-domain of a function are subsets R9st of all real numbers). It is called a real valuedfunction or in short a real function.

EXAMPLES:

(i) If for non-zero x, a f(x) +b f1

x

=1

x-5, where a b≠ then find f(x).

(ii) Let g : R R→ be given by g(x) = 4x + 3. If ( ) .... ( ),n

n times

g x gogog og x−

= show that

gn(x) = 4nx + (4n-1).If g-n(x) denotes the inverse of gn (x), prove that ( ) 4 (4 1)n n ng x x− − −= + −

for all x ∈ R.

DOMAIN: Generally real functions in calculus are described by some formula and their domains are notexplicitly stated. In such cases to find the domain of a function f (say) we use the fact that domain is the setof all real numbers x for which f (x) is a real number.

In other words, determining the domain of a function f means finding all real numbers x for whichf(x) is real. For example, if f(x) = 2 ,x− then f(x) is real for allx ≤ 2. For x > 2, f(x) is not real. So, domain of f(x) is the set of all real numbers less than or equal to 2 i.e. (-∞ , 2]

EXAMPLE: Find the domain and range of function1

( )2 cos3

f xx

=−

.

SOME STANDARD REAL FUNCTIONS

CONSTANT FUNCTION: Let k be a fixed real number. Then a function f(x) given by f(x) =k for allx R∈ is called a constant function.

GREATEST INTEGER FUNCTION: For any real number x, we denote [x], the greatest integer less thanor equal to x.

PROPERTIES OF GREATEST INTEGER FUNCTION:If n is an integer and x is any real number between n and n+1, then the greatest integer function has thefollowing properties :(i) [-n] = -[n] (ii) [x + n] = [x] + n (iii) [-x] =-[x]-1

(iv) [x]=[-x]=1,

0 ,

if x Z

if x Z

− ∉ ∈

(v)2[ ] ,

2[ ] 1,

x if x Z

x if x Z

∈ + ∉

(vi) [x] ≥ n ⇒ x ≥ n, where n ∈ Z (vii) [x] ≤ n ⇒ x < n + 1, n∈ Z(viii) [x] > n ⇒ x ≥ n + 1, n∈ Z (ix) [x] < n ⇒x < n, n∈ Z

(x) [x+y] = [x] + [y=X-[X]] for all x, y ∈ R.

(xi) [x] +1 2 1

....n

x x xn n n

− + + + + + + =[nx], n ∈ N

(1) If [x] and [x] denote respectively the fractional and integral parts of a real numberx. Solve the equation 4[x] = x+[x]

(2) I f[x] and [x] denote the fractional and integral parts of x and (x) is defined as

follows2[ ] [ ], 0

( )[ ] 3[ ], 0

x x xx

x x x

− < + ≥

then solve the equation : (x) = x + {x}

SIGNUM FUNCTION:

The function defined by f(x) =| |

, 0

0 , 0

xx

xx

≠ =

(Or) f(x) =

1, 0

0 , 0

0 , 0

x

x

x

> = =

is called signum function.

RECIPROCAL FUNCTION: The function that associates each nonzero real number x to its reciprocal 1/xis called the reciprocal function.

LOGARITHMIC FUNCTION: If ‘a’ is a positive real number, then the function that associates everypositive real number to loga x i.e. f(x) = loga x is called the logarithmic function.EXPONENTIAL FUNCTION: If a is positive real number, then the function which associates every realnumber x to ax i.e. f(x) = ax is called the exponential function.

SQUARE ROOT FUNCTION: The function that associates every positive real number x to + x is called

the square root function, i.e., f(x) = + x .

POLYNOMIAL FUNCTION: A function of the form f(x) = aoxn +a1 xn-1 +…+an-1 x+an, where ao, a1, a2,

…..an are real numbers, ao ≠ 0 and n N∈ , is called polynomial function of degree n.

The domain of a polynomial function is always R.

RATIONAL FUNCTION: A function of the form f(x) =( )

,( )

P x

q xwhere p(x) and q(x) are polynomials and

q(x) ≠ 0, is called a rational function.

SUM Let f and g be two real functions with domain D1 and D2 respectively. Then, we define their sum f + gas that function from D1 ∩ D2 to R which associates each 1 2x D D∈ ∩ to the number f(x) + g(x).

Thus, f+g: D1 ∩ D2 → R such that (f+g) (x) =f(x) + g(x) for all 1 2x D D∈ ∩ .Similarly, we define the difference, product and quotient as follows:

DIFFERENCE f-g : D1 ∩ D2 → R such that (f-g) 9x) = f(x) –g(x) for all x ∈ D1 ∩ D2

PRODUCT 1 2:fg D D R∩ → such that (fg) (x)=f9x0 g(x) for all 1 2x D D∈ ∩

QUOTIENT 1 2: { | ( ) 0}f

D D x g x Rg

∩ − = → such that( )

( )( )

f f xx

g g x

=

for all 1 2 { | ( ) 0}.x D D x g x∈ ∩ − =

SCALAR MULTIPLE for any real number c, the function cf is defined by

(cf) (x) = c.f(x) for all x ∈ D1.

REMARK Note that the above operations are defined here are true only for real functions. For generalfunctions from one set to another, these do not make sense.

COMPOSITION OF FUNCTIONS: Let f and g be two functions with domain D1 and D2 respectively. Ifrange (f) ⊂ domain g (g), we define gof by the rule

(gof) (x) = g(f(x)) for all x ∈ D1.

Also, if range (g) ⊂ domain (f), we define fog by the rule (fog) (x) = f(g(x)) for all

2x D∈It follows from the above discussion that if f(x) and g(x) are two real functions with domains D1 and

D2 respectively. Then(i) Domain of 1 2( )f g D D± = ∩ (ii) Domain of 1 2( )fg D D= ∩

(iii) Domain of 1 2 { | 9 ) 0}f

D D x g xg

= ∩ − =

(1) For what real values of ‘a’ does the range of the function2

1( )

1

xf x

a x

−=− +

not contain any values

belonging to the interval [-1, -1/3] ?

(2) For what real values of ‘a’ does the range of the function f(x) =2

1

1

x

x a

−− −

not attain any value from the

interval [-1, 1]?

Fin the domains of definition of the following functions:

PERIODIC FUNCTIONS:

PERIOD If f(x) is a periodic function, then the smallest positive real number T is called the period orfundamental period of function f(x) if.

F(x+T) = f(x) for all .x R∈(1) Prove that the function (x) = x-[x] is a periodic function. Also find its period.(2) Let f(x) be a real valued function with domain R such that

f(x + p) = 1+[2-3 f(x) + 3 (f(x))2 – (f(x))3]1/3 hold good for all .x R∈ and some positive constant p, thenprove that f(x) is a periodic function.

SOME USEFUL RESULTS ON PERIODIC FUNCTIONS

RESULT 1 If f(x) is a periodic function with periodic. T and a, b, ∈ R such that 0a ≠ , then af(x) + b isperiodic with period T.

RESULT 2 If f(x) is a periodic function with period T and a, b ∈ R such that a ≠ 0, then f(ax+b0 is periodicwith period T |a|.

RESULT 3 Let f(x) and g(x) be two periodic functions such that :

Period of f(x) = ,m

nwhere m, n ∈ N and m, n are co-prime.

and,

Period of g(x) =r

s, where r N∈ and s N∈ are coprime 3

Then, (f+g) (x) is periodic with period T given by T =( , )

,( , )

LCM of m r

HCF of n sprovided that there does not

exist a positive number k < T for which f(k+x) = g (x) and g(k=x)=f(x), else k will be the period of (f+g)(x).

EXAMPLE Prove that f(x) = sin-1 (sinx) is a periodic function

EVEN FUNCTIONS A function f(x) is said to be an even function if f(-x) = f(x)for all x.

ODD FUNCTION A function f(x) is said to be an odd function if (-x) = -f(x) for all x.(1) If f is an even function defined on the interval [-5, 5], then find the real values of

x satisfying the equation f(x) = f1

2

x

x

+ +

.

(2) Extend f(x) = x2 + x defined in [0, 3] onto the interval [-3, 3] so that f(x)(i) even (ii) odd.

IIT-JEE Mathematics

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