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Preface

About the book


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IIT - MATHS

SET - 3


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INDEX

1. VECTOR ALGEBRA...........................................................................................

2. MATRIX AND DETERMINANTS .......................................................................

3. QUADRATIC EQUATIONS ...............................................................................

4. INDEFINITE INTEGRATION ...........................................................................

5. DEFINITE INTEGRATION ...............................................................................

6. AREAS UNDER CURVES ..................................................................................

7. DIFFERENTIAL EQUATIONS..........................................................................

2

24

52

78

106

128

158


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IIT-MATHS-SET-III

1 VECTOR ALGEBRA


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3

VECTORS_THEORY

A scalar is a quantity, which has only magnitude but does not have a direction. For ILLUS-

TRATION time, mass, temperature, distance and specific gravity etc. are scalars.

A Vector is a quantity which has magnitude, direction and follow the law of parallelogram

(addition of two vectors). For ILLUSTRATION displacement, force, acceleration are vec-

tors.

(i) There are different ways of denoting a vector : a

or a or a are different ways.

We use for our convenience a, b, c

etc. to denote vectors, and a, b, c to denote their

magnitude. Magnitude of a vector a

is also written as a

.

(ii) A vector a

may be represented by a line segment OA and arrow gives direction of this vector..

Length of the line segment gives the magnitude of the vector.

AO

Here is the initial point andis the terminal point of OA

O

A

ADDITION OF TWO VECTORS

Let OA a , AB b

and OB c

.

Here c

is sum (or resultant) of vectors a

and b

. It is to be noticed that the initial point of

coincides with the terminal point of and the line joining the initial point of to the terminal point

of represents vector in magnitude and direction.

a

b

ba c =

OA

TYPE OF VECTORS

(i) Equal Vectors

Two vectors are said to be equal if and only if they have equal magnitudes and same direction.


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IIT-MATHS-SET-III

A B

C DAB

As well as

(ii) Zero Vector (null vector)

A vector whose initial and terminal points are same, is called the null vector. For ILLUSTRA-

TION.

Such vector has zero magnitude and no direction, and denoted by 0

.

AB BC CA AA

orAB BC CA 0

C

BA

(ii i) Like and Unlike Vectors

Two vectors are said to be

(i) Like, when they have same direction.

(ii) Unlike, when they are in opposite directions.

and are two unlike vectors as their directions are opposite, and are like vectors.

(iv) Unit Vector


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VECTORS_THEORY

A unit vector is a vector whose magnitude is unity. We write, unit vector in the direction of as

. Therefore = .

(v) Parallel vectors

Two or more vectors are said to be parallel, if they have the same support or parallel support.

Parallel vectors may have equal or unequal magnitudes and direction may be same or oppo-site. As shown in figure

a

b

c

A

BC

D

(vi) Position Vector

If is any point in the space then the vector is called position vector of point , whereOis the

origin of reference. Thus for any points A and B in the space,

(vii) Coinitial vectors

Vectors having same initial point are called coinitial vectors. As shown in figure:

Here and are coinitial vectors.

O

d a

cC

D

b

SOME PROPERTIES OF VECTORS

(i) a b b a

(Vector addition is commutative)


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IIT-MATHS-SET-III

(ii) a b c a b c

(Vector addition is associative)

(iii) a b a b

, equality holds when a

and b

are like vectors

(iv) a b a b

, equality holds when a

and b

are unlike vectors

(v) m na mna n ma

(where m , n are scalars)

(vi) m a b ma mb

(where m , n are scalars)

(vii) m a b ma mb

(where m is a scalar)

COLLINEAR VECTORS

Two vectors are said to be collinear if and only if there exists a scalar m such that a mb

.

Thus

(i) any vector a

and zero vector are always collinear..

(ii) like and unlike vectors are collinear.

Note that xa yb 0 x y 0

if and only if a

and b

are noncollinear. Thus repre-

sentation of any vector as a linear combination of noncollinear vectors a

and b

is unique.

COPLANAR VECTORS

Three vectors a, b , c

are coplanar if there exists a relationxa yb zc

= 0

(where x, y, z are scalars, not all zero)

Thus,

(i) any two vectors a

and b

and a zero vector are always coplanar..

(ii) if any two of a

, b

and c

are collinear, then a

,b

and c

are coplanar..

(iii) there exists a plane which can contain all coplanar vectors.

Note thatxa yb zc 0

x = 0, y = 0, z = 0 if and only if vectors a

, b

and c

are

noncoplanar.

Any vector r , coplanar with noncollinear vectors a

and b

, can be expressed as a linear

combination of vectors a

and b

uniquely..

i.e., for same scalars mand n


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VECTORS_THEORY

B

O A

r

b

ma a

nb

Any vector r in space can be written as a linear combination of three noncoplanar vectors

a

,b

and c

uniquely..

i.e., r la mb nc

for some scalars l , m and n .

COLLINEARITY AND COPLANARITY OF POINTS

(i) The necessary and sufficient condition for three points with position vectors and to be

collinear is that there exist scalars x, y, z, not all zero, such that , where x + y + z = 0.

(ii) The necessary and sufficient condition for four points with position vectors and to be copla-

nar is that then exist scalars x, y, z and u, not all zero, such that , where x + y + z + u = 0.

SECTION FORMULA

LetA,Band Cbe three collinear points in space having position vectors a, b

and r

.

LetAC n

CB m

or, m AC nCB

or, m AC nCB

. . . (i)

(As vectors are in same direction)

Now,OA AC OC AC r a

. . . (ii)

. . . (iii)

Using (i), we get r =

ma nb

m n

ORTHOGONAL SYSTEM OF UNIT VECTORS

Let OX, OYand OZbe three mutually perpendicular straight lines. Given any point

P(x, y, z)in space, we can construct the rectangular parallelopied of whichOPis a diagonal and OA

=x, OB = y, OC = z.


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IIT-MATHS-SET-III

HereA, B, Care (x, 0, 0), (0, y, 0)and (0, 0, z)respectively andL, M, Nare (0, y, z),

(x, 0, z)and (x, y, 0)respectively.

Let i , j , kdenote unit vectors along OX, OY and OZ respectively..

We have r OP xi yj zk

as OA xi , OB yj

and OC zk

.

ON OA AN

OP ON NP

So, OP OA OB OC NP OC,AN OB

2 2 2r | r | OP x y z

2 2 2

r xi yj zk r

| r | x y z

= li mj nk

r lr i mr j nr k

2 2 2

xl cos

x y z

(where is the angle between OPand xaxis)

2 2 2

ym cos

x y z

( is the angle between OPand yaxis)

2 2 2

zn cos

x y z

( is the angle between OPand zaxis)

l,m,n are defined as the direction cosines of the lineOPand x, y, z are defined as direction ratios

of the line OP.

If P (x1, y

1, z

1) and Q (x

2, y

2, z

2) then PQ

= (x2 x

1) i + (y2 y1)

j + (z2 z1)k


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VECTORS_THEORY

Therefore PQ = 2 2 2

2 1 2 1 2 1x x y y z z

Hence direction ratios of the line through P and Q are x2 x

1, y

2 y

1and z

2 z

1and its

direction cosines are2 1 2 1x x y y,PQ PQ

and

2 1z z

PQ

.

SOME PROPERTIES OF DIRECTION COSINES AND RATIOS

(i) lr,mr,nrare the projection of r on x, y and zaxis.

(ii) r = l i m j n k

(iii) 2 2 2l m n = 1

(iv) If a, b and c are three real numbers such thatl m n

a b c , then a, b, c are the direction

ratios of the line whose direction cosines are l ,m and n .

SCALAR PRODUCT OF TWO VECTORS (DOT PRODUCT)

The scalar product, a.b of two nonzero vectors a

and b

is defined as a b

cos,

where is angle between the two vectors, when drawn with same initial point.

Note that 0 .

If at least one of a

andb

is a zero vector, then a.b

is defined as zero.

PROPERTIES

(i) a. b b . a

(scalar product is commutative)

(ii) 22 2

a a.a a a

(iii) ma .b m a .b a. mb

(where m is a scalar)

(iv)1 a. b

cosa . b

(v) a.b 0

Vectors a

and b

are perpendicular to each other..

[ a

, b

are nonzero vectors].

(vi) i . j = j.k k .i = 0

(vii) a. b c a.b a.c


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IIT-MATHS-SET-III

(viii) 22 2 2

a b . a b a b a b

(ix) Let1 2 3 1 2 3

a a i a j a k , b b i b j b k

Then 1 2 3 1 2 3 a. b a i a j a k . b i b j b k

= 1 1 2 2 3 3a b a b a b

* Algebraic projection of a vector along some other vector

a.b a.bON OBcos b

| a |a b

VECTOR (CROSS) PRODUCT

The vector product of two nonzero vectors a

and b

, whose module are a and b respec-

tively, is the vector whose modulus is ab sin , where 0 is the angle between

vectors a

and b

. Its direction is that of a vector n

perpendicular to both a

and b

, such

that a, b,n

are in righthanded orientation.

By the righthanded orientation we mean that, if we turn the vector a

into the vector b

through the angle, then n

points in the direction in which a right handed screw would move

if turned in the same manner.

Thus a b a b

sin n

If at least one of a

and b

is a zero vector, then a b

is defined as the zero vector..

PROPERTIES

(i) a b b a

(ii) ma b m a b a mb

(where mis a scalar)


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VECTORS_THEORY

(iii) a b 0

vectors a

and b

are parallel. (provided a

and b

are nonzero vec-

tors).

(iv) i i j j k k 0

.

(v) i j k j i , j k i k j ,k i j i k .

(vi) a b c a b a c

.

(vii) Let a

= 1a i + 2 3

a j 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

.

= 2 3 3 2 3 1 1 3 1 2 2 1 i a b a b j a b a b k a b a b

(viii) sin =

a b

a b

. (Note : we cannot find the value of by using this formula)

(ix) Area of triangle =1 1 1

ap ab sin a b2 2 2

(x) Area of parallelogram = ap ab sin a b

.

SCALAR TRIPLE PRODUCT

The scalar triple product of three vectors a, b

and c

is defined as a b .c

Let a

= a1i

+ a2j + a3k

, b

= b1i

+ b2 j + b3k

, c

= c1 i + c2 j + c3k

.


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IIT-MATHS-SET-III

Then2 3 1 3 1 2

1 2 32 3 1 3 1 2

1 2 3

i j ka a a a a a a b a a a i j k b b b b b b

b b b

1 2 3

2 3 1 3 1 21 2 3 1 2 3

2 3 1 3 1 21 2 3

a a a

a a a a a aa b .c c c c b b bb b b b b b

c c c

Therefore a b .c b c .a c a .b b a .c c b .a a c .b

Note that a b .c b c .a a. b c

, hence in scalar triple product dot and cross are inter-

changeable. Therefore we denote a b .c

by a b c

.

PROPERTIES

(i) a b . c

represents the volume of the parallelopied, whose adjacent sides are rep-

resented by the vectors a,b

and c

in magnitude and direction. Therefore three vectors

a,b , c

are coplanar if and only if a b c

= 0 i.e.,

1 2 3

1 2 3

1 2 3

a a a

b b b

c c c= 0

(ii) Volume of the tetrahedron =1

a b c6

.

(iii) a b c d a c d b c d

(iv) a a b

= 0.

VECTOR TRIPLE PRODUCT

The vector triple product of three vectors a,b

and c

is defined as a b c

. If at least

one of a,b

and c

is a zero vector or b

and c

are collinear vectors ora

is perpendicular

to both b

and c

, only then a b c 0

. In all other cases a b c

will be a nonzero

vector in the plane of noncollinear vectors b

and c

and perpendicular to the vectora

.

Thus we can take a b c b c

, for some scalarsand. Since a a b c

,

a. a b c 0 a. b a. c 0


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VECTORS_THEORY

a. c , a. b ,

for same scalar .

Hence a b c a. c b a.b c

, for any vectors a,b

and c

satisfying the

conditions given in the beginning.

In particular if we take, a b i , c j

, then = 1.

Hence a b c a. c b a. b c

RECIPROCAL SYSTEM OF VECTORS

Let a,b

and c

be a system of three noncoplanar vectors. Then the system of vectors

a , b

and c

which satisfies a.a b.b c .c

= 1 and

a. b a. c b. a b . c c . a c . b

= 0, is called the reciprocal system to the vectors

a,b , c

. In term of a,b , c

the vectors a ,b , c

are given by a

=

b c c a a b,b ,c

a b c a b c a b c

.

PROPERTIES

(i) a.b a.c b.a b.c c.a c.b = 0

(ii) The scalar triple product [a b c] formed from three noncoplanar vectors a, b, c is

the reciprocal of the scalar triple product a b c formed from reciprocal system.

ANGLE BETWEEN TWO LINES

Let the vector equations of two lines be r a b

and r c d

. These two lines are

parallel to the vectors 1 1 1 b a i b j c k

and 2 2 2 d a i b j c k

respectively. Therefore,

angle between these two lines is equal to the angle betweenb

andd

. Thus, if is the angle

between the lines, then

b. dcos

b d

.

InCartesian Form it is given as 1 2 1 2 1 2cos l l m m n n

1 2 1 2 1 2

2 2 2 2 2 21 1 1 2 2 2

a a b b c c

a b c a b c

, where 1 1 1l ,m ,n and 2 2 2l ,m ,n are direction cosines

and 1 1 1a ,b ,c and 2 2 2a ,b ,c are direction ratios of the given lines.

* If the lines are perpendicular, then b.d

= 0. (vector form)


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IIT-MATHS-SET-III

i.e., 1 2 1 2 1 2l l m m n n = 0 or 1 2 1 2 1 2a a b b c c = 0 (Cartesian form)

* If the lines are parallel, then b

andd

are parallel, therefore b

= d

for some scalar .

i.e.,1 1 1

2 2 2

a b c

a b c .


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VECTORS_THEORY


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IIT-MATHS-SET-III

ASSIGNMENT


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VECTOR ALGEBRA

MULTIPLE ANSWER QUESTIONS

1. The vectors li+j+ 2k, i+ lj kand 2ij+ lkare coplanar if

a)= -2 b) = 1 + 3 c) 1 - 3 d) = 0

2. The values of x for which the angle between the vectors a = xi 3j kand b = 2xi+ xj kis acute, and

the angle between the vector band the axis of ordinates is obtuse, are

a) 1, 2 b) 2, -3 c) all x < 0 d) all x > 0

3. Let a= 2ij+ k, b= i+ 2j kand c= i+j 2kbe three vectors. A vector in the plane of band c

whose projection on ais of magnitude 3/2 is

a) 2i+ 3j 3k b) 2i+ 3j+ 3k c) 2ij+ 5k d) 2i+j+ 5k

4. Let a= 4i+ 3jand b be two vectors perpendicular to each other in xy-plane. The vectors c in the same

plane having projections 1 and 2 along a and care

a) -3

2i+

2

11j b) 2ij c) -

5

2i+

5

11j d)

3

2i+

2

11j

5. Let the unit vectors Aand Bbe perpendicular and the unit vector Cbe inclined at an angle q to both A

and B. If C = aA+ bB+ g(Ax B) then

a)= b)

2

= 1 - 2

2

c)

2

= - cos 2 d)

2

=

1 cos 2

2

6. The point of intersection of the lines l1: r(t) = (i 6j+ 2k) + t(i+ 2j+ k)

l2: R(u) = (4j+ k) + u (2i+j+ 2k)

a) at the tip of r(7) b) at the tip of R(4) c) (8, 8, 9) d) at the tip of R(2)

7. If a, bcare three unit vectors such that ax (bxc) =2

1band cbeing non parallel then

a) angle between aand bis p/2 b) angle between aand cis p/4

c) angle between aand cis p/3 d) angle between aand bis p/3

8. If a, band cbe non-coplanar unit vectors equally inclined to one another at an acute angle q. Ifax b+

bx r= pa+ qb+ rcthen

a) p = r b) p =

cos21

cos2q,

cos21

1

c) r = cos21

1d) p =

cos21

cos2q

9. The vectors ai + 2aj 3ak, (2a + 1) i+ (2a + 3)j+ (a + 1) kand (3a + 5) i+ (a + 5)j+ (a + 2) kare

non-coplanar for a in

a) {0} b) (0, ) c) (-, 1) d) (1, )

SECTION-B


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IIT-MATHS-SET-III

10. If K is the length of any edge of a regular tetrahedron then the distance of any vertex from the opposite

face is

a) 3/2 K b)2

3K2 c)

2

3K d) 3 K

11. Two sides of a triangle are formed by the vectors a = 3i + 6j 2k and b = 4i j + 3k. Acute angles ofthe triangle are

a) cos-17

75b) cos-1

26

27c) cos-1

3

15d) cos-1

2

3

12. If the unit vectors aand bare inclined at an angle 2 q and < 1, then if 0 q p, q lies in the interval

a) 0,6

b)5

,6

c) ,6 2

d)5

,2 6

13. If (ax b) x (cx d) = ha+ kb= rc+ rdwhere a and b are non-collinear and c and d are also non-collinear

then

a) h = [b c d] b) k = [a c d] c) r = [a b d] d) s = [a b c]


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VECTOR ALGEBRA

PASSAGE TYPE QUESTIONS

PASSAGE 1:

Let C: r(t) = x(t) i+ y(t)j+ z(t) k

Be a differentiable curve i.e h

hhtlim0x

rr

exist for all t. The vector

R(t) = x(t)i + y(t)j + z(t) k

If not O, is tangent to the curve C at the point P(x((t), y(t), z(t)) and r(t) points in the direction of

increasing t.

1. The point P on the curve r(t) = (1 2t) i+ t2j+ 2e(t-1)k

at which the tangent vector r (t) is parallel to the radius vector r(t) is

a) (-1, 1, 2) b) (1, -1, 2) c) (-1, 1, 2) d) (1, 1, 2)

2. A parametrized tangent vector to r(t) = ti+ t2j+ t3kat (2, 4, 8) is

a) R(u) = 2i+ 4j+ 8k+ u(i+j+ 4k) b) R(u) = i+ 2j+ 4k+ u(i+ 4j+ 12k)

c) R(u) = i+ 4j+ 12k+ u(2i+ 4j+ 8k) d) R(u) = 2i+ 4j+ 8k+ u (i+ 4j+ 12k)

3. The tangent vector to r(t) = 2t2i+ (1 t)j+ (3t2+ 2) kat (2, 0, 5) is

a) 4i+j 6k b) 4ij+ 6k c) 2ij+ 6k d) 2i+j 6k

PASSAGE 2:

Equation of a line can be obtained as the intersection of two planes, or passing through a point and

parallel to given plane. Similarly equation a plane can be obtained having different condition e.g. passing through

three points or through a point and perpendicular to two planes.

4. The line through the point c, parallel to the plane r.n= 1 and perpendicular to the line, r= a+ tbis

a) r= c+ tax n b) r= c+ tbx n c) r= c+ tn d) r= a+ t(cx n)

5. The line through the point aand parallel to the planes r.n1= q

1, r.n

2= q

2is

a) r= a+ tn1

b) r= a+ t(n1 n

2) c) r= a+ tn

2d) r= a+ t(n

1x n

2)

6. The plane which passes through the two points aand band is perpendicular to the plane r.n= q is

a) r.((b a) x n) = q b) r. (a b) = q

c) r.((b a) x n) = [abn] d) r.((b a) xn) = [a n b]

7. The plane which passes through aand is perpendicular to the plane r.n= q and is parallel to the line r=

b+ tcis

a) r.b= [a n c] b) [r n c] = [a n c] c) r.a= [b n c] d) [r c n] = [a n c]

SECTION-C


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IIT-MATHS-SET-III

SECTION-D

MATCHING TYPE QUESTIONS

1. If aand bare two units vectors inclined at angle a to each other then

i) ba < 1 if a)3

2 < <

ii) baba if b) /2 <

iii) 2 a b c)= /2

iv) ba < 2 d) 0 q < p/2

2. a, b, c, dare given vectors. Match solution of equation in coloumn 1 to its solution in column 2.

i) rx a+ (r. b) c= d a) r= c-ba

ac

.

.b

ii) r= rx a+ b b) r=aa

ra

.

.a+ ax

iii) rx b= cx b, r. a= 0 c) r= - 21

b(ax b) + yb, y is a parameter

iv) rxb= a, where a, b are such that ais perpendicular to b

= 21 .

1x

a ba b a b

a

3. Given two vectors a= i+j- k, b= ij+ k, c= i+ 2j k

i) a vector perpendicular to the vector aand coplanar with aand b a) (1/ 2 ) (-i+ 2j+ 3k)

ii) a vector perpendicular to aand the vector in (i) b) 1/ 2 (i+ k)

iii) a vector perpendicular to band c c)6

1(2i+j+ k)

iv) a vector perpendicular to aand a+ c d) (1/ 2 ) (j+ k)

4. The area / volume of

i) triangle with vertices whose position vector w.r.t O is i + 2j + 3k, 2i j k, i + j - k a) 3 /2

ii) tetrahedra with vertices O, i+j k, ij+ k, -i+j+ k b) 2/3

iii) tetrahedra with vertices i+k, 2ij, i+ 2j+ 5k, i+ 2j+ k c) 89 /2

iv) triangle with vertices i,j, i+j+ k d) 6

5. Let a, b, cbe any three vector then match the following vectors

i) ax (b+ c) + bx (c+ a) a) [c b a] a

ii) ax (bx c) + bx (cx a) b) [a c b] b

iii) (axb) x (cxb) c) (ax b) x c

iv) (ax b) x (cx a) d) (a+ b) x c


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VECTOR ALGEBRA

WRITE THE ANSWER

1. If d= (ax b) + (bx c) + r(cx a),[a b c] = 1/8 and d. (a+ b+ c) = 8 then + + r is equal

to.

2. b= 4i+ 3jand cbe two vectors perpendicular to each other in the xy-plane. If ri, i = 1,2

........n are the vectors in the same plane having projections 1 and 2 alongband crespectively

then

n

1i

2ir is equal to .

3. If A= (1, 1, 1) and C= (0, 1,-1) are given vectors andBis a vector satisfying Ax B= Cand

A.B= 3 then 9 2B is equal to.

4. If [b c d] = 24 and (ax b) x (cx d) + (ax c) x (dx b) + (ax d) x (bx c) + ka= 0 then k is

equal to.

5. Suppose that a, b, cdo not lie in the same plane and are non zero vectors such that

a = 1, b - 2, c = 2, a. b= 1, b. c= 2 and the angle between a band b cis p/6. If d

is any vector such that d. a= d. b= d. cand 2d =2

k for any scalar a, then k is equal

to.

6. If a= i+ 2i 3k, b= 2i+j kand uis a vector satisfying axu=axband a. u= 0 then 2

2u is equal to.

7. If the vectors ai + j + k, i + bj + k and i + j + ck, )a b c 1) are coplanar, find the value of

c1

1

b1

1

a1

1

is

8. Anon zero vector ais parallel to the line of intersection of the plane determined by the vectors

i, i+jand the plane determined by the vectors ij, i- k. If the acute angle between a and the

vector i 2i+ 2kis q find 2 cos q

9. Let OA= a, OB= 10a+ 2band OC = b, where O, A and C are non-collinear points. Let p

denote the area of the quadrilateral OABC, and let q denote the area of the parallelogram

with OAand OCas adjacent sides. If p = kp find K.

10. Let a= ij, b= i+ 2j+ 2k, c= 2i+j+ 2kand d= 2ij+ k. If p is the shortest distance

between the lines

r= a+ t band r= c+ p dfind 2p2

11. If a, c, dare non-coplanar vectors satisfyingd.(ax (bx(cx d))) = k [a c d] and b. d= 14 find

k.

SECTION-E


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IIT-MATHS-SET-III

12. A particle is displacement from the point whose position vector is 5i 7j 7kto the point

whose position vector is 6i+ 2j 2kunder the action of constant forces 10ij+ 11k, 4i+

5j+ 6kand 2i+j 9k. Find the total work done .

13. In a triangle ABC, a point P is taken on the side AB such that AP: BP such that CQ : BQ = 2

: 1.If R is the point of intersection of lines AQ and CP. Suppose that the area of the triangle

ABC is D, if it is known that the area of triangle BRC is one unit. Find the value of 4 D

14. ABCD is a regular hexagon. If AB = 4 units find FCEBAD

15. In a DABC, the median CM is perpendicular to the angle bisector AL and CM and CM : AL

= 1: 3 find 85 cos A.

SECTION-B KEY

SECTION-C KEY

1 2 3 4 5 6 7 8

a,b,c b,c a,c b,c a,b,c,d a,b,c a,c a,b,c

9 10 11 12 13

b,d c a,b a,b b,c

1 2 3 4 5 6 7

a d b b d c b


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VECTOR ALGEBRA

SECTION-E KEY

1 2 3 4 5 6 7 8

64 20 33 48 3 5 1 1

9 10 11 12 13 14 15

6 1 14 97 17 16 77

SECTION-D KEY

1. (i) (a), (ii) (c), (iii) (b), (iv) (d)

2. (i) (b), (ii) (d), (iii) (a), (iv) (c)

3. (i) (c), (ii) (d), (iii) (a), (iv) (b)

4. (i) (c), (ii) (b), (iii) (d), (iv) (a)

5. (i) (d), (ii) (c), (iii) (b), (iv) (a)


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IIT - MATHS - SET - III

2 MATRIX ANDDETEMINANTS


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MATRICES AND DETERMINANTS

Matrix

A system of m n numbers arranged in the form of an ordered set of m rows and n columas

is called an m n matrix. It can be read as m by n matrix.It is represents as A = [aij]

m nand

can be written in expanded form as

11 12 1n

21 22 2n

m1 m2 mn

a a . . . aa a . . . a

Aa a . . . a

e.g.,

2 1 0

1 1 3

6 5 1

is a 3by 3 matrix.

DIFFERENT TYPES OF MATRICES

(i) Square Matrix: A matrix for which the number of rows is equal to the number of

columns (each equal to n) is called a square matrix of order n.

e.g. A =

1 0 2 3

2 1 4 5

3 2 4 1

1 0 0 2

is a square matrix of order 4.

(ii) Null Matrix: The matrix whose all elements are zero is called null matrix or zero matrix.

It is usually denoted by O.

(iii) Identity Matrix :A square matrix in which all the elements along the main diagonal

(elements of the from aii) are unity is called an identity matrix or a unit matrix. An identity

matrix of order n is denoted by In.

e.g. I3=

1 0 0

0 1 00 0 1

(iv) Scalar matrix : A matrix whose diagonal elements are all equal and other entries are

zero, is called a scalar matrix

e.g.k 0 0

A 0 k 00 0 k

(v) Triangular Matrix: A square matrix whose elements above the main diagonal or below

the main diagonal are all zero is called a triangular matrix.


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Note: (i) [aij]

n nis said to be upper triangular matrix if i > j aij= 0,

(ii) [aij]

n nis said to be lower triangular matrix if i < j aij= 0.

e.g.

1 2 1 3

0 2 1 0

0 0 3 10 0 0 3

is an upper triangular matrix.

(vi) Diagonal Matrix: A square matrix of any order with zero elements every where, ex-

cept on the main diagonal, is called a diagonal matrix.

e.g.

3 0 0

0 5 0

0 0 1

is a diagonal matrix of order 3.

(vii) Row Matrix: A matrix which has only one row and n columns is called a row matrix of

length n

e.g., [2 1 3 0]14

is a row matrix of length 4.

(viii) Column Matrix: A matrix which has only one column and m rows is called a column

matrix of length m.

e.g.

13

x2

1

is a column matrix of length 4.

(ix) Sub Matrix: A matrix obtained by omitting some rows or some columns or both of a

given matrix A is called a sub matrix of A.

e.g., If

2 0 4

A 5 6 8

3 2 29

, then 2 05 6

is a submatrix of A which is obtained by omitting

third row and third column of A.

EQUALITY OF TWO MATRICES

Two matrices A = [aij] and B = [b

ij] are said to be equal if they are of the same order and their

corresponding elements are equal. If two matrices A and B are equal, we write A = B.

ADDITION OF MATRICES

If A and B are two matrices of the same order m n, then their sum is defined to be the matrix

of order m n obtained by adding the corresponding elements of A and B.


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e.g. If

11 12 13 11 12 13

21 22 23 21 22 23

31 32 33 31 32 33

a a a b b b

A a a a and B b b b

a a a b b b

,

then

11 11 12 12 13 13

21 21 22 22 23 23

31 31 32 32 33 33

a b a b a b

A B a b a b a b

a b a b a b

PROPERTIES OF MATRIX ADDITION

(i) Matrix Addition is Commutative

If A and B are two m n matrices, then A + B = B + A.

(ii) Matrix addition is associative

If A, B, C are three matrices, each of the order m n, then (A + B) + C = A + (B + C).

(iii) Existence of additive identity

If O is the m n null matrix, then A + O = A = O + A for every m n matrix A. O is called

additive identity.

MULTIPLICATION OF A MATRIX BY A SCALAR

If A = [aij]

m nand is a scalar, then A = [aij]m n

e.g.

2 1 3

A 2 5 1

0 1 2

3A =6 3 96 15 3

0 3 6

.

MULTIPLICATION OF TWO MATRICES

Let A = [aij]

m n and B = [b

jk]

npbe two matrices such that the number of columns in A is equal

to number of rows in B. Then the m p matrix C = [cik]

m p,

where cik=

n

ij jk j 1 a .b (where i = 1, 2, 3 ... m, k = 1, 2, 3 ... p), is called the product of thematrices A and B. We have

11 12 1p 11 12 1p11 12 1n

21 22 2p 12 22 2p21 22 2n

n1 n2 np m1 m2 mpm1 m2 mn m n n p m p

b b ... b c c ... ca a ... a

b b ... b c c ... ca a ... a

... ... ... ... ... ... ... ...... ... ... ...

b b ... b c c ... ca a ... a

,

where Cij

= ai1

b1j

+ ai2

b2j

+ ... + ain

bnj

=n

ik kj

k 1

a b


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Properties of Matrix Multiplication

(i) Matrix multiplication is associative

i.e., (AB)C = A(BC), A, B and C are m n, n p and p q matrices respectively.

(ii) Multiplication of matrices is distributive over addition of matrices

i.e., A(B + C) = AB + AC

(iii) Existence of multiplicative identity of square matrices.

If A is a square matrix of order n and In is the identity matrix of order n, then

A In = I

nA = A.

(iv) Whenever AB and BA both exist, it is not necessary that AB = BA.

(v) The product of two matrices can be a zero matrix while neither of them is a zero matrix.

e.g., If A =0 1 1 0 0 0

and B then AB0 0 0 0 0 0

, while neither A nor B is a null ma-

trix.

(vi) In the case of matrix multiplication of AB = 0, then it doesnt necessarily imply that A =

0 or B = 0 or BA = 0.

TRACE OF A MATRIX

Let A be a square matrix of order n. The sum of the diagonal elements of A is called the traceof A.

Trace (A) =n

ii 11 22 nmi 1

a a a ... a .

TRANSPOSE AND CONJUGATE OF A MATRIX

The matrix obtained from any given matrix A, by interchanging rows and columns, is called the

transpose of A and is denoted by A or AAT.

e.g., If

3 2

1 2

A 4 5

7 8

, then2 3

1 4 7A

2 5 8

Properties of Transpose of a matrix

(i) (A ) A (ii) (A B) A B

(iii) ( A) A , being any scalar (iv) (AB) B A


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Conjugate of a Matrix

The matrix obtained from any given matrix A, containing complex number as its elements, on

replacing its elements by the corresponding conjugate complex numbers is called the conju-

gate of A and is denoted by A .

e.g.,1 2i 2 3i 3 4i

A 4 5i 5 6i 6 7i

8 7 8i 7

,then1 2i 2 3i 3 4i

A 4 5i 5 6i 6 7i

8 7 8i 7

Properties of Conjugate of a matrix

(i) A A (ii) A B A B

(iii) A A, being any scalar (iv) (AB) A B

Transpose Conjugate of a Matrix

The transpose of the conjugate of a matrix A is called transposed conjugate of A and is

denoted by A . The conjugate of the transpose of A is the same as the transpose of the

conjugate of A

i.e., A A A

If A = [aij]

m n, then ji n mA [b ]

, where bji= ija

i.e., the (j, i)thelement of A = the conjugate of (i, j)th element of A.

e.g., If

1 2i 2 3i 3 4i 1 2i 4 5i 8

A 4 5i 5 6i 6 7i , then A 2 3i 5 6i 7 8i

8 7 8i 7 3 4i 6 7i 7

Properties of Transpose Conjugate of a matrix

(i) (A ) A (ii) (A B) A B

(iii) (kA) kA , k being any scalar (iv) (AB) B A

SPECIAL MATRICES

Symmetric Matrix

A matrix which is unchanged by transposition is called a symmetric matrix. Such a matrix is

necessarily square

e.g.,

2 1 3

1 4 1

3 1 5

Thus if A = [aij]

m nis a symmetric matrix then m = n, a

ij= a

jii.e., A A .


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Skew Symmetric Matrix

A square matrix A = [aij] is said to be skew symmetric, if a

ij= a

jifor all i and j

e.g.

0 2 3 1

2 0 4 3

3 4 0 1

1 3 1 0

Thus if A = [aij]

m nis a skew symmetric matrix, then m = n, a

ij= a

ji i.e., A A .

Obviously diagonal elements of a square matrix are zero.

Orthogonal Matrix

A square matrix A is said to be orthogonal, if AA = A A I , where I is a unit matrix.

Note: (i) If A is orthogonal, then A is also orthogonal.

(ii) If A and B are orthogonal matrices then AB and BA are also orthogonal matrices.

Unitary Matrix

A square matrix A is called unitary matrix if AA A A I .

Idempotent Matrix: A square matrix A is called idempotent provided it satisfies the relation A2=

A.

e.g. The matrix2 2 4

A 1 3 41 2 3

is idempotent as

22 2 4 2 2 4 2 2 4

A 1 3 4 1 3 4 1 3 41 2 3 1 2 3 1 2 3

Involutary Matrix

A matrix A such that A2= I, is called involutary matrix.

Nilopotent Matrix

A square matrix A is called a nilpotent matrix, if there exists a positive integer m such that Am

=O. If m is the least positive integer such that Am= O, then m is called the index of the

nilpotent matrix A.


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DETERMINANT

Equations a1x + b

1y = 0 and a

2x + b

2y = 0 in x and y have a unique solution if and only if

a1b

2a

2b

1 0. We write a1b2 a2b1as

1 1

2 2

a b

a band call it a determinant of order 2.

Similarly the equations a1x + b

1y + c

1z = 0, a

2x + b

2y + c

2z = 0 and a

3x + b

3y + c

3z = 0 have

a unique solution if a1(b

2c

3 b

3c

2) + b

1(a

3c

2 a

2c

3) + c

1(a

2b

3 a

3b

2) 0

i.e., 1 1 1

2 2 2

3 3 3

a b c

a b c 0

a b c

The number ai, bi, ci (i = 1, 2, 3) are called the elements of the determinant.

The determinant obtained by deleting the ith row and jth column is called the minor of the

element at the ith row and jth column. We shall denote it by Mij. The cofactor of this element

is (1)i+j Mij, denoted by C

ij.

Let A = [aij]

33be a matrix, then the corresponding determinant (denoted by det A or | A |) is

11 12 13

21 22 23

31 32 33

a a a

a a a

a a a

.

It is easy to see that | A | = a11

C11

+ a12

C12

+ a13

C13

(we say that we have expanded the

determinant | A | along first row). Infect value of | A | can be obtained by expanding it along any

row or along any column. Further note that if elements of a row (column) are multiplied to the

cofactors of other row (column) and then added, then the result is zero.

PROPERTIES OF DETERMINANTS

(i) The value of a determinant remains unaltered, if its rows are changed into columns and the

columns into rows.

e.g.,

1 1 1 1 2 3

2 2 2 1 2 3

3 3 3 1 2 3

a b c a a a

a b c b b b

a b c c c c

. Thus any property true for rows will also be true for col-

umns.

(ii) If all the elements of a row (or column) of a determinant are zero, then the value of the

determinant is zero.


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e.g.,

1 1 1 1 1

2 2

3 3 3 3 3

0 b c a b c

0 b c 0, 0 0 0 0

0 b c a b c

(iii) If any two rows (columns) of a determinant are identical, then the value of the determi-

nant is zero.

e.g.,

1 1 1

2 2 2

3 3 3

a a c

a a c 0

a a c

(iv) The interchange of any two rows (columns) of a determinant results in change of its sign

i.e,

1 1 1 1 1 1

2 2 2 2 2 2

3 3 3 3 3 3

a b c b a c

a b c b a c

a b c b a c

(v) If all the elements of a row (column) of a determinant are multiplies by a non-zero con-

stant, then the determinant gets multiplied by that constant.

e.g.,1 1 1 1 1 1

2 2 2 2 2 2

3 3 3 3 3 3

a kb c a b c

a kb c k a b c

a kb c a b c

and1 1 1 1 1 1

2 2 2 2 2 2

3 3 3 3 3 3

a b c ka kb kc

k a b c a b c

a b c a b c

(vi) If each element of a row (column) of a determinant is a sum of two terms, then determi-

nant can be written as sum of two determinant in the following way:

1 1 1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3 3 3

a b c d a b c a b d

a b c d a b c a b d

a b c d a b c a b d

In general

n n n

r 1 r 1 r 1n

2 2 2 2 2 2r 1

3 3 3 3 3 3

f (r) g(r) h(r)f (r) g(r) h(r)

a b c a b c

a b c a b c

(vii) The value of a determinant remains unaltered under a column operation of the form

i i j k C C C C ( j, k i) or a row operation of the form

i i j k R R R R ( j,k i).


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e.g.

1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3

a b c a b 2a 3c c

a b c a b 2a 3c c

a b c a b 2a 3c c

, obtained after C2 C2+ 2C1+ 3C3.

(viii) Product of two determinants

1 1 1 1 2 3

2 2 2 1 2 3

3 3 3 1 2 3

a b c

a b c m m m

a b c n n n

l l l

1 1 1 1 1 1 1 2 1 2 1 2 1 3 1 3 1 3

2 1 2 1 2 1 2 2 2 2 2 2 2 3 2 3 2 3

3 1 3 1 3 1 3 2 3 2 3 2 3 3 3 3 3 3

a b m c n a b m c n a b m c n



l l l

l l l

l l l

(row by column multiplication)

1 1 1 2 1 3 1 1 1 2 1 3 1 1 1 2 1 3

2 1 2 2 2 3 2 1 2 2 2 3 2 1 2 2 2 3

3 1 3 2 3 3 3 1 3 2 3 3 3 1 3 2 3 3

a b c a m b m c m a n b n c n



l l l

l l l

l l l

(row by row multiplication)

We can also multiply determinants column by row or column by column.

(ix) Limit of a determinant

Letx a x a x a

x a x a x a x a

x a x a x a

lim f(x) lim g(x) lim h(x)f (x) g(x) h(x)

(x) (x) m(x) n(x) , then lim (x) lim l(x) lim m(x) lim n(x) ,

u(x) v(x) w(x) lim u(x) lim v(x) lim w(x)

l

provided each of nine limiting values exist finitely.

(x) Differentiation of a determinant

Let

f (x) g(x) h(x)

(x) l(x) m(x) n(x) ,

u(x) v(x) w(x)

then

f (x) g (x) h (x) f (x) g(x) h(x) f (x) g(x) h(x)

(x) (x) m(x) n(x) (x) m (x) n (x) (x) m(x) n(x)u(x) v(x) w(x) u(x) v(x) w(x) u (x) v (x) w (x)

l l l


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(xi) Integration of a Determinant

Let

f (x) g(x) h(x)

(x) a b c

l m n

, where a, b, c, l, m and n are constants,

then

b b b

a a ab

a

f (x)dx g(x)dx h(x)dx

(x)dx a b c

m n

l

Note that if more than one row (column) of (x) are variable, then in order to find

b

a

(x)dxfirst we evaluate the determinant (x) by using the properties of determinants and then weintegrate it .

SPECIAL DETERMINANTS

(i) Skew symmetric Determinant

A determinant of a skew symmetric matrix of odd order is zero.

e.g.,

0 b c

b 0 a 0c a 0

(iii) Circulant Determinant

A determinant is called circulant if its rows (columns) are cyclic shifts of the first row (columms).

e.g.,

a b c

b c a

c a b

. It can be show that its value is (a3+ b3+ c3 3abc) .

(iv)2 2 2

1 1 1

a b c (a b) (b c) (c a)

a b c

(v)3 3 3

1 1 1

a b c (a b) (b c) (c a) (a b c)

a b c


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(iv)2 2 2

3 3 3

1 1 1

a b c

a b c= (a b) (b c) (c a) (ab + bc + ca)

INVERSE OF A SQUARE MATRIX

Let A be any nrowed square matrix. Then a matrix B, if exists, such that AB = BA = In, iscalled the inverse of A. Inverse of A is usually denoted by A1 (if exists).

We have |A| In= A(adjA)

|A| AA1= (adjA). Thus the necessary and sufficient condition for a square matrix A to

possess the inverse is that |A| 0 and then AA1 =Adj(A)

| A |A square matrix A is called non-

singular if |A| 0. Hence a square matrix A is invertible if and only if A is non-singular.

Properties of Inverse of a Matrix

(i) Every invertible matrix possesses a unique inverse.

(ii) If A and B are invertible matrices of the same order, then AB is invertible and

(AB)1= B1 A1.

(iii) If A is an invertible square matrix, then AT is also invertible and (AT)1= (A1)T.

(iv) If A is a non-singular square matrix of order n, then |adjA| = |A|n1

(v) If A and B are non-singular square matrices of the same order, then

adj (AB) = (adj B) (adj A)

SYSTEM OF LINEAR SIMULTANEOUS EQUATIONS

Consider the system of linear non-homogenoeus simultaneous equations in three unknowns x,

y and z, given by a1x + b

1y + c

1z = d

1, a

2x + b

2y + c

2z = d

2and a

3x + b

3y + c

3z = d

3,

Let

1 1 1 1

2 2 2 2

3 3 3 3

a b c x d

A a b c , X y , B d ,a b c z d

Let | A | =

1 1 1 1 1 1

2 2 2 x 2 2 2

3 3 3 3 3 3

a b c d b c

a b c , d b c

a b c d b c

, obtained on replacing first column of by

B.


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Similalry let1 1 1 1 1 1

y 2 2 2 z 2 2 2

3 3 3 3 3 3

a d c a b d

a d c and a b d

a d c a b d

.

It can be shown that AX = B, x y zx , y. , z .

(i) Determinant Method of Solution

We have the following two cases :

Case I

If 0, then the given system of equations has unique solution, given by

x y zx / , y / and z / .

Case II

If 0, then two sub cases arise:

(a) at least one of x , y zand is non-zero, say x 0. Now in x. x , L.H.S. is zero

and R.H.S. is not equal to zero. Thus we have no value of x satisfying x. x . Hence given

system of equations has no solution.

(b) x y z 0. In the case the given equations are dependent. Delete one or two

equation from the given system (as the case may be) to obtain independent equation(s). Theremaining equation(s) may have no solution or infinitely many solution(s). For example in x +

y + z = 3, 2x + 2y + 2z = 9, 3x + 3y + 3z = 12, x y z 0 and hence equations are

dependent (infact third equation is the sum of first two equations). Now after deleting the third

equation we obtain independent equations x + y + z = 3, 2x + 2y + 2z = 9, which obviously

have no solution (infact these are parallel planes) where as in x + y + z = 3, 2x y + 3z = 4,

3x + 4z = 7, x = y z 0 and hence equations are dependent (infact third equation

is the sum of first two equations). Now after deleting any equation (say third) we obtain

independent equations x + y + z = 3, 2x y + 3z = 4, which have infinitely many solutions

(infact these are non parallel planes) For let z = R, then7 4

x3

and

2y

3

. Hence

we get infinitely many solutions.

(ii) Matrix Method of Solution

(a) 0, then AA1 exists and hence AX = B AA1(AX) = A1B x = AA1B and

therefore unique values of x, y and z are obtained.

(b) If 0, then from the matrix [A : B], known as augmented matrix (a matrix of order


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3 4). Using row operations obtain a matrix from [A : B], whose last row corresponding to

A is zero (which is possible as 0 ). If last entry of B in this matrix is non-zero, then the

system has no solution else the given equations are dependent. Proceed further in the same

way as in the case of determinant method of solution discussed earlier.

Aliter of (ii (b)) : We have AX = B ((adj A)A)X = (adj A)B X = (adj A)B.

If = 0, then X = 03 1, zero matrix of order 3 1. Now if (adj A)B = 0, then the system

AX = B has infinitily many solution, else no solution.

Note : A system of equation is called consistent if it has a least one solution. If the system has no

solution, then it is called inconsistent.

Illustration 8:

Solve the system of equations

x + 2y + 3z = 1

2x + 3y + 2z = 2

3x + 3y + 4z = 1

with the help of matrix inversion.

Solution :

The given system of equations in the matrix form can be written as

1 2 3 x 1

2 3 2 y 2

3 3 4 z 1

AX = B

where

1 2 3 x 1

A 2 3 2 ,X y and B 2

3 3 4 z 1

.

Now |A| = 1(12 6) 2 (8 6) + 3(6 9)

= 6 4 9 = 7 0.

Hence the given system has unique solution.

Let C be the matrix of cofactors of elements in |A|. then

11 12 13

21 22 23

31 32 33

C C C

C C C CC C C


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Here11 23

12 31

13 32

21 33

22

3 2 1 2C 6 ; C 3

3 4 3 3

2 2 2 3C 2 ; C 5

3 4 3 2

2 3 1 3C 3 ; C 43 3 2 2

2 3 1 2C 1 ; C 1

3 4 2 3

1 3C 5

3 4

6 2 3

C 1 5 3

5 4 1

Adj A =6 1 5

C 2 5 4

3 3 1

1

6 1 5AdjA 1

A 2 5 4| A | 7

3 3 1

6 / 7 1/ 7 5/ 7

2 / 7 5/ 7 4 / 7

3/ 7 3/ 7 1/ 7

16 / 7 1/ 7 5 / 7 1

A B 2 / 7 5 / 7 4 / 7 2

3/ 7 3 / 7 1/ 7 1

x 3 / 7

y 8 / 7

z 2 / 7

( AA1B = X)

x = 3/7, y = 8/7, z = 2/7

SYSTEM OF LINEAR HOMOGENEOUS SIMULTANEOUS EQUATIONS


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39


Consider the system of linear homogeneous simultaneous equations in three unknowns x, y

and z, given by a1x + b

1y + c

1z = 0, a

2x + b

2y + c

2z = 0 and a

3x + b

3y + c

3z = 0.

In this case, system of equations is always consistent as x = y = z = 0 is always a solution. If

the system has unique solution (the case when coefficient determinant 0), then x = y = z =

0 is the only solution (called trivial solution). However if the system has coefficient determinant

= 0, then the system has infinitely many solutions. Hence in this case we get solutions otherthan trivial solution also and we say that we have non-trivial solutions.


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40

IIT-MATHS-SET-III

ASSIGNMENT


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41

MATRICES

ILLUSTRATION : 01

With 1 2, as cube roots of unity, inverse of which of the following matrices exists?

a)

2

1b)

1

12

c)

1

2

2

d) None of these.

Solution :

Ans:(d)

01

01

10

1

2

22

2

,, Hence inverse does not exist.

ILLUSTRATION : 02

If A =

1629

835

432

, then trace of A is,

a) 17 b) 25 c) 8 d) 15

Solution :

Ans:(d)

[sum of leading diagonal elements is called Trace of matrix] 151632

ILLUSTRATION : 03

If A is an orthogonal matrix, then

a) |A| =0 b) |A|= 1 c) |A| = 2 d) None of these.

Solution :

Ans:(b)

A A' I

A A' | I |

| A | | A | 1 | A' | | A| for any square matrix

| A| 1

ILLUSTRATION : 04

WORKEDOUT ILLUATRATION


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IIT-MATHS-SET-III

If A =

2 2 4

1 3 4

1 2 x

is an idempotent matrix then x =

a) -5 b) -1 c) -3 d) -4

Solution :

Ans:(c)

Here 2A A

2

2 2 16 4x 2 2 4

1 3 16 4 x 1 3 4

4 x 8 2x 12 x 1 2 x

On comparing 16 4x 4 x 3 ILLUSTRATION : 05

If A is non-singular matrix, then Det 1A

a) Det 21

A

b) 21

Det A c)1

DetA

d) 1

Det A

Solution :

Ans:(d)

det 1 1 1 1

AA det I det A det A 1 det Adet A

ILLUSTRATION : 06

The rank of

1 2 3

2 4 6

3 6 9

is equal to

a) 1 b) 2 c) 3 d) None of these.

Solution :

Ans:(a)

1 2 3

A 2 4 6

3 6 9


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MATRICES

21

31

R 2 1 2 3

~ 0 0 0 p A 1

R 3 0 0 0

ILLUSTRATION : 07

If2 1

A A I 0, then A

a) A I b) I A c) I A d) None of these

Solution :

Ans:(c)

2A A I 0 1 2 1A A A I A .0 1 1A A A A I 0

1 1A I A 0 A A I

ILLUSTRATION : 08

If the matrix A=

8 6 2

6 7 4

2 4

is singular, then =

a) 3 b) 4 c) 2 d) 5

Solution :

Ans:(a)

If matrix A is singular. Then |A| =0

8 6 2

| A| 6 7 4 0

2 4

8 7 16 6 6 8 2 24 14 0 3

ILLUSTRATION : 09

If A is a 3 x 3 matrix and det 3 A k det A , k a) 9 b) 6 c) 1 d) 27

Solution :

Ans:(d)

3det 3A k det A

3 det A k det A

k 27

ILLUSTRATION : 10


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44

IIT-MATHS-SET-III

The equations 2x y 5, x 3 y 5, x 2 y 0 have

a) no solution b) one solution c) two solutions d) infinity many solutions

Solution :

Ans:(b)

2x y 5 .........( i )

x 3 y 5 .........( ii )

x 2 y 0 .........( iii )

Solving 1 & 2 , we get x 2 & y 1

Which is satisfied (3).


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45

MATRICES

1. If A =

11-2

221-

1-21

then det adjAadj is

a) 117 b) 217 c) 317 d) 417

2. If A is an invertible matrix, which of the following is not true ?

a) If A is symmetric so is A-1 b)If A is a scalar matrix so is A-1

c)If A is a triangular matrix so is A-1 d) If |A| equals 2, so does |A-1|

3. If the matrix

1053

842

231

is singular then l is

a)-2 b) 4 c) 2 d) -4

4. If A = ,cossin-

sincos

then 3A =

a)

33

33

cossin

sincosb)

cos3sin3-

sin33cosc)

cos3-sin3-

sin33cosd)

cos3-sin3-

sin3-3cos

5. If A and B are two square matrices such that B = -A-1

BA, then (A + B)2

is equal toa) 0 b) A2+ B2 c) A2+ 2AB + B2 d) A + B

6. Let A be a matrix of order 3 and let D denote the value of determinant A. Then det. (-2A)

a) -8D b) -2D c) 2D d) 8D

7. The element in the first row and third column of the inverse of the matrix

100

210

3-21

is :

a) 2 b) 0 c) 1 d) 7

8. The matrix

254-

5-23

43-2

is:

a) symmetric b) skew symmetric c) non- singular d) singular

SECTION A

ONE ANSWER TYPE QUESTIONS


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46

IIT-MATHS-SET-III

9. If A =

200

020

002

then 5A =

a) 5A b) 10A c) 16A d) 32A

10. If A =

2

2and |A3| = 125, then a is equal to

a) 1 b) 3 c) 4 d) 5

11. If A =

13

01B,

11

0, then value of a for which AA2= B is

a) 1 b) 1 c) I d) no real values

12. Determinant of a skew symmetric matrix of odd order is

a) zero 2) positive c) 1 d) a non zero perfect square

13. If A is a square matrix, then adj. AT (adjA)Tis equal to :

a) 2 |A| b) 2A c) unit matrix d) null matrix

14. If I =

ab

b-a

1tan-

an t1

1tan

tan-11

then :

a) a =1, b =1 b) a = cos2q, b = sin2q c)a = sin2q, b = cos2q d) none of the above

15. If A is a square matrix of order n x n, then adj(adjA) is equal to :

a) AA n 1|| b) AA n|| c) AA n 2|| d) None of the above

16. The value of xfor with [ 1 1 x]

1

1

1

012

120

201

= 0 is :

a) 2 b) -2 c) 3 d) -3

17. If D = diag. n

dddd ...,.........,, 321 where 0id for all ,.......,.........2,1 ni then D-1is equal to

a) diag. 11211 ................ nddd b) D c) nI d) 0

18. If A = diag. n

dddd ...........,, 321 then nA is equal to :

a) 1A b) nI c) nnnn ddddiag ....,.........,. 21 d) none of the above


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MATRICES

19. If A, B are two square matrices such that AB = A and BA = B, then :

a)A, B are idempotent b) only A is idempotent

c) only B is idempotent d) none of the above

20. If A =

11

11and Nn then AAnis equal to :

a) An2 b) An 12 c) nA d) none of these

21. If the system of homogeneous equations x + ay = 0, az + y = 0 and ax + z = 0 has infinite number ofsolutions, then the value of a is

a) 1 b) 1 c) 0 d) no real values

22. If a matrix A is symmetric as well as skew symmetric, then :

a) A is a diagonal matrix b) A is a null matrix

c) A is a unit matrix d) A is a triangular matrix

23. If A, B are symmetric matrices of the same order, then the matrix AB BA is :

a) 0 b) symmetric c) I d) skew - symmetric

24. The inverse of a symmetric matrix is

a) symmetric b) skew symmetric c) a diagonal matrix d) none of the above

25. If A is symmetric matrix and n N, then Anis

a) symmetric b) skew symmetric c) a diagonal matrix d) none of the above

26. Let A be a square matrix. Then which of the following is not a symmetric matrix :

a)A + A1 b) AA1 c) A1A d) A A1

27. If A =

cosxsinx-

sinxcosxand A (adjA) = k

10

01then the value of kis :

a) xx cossin b) 1 c) 2 d) 3

28. Each diagonal element of a Hermitian matrix is

a) a real number b) an imaginary number

c) a non-zero real number d) none of the above

29. Let A =

3-1-2-

625

311

then A is :

a) scalar b) diagonal c) nilpotent d) idempotent


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IIT-MATHS-SET-III

30. If A = ija is a scalar matrix, then trace of A is :

a) n

i

n

j

ija b) n

i

ija c) n

j

ija d)n

i

iia

31. If

costsint-

sintcosttR then R(s) R(t) equals :

a) R ( s + t ) b) R( s t) c) R (s) + R( t) d) None of the above

32. A and B are square matrices of order n x n, then (A B)2is equal to

a) 22 2 BABA b) 22 BA c) 22 2 BBAA d) 22 BBAABA

33. If A is a singular matrix, then adj A is :

a) singular b) non-singular c) symmetric d) not defined

34. If A and B are square matrices of the same type, then :

a) A + B = B + A b)A + B = A B c) A B = B A d) AB = BA

35. If in

333

222

111

cb

cb

cb

a

a

a

the co-factor ofra is rA , then 332211 AcAcAc is

a) 0 b) D c) D d) D2

36. If A =

2

2

sinsincossincoscos and B =

2

2

sinsincossincoscos are two matrices such that the

product AB is a null matrix, hen a - b is :

a) 0 b) an odd multiple of2

c) multiple of p d) none of the above

37. If A =

cossin-

sincosthen

n

n An

1lim is

a) a zero matrix b) an identity matrix c)

01-

10d) none of the above

38. Let

100

0cossin

0sin-cos

F where a R. Then 1F is equal to

a) 1F b) F c) 2F d) none of the above

39. If

2312

A

3-523-

=

1001

then the matrix A equals :


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MATRICES

a)

01

11b)

10

11c)

11

01d)

11

10

40. Let A be an invertible matrix and suppose that the inverse of 7A is

7-4

21-then the matrix A is :

a)

14

21b)

1/72/7

4/71c)

12

41d)

1/74/7

2/71

41. The value ofxfor which the matrix

x xx

x xx

xxx

is singular is :

a) 3x b) 3

x c) x d)0

42. If xf = 542 xx then Af , where A ==

122

212

221

equals.

a) O b)I c) I d) 2I

43. The value of the determinant A, A =

1sin-1-

sin1sin-

1sin1

lies in the interval

a) [0, 4] b) [0, 2] c) (2,4) 4) (2,3)

44. If the matrix AB = O and one of them is non singular then

a) A = O or B = O b) A = O and B = O

c) it is not necessary that either A = O or B = O d) A O, B O

45. The number of solutions of the equations 132 xx , ,22 31 xx 21 2xx =0

a) zero b) one c) two d) infinite

46. Let a., b,c be positive real numbers. The following system of equations in x,y and z

12

2

2

2

2

2

c

z

b

y

a

x, 12

2

2

2

2

2

c

z

b

y

a

x, 12

2

2

2

2

2

c

z

b

y

a

xhas :

a) no solution b) unique solution

c) infinitely many solutions d) finitely many solutions


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IIT-MATHS-SET-III

47. The matrix

011-7

1105-

7-50

is known as

a) symmetric matrix b) diagonal matrix

c) upper triangular matrix d) skew symmetric matrix

48. If 5A = O such that IAn for 1 n 4, then 1

AI equals :

a) A4 b) A3

c) I +A d) 432 AAAAI

49. Let A =

cossin

sincosLet

2x2

n bijA .Define2x2n

n

nbijlimAlim

. Then

nlim

n

An

equals

to

a) zero matrix b) unitary matrix c)

01

10d) limit does not exist

SECTION B

ONE ANSWER TYPE QUESTIONS

1. If A =

111

111111

then

a) A3= 9A b) A3= 27A

c) A + A = A2 d) A-1does not exist

2. For all values of l, the tank of the matrix A =

212491

688

541

2

a) for l = 2, r (A) = 1 b) for l = -1, r(A) = 2

c) for l 2, -1, r(A) = 3 d) none of these


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MATRICES

KEY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

b b b c b a b c d b d a d b c

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

b a c a b a b d a a d b d a a

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

d d a a d b a b a a b a c a d

46 47 48 49

b d d a

KEY

1 2

a,c,d a,b,c


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53

QUADRATIC EQUATION

Basic concepts

An equation of the form ax2+ bx + c = 0, where a 0 and a, b, c are real numbers, is called

a quadratic equation over reals.

The numbers a, b and c are called the coefficients of the quadratic equation. A root of the

quadratic equation is a number (real or imaginary) such that 2a + b + c = 0.

The roots of the quadratic equation are given by

a2

ac4bbx

2

The quantity D (= b2 4ac) is known as the discriminant of the equation.

Basic Results

(i) The quadratic equation has real and equal roots if and only if D = 0

(ii) The quadratic equation has real and distinct roots if and only if D > 0

(iii) The quadratic equation has complex roots with non-zero imaginary parts if and only if

D < 0.

(iv) If p + iq (p and q being real) is a root of the quadratic equation, where 1i , then

p iq is also a root of the quadratic equation.

(v) If qp is an irrational root of the quadratic equation, then p qis also a root of the

quadratic equation provided that all the coefficients are rational, q not being a perfect square.

(vi) The quadratic equation has rational roots if D is a perfect square of a rational number

and a, b, c are rationals.

(vii) If a = 1 and b, c are integers and the roots of the quadratic equation are rational, then the

roots must be integers.

(viii) If the quadratic equation is satisfied by more than two distinct numbers (real or imagi-

nary), then it becomes an identity i.e., a = b = c = 0

(ix) Let and be two roots of a given quadratic equation. Then + =a

band

a

c .

(x) A quadratic equation, whose roots are and can be written as (x ) (x ) =

0 i.e., ax2+ bx + c a(x - ) (x ).

METHODS OF INTERVALS (wavy curve method)

In order to solve inequalities of the form

0xQxP,0

xQxP , where P(x) and Q(x) are polynomials, we use the following method:


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54


If x1and x

2(x

1< x

2) are two consecutive distinct roots of a polynomial equation, then within this

interval the polynomial itself takes on values having the same sign. Now find all the roots of the

polynomial equations P(x) = 0 and Q(x) = 0. Ignore the common roots and write

m321n321

x.....xxx

x.....xxxxf

xQ

xP

,

Where a1, a

2, . . . . . a

n, b

1, b

2, . . . . . , b

mare distinct real numbers. Then f(x) = 0 for x = a

1,

a2, . . . . . , a

nand f(x) is not defined for x = b

1, b

2, . . . . . , b

m. Apart from these (m + n) real numbers

f(x) is either positive or negative. Now arrange a1, a

2, . . . . . , a

n, b

1, b

2, . . . . . , b

min an

increasing order say c1, c

2, c

3, c

4, c

5, . . . . . , c

m+n. Plot them on the real line. Draw a curve

starting from right of cm+n

along the real line which alternately changes its position at these

points. This curve is known as the wavy curve.

The intervals in which the curve is above the real line will be the intervals for which f(x) is positive

and the intervals in which the curve is below the real line will be the intervals in which f(x) is

negative.

QUADRATIC EXPRESSION

The expression ax2+ bx + c is said to be a real quadratic expression in x where a, b, c

are real and a 0,

Let f(x) = ax2+ bx + c, where a, b, c R (a 0).

f(x) can be re-written as f(x) =

2

2

2

22

a4

D

a2

bxa

a4

bac4

a2

bxa , where

D = b2 4ac is the discriminant of the quadratic expression.

Therefore y = f(x) represents a parabola whose axis is parallel to the y-axis, with vertex at A

a4

D,

a2

b.

Note that if a > 0, the parabola will be concave upwards and if a < 0 the parabola will be

concave downwards and it depends on the sign of b2- 4ac that the parabola cuts the x-axis

at two points

(b2 4ac > 0), touches the x-axis (b24ac = 0) or never intersects with the x-axis (b2- 4ac 0 and b24ac < 0


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QUADRATIC EQUATION

f(x) > 0 x R

x x

y = f(x)

b/2a

In this case the parabola always remains concave upwards and above the x-axis

(ii) a > 0 and b2 4ac = 0

f(x) 0 x Rx x

y = f(x)

b/2a

In this case the parabola touches the x-axis and remains concave upwards.

(iii) a > 0 and b2 4ac > 0.

Let f(x) = 0 has two real roots and ( < ).

Then f(x) > 0 x( , ) ( , ),

f(x) < 0 x( , ) and f(x) = 0 for x { , } .x x

b/2a

y = f(x)

In this case the parabola cuts the x-axis at two points and and remains concave up-

wards.

(iv) a < 0 and b2 4ac < 0

x x

b/2a

y = f(x)

f(x) < 0 xR.

In this case the parabola remains concave downwards and always below the x-axis.

(v) a < 0 and b2 4ac = 0

f(x) 0 xR.

x x b/2a

y = f(x)


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56


In this case the parabola touches the x-axis and remains concave downwards.

(vi) a < 0 and b2 4ac > 0

Let f(x) = 0 have two real roots and ( < ).

Then f(x) < 0 x( , ) ( , ),

f(x) > 0 x( , ) and f(x) = 0 for x { , } .x

x

y = f(x)

b/2a

In this case the parabola cuts the x-axis at two point and and remains concave down-

wards.

Notes:(i) if a > 0, then minima of f(x) occurs at x = b/2a and if a < 0, then maxima of f(x)

occurs at x = b/2a

(ii) If f(x) = 0 has two distinct real roots, then a.f(d) < 0 if and only if d lies between the roots

and a.f(d) > 0 if and only if d lies outside the roots.

4. INTERVALS OF ROOTS

In some problems we want the roots of the equation ax2+ bx + c = 0 to lie in a given interval.

For this we impose conditions on a, b and c. Since a 0, we can take f(x) =a

cx

a

bx 2 .

(i) Both the roots are positive i.e., they lie in (0, ) if and only if roots are real, the sum of

the roots as well as the product of the roots is positive.

+ =b

0a

and = ac > 0 with b2 4ac 0.

Similarly, both the roots are negative i.e., they lie in ( , 0) if and only if roots are real, the

sum of the roots is negative and the product of the roots is positive.

i.e., 0a

b

and =

a

c> 0 with b2 4ac 0.

(ii) Both the roots are greater than a given number k if and only if the following three conditions

are satisfied:

D 0, ka2

b

and f(k) > 0.


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57

QUADRATIC EQUATION

kx xb/2a

kx xb/2a

(iii) Both the roots are less than a given number k if and only if the following conditions are

satisfied:

D 0,a2

b < k and f(k) > 0.

(iv) Both the roots lie in a given interval (k1, k

2), if and only if the following conditions are

satisfied:

D 0, k1< a2b

< k2and f(k

1) > 0, f(k

2) > 0.

k1 k2x xb/2a

k1 k2x xb/2a

(v) Exactly one of the roots lies in a given interval (k1, k

2) if and only if f(k

1). f(k

2) < 0.

xk

1

k2

x

x

k1

k2x

(vi) A given number k lies between the roots if and only if f(k) < 0.

xk

x

In particular, the roots of the equation will be of opposite signs if and only if 0 lies between the

roots

f(0) < 0. QUADRATIC INEQUATIONS

Let f(x) = ax2 + bx + c be a quadratic expression. Then inequations of the type f(x) 0

or f(x) 0 are known as quadratic inequations. The study of these can be easily done by

taking the corresponding quadratic expression and by applying the basic results of quadratic

expression.

CONDITION FOR COMMON ROOT(S)

Let ax

2

+ bx + c = 0 and dx

2

+ ex + f = 0 have a common root (say). Thena + b + c = 0 and 0fed 2


62/185

58


Solving for 2 and , we getbdae

1

afdccebf

2

i.e.,bdae

afdcand

bdae

cebf2

(dc af)2 = (bf ce) (aebd)

which is the required condition for the two equations to have a common root.

Note: Condition for both the roots to be common isf

c

e

b

d

a

THEORY OF POLYNOMIAL EQUATIONS

Consider the equation

anxn+ a

n1xn1 + a

n2xn2 .... + a

1x + a

0= 0 ... (1)

(a0, a

1, .... , a

nare real coefficients and a

n 0)

Let 1 , 2 , ......., n be the roots of equation (1), Then

anxn+ a

n1xn1 + a

n2xn2 + ... + a

1x + a

a an )x( 1 (x 2 ) ... (x )n ... (2)

Comparing the coefficients of like powers of x, we get

1 =n 1

1 2 3 nn

a.....

a

1 2

n

2nn1n32413121

a

a......

...............

n

rnr

n2rn1rnr21a

a)1(...........

n

0nn21

a

a)1(.....

In general n

ini

i21a

a)1(....

e.g., If , , and are the roots of ax4+ bx3+ cx2+ dx + e = 0, then

a/b

a/c

a/d

e

a


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59

QUADRATIC EQUATION

Remarks

A polynomial equation of degree n has n roots (real or imaginary).

If all the coefficients are real then the imaginary roots occur in pairs i.e., number of

imaginary roots is always even.

If the degree of a polynomial equation is odd then the number of real roots will also beodd. It follows that at least one of the roots will be real.

If is a repeated root, repeating r times of a polynomial equation f(x) = 0 of degree n

i.e., f(x) = )x(g)x( r , where g(x) is a polynomial of degree nr and g )( 0, then

f( ) = )(f = )(f = ... = f (r1) )( = 0 and )(fr 0 and vice versa.

Thus polynomial in x of degree n can be factorized into a product of linear/quadratic

form.

Remainder Theorem

If we divide a polynomial p(x) in x by (x ), then remainder obtained is p( ). Note

that if ,0)(p then x is a factor of p(x).

If a polynomial of degree n has n + 1 roots say x1, x2, ... xn+1, xi xj if i j, then the

polynomial is identically zero. i.e., p(x) 0.

(In other words, the coefficients a0, ... a

n are all zero).

222 < 0, which is not possible if all , and are reals. So atleast one root

i s

non-real. As imaginary roots occurs in pair, given cubic equation has two non-real roots and

one real root.


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60

IIT-MATHS-SET-III

3-A QUADRATIC EQUATIONSASSIGNMENT


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61

QUADRATIC EQUATIONS

WORKEDOUT ILLUATRATION

ILLUSTRATION : 01

If a and b 0 are the roots of the equation 02 baxx , then the least value of Rxbaxx 2 is

a) 49

b) 49

c) 41

d) 41

Ans: (b)

Solution :

Since a and b are the roots of the equation.

2x ax b 0

Therefore, a b a and ab = b

Now, ab b a 1 b 0 a 1 b 0

Putting a=1 in a+b =-a, we get b=-2

Since 2y x ax b is a parabola opening upward.

So, minD

y4

[ Using :2

min

Dy for y ax bx c

4a ]

=2a 4b 9

4 4

ILLUSTRATION : 02

If , are the roots of the equation 2ax bx c 0 , then the value of the determinant.

1 cos cos

cos 1 cos

cos cos 1

is

a) sin b) sin sin c) 1 cos d) None of these

Ans: (d)

Solution :

We have

1 cos cos

cos 1 cos

cos cos 1

=

cos sin 0

cos sin 0

1 0 0

cos sin 0

cos sin 0

1 0 0

= (0) (0) = 0 for all values of ,


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ILLUSTRATION : 03

Let2 42 2a cos i sin , a a a

7 7

& 3 5 6a a a . Then the equation whose roots are , is

a) 2x x 2 0 b) 2x x 2 0 c) 2x x 2 0 d) 2x x 2 0

Ans: (d)

Solution :

We have :2 2

a cos i sin ,7 7

7a

72 2

cos i sin cos 2 i sin 2 1 0i 17 7

Now + = 2 3 4 5 6 a a a a a a

6 71 a a a a 1a

1 a 1 a 1 a

1 1 7a 1

and . 2 4 3 5 6 4 3 2 3a a a a a a a 1 a a 1 a a

= 4a 2 3 3 4 3 5 6 1 a a a a a a a a

= 4a 2 3 4 5 6 1 a a 3a a a a

= 4 5 6 7 8 9 10a a a 3a a a a

= 2 3 4 5 6 3 a a a a a a

7 8 7 9 7 2 2 10 7 3 3a 1 a a a a, a a a a and a a a a

=6 71 a a a a 1

3 a 3 31 a 1 a 1 a

= 3 1 2 So, the required equation is

2 2x x 0 or x x 2 0

ILLUSTRATION : 04

Let ,be the roots of 2ax bx c 0 ; , be the roots of 2px qx r 0 ; and 1 2D ,D the respective

discriminants of these equations. If , , and are in A.P. then 1 2D : D =

a)

2

2

a

b b)

2

2

a

p c)

2

2

b

q d)

2

2

c

r


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Ans: (b)

Solution :

We haveb c q r , . , and

a a p p.

Now , , , are in AP

2 2 2 2

4 4

2 2 2 2

2 2 2 2

b 4c q 4r b 4ac q 4rp

a a p p a p

1 2

2 2

D D

a p

2

1

2

2

D a

D p

ILLUSTRATION : 05

If every pair from among the equations

2 2

x px qr 0, x qx rp 0 and

2

x rx pq 0 has acommon root, then the sum of the three common roots is

a) 2 p q r b) p q r c) p q r d) pqr

Ans: (b)

Solution :

The given equations are

2 2x px qr 0; x qx rp 0 and 2x rx pq 0

Let ,be the roots of i : , be the roots of (ii) and , be the roots of (iii). Since , is a

common root of i and (ii).

2 p qr 0 and 2 q rp 0

p q r q p 0 r

Now, ,= qr r = r = qr r = q

Since , and are roots of (ii). Therefore,

= rp r = rp = p

++ = q+r+p = p+q+r..

Note: ++ can also be equal to -1/2 (p+q+r) and 0.

ILLUSTRATION : 06

If a Z and the equation x a x 10 1 0 has integral roots, then the values of a are

a) 10,8 b) 12,10 c) 12,8 d) None of these.

Ans: (c)


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Solution :

Since a and x are integers. Therefore, x a x 10 1 0

x a x 10 1

x a 1 and x 10 1 or x a 1 and x 10 1

x 9 and a 8 or x 11 and a 12

a=8 or a=12.

ILLUSTRATION : 07

If the equation 2

3x

1

p27 3 15 x 4 0

has equal roots , then p=

a) 0 b) 2 c) -1/2 d) None of these.

Ans: (d)

Solution :

The given equation will have equal roots iff

Disc = 0

2

1/ p 1/ p27 3 15 144 0 27 3 15 12

1/ p 1/ p 1/ p 1 127 3 27 or 3 3 1 or 3 0 or 2

9 p

But 1/p cannot be zero. So, p = -1/2.

ILLUSTRATION : 08

The number of solutions of the equation x x105 5 log 25, x R

is

a) 0 b) 1 c) 2 d) infinitely many

Ans: (a)

Solution :

LHS of the given expression, being the sum of a number and its reciprocal, is greater than or equal to 2 whereas

RHS is less than 2. So, the given equation has no solution.

ILLUSTRATION : 09

The integer kfor which the inequality 2 2x 2 4k 1 x 15k 2k 7 0 is valid for any x, is

a) 2 b) 3 c) 4 d) none of these

Ans: (b)


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QUADRATIC EQUATIONS

Solution :

Let f x 2 2x 2 4k 1 x 15k 2k 7 0 . Then,

f x 0 Disc 0 2coeff . of x 0

2 24 4k 1 4 15k 2k 7 0

2k 6k 8 0 2 k 4.

ILLUSTRATION : 10

The condition that 3 2x px qx r 0 may have two of its roots equal to each other but of opposite signs

is

a) r = pq b) 3r 2 p pq c) 2r p q d) None of these.

Ans: (a)Solution :

Let ,, be the roots of the given equation such that=-. Then.

+ + = p = p.

Since is a root of the given equation, so it satisfies the equation i.e,

3 2p q r 0 3 3p p pq r 0 r pq

ILLUSTRATION : 11

The number of real roots of 4 4

6 x 8 x 16 is

a) 0 b) 2 c) 4 d) None of these.

Ans: (b)

Solution :

Let y=7-x. Then the given equation becomes

4 4 4 2

y 1 y 1 16 y 6 y 7 0

2 2 2y 1 y 7 0 y 1 0 2y 7 0

y 1

7 x 1 x 6 ,8

ILLUSTRATION : 12

The value of a for which the equation 2 21 a x 2ax 1 0 has roots belonging to (0,1) is


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a)1 5

a2

b) a 2 c)

1 5a 2

2

d) a 2

Ans:(b)

Solution :

Let f x 2 21 a x 2ax 1 0 then, f x 0 has roots between 0 and 1 if.

(i) Disc 0 (ii) 21 a f 0 0 and 21 a f 1 0

Now, Disc 0 2 24a 4 1 a 0 , which is always true.

21 a f 0 0 21 a 0

2a 1 0 a 1 or a 1

and 2 2 21 a f 1 0 1 a 2a a 0

a a 1 a 1 a 2 0

a 1 or a 2 or 0 a 1

From (I) and (ii) , we get: a 1 or a 2 .

ILLUSTRATION : 13

If the product of the roots of the equation is 31, then the roots of the equation are real for k equal to

a) 1 b) 2 c) 3 d) 4

Ans: (d)

Solution :

Produce of roots = 31

2log k2e 1 31

2 2 22k 1 31 2k 32 k 16 k 4.

But k> 0. Therefore, k=4.

Now, Disc = 2 2log k 2 2log k 28k 8e 4 8k 8e 4 8k 4 0 k

Hence, k = 4.

ILLUSTRATION : 14

The roots of the equation 2 2x 15 x 15

a b a b 2a,

where 2a b 1 , are


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QUADRATIC EQUATIONS

a) 2, 3 b) 4, 14 c) 3, 5 d) 6 , 20

Ans: (b)

Solution :

We have,

22

a b a b a b 1

a b a b 1a b a b a b

So, by putting 2x 15

a b y,

the given equation becomes.

21y 2a y 2ay 1 0

y

2 2 2

y a a 1 y a a 1

y a b 2a 1 b

2x 15

y a b a b a b ,a b

2x 15 1 or 2x 15 1 x 4, x 14

ILLUSTRATION :15

The value of 8 2 8 2 8 2 8 is

a) 10 b) 6 c) 8 d) 4.

Ans:(d)

Solution :

Let x = 8 2 8 2 8 2 8 . Then,

2 2x 8 2x x 8 2x x 2x 8 0

x 4 . x 0

ILLUSTRATION : 16

The harmonic mean of the roots of the equation 25 2 x 4 5 x 8 2 5 0 is

a) 2 b) 4 c) 7 d) 8

Ans: (b)

Solution :

Let ,be the roots of the given equtaion. Then,


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4 5 8 2 5and

5 2 5 2

Let H be the H.M. of a and b. Then.

2 16 4 5H 4

4 5

ILLUSTRATION : 17

In a traingle PQR, R / 2 . If tan (P/2) and tan (Q/2) are the roots of the equation 2ax bx c 0

a 0 . then

a) a+b=c b) b+c=0 c) a+c=b d) b=c

Ans: (a)

Solution :

R / 2 P

P Q2 2

Q

2 4

P Q tan P / 2 tanQ / 2

tan tan2 2 4 1 tan P / 2tanQ / 2

b

a 1c

1a

P Qtan tan

2 2

b / a and

P Q ctan tan

2 2 a

c b1

a a

a c b

a b c

ILLUSTRATION : 18

If the roots of the equation 2 2x 2ax a a 3 0 are real and less than 3, then

a) a0 and f 3 0


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QUADRATIC EQUATIONS

2 2 2a a a 3 0 and a 5a 6 0

a 3 and a 2 a 3 0

a 3 and a 2 or a 3

a 2 .

1. If and are the roots of 012 xx then 1313 32 and 3131 23 are the roots of :

a) 012 xx b) 0752 xx c) 0752 xx d) 0752 xx

2. The set of values ofxwhich satisfy 8325 xx and 412

x

xis :

a) 3,2 b) 3,21, c) 1, d) (1,3)

3. If x, y and z are real numbers, then xyzxyzzyx 23694 222 is always

a) Positive b) non-positive c) zero d) non-negative

4. If ris the ratio of the roots of the equation ,02 cbxax then

r

r21

a) 1 b) acb 2 c) b2/ac d) acb 42

5. The greatest value of the expression124

12 xx

is :

a) 4/3 b) 5/2 c) 13/14 d) None of these

6. If the product of the roots of the equation 01222 log22 kekxx is 31, then the roots of the

equation are real for k equal to

a) 1 b) 2 c) 3 d) 4

7. If the sum of the squares of the roots of the equation 0sin12sin2 xx is least then

a) /4 b) /3 c) /2 d) /6

8. If 12 xx is factor of dcxbxax 23 , then the real root of 023 dcxbxax is

a) -d/a b) d/a c) a/d d) None of these

9. The values of a for which the quadratic equation 04182 232 aaxaax posses roots of

opposite signs are given by

a) a > 0 b) a > 5 c) 4 < a < 8.5 d) 0 < a < 4

SECTION A - SINGLE

ANSWER TYPE QUESTIONS


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10. The number of positive integral solutions of

0

725

24365

432

xx

xxxis

a) 4 b) 3 c) 2 d) 1

11. The maximum and minimum values of 32

9142

2

xx

xx

are

a) 3,1 b) 4 5 c) 0, - d) -

12. A solution of the equation

dcx

dxcx

bax

bxax

is

a) bacddcab )( b) dcabbacd

c) cabddbac d) None of the above

13. The integer k for which the inequality 0721514222

kkxkx is valid for anyxis

a) 2 b) 3 c) 4 d) None of these

14. If the quadratic expression 010a3ax2x2 , Rx then

a) a > 5 b) |a+ < 5 c) -

Iit 3 Unlocked

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