· PDF file · 2018-02-112018-02-11 · ... cos2A + cos2B−cos2C =...

Winglish Tuition Centre Puduvayal 1 +1 Maths Q & A

Mythili Publishers, Karaikkudi. 8300121037

PROVE BY FACTOR METHOD OF DETERMINANTS

1. ))()((

1

1

1

2

2

2

accbba

cc

bb

aa

2. ))()()((

1

1

1

3

3

3

cbaaccbba

cc

bb

aa

3. ))()()((

1

1

1

32

32

32

cabcabaccbba

cc

bb

aa

4. ))()()((

2

2

2

cbaaccbba

ccba

bbac

aacb

5. ))()()((222 cabcabaccbba

abcabc

cba

cba

6. )((2

)(

)(

)(

222

222

222

cbaabc

bacc

bacb

aacb

7. )2()( 2 axax

xaa

axa

aax

8. )9()1(

432

522

531

2

xx

x

x

x

9. abc

baacbc

abaccb

bacacb

8

10. 0

cxba

cbxa

cbax

(Solve )

PROPERTIES OF DETERMINANTS

11. 0

/1

/1

/1

2

2

2

baabc

accab

cbbca

12. 0

22

22

22

baabba

accaac

cbbccb

13. ))()((2 accbba

cbaab

acbac

bccba

14. )c+ ( 2222

2

2

2

ba

cbcac

bcbab

acaba

15. 222

2

2

2

c4 ba

cbcac

bcbab

acaba

16. 3)(

22

22

22

cba

baccc

bacbb

aacba

17.

cbaabc

a

b

a111

1

111

111

111

18. 3333 cbaabc

bac

acb

cba

19. abccba

baaccb

accbba

cba

3333

20. 2))(( zxzyx

zyyx

xzxz

yxzy

21. 0

1

1

1

32

32

32

zzz

yyy

xxx

Show that xyz= 1

PRODUCT OF DETERMINANTS

www.Padasalai.Net www.TrbTnpsc.com

http://www.trbtnpsc.com/2017/06/latest-11th-study-materials-tamil-medium-english-medium.html

www.Padasalai.Net



VECTORS

1. a) Show that given vectors are Coplanar

kjkjikji 2,432,32

b) Show that given vectors are Coplanar

kjikjikji 5203,987,765

c) Show that given vectors are Coplanar

kjkjikji 57,2,3

d) S.T. points given by vectors are coplanar

kjikjkji 493,,54

2. a) Show that points whose p.v given are collinear

kikjikji 7,32,532

b) Show that points whose p.v given are collinear

kjkjikji 107,432,32

c) Show that points whose p.v given are collinear


d) Show that points whose p.v given are collinear

cbacbacba 774,32,32

e) Show that points whose p.v given are collinear

bacbacba 138,32,32

f) Show that points whose p.v given are collinear

cacbacba 7,32,532

g) Find m if given vectors are collinear

jmibjia 6and32

3. a) Show that Vertices of triangle whose p.v is

given form equilateral traiangle


b) Show that Vertices of triangle whose p.v is

given form equilateral traiangle


c) Find centroid of triangle whose p.v of vertices

kjikjikji 23,32,32

d) Find length of sides if vertices p.v of triangle

kjikjikji 653,2,432

4. ST given vectors form right angled triangle

kjikjikji 53,443,2

5. a) Find Magnitude and directional cosines of

kji 72

b) Find Magnitude and dc of sum of vectors

kjikji 345,773

c) Find Magnitude and dc of sum of vectors

kjikji 425,73

6. a) Find Unit vectors in the direction of

kji 1243

b) Find Unit vectors in the direction of

ji 3

7. a) Find Unit vectors parallel to

ji 43

b) Find Unit vectors parallel to sum of

kjkji 22,853

c) Find Unit vectors parallel to

cba 423 where

kjickjibkjia 2,342,43

d) Find Unit vectors parallel to

ji2

whose magnitude 5 units

BINOMIAL THEOREM

1. a) Find the coefficient of x5

17

3

1

xx

b) Find the coefficient of x5

111

xx

2. a) Find the constant term 10

2

2

xx

b) Find the term independent of x 12

2 12

xx

c) Find term independent of x 92

23

34

xx

d) Find the term independent of x 9

2

99

cxx

3. a) Find middle term in 82

32

3

xx

b) Find middle term in 16

bx

xb

c) Find middle term in 16

x

xa

d) Find middle term in 13)2yx



www.Padasalai.Net



e) Find middle term in 17

2

2

xx

BINOMIAL SERIES

4. Find first four terms in the expansion of

34

425

36

1)

)2(1

)

)1())41()

xd

xc

xbxa

5. a) Find coefficient of x8 in 21

)21(

x

b) Find 5th term in the expansion 211

3)21( x

c) Find (r+1)th term in the expansion 4)1( x

6. Evaluate correct to 2 decimal places

333

128

1)31003)2126)1

7. a) If x is large show that 2

3 33 3 136

xxx

b) If x is large show that x

xx8

925 22

c) If x is small show that 2

111 2x

xxx

8. a) Show that

..11

!21

11

12)1(2

xxn

nxx

nx nn

b) Show that

....1

12.1

)1(111

2

xnn

xnxn

MATHEMATICAL INDUCTION

9. Prove by mathematical induction that

a) n2 +n is even

b) (2n+1)(2n - 1) is odd


nnh

nnng

nnnf)

nne

nnnd

nnnc

nnnnb

nnna

21

121

.....21

21

21

)

)1(24.....1284)2

)13()23(.....741

)12(......531)

)1(2.....642)4

)1(....321)

6)12)(1(

.....321)

2)1(

.....321)

32

2

223333

2222


a) 2n - 1 is divisible by 7 for all natural numbers n

b) 52n - 1 is divisible by 24 for all natural numbers n

c) 102n - 1+1 is divisible by 11

d) n(n + 1) (n + 2) is divisible by 6

e) 353 23 n + n + = nSn is divisible by 3 for all

f) 11672 n - n is divisible by 64

g) b - aby divisible isnn ba

TRIGNOMETRY

1. If A, B are acute angles,

a) If sinA = 53

; cos B= 1312

, find cos(A + B)

b) If sinA = 31

, sinB = 41

find sin (A + B)

2. If (α + β) and (α - β) are acute,

a) If cos(α+β) = 54

; sin (α − β) = 135

find tan 2α

3. a) If tanA = 65

, tan B = 111

show that A+B = 45°

b) If tan α = 21

, tan β =31

, show that α+β = 4

4. If A + B = 45°, show that

a) (1 + tanA)(1+tanB) = 2

b) (cotA−1) (cotB − 1) = 2

c) Hence deduce the value of tan 22½ °

5. If tanα = 13 and tan β = 17 show that 2α+β =4

6. Ifx

x1

+ = cos 2 prove that

xx

121

= cos2 2

7. a) Show that sin 20° sin40° sin80° = 83

b) Prove that sin20° sin40° sin60° sin80° = 163

c) Show that cos20° cos40° cos80° = 81

d) Prove that cos20°cos40° cos60° cos80° = 161

8. If A + B + C = π, prove that

a) sin2A + sin2B + sin2C = 4sinA sinB sinC

b) cos2A + cos2B−cos2C = 1−4sinA sinB cosC

c) sin2A − sin2B + sin2C = 4 cosA sinB cosC


a) CBAC= BA+ cossinsin21coscoscos 222

b) cosC cosB2cosA + 2 = Csin + Bsin +A sin 222

c)2

sin2

sin2

sin212

sin2

sin2

sin 222 CBA-=

C+

B+

A

10. If A + B + C = 90°, show that

Bcot A cot sin2C - sin2B +sin2A sin2C + sin2B +sin2A



www.Padasalai.Net



TRIGNOMETRIC EQUATION

I. Solve :

1. cos2x + sin2x + cosx = 0

2. 2cos2θ + 3sinθ = 0

3. sin2θ − 2cosθ + ¼ = 0

4. 2sin2x + sin22x = 2

5. tan2θ + (1 − 3 ) tanθ − 3 = 0

6. tan2x = tanx

7. 2tanθ − cotθ = − 1

II. Solve :

1. sin3x = sinx

2. sin 4x + sin2x = 0

3. sin2x + sin6x + sin4x = 0

4. cosx + cos2x + cos3x = 0

5. sin2x + sin4x = 2sin3x

III. Solve :

1. 3 sin x + cosx = 2

2. sinθ + cosθ = 2

3. sinθ − cosθ = − 2

4. 2 secθ + tanθ = 1

5. cosecθ − cotθ = 3

INVERSE TRIGONOMETRIC FUNCTION

I. Prove that :

1. 92

tan131

tan71

tan 111

2. 1127

tan53

tan54

cos 111

3. 4

tantan 11

nmnm

nm

4. 42

1tan

21

tan 11

xx

xx

5. 32

1cot

12

tan2

12

1

xx

xx

6. 74

tan1tan1tan 111 xx

7.

zxyzxyzzyx

zyx1

tantantantan 1111

8. Solve: π

x = x + 4

3tan2tan 11

II. Prove that :

1. 22111 11[sinsinsin xyyxyx

2. 22111 11[sinsinsin xyyxyx

3. 22111 11[coscoscos yxxyyx

4. 22111 11[coscoscos yxxyyx

I. In a triangle ABC prove that

1. a sinA − b sinB = c sin(A − B)

2. ∑ a sin (B − C) = 0

3. a3 sin(B − C) + b3 sin(C − A) + c3 sin(A − B) = 0

4. 2

2

c2

sinsin ba

(A +B)(A - B)

5. 2

sin2

cosA

acbCB

6. 0sin

sinsin

sinsin

sin 222

CB)(Ac

BA)(Cb

AC)(B a

7. If a cosA = b cosB show that the triangle is either

an isosceles triangle or right angled triangle?

II. In a triangle ABC prove that

8. a(b cosC − c cosB) = b2 − c2

9. 2A

cos c)(b+2A

sin c) + (b = a 22222

10. abc

cbac

Cb

Ba

A2

coscoscos 222

III. In a triangle ABC prove that

11. ∑ a(b2 + c2) cosA = 3abc

12. ∑ (b + c) cosA = a + b + c

13. ∑ a(sinB − sinC) = 0

IV. In a triangle ABC prove that

14. 222

222

tantan

acbbac

BA

Miscellenous

1. If cosθ + sinθ = 2 cosθ, show that

cosθ − sinθ = 2 sinθ

2. Prove that (1+tanA + secA) (1+cotA − cosecA)= 2

3. If tanθ + sinθ = p, tanθ − sinθ = q and p > q then

show that p2 − q2 = 4 pq

4. If tanθ + secθ = x, show that

2tanθ = x

x1

, 2secθ =x

x1

, 11

sin2

2

xx


a) tanA + tanB + tanC = tanA tanB tanC

b) tan2A + tan2B + tan2C = tan2A tan2B tan2C

6. Prove that 2

tan cossin + 1 cossin + 1

7. If tanθ = 3 find tan3θ

8. If sinA = 53

find sin3A

9. Prove that: sinA + sin(120°+A) + sin(240°+A) = 0

10. Prove that cosA + cos(120°+A)+cos(120° −A) = 0



www.Padasalai.Net



STRAIGHT LINES

I. Perpendicular Distance

1. Find length of the perpendicular from (2, − 3)

to the line 2x − y + 9 = 0

2. Find co-ordinates of the points on the straight

line y = x + 1 which are at a distance of 5 units

from the straight line 4x − 3y + 20 = 0

3. Find points on y-axis whose perpendicular

distance from 4x − 3y − 12 = 0 is 3

4. Find the length of the perpendicular from (3,

2) to the straight line 3x + 2y + 1=0.

5. Find the distance of the line 4x − y = 0 from

(4, 1) along the straight line making 135° with

the positive direction of the x-axis.

II. Find distance between parallel lines

1. 2x + y − 9 = 0 and 4x + 2y + 7 = 0

2. 2x + 3y − 6=0 and 2x + 3y + 7 = 0.

III. Find the equations of

1. Medians of the triangle formed by the points

(2, 4), (4, 6) and (− 6, − 10).

2. Diagonals of a quadrilateral whose vertices are

(1, 2), (− 2, − 1), (3, 6) and (6, 8)

IV. Equation of Straight Line

1. passing through point (1, 2) and making

intercepts on the co-ordinate axes which are in

the ratio 2 : 3.

2. through the point (2, 2) and having intercepts

whose sum is 9.

3. passes through the point (− 1, 3) whose

intercept on the x-axis is 3 times its intercept

on the y-axis

4. which cut off intercepts on the axes whose

sum and product are 1 and − 6 respectively.

V. Angle between two straight lines

1. Find the angle between the straight lines

2x + y = 4 and x + 3y = 5

2. Show that the angle between

3x + 2y = 0 and 4x − y = 0 is equal to angle

between 2x + y = 0 and 9x + 32y = 41

3. Show that the triangle whose sides are

y = 2x + 7, x − 3y − 6 = 0 and x + 2y = 8

is right angled. Find its other angles.

4. Show that straight lines form isosceles triangle

4x −3y −18 = 0, 3x−4y +16 = 0, x + y − 2 = 0

5. Show that straight lines form isosceles triangle

3x+ y +4 = 0, 3x+4y −15 = 0, 24x −7y−3 = 0.

VI. Concurrency

1. Show that the straight lines are concurrent

3x + 4y = 13; 2x −7y + 1 = 0 and 5x − y = 14.

2. Show that the straight lines are concurrent

3x + 4y=13; 2x − 7y + 1 = 0 and 5x − y = 14

3. Find ‘a’ for which straight lines are concurrent

3x + y + 2 = 0,2x − y+3 = 0 and x + ay − 3 = 0


x−6y+a = 0, 2x+3y+4 = 0 and x + 4y + 1= 0


x + y − 4 = 0, 3x + 2 = 0 and x − y + 3a = 0

6. If the given equations are concurrent,

ax+by + c = 0, bx+cy+a = 0 and cx + ay +b = 0

Show that a3 + b3 + c3 = 3abc

VII. Co-ordinates of orthocentre of the triangle

1. Find the co-ordinates of the orthocentre of the

triangle whose vertices are the points

(− 2, − 1), (6, − 1) and (2, 5)

2. Find the co-ordinates of orthocentre of the

triangle formed by the straight lines

x − y − 5 = 0, 2x − y − 8 = 0 and 3x − y−9 = 0



x +y −1 = 0, x +2y − 4 = 0 and x + 3y − 9 = 0



x + 2y = 0, 4x + 3y = 5 and 3x + y = 0.

PAIR OF STRAIGHT LINES

I. Sum &product of slopes of pair of straight lines

1. Slope of one of the straight lines of

ax2 +2hxy + by2 = 0 is thrice that of other,

show that 3h2 = 4ab

2. Slope of one of the straight lines

ax2 + 2hxy + by2 = 0 is twice that of the other,

show that 8h2= 9ab.

II. Angles between pair of straight lines

1. Find the angle between the straight lines

x2+ 4xy + 3y2 = 0

2. Find he angle between pair of straight lines

(a2 − 3b2) x2 + 8ab xy + (b2 − 3a2)y2 = 0

3. If angles between pair of straight lines

ax2 + 2hxy + by2 =0 is 60°

Show that (a + 3b) (3a + b) = 4h2

III. Condition for a Pair of Straight Lines



www.Padasalai.Net



1. If the given equation represents a pair of

perpendicular straight lines, find a and c.

ax2 + 3xy − 2y2 − 5x + 5y + c = 0

2. Show that given equation represents a pair of

straight lines. Find angle between them

x2 − y2 + x − 3y − 2 = 0

3. Show that the given equation represents a pair

of straight lines.

3x2 + 10xy + 8y2 + 14x + 22y + 15 = 0

Show that angle between them is tan−1

112

IV. Separate equation of the straight lines.

1. S.T the eqn. represents pair of straight lines

3x2 + 7xy + 2y2 + 5x + 5y + 2 = 0

Find separate equation of straight lines.

2. S.T the eqn. represents a pair of parallel lines

4x2 + 4xy + y2 − 6x − 3y − 4 = 0

Find distance between them.

3. S.T the eqn. represents a pair of parallel lines

9x2 + 24xy + 16y2 + 21x + 28y + 6 = 0

Find the distance between them.

4. For what value of k does the eqn represents a

pair of straight lines?

12x2 + 7xy + ky2 + 13x − y + 3 = 0

Also write the separate equations.

5. Find k such that the equation represents a pair

of straight lines.

12x 2 + 7xy − 12y − x + 7y + k = 0

Find i) separate equations of straight lines

ii) angle between them.

6. If the equation represents a pair of straight

lines, find the value of c.

12x2 − 10xy + 2y2 + 14x − 5y + c = 0

Find i) separate equations of straight lines

ii) angle between them.

V. Combined equation of the straight lines.

1. Find the combined equation of the straight

lines whose separate equations are

x + 2y − 3 = 0 and 3x − y + 4 = 0

2. Find the combined equation of the straight

lines whose separate equations are

x + 2y − 3 = 0 and 3x + y + 5 = 0

3. Find combined equation of straight lines

through origin, one of which is parallel to and

the other is perpendicular to 2x + y + 1 = 0

CIRCLE

I. Find the equation of the circle if

1. centre and radius are (2, − 3) and 4.

2. centre of the circle is (7, − 3)

and area of a circle is 16π square units.

3. centre is (− 4, 5)

circumference is 8π units.

4. centre (2, − 3) and radius 3.

Show that it passes through (2, 0)

5. centre (1, − 2)

passing through the point (4, 1)

6. centre at (2, 3).

passing through the point (1, 2)

7. Two diameters of with radius 5 units are

x + 2y = 7, 2x + y = 8

8. Extremities of a diameter are

(2, − 3) and (3, 1)

9. Described on the line joining the points

(1, 2) and (2, 4) as its diameter.

II. General equation of the circle

1. Find the centre and radius of the circle

3x2+3y2−2x+6y − 6 = 0

2. Find the centre and radius of the circle

(x − 3) (x − 5) + (y − 7) (y − 1) = 0

3. Find the values of a and b if the equation

represents a circle

(a − 4)x2 + by2 +(b −3)xy + 4x + 4y−1 = 0

4. Find the values of a and b if the equation

represents a circle

(a −2)x2 + by2 + (b −2)xy +4x + 4y −1= 0

Write down resulting equation of the circle

5. Find circumference and area of the circle

x2 + y2 − 6x − 8y + 15 = 0

III. Parametric Equations

6. Find the parametric equations of the circle

x2 + y2 = 16

7. Find the parametric equation of the circle

4x2 + 4y2 = 9

8. Find the cartesian equation of the circle

x = 2 cos θ, y = 2 sin θ

9. Find the cartesian equation of the circle

x = ¼ cosθ, y = ¼ sin θ and 0 ≤ θ ≤ 2π

IV. Equation of circle passing through 3 points

1. Find the equation the circle passing through

the points (0,1), (2,3) and (−2, 5).



www.Padasalai.Net



2. Find the equation of the circle passing through

the points (1, 0), (0, −1)and (0, 1).

3. Find the equation of the circle passing through

the points (1, 1), (2, −1)and (3, 2).

V. Equation of circle having centre on St.Line

1. Find equation of the circle passing through the

points (0, 1), (2, 3) and having the centre on

the line x − 2y + 3 = 0

2. Find the equation of the circle that passes

through the points (4, 1) and (6, 5) and has its

centre on the line 4x + y = 16.

3. Find equation of the circle whose centre is on

the line x = 2y and which passes through the

points (− 1, 2) and (3, − 2).

TANGENTS

1. Find the length of the tangent

from (2, 3) to the circle x2+y2−4x−3y +12= 0.

2. Find the length of the tangent

from (1, 2) to the circle x2 +y2 −2x +4y +9 = 0

3. Find the equation of the tangent to the circle

x2+y2=25 at (4, 3)

4. Find the equation of tangent to

x2 + y2 − 4x + 4y − 8 = 0 at (− 2, − 2)

5. Find the equation of tangent to

x2 + y2 − 4x + 8y − 5 = 0 at (2, 1).

6. Find length of chord intercepted by the circle

x2 + y2 − 2x − y + 1 = 0 and the line x − 2y = 1

7. Find length of chord intercepted by the circle

x2 +y2 − 14x + 4y + 28 = 0 and line x −7y+4= 0

8. Find the value of p if the line

3x+4y −p = 0 is tangent to circle x2+y2 = 16.

9. Find the value of p if the line

3x + 4y − p = 0 is a tangent to x2 + y2 − 64 = 0

10. Determine whether the point lie outside/inside

(2, 3) the circle x2 + y2 − 6x − 8y + 12 = 0.


(7, − 11) the circle x2 + y2 − 10x = 0?


(− 2, 1), (0, 0) (4, − 3)

circle x2 + y2 − 5x + 2y − 5 = 0

13. Find the equation of the circle which has its

centre at (2, 3) and touches the x-axis.

FAMILY OF CIRCLES

I. Circles touching each other:

1. Show that the circles touch each other.

x2 + y2 − 4x + 6y + 8 = 0

x2 + y2 − 10x − 6y + 14 = 0

2. Show that the circles touch each other.

x2 + y2 − 2x + 6y + 6 = 0

x2 + y2 − 5x + 6y + 15 = 0.

3. Show that each of circles touches other two

x2 + y2 + 4y − 1 = 0,

x2+ y2 + 6x + y + 8 = 0

x2 + y2 − 4x − 4y − 37 = 0

II. Concentric circles:

1. Find equation of circle concentric with circle

x2 + y2 − 4x − 6y − 9 = 0

and passing through the point (− 4, − 5).


x2 + y2− 8x + 12y + 15 = 0

and passes through the point (5, 4)


x2 + y2 − 2x − 6y + 4 = 0

and having radius 7.

III. Orthogonal circles:

1. Prove that the given circles are orthogonal.

x2 + y2 − 8x + 6y − 23 = 0

x2 + y2 − 2x − 5y + 16 = 0

2. Prove that the given circles are orthogonal.

x2 + y2 − 8x − 6y + 21 = 0

x2 + y2 − 2y − 15 = 0

III. Equation of Orthogonally touching circles:

1. Find circle which cuts orthogonally each of

x2 + y2 + 2x + 4y + 1 = 0,

x2+y2 − 4x+3 = 0

x2 + y2 + 6y + 5 = 0

2. Find circle which cuts orthogonally each of x2 + y2 + 2x + 17y + 4 = 0,

x2 + y2 + 7x + 6y + 11 = 0

x2 + y2−x + 22y + 3 = 0

3. Find circle which cuts orthogonally each of x2 + y2 = 9 , x2 + y2 − 2x + 8y − 7 = 0

which passes through the point (1, 2)

4. Find circle which cuts orthogonally each of x2 + y2 + 5x − 5y + 9 = 0

x2 + y2 − 2x +3y − 7 = 0

which passes through (1, − 1)

5. Find circle which cuts orthogonally each of x2 + y2 − 8x − 2y + 16 = 0

x2 + y2 − 4x − 4y − 1 = 0

which passes through (1, 1)



www.Padasalai.Net



SPECIAL TYPES OF SEQUENCES AND SERIES

1. Find the nth partial sum of the series

i)

13

1

nn

ii)

12

1

nn

iii)

1

5n

n

2. Find the single A.M between

i) 7 and 13 ii) 5 and − 3 iii) (p + q) and (p−q)

3. i) Find n arithmetic means between a and b and

find their sum

ii) Insert four A.Ms between − 1 and 14.

iii) Find five arithmetic means between 1 and 19

iv) Find six arithmetic means between 3 and 17

4. i) Find n geometric means between a and b and

find their product.

ii) Find 5 geometric means between 576 and 9.

5. i) 5th and 12th terms of H.Pare 12 and 5

Find 15th term

ii) 1st and 2nd terms of H.P are 31

and 51

Find 9th term.

iii) If pth and qth terms of a H.P are q and p

Show that (pq)th term is 1.

6. Three numbers form a H.P. Sum of numbers is 11

sum of the reciprocals is one. Find the numbers.

7. If a, b, c are in H.P., prove that 2

cbcb

abab

8. If a, b, c are in A.P. and a, mb, c are in G.P then

prove that a, m2b, c are in H.P

9. If b is G.M of a and c and x is A.M of a and b and

y is A.M of b and c, prove that 2yc

xa

10. Difference between two positive numbers is 18,

and 4 times their G.M is equal to 5 times their

H.M. Find the numbers.

11. If the A.M between two numbers is 1, prove that

their H.M is the square of their G.M.

12. If b is A.M of a and c and (b − a) is G.M of a

and c − a, show that a : b : c = 1 : 3 : 5

13. If a, b are two different positive numbers then

prove that i) A.M., G.M., H.M. are in G.P.

ii) A.M > G.M > H.M

PARTIAL FRACTIONS

I. Linear factors, none of which is repeated

1. 23

732 x + – x

x + 2.

1) + ( 1)(1

xx

3. 2 x56

1 -7x x +

4. )) (x + (x

x + 14

42

5. 3) ( 2)( 1) (

12

xxx+ x + x

II. Linear factors, some of which are repeated

6. 22) +(x 1) (x

9

7. 2) +( 1)(

12x x

8. 21)( 2) + (

2

xx

x 9.

3) +(x 2) (1 +

2x x

10. 2)( 2 + 6 -

2

2

x + xxx

11. 3

2

)2(7 5 2

xxx

III. Not factorable into linear factors

12. )1( )6(

922

2

x + + x + x x x

13. )1)(2(

32

2

xx

x

14. )1)(1(

22

xx

x 15.

)23)(12(6257

2

2

xxx

xx

IV. improper fraction.

16. 651

2

2

x + x + x + x

17. 121 + +

2

2

xx xx

COMPOSITION OF FUNCTIONS

1. Let A = {1, 2}, B = {3, 4} and C = {5, 6} and

f:A→ B and g : B → C such that

f(1) = 3, f(2) = 4, g(3) = 5, g(4) = 6.

Find gof.

2. f : R → R, g : R → R are defined by

f(x) = x2 + 1, g(x) = x − 1.

Show that fog ≠ gof.

3. f, g : R → R be defined by

f(x) = 2x + 1, g(x) =2

1x

Show that (fog) = (gof).

4. f, g : R → R, defined by

f(x) = x + 1 , g(x) = x2,

Find : i) (fog) (x) ii) (gof) (x)

iii) (fof) (x) iv) (gog) (x)

v) (fog) (3) vi) (gof) (3)

5. f : R → R be a function defined by

f(x) = 2x + 1.



www.Padasalai.Net



Find f −1

6. Let f : R → R be defined by

f(x) = 3x + 2.

Show that fof−1 = f−1of = I

7. f, g : R→R are defined by

f(x) = x+1, g(x)=x2.

Find f+g, f − g, fg, gf

, 2f, 3g.

8. f, g : R → R, defined by

f(x) = x + 1; g(x) = x2,

Define i)(f+g) (x) ii) gf

(x)

iii)(fg) (x) iv)(f−g)(x)

v) (gf) (x)

QUADRATIC INEQUATIONS

I. Solve the inequality

1) 4x2 − 25 ≥ 0 2) 4 − x2 < 0 3) x2≤ 9

II. Solve the inequality

1) x2 − 7x + 6 > 0 2) − x2 + 3x − 2 > 0

3) x2 − 3x − 18 > 0 4) x2 + x − 12 < 0

5) 7x2 − 7x − 84 ≥ 0

III. Solve the quadratic inequation

1) 64x2 + 48x + 9 < 0 2) 2x2− 3x + 5 < 0

3) 2x2 − 12x + 50 ≤ 0

IV. Solve the quadratic inequation

1) 011

xx

2) 2123

xx

3) 112

x

x

4) 343

541

xx

xx

5) 233

132

xx

xx

V. If x is real (x R), prove that

1. Range of

771

4343

2

2

, is x + + x

x + x f(x) =

2. Range of

331

4242

2

2

, is x + + x

x + x f(x) =

3. 727134

2

2

xxxx

can't have any value between 5

and 9

4. 952 xx

x lies between 111

and 1.

LIMIT OF A FUNCTION

I. Evaluate

1. 11

1

lim 3

xx

x 2.

xx

x

1)1(

0

lim 4

3. ax

ax

ax

nm

lim 4.

1

1

1

lim 3

x

x

x

5. h

xhx

x

22)(

0

lim

6. Find positive integer n such that

10833

3

lim

xx

x

nn

7. Find positive integer n such that

3222

2

lim

xx

x

nn

II. Evaluate

1. x

x

x

sin

0

lim

2.

2

cos1

0

lim

3.

23

2sin

0

lim

xx

x

4.

1sin11

0

lim

xx

x

5. xx

x

sinsin

0

lim

6.

xxaxa

x

)sin()sin(

0

lim

III. Evaluate

1. x

e

x

x 1

0

lim

2.

33

lim 3

xee

x

x

3. x

e

x

x

tan1

0

lim tan

4.

xxe

x

x 1sin

0

lim

IV. Evaluate

1. x

x

x

)1log(

0

lim

2.

1log

1

lim

xx

x

V. Evaluate

1. x

a

x

x 1

0

lim

2.

)11(

12

0

lim

xx

x

3. xx

xx 65

0

lim

4.

)11(

12

0

lim

xx

x

6. Compute x

ba

x

xx

0

lim

Hence evaluate xx

xx 65

0

lim

VI. Evaluate

1. 5

11

lim

n

nx 2.

3

33lim

x

xx

x

3. xxx

sec3

2

)cos1(lim



www.Padasalai.Net



DIFFERENTIATION TECHNIQUES

1. Find dxdy

if

xy

xy

xxy

xxyxyxyxy

log)71

)61

)5

)4)3)2)1

3

5

2. 1. Find dxdy

if y = x3−6x2+7x + 6.

2. If f(x) = x3−8x+10, find f′(x) and f′(2), f′(10).

3. If for f(x) = ax2+bx+12,f′(2)=11, f′(4) = 15

Find a and b.

3. Product rule for differentiation

x x e

xexxe x

x x + ec x x -

xlog exxxec x

xbaxx

xxxe

xxx e

x

xx

x

x

nx

cotlog)12

log)11sin10)

)cos5sin2( )cos4sec3(9)

x)cos 5 +sin x (2 x)cosec 4 -secx (38)

)( 2) + 7 + ()7cotcos6)

cosx) 2(1 )sin()62) + ( 1)5)(

3) + (2 1) - (4)4cos)3

log)2tan)1

2

2

22

2

5. Differentiate using quotient rule.

1tan1tan

)10cossincossin

)9

logcossin

)82log

2log)7

cossincossin

)6

1

1)5

log)4

sinlog

)344

)25432

)1

2

2

2

2

2277

xx

xxxxxx

xxxex

xx

xxxxxx

x

x

e

xxx

xx

xx

x

x

6. Derivative of a composite function (Chain rule)

)(sin )15)( log )41 )()13

)12)11sin)102sin)9

cos)8cos)7)(logtan)6cot1)5

)log(sin)4)(sin)3)2)(logsin)1

2

sin)sin(log23

2

sin

2

ax+bax+bax+b

eexx

xxxx

xbaxex

nn

xx

x

7. Derivatives of inverse functions

)(tancot)(cottan.7

11

sin.611

cos.5)(tan.4

)(logtan.3cot.22(sin.1

11

111

1121 2

xx

xx

xx

e

x) (ex)+x

x

x--

8. Logarithmic Differentiation

xxx

xxx

xxx

xxxx

1

2

sinlog1sin

sintan2

)(log)7)(tan)6)5

sin)4)3)2)1

74

)2)(2()12

3)(3(2)1(

)11log

)cos(sin)10

)(sin)9)12()8

2

2

sin12

xx

xx

xxxx

xe

ex

xxxx

x

x

xxx

9. Differentiation of parametric functions

3

2

3

2

33

33

13

,13

)9

44)8

2)7

sin)2

tanlog(cos)6

2sinsin22coscos25)

). cos (1 a =y ),sin + ( 4)

tansec3)

sincos)2

sincos)1

tat

yt

atx

tt, y =x =

at, y = x = at

,y = a + x = a

, y = x =

x = a

y = b x = a

tt, y = a x = a

, y = b x = a

10. Differentiation of implicit functions

xy

x + yyx

m + nnm

x- ye

= yx

(x + y)xy = = e + ee

= (x + y) y x

= x + yexy + x (xy) xy =

x = x + x - y y

= xx + y x + y

y) = (xx + y) +

yx + + + x

y+y

yxa = + yx yy = x

b

y

a

x. = xy+ y + x

.14

100.13.12

.11

.10tan.9

02costan.8

1cotsec)1(.7

1tan(tan.6

0 = sintan1

.5

4.4sin.3

1.2648.1

2

2

22

22

2

33244

2

2

2

233

15. If 022222 hxy+c = fy+gx+ ++byax

Prove that 0fhx + by + gax + hy +

dxdy

11. Higher order Derivatives.

1. Find the second order derivative of:

i) log (log x) ii) x2 + 6x + 5 iii) x sinx

iv) cot−1x . v) x3 + tan x.

2. Find the third order derivatives of

i) log (cosx) ii) y = x2 iii) x2 + cotx

iv) emx + x3 v) x cos x .



www.Padasalai.Net



3. If y = A cos4x + B sin 4x

Show that y2+16y = 0

4. If y = 500 e7x + 600e −7x,

show that 492

2

dx

yd

5. If y = etan−1x

prove that (1 + x2)y2 + (2x − 1)y1 = 0

6. If y = log (x2 − a2),

prove that y3 =

33 )(

1)(

12

axax

7. If y = sin (ax +b),

prove that y3 = a3 sin(ax+b+ )2

3

8. If x = sin t; y = sin pt

show that 0)1( 22

22 yp

dxdy

xdx

ydx

9. If x = a (cos θ + θ sin θ), y = a (sin θ − θ cos θ),

show that 32

2sec

dx

yda

10. If y = (x3 − 1),

prove that x2 y3 − 2xy2 + 2y1 = 0

11. If y = cos (m sin−1x),

prove that (1− x2)y3−3xy2 + (m2 − 1)y1= 0

12. If y = eax sin bx,

prove that 0)(2 222

2 yba

dxdy

adx

yd

INTEGRAL CALCULUS

1. Integrate the following with respect to x.

xexx

xxx

xxx

xxx

1)12

1)11)10

1)9

1)8

1)7

)6)5)4

)3)2)1

3 42

55

3 4771

10

25

116


dxecx

dxx

dxxx

dx

cos1

)4sin

1)3

sincot

)2xcos

sinx )1

2

2



dxx

dxx

dxxdx

cos11

cos11

cossin 22


dxxdx 33 cossin


dxxxdxxx

dxxxdxqxpx

dxxxdxnxmx

2sin10sin4sin2cos

cos3coscoscos

5cos7sincossin

7. Integrate the following

dxx

x + x + xdx

xxx

2

232 23415


dxeee

dxe

ex

xx

x

x

21 22


12

12 243

xxx

dxx

x


x

xx

x

xx

c

ba

6

32 1111


43 xx

dx

caxbax

dx


32)12(1)1( xxdxxx


)9)(2)(2(

1

)2)(1(

9)3(2(

1

2

2

2

xxx

x

dxxx

dxxx

x

METHOD OF SUBSTITUTION


i. a) dxnmxlx

mlx

2

2 b) dx

xxx

512

2

c) dxx

x sin1

cos d) dxe

ex

x

5



www.Padasalai.Net



ii. a) xtan b) xcot

c) dxxsec d) dxecxcos

iii. a) dxxx

x

seclog

tan1 b) dxx

x )(sinlog

cot

c) dxeex

exexe

xe

11 d) dx

ex

exxe

xe

11

e) dxxx log

1 f) dxxx

1

g) dxxbxa

x 22 sincos

2sin


i. a) dxx

e x

b) dxx

e x

2

tan

cos

ii a) dxx

e x

2

sin

1

1

b) dxx

e xm

2

tan

1

1

iii a) dxee xx4log3 b) dxex

ax 1

c) dxex x545


i. a) dxxx 41716 )1( b) dxx

x 1025

24

)1(

c) dx)x+xx 42 52)(14( d) dxxx

2)(log

e) dxxx cossin15 f) dxx 7sin

g) dxx

xxx

4)log)(1(5h) dx

xx

log

ii. a) dxxxx 53)32( 2 b) dxxx

x

653

562

c) dxx

x sin

cos d) dxxx sectan

e) dxxx

x cossin

tan f) dxx

xx

4

21

1

)(sin

iii) a) dxaxx m )( b) dxxx 152 )2(

c) dxxx 16)1( d)

dx

cbax

bax102 )(

24

iv) a) dxxx 14)32( b) dxxx 32)1(

c) dxxx 12)53( d) dxxx )1)1( 2

4. Integrate the following (Miscelleanous)

a) dxx

x 32

15

1 b) dx

ee

eexx

xx

22

c) dxx

x

sin d) dxax

x )sin(

sin

e) dxax

x )cos(

cos f) dx

xx

tan1tan1

g) dxxx

2sintanlog

h) dxx sectan3

INTEGRATION BY PARTS

I. Integrate the following

1. dxx log 2. dxxx log

II. Integrate the following

1. dxxex 2. dxxe-x

3. dx ex x25 4. dxex x

23

5. dxex x 22 6. dxex x 32

7. dxe x 8. dxx e x 32

9. dx

x

xex

--

2

1sin1

1)(sin

III. Integrate the following

1. dxx- 1sin 2. dxx- 1tan

3. dxxx - 1tan 4. dxx

xx -

2

1

1

sin

IV. Integrate the following

1. dx x x sin 2. dxxx cos

3. dxxx 2cos 4. x dxx x tansec

5. dxxx - )(sin 21 6. dx xx 3cos2

7. dxxx 2cos2 8. dxxxx 2cos3sin

9. dxxx x 2cos5cos 10.

11. x dxecx 2cos 12. dxxx 2sin

13. dxxx 2sec 14. x dxx 2tan


1. dxbx eax cos 2. dxx ex cos

3. dxx ex 2cos 4. dxxxe x 2sin5cos4

5. dxxe x- 33 cos 6. dx bx eax sin

7. dxx e x 3sin2 8. dxx e x 2sin3


1. dxx 2sec3 2. dxx 3sec

3. dxxec3cos

V. Integrate the following (Miscelleanous)



www.Padasalai.Net



1. dxx

x - x-

2

31

313

tan 2. dxxx-

2

1

12

tan

3. dxx x 5

STANDARD INTEGRALS


1. 291 xdx 2.

16

12x

dx

3. 252 + xdx 4. 16)2( 2 + x +

dx

5. 4)53( 2 + x +

dx


1. 291 xdx 2. 241 x

dx

3. 216 xdx 4. 2)3(9 x

dx

5. 2)14(7 x +

dx


1. 252 x

dx 2. 4)2( 2x

dx

3. 9)12( 2x

dx 4. 16)12( 2 x +

dx

5. 7)53( 2 x +

dx


1. 225 x

dx 2.

161

2x

dx

3. 2161 x

dx 4. 216 xdx

5. 24 x

dx 6. 2)1(25 x

dx

7. 2)32(11 x +

dx

V. Integrate the following

1. 12 + x

dx 2. 169 2x

dx

3. 4)52( 2 + x +

dx 4. 6)53( 2+x

dx

VI. Integrate the following

1. 92x

dx 2. 254 2x

dx

3. 912x

dx 4. 15)1( 2x+

dx

5. x +

dx

16)32( 2

VII. Integrate the following

1. dxx 294 2. dx x 24

3. dx x + 2)2(25

VIII. Integrate the following

1. dxx 2516 2

IX. Integrate the following

1. dx + x 169 2 2. dx + x 21

3. dx + x + 4)1( 2 4. dx + )x + 912( 2

SQUARE COMPLETION FORMULA


1. 752 x + + xdx 2. 1372 2 x++x

dx

3. 1372 2 x + + xdx 4. 1069 2 x + + x

dx


1. 572 x + xdx 2. 10133 2 x+x

dx

3. 21 x + x dx 4. 2965 x x

dx

5. 332 x + xdx 6. 10133 2 x x

dx


1. 100162 x + + x

dx 2. 1032 x++x

dx

3. 2652 x + + x

dx 4. 1242 x + x

dx

5. 2082 x + x

dx


1. 289 xx +

dx 2. 26 xx

dx

3. 22518 xx

dx 4. 21 x + x

dx

5. 28 x x

dx

V. Integrate the following

1. dx)) (x - (x + 21 2. dxx+ x 642

3. dx x + +x 142 4. dxx+x 1032

VI. Integrate the following

1. dxxx + 2584 2. dx+xx( )1)(2



www.Padasalai.Net



3. dxx + 2)13(169 4. dx xx 22

5. dxxx 231 6. dx+xx )3)(2(

PARTIAL FRACTION FORMULA


1.

dxx + + x

x83

342

2. dx+ x + xx +

123

2

3.

dxxx

x2

252

4.

dx + x + x

x1

232

5.

dxx + + x x

3213

2 6.

dx

+ x + x x

3212

2

7.

dx x x

x 21

1 8. dxx + + x

x + 13

142


1. dx+x+x

x +

32

132

2. dx

x + x

+x28

1

3.

dx

x+x

x

12

342

4. dx

x+x

x+226

2

5. dx

xx

x - 2710

32 6. dxx++x

x +

743

232

7. dx

xx

x +

)5)(4(

76 8. dx

- x + x

11

PROBABILITY

1. In a single throw of two dice, find the probability of obtaining

i) sum of less than 5

ii) sum greater than 10,

iii) a sum of 9 or 11.

2. Three coins are tossed once. Find the probability of getting

i) exactly two heads

ii) atleast two heads

iii) atmost two heads.

3. A single card is drawn from a pack of 52 cards. What is the probability that the card is

i) a jack or king

ii) 5 or smaller

iii) either queen or 7.

4. A bag contains 5 white and 7 black balls. 3 balls

are drawn at random. Find the probability that

i) all are white

ii) one white and 2 black.

5. In a box containing 10 bulbs, 2 are defective. What is the probability that among 5 bulbs chosen

at random, none is defective.

6. 4 mangoes and 3 apples are in a box. If two fruits are chosen at random, the probability that

i) one is a mango and the other is an apple

ii) both are of the same variety.

7. What is the chance that

i) non-leap year ii) leap year

should have fifty three Sundays?

8. An integer is chosen at random from the first fifty

positive integers. What is the probability that the integer chosen is a prime or multiple of 4.

9. Three letters are written to three different persons and addresses on three envelopes are also written.

Without looking at the addresses, what is the probability that

i) all letters go into right envelopes,

ii) none goes into right envelopes

10. A cricket club has 15 members, of whom only 5

can bowl. What is the probability that in a team of 11 members atleast 3 bowlers are selected?

11. Out of 10 outstanding students in a school there are 6 girls and 4 boys. A team of 4 students is

selected at random for a quiz programme. Find the probability that there are atleast 2 girls.

SOME BASIC THEOREMS ON PROBABILITY

1. a) P(A)=0.36,P(A or B) = 0.90,P(A and B)= 0.25

Find i) P(B), ii) P( A ∩ B )

b) P(A) = 0.28, P(B) = 0.44,

Find i) P( A ) ii) P(A B)

iii) P(A ∩ B ) iv) P( A ∩ B )

c) P(A) = 0.5, P(B) = 0.6 and P(A ∩ B) = 0.24.

Find i) P(A B) ii) P( A ∩ B)

iii) P(A ∩ B ) iv) P( A B )

v) P( A∩ B )

d) P(A) = 0.35, P(B) = 0.73 and P(A ∩ B) = 0.14,

Find i)P(AB) ii) P( A ∩B)

iii) P(A∩ B ) iv) P( AB )

v) P( A ∩ B )

2. A card is drawn at random from a well-shuffled

deck of 52 cards. Find the probability of drawing

i) a king or a queen

ii) a king or a spade

iii) a king or a black card



www.Padasalai.Net



3. The probability that a girl will get an admission in

IIT is 0.16, the probability that she will get an

admission in Government Medical College is

0.24, and the probability that she will get both is

0.11. Find the probability that

i) She will get atleast one of the two seats

ii) She will get only one of the two seats

4. A die is thrown twice. Let A be the event. “First

die shows 4’ and B be the event, ‘second die

shows 4’. Find P(A B).

5. The probability of an event A occurring is 0.5 and

B occurring is 0.3. If A and B are mutually

exclusive events, then find the probability of

neither A nor B occurring

6. A card is drawn at random from a deck of 52

cards. What is the probability that drawn card is

i) a queen or club card

ii) a queen or a black card

7. The probability that a new ship will get an award

for its design is 0.25, the probability that it will get

an award for the efficient use of materials is 0.35,

and that it will get both awards is 0.15. What is the

probability, that

i) it will get atleast one of the two awards

ii) it will get only one of the awards

CONDITIONAL PROBABILITY:

1. A coin is tossed twice. Event E = Head on first

toss, F = head on second toss.

Find i) P(E ∩ F) ii) P(E F) iii) P(E/F)

iv) P( E /F) v) Are E and F independent ?

2. A and B are two independent

a) P(A) = 0.5 and P(A B) = 0.8. Find P(B).

b) If P(A) = 0.4, P(B) = 0.7 and P(B / A) = 0.5

Find P(A / B) and P(A B).

c) P(A) = 2/5, P(B) =3/4, A B = (sample space)

Find the conditional probability P(A / B).

d) P(A B) = 0.6, P(A) = 0.2 find P(B)

e) P(A B) = 5/6, P(A ∩ B) = 1/3, P( B ) = 1/2

show that A and B are independent.

f) P(A) = 0.25, P(B) = 0.48,

Find i) P(A ∩ B) ii)P(B / A) iii)P( A ∩ B )

g) P(A) = 0.50, P(B) = 0.40 and P(A ∩ B) = 0.20.

Verify i) P(A / B) = P(A), ii) P(A/ B ) = P(A)

iii) P(B / A) = P(B) iv) P(B / A ) = P(B)

h) P(A) = 0.3, P(B) = 0.6 and P(A ∩ B) = 0.25

Find i) P(AB) ii) P(A/B) iii) P(B/ A )

iv) P( A / B) v) P( A / B )

i) P(A) = 0.45 and P(A B) = 0.75.

Find P(B) if i) A and B are mutually exclusive ii) A and B are independent events

iii) P(A / B) = 0.5

iv) P(B / A) = 0.5

3. a) X speaks truth in 95 percent of cases, and Y in

90 percent of cases. In what percentage of cases are they likely to contradict each other in stating

the same fact.

b) A speaks truth in 80% cases and B in 75%

cases. In what percentage of cases are they likely to contradict each other in stating the same fact?

4. Two cards are drawn from a pack of 52 cards in succession.

a) Find the probability that both are kings when

(i) The first drawn card is replaced

(ii) The card is not replaced

b) What is the probability of getting two jacks if

i) first card is replaced before the second is drawn ii) not replaced before second card is drawn.

c) What is the probability of drawing

i) a red king

ii) a red ace or a black queen.

5. One bag contains 5 white and 3 black balls.

Another bag contains 4 white and 6 black balls. If

one ball is drawn from each bag, find the

probability that (i) both are white (ii) both are

black (iii) one white and one black.

6. A husband and wife appear in an interview for two

vacancies in the same post. The probability of

husbands’ selection is 1/6 and that of wife’s

selection is 1/5. What is the probability that

i) both of them will be selected

ii) only one of them will be selected

iii) none of them will be selected



www.Padasalai.Net



7. A problem is given to 3 students

a) Whose chances of solving it are 12 , 1and 25

What is the probability that

i) problem is solved

b) Whose chances of solving it are 1/2, 1/3 and 1/4

What is the probability that

i) the problem is solved

ii) exactly one of them will solve it.

8. A year is selected at random. What is the

probability that

i) it contains 53 Sundays

ii) it is a leap year contains 53 Sundays

9. For a student the probability of getting admission

in IIT is 60% and probability of getting admission

in Anna University is 75%. Find probability that

i) getting admission in only one of these

ii) getting admission in atleast one of these.

10. A can hit a target 4 times in 5 shots, B 3 times in

4 shots, C 2 times in shots, they fire a volley.

What is the chance that the target is damaged by

exactly 2 hits?

11. Two thirds of students in a class are boys and rest

girls. It is known that the probability of a girl

getting a first class is 0.75 and that of a boys is

0.70. Find the probability that a student chosen at

random will get first class marks.

TOTAL PROBABILITY OF AN EVENT

1. An urn contains 10 white and 5 black balls. While another urn contains 3 white and 7 black balls.

One urn is chosen at random and two balls are drawn from it. Find the probability that both balls

are white.

2. a) A factory has two machines I and II. Machine I produces 30% of items of the output and Machine II produces 70% of the items. Further 3% of items

produced by Machine I are defective and 4% produced by Machine II are defective. If an item is

drawn at random

i) Find the probability that it is a defective item

ii) If it is defective, what is the probability that it was produced by Machine-II.

b) In a factory, Machine-I produces 45% of the

output and Machine-II produces 55% of the

output. On the average 10% items produced by I

and 5% of the items produced by II are defective.

An item is drawn at random from a day’s output.

i) Find the probability that it is a defective item

ii) If it is defective, what is the probability that it

was produced by Machine-II.

c) A factory has two Machines-I and II.

Machine-I produces 25% of items and Machine-II

produces 75% of the items of the total output.

Further 3% of the items produced by Machine-I

are defective whereas 4% produced by Machine-

II are defective. If an item is drawn at random

i) what is the probability that it is defective?

3. The chances of X, Y and Z becoming managers of

a certain company are 4 : 2 : 3. The probabilities

that bonus scheme will be introduced if X, Y and

Z become managers are 0.3, 0.5 and 0.4

respectively. If the bonus scheme has been

introduced, what is the probability that Z is

appointed as the manager.

4. A consulting firm rents car from three agencies

such that 20% from agency X, 30% from agency

Y and 50% from agency Z. If 90% of the cars

from X, 80% of cars from Y and 95% of the cars

from Z are in good conditions what is the

probability that the firm will get a car in good

condition? Also If a car is in good condition, what

is probability that it has came from agency Y?

5. Bag A contains 5 white, 6 black balls and bag B

contains 4 white, 5 black balls. One bag is

selected at random and one ball is drawn from it.

Find the probability that it is white.

6. There are two identical boxes containing respectively 5 white and 3 red balls, 4 white and 6

red balls. A box is chosen at random and a ball is drawn from it

i) find the probability that the ball is white

ii) if white probability that it is from first box?

7. Three urns each containing red and white chips as

given below.

Urn I : 6 red 4 white Urn II : 3 red 5 white

Urn III : 4 red 6 white

An urn is chosen at random and a chip is drawn from urn. i) Find probability that it is white.

ii) If chip is white find probability that it is from urn II



www.Padasalai.Net

· PDF file · 2018-02-112018-02-11 · ... cos2A + cos2B−cos2C =...

Documents

Transcript of · PDF file · 2018-02-112018-02-11 · ... cos2A + cos2B−cos2C =...