Class XII Mathematics Vector Algebra and 3D

1

VECTOR ALGEBRA & 3D Weightage 17 Marks

SYLLABUS:

1. VECTOR ALGEBRA

Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction

ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors),

position vector of a point, negative of a vector, components of a vector, addition of vectors,

multiplication of a vector by a scalar, position vector of a point dividing a line segment in a

given ratio. Scalar (dot) product of vectors, projection of a vector on a line. Vector (cross)

product of vectors. Scalar triple product of vectors

SUMMERY OF QUESTIONS (YEAR WISE)

YEAR VSA (1 Mark) SA (4 Marks) LA (6 Marks) Total Marks

2009 3 1 0 7

2010 2 1 0 6

2011 2 1 0 6

2012 2 1 0 6

2013 2 1 0 6

PREVIOUS YEARS QUESTIONS

2009

1. Find projection of a

on b

if . 8 2 6 3a b and b i j k

1

2. Write a unit vector in the direction of 2 6 3a i j k

1

3. Find the value of p for which 3 2 9 3a i j k and b i pj k

are parallel vectors. 1

4. If a b c d

and a c b d

, show that a d

is parallel to b c

. 4

2010

1. Write a vector of magnitude 15 units in the direction of vector 2 2i j k

1

2. What is the cosine of the angle which the vector 2 i j k

makes with y-axis. 1

3. Find the position vector of the point R which divides joins of points P and Q whose position

vectors are 2a b

and 3a b

in 1:2 internally. Also show that P is the mid point of RQ. 4

2011

1. For what value of ‘a’ the vectors 2 3 4i j k

and 6 8a i j k

are collinear. 1

2. Find the direction cosines of the vector 2 5i j k

. 1

3. Find the unit vector perpendicular to each of the vectors a b

and a b

where

3 2 2a i j k

and 2 2b i j k

4

2012

1. Write a vector of magnitude 15 units in the direction of vector. 1

2. What is the cosine of the angle which the vector 2 i j k

makes with y-axis? 1

3. Find the position vector of the point R which divides joins of points P and Q whose position

vectors are 2a b

and 3a b

respectively externally in the ratio 1:2 externally.

Also show that P is the mid point of RQ. 4

2013

2

1. If a unit vector a

makes angles 3

with i ,

4

with j and an acute angle with k

, find

the value of . 1

2. If 2a x i j z k

and 3b i j k

are two equal vactors , then write value of x + y + z 1

3. If a

and b

are two vectors such that a b a

, then prove that 2 a b

is perpendicular

to vector b

4

Topic Wise Distribution of questions

Topic 2009 2010 2011 2012 2013

Introduction 2 6 2 6 1

Scalar Product 1 0 0 0 4

Vector Product 4 0 4 0 0

Triple Scalar Product ----------- ----------- ----------- ----------- -----------

Important results

1. For point A ( x, y, z), OA

= x i y j z k

2. For points A 1 1 1, ,x y z and B 2 2 2, ,x y z ; AB

= 2 1x x i

+ 2 1y y j

+ 2 1z z k

3. If kcjbiar

, then r

= 2 2 2x y z

4. Two vectors a

and b

are collinear if a

= k b

. Points A, B and C are collinear if AB

= k AC

5. a

aa

; a

= .a a

6. nm

OAnOBmOP

(Internally) ;

nm

OAnOBmOP

(externally);

2

OAOBOP

(mid point)

7. For OP x i y j z k

vector components are x i

, y j

and z k

& scalar components are x, y & z.

8. For a vector kcjbiar

a, b, c are direction ratios.

For unit vector knjmilr

l, m, n are direction cosines.

222 cbar

and 122 nml

9. For direction ratios a, b and c, direction cosines can be calculated as

2 2 2

al

a b c

,

2 2 2

bm

a b c

and

2 2 2

cn

a b c

DOT (SCALAR ) PRODUCT:

1) abba

.. ; Also cabacba

..).(

2) 22 . aaaa

; 1... kkjjii

3) For two perpendicular vectors ;,ba

0. ba

; 0... ikkjji

4) Projection of .a

on b

=

b

ba

.

; Projection vector of a

on b

= b

b

ba

..

2

5 ) For vectors kcjbiabkcjbiaa

222111 & ; 212121. ccbbaaba

6) To find angle between two vectors 2

2

2

2

2

2

2

1

2

1

2

1

212121.cos

cbacba

ccbbaa

ba

ba

;

Condition for two vectors to be perpendicular 0212121 ccbbaa

3

Condition for two vectors to be parallel ba

. OR 2

1

2

1

2

1

c

c

b

b

a

a

7) bababa

.22

22

; 22. bababa

accbbacbacba

.2.2.222

22

VECTOR (CROSS) PRODUCT

ba

= a. b sin n

, with be the angle between the two vectors and n

the unit vector

perpendicular to both the vectors.

1) abbaabba

; Also cabacba

)(

2) 0,0 kkjjiiaa

3) Magnitude of cross-product sin.. baba

4) Unit vector perpendicular to the plane containing vectors a

andb

, ba

ban

Vectors of magnitude ‘k’ perpendicular to a

andb

is given by ba

bak

5) For kcjbiabkcjbiaa

222111 & ;

222

111

cba

cba

kji

ba

6) Geometrically vector product represents the area of the parallelogram whose sides are

represented by the two vectors i.e. Area of parallelogram with consecutive sides represented

by ba

& ; Area = ba

7) Area of a triangle with sides ba

& ; Area = 2

1ba

8) Area of parallelogram with diagonals represented by 21 ,dd

; Area = 2

121 dd

9) Lagrange’s Identity 2222

.. bababa

SCALAR TRIPLE PRODUCT: For vectors cba

&, , scalar triple product is cba

. OR cba

. and

is denoted by [ a

b

c

]

Properties:

1. For a parallelepiped having its conterminous edges represented by vectors cba

&,

Volume of parallelepiped = [ a

b

c

]

2. Three vectors cba

&, are coplanar iff [ a

b

c

] = 0

3. Points A, B, C and D are coplanar if [ ADACAB ] = 0

4. For vectors kcjbiabkcjbiaa

222111 & and kcjbiac

333

[ a

b

c

] =

333

222

111

cba

cba

cba

5. [ a

b

c

] = - [ b

ca

], 0aba

MLM

4

ONE MARKER

1. For the following points A, B find the vector AB

as well as their magnitudes

A (2, -1, 3), B (0, -5, 2)

2. Find the unit vector along the vectors 4 2i j k

3. Find x so that x i j k

is a unit vector.

4. Find the vector along the direction of the vector 3 2 6i j k

and of magnitude 5

5. Find the scalar and vector components of the vector AB

with A(1, 0, 1), B(2, -2, 4).

6. Find the direction ratios and direction cosines of the line passing through the points

(1, 2, 4) and (-5, 2, 3).

7. Find the scalar product of the vectors 3 5 2i j k and b i j k

8. If , 2 2 4a i j k b i j k and c i j k

find a unit vector parallel to the vector

2 3a b c

.

9. Find the position vector of the point which divides joins of points 2 3a b

and 2a b

in 1:3 internally

and externally using vectors.

10. Find the position vector of the point dividing the join of A ( -1, 2, 4) , B ( 4, -3, 4) in (1) 1:3

internally (2) 1:3 externally using vectors.

11. Find so that kjia

923 is (1) perpendicular (2) parallel to kjib

3

12. Find a

and b

, if . 8 8a b a b and a b

13. For bafindkibkjia

2;3,23 .

14. Show that 0a b c b c a c a b

15. Evaluate . . .i j k j i k k i j

16. Find the angle between the vectors ba

& if .a b a b

.

17. Find the angle between the vectors kjibandkjia

84623

FOUR MARKERS

18. Show that the points A (2, -1, 3), B (3, -5, 1) and C (-1, 11, 9) are collinear.

19. Show that points A, B and C with p.v. 3 4 4i j k

. 2i j k

and 3 5i j k

forms a right triangle.

20. If , ,a b c

are mutually perpendicular unit vectors and. 0a b c

, show that a b c

3 .

21. If , ,a b c

are mutually perpendicular vectors of equal magnitude, show that , ,a b c

are equally inclined

to ( )a b c

.

22. If 2 2 3 , 2 3a i j k b i j k and c i j

, find so that a b

is perpendicular to c

.

5

23. Find projection & projection vector of a

on b

for 5 3 4 , 6 8a i j k b i j k

24. For 3 & 2 3a i j b i j k

, express b

in the form of two vectors 1b

, 2b

such that1b

is parallel to

a

and 2b

ia perpendicular to a

.

25. If ,7,5,3,0 cbacba

find the angle between ba

& .

26. For vectors , ,a b c

if 3, 2, 2,a b c

and. 0a b c

, Find . . .a b b c c a

.

27. Find a vector of magnitude 9 and perpendicular to both the vectors .2&32 kikji

28. Find the area of (i) parallelogram (ii) triangle whose adjacent sides are represented by

2 .i j k and i j k

29. Given bafindbaba

;12.&2,10 .

30. Let 4 2 , 3 2 7a i j k b i j k

and 2 4c i j k

. Find a vector d

which is

perpendicular to both &a b

and . 15c d

.

31. If a b c d

and a c b d

, show that a d

is parallel to b c

.

32. Find the volume of the parallelepiped whose sides are given by

kjikjikji

357&375,573

33. Show that points (6,-7,0) ,(16,-29,-4), ( 0,3,-6) and (2,5,10) are coplanar.

34. Find so that points kjiandkjikji

5332,2 .are coplanar.

35. Show that cbaaccbba

2

36. Show that 0 accbba

List of important questions and examples (NCERT)

EXAMPLES: 8,9, 11, 12, 14,15,19,20,26,28,29,30

Ex.10.2: 2,3,5,8,10,11,12,14,15,16,17

Ex. 10.3: 2,4,5,6,10,111,13,15,16

Ex.10.4: 2,3,5,7,9,10,12

MISC: 5,6,7,8,9,11,12,13,14,18,19

6

THREE – DIMENSION GEOMETRY

SYLLABUS

Direction cosines and direction ratios of a line joining two points. Cartesian and vector

equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian

and vector equation of a plane. Angle between (i) two lines, (ii) two planes. (iii) a line and

a plane. Distance of a point from a plane.



2009 ------- 1 1 10

2010 1 1 1 11

2011 1 1 1 11

2012 1 1 1 11

2013 1 1 1 11


2009

1. Find p so that lines are perpendicular 2

3

2

147

3

1

z

p

yx ;

5

6

1

5

3

77 zy

p

x

4

2. Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of 6

the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.

2010

1. Write vector equation of the line 5 4 6

3 7 2

x y z 1

2. Find the Cartesian equation of the plane passing through the points A(0,0,0) , B (3, -1, 2) and

parallel to line 4 3 1

1 4 7

x y z

4

3. The points A ( 4, 5, 10) , B(2,3,4)and C (1,2,-1) are three vertices of a parallelogram ABCD.

Find the equations of the sides AB and BC and also find the co-ordinates of point D. 6

2011

1. Write the intercept cut off by the plane 2x + y – z = 5 on x-axis. 1

2. Find the angle between the following pair of lines

2 1 3

2 7 3

x y z

2 2 8 5

1 4 4

x y z

and check whether the lines are parallel or perpendicular. 4

3. Find the equation of plane which contains the lines of intersection of the planes

.( 2 3 ) 4 0r i j k

; .(2 ) 5 0r i j k

and which is perpendicular to the plane

.(5 3 6 ) 8 0r i j k

6

2012

1. Write the vector equation of the following line:

5 4 6

3 7 2

x y z 1

2. Find the Cartesian equation of the plane passing through the points A(0, 0, 0) and B(3, –1, 2) and

parallel to the line 4 3 1

1 4 7

x y z

4

7

3. The points A(4, 5, 10), B(2, 3, 4) and C(1, 2, –1) are three vertices of a parallelogram ABCD.

Find the vector equations of the sides AB and BC and also find the coordinates of point D. 6

2013

1. Find the Cartesian equation of the line which passes through the point (-2, 3, -5) and is parallel

to the line 3 4 8

3 5 6

x y z 1

2. Find the coordinates of the point, where the line 2 1 2

3 4 2

x y z intersect the plane

x – y + z – 5 = 0. Also find the angle between the line and plane. 4

OR

Find the equation of plane which contains the lines of intersection of the planes

.( 2 3 ) 4 0r i j k

; .(2 ) 5 0r i j k

and which is perpendicular to the plane

.(5 3 6 ) 8 0r i j k

3. Find the vector equation of the plane passing through the points with position vectors 2i j k

;

2i j k

and 2i j k

. Also find the point of intersection of this plane and the line

3 (2 2 )r i j k i j k

6


Topic 2009 2010 2011 2012 2013

St Line 4 7 4 7 5

Plane 6 4 7 4 6

St line & Plane ------------ ------------ ------------ ------------ ------------

Important results

STRAIGHT LINE

1. Line passing through point A (P.V. a

) and parallel to m

is mar

Cartesian form: A (111 ,, zyx ) & a, b, c be D. R. of vector parallel to line

c

zz

b

yy

a

xx 111

2. A general point on the line is

P ( 111 ,, zcybxa )

3. Line passing through two points A (P.V. a

) & B (P.V.b

) is abar

Cartesian form: For points A ( 111 ,, zyx ) and B ( 222 ,, zyx ) 12

1

12

1

12

1

zz

zz

yy

yy

xx

xx

4. For lines having angle between them

111 mar

, 222 mar

OR 1

1

1

1

1

1

c

zz

b

yy

a

xx

;

2

1

2

1

2

1

c

zz

b

yy

a

xx

2

2

2

2

2

2

2

1

2

1

2

1

212121coscbacba

ccbbaa

=

mm

mm

1

21.

For perpendicular lines 0. 21212121 ccbbaamm

For parallel lines 2

1

2

1

2

1

21 ;c

c

b

b

a

amm

5. Skew lines in space are lines which are neither parallel nor intersecting.

6. Shortest distance between two skew lines.

For two lines 111 mar

; 222 mar

8

S. D. = 2 1 1 2

1 2

( ).( )a a m m

m m

OR S.D. =

2 1 2 1 2 1

1 1 1

2 2 2

2

1 2 2 1

x x y y z z

a b c

a b c

b c b c

Condition for two lines to intersect 2 1 1 2( ).( ) 0a a m m

For parallel lines mar 11

; mar 22

S. D. =

2 1( )a a m

m

PLANE

1. General equation of plane is a x + b y +c z + d. = 0 Here a, b, c is the d. r. of normal

(perpendicular) to plane.

Vector Form 0. dnr

, here n

is normal (perpendicular) to plane.

2. Plane through one point A (111 ,, zyx ) with p.v. a

and normal to vector kcjbia

(Direction ratios a, b, c) 0.;0111 narzzcyybxxa

OR nanr

..

3. Equation of plane through three points A 111, zyx ; B 222 , zyx ; C 333 ,, zyx

0

13121

13121

13121

zzzzzz

yyyyyy

xxxxxx

OR 0 acabar

4. Equation of plane having intercepts a, b, c on the axes 1c

z

b

y

a

x

5. Plane perpendicular to unit vector n

and at a distance of p from origin

Cartesian Form: l x + m y + n z = p, here l, m, n are d. cosines of normal to plane.

6. Equation of plane through a point A (x 1, y 1, z1) and

(a) perpendicular to planes a 1x + b 1 y + c 1 z + d 1= 0 ; a 2x + b 2 y + c 2 z + d 2= 0 or

(b) parallel to two vectors with direction ratios kcjbia

111 kcjbia

222 or

(c) parallel to two lines having direction ratios a1, b1, c1 & a2, b2, c2

0

222

111

111

cba

cba

zzyyxx

7. Equation if plane through two points A (x 1, y 1, z1) , B (x 2, y 2, z 2) and

(a) perpendicular to plane a 1x + b 1 y + c 1 z + d 1= 0 OR (b) parallel to one vector kcjbia

or

(c) parallel to line having D.R.’s a, b, c

0121212

111

cba

zzyyxx

zzyyxx

pnr

.

9

8. Equation of plane that contains the lines 1

1

1

1

1

1

c

zz

b

yy

a

xx

&

2

1

2

1

2

1

c

zz

b

yy

a

xx

is 0

222

111

111

cba

cba

zzyyxx

Two lines 1

1

1

1

1

1

c

zz

b

yy

a

xx

&

2

1

2

1

2

1

c

zz

b

yy

a

xx

are coplanar if

0

222

111

121212

cba

cba

zzyyxx

9. Equation of plane that contains parallel lines c

zz

b

yy

a

xx 111

&

c

zz

b

yy

a

xx 111

is 0121212

111

cba

zzyyxx

zzyyxx

10. Equation of plane through the line of intersection of planes

0:&0: 2222211111 dzcybxadzcybxa is 021

11. Distance of a point from a plane:

From point A (p.v. a

) From point A (111 ,, zyx )

and plane 0. dnr

and the plane a x + b y + c z + d = 0

p = n

dna

.

p = 222

111

cba

dzcybxa

12. Angle between the planes: If be the angle between the planes

0&00.&0. 222211112211 dzcybxadzcybxaORdnrdnr

2

2

2

2

2

2

2

1

2

1

2

1

212121coscbacba

ccbbaa

=

nn

nn

1

21.

For perpendicular planes 0. 21212121 ccbbaann

;

For parallel planes 2

1

2

1

2

1

21 ;c

c

b

b

a

ann

LINE AND PLANE

1. Angle between line and plane: For line 0.& dnrplanemar

OR For line 1

1

1

1

1

1

c

zz

b

yy

a

xx

& Plane 02222 dzcybxa

2

2

2

2

2

2

2

1

2

1

2

1

212121.

cbacba

ccbbaa

mn

mnSin

For line to be parallel to plane 0. 212121 ccbbaamn

For line to be perpendicular to plane 2

1

2

1

2

1;c

c

b

b

a

amn

10

2. Equation of line perpendicular to plane a x + by + c z + d = 0 and passing through the point

( x1, y1, z1) is c

zz

b

yy

a

xx 111

IMPORTANT QUESTIONS (THREE DIMENSIONS)

1. Find the equation of line in cartesian and vector form passing through point kji

432

and parallel to kji

543 .

2. Find equation of line through (2, -1, 1) and parallel to the line whose equation is

3

2

7

1

2

3

zyx.

3. Find vector and Cartesian equation of line passing through the points (2, 3, 4) & ( -1, 2, 3)

4. Find the equation of x –axis.

5. Find the point where line 4

5

3

2

2

1

zyx meets plane 2 x + 4 y – z = 3.

6. Shown that lines intersect jikjir

3)( ; kikir

32)4(

7. Find the point of intersection of the lines kjikjir

432)32( ;

kjijir

25)4(

8. Find the angle between the lines kjikjir

2)32( ;

kjijir

453)9(

9. Find p so that lines are perpendicular 2

3

2

147

3

1

z

p

yx ;

5

6

1

5

3

77 zy

p

x

10. Find p so that lines are parallel kjikjir

2)32( ; kpjijir

33)9(

11. Find the distance to the line 2

7

2

7

3

6

zyx from the point (1, 2, 3)

12. Find the image in the line kjikjir

11410)8211( of the point (2, -1, 5)

13. Find the shortest distance between the lines

kjikjir

)2( ; kjikjir

22)2(

14. Find the equation of plane in Cartesian form 010)23.( kjir

15. Find the equation of the plane in vector form 2 x + 3 y – 10 = 0

16. Find the equation of plane passing through points (2, 2, -1), (3, 4, 2), (7, 0, 6)

17. Find the equation of plane passing through the point ( 1, 0, -2) and perpendicular to the planes

2x + y –z -2 = 0, x – y – z – 3 = 0

18. Find the plane passing through the point (2, 3, 4) and parallel to the vectors kji

3 and

kji

232 .

11

19. Find the plane passing through the point (2, 1, 4) and parallel to the lines 4

6

3

1

2

1

zyx and

4

7

1

6

3

2

zyx.

20. Find the equation of plane passing through the points ( -1, 2, 3), (2, 3, 7) and perpendicular to the

plane 3x – 2 y + 2 z + 10 = 0

21. Find the equation of plane at a distance of 3 from origin and normal to kji

43 .

22. Find the plane passing through the point (1,2,3) and normal to vector kji

3 .

23. Find the plane through (2, -1, 3) and parallel to plane 3x – y + 4 z + 6 = 0.

24. The foot of perpendicular from origin to a plane is (2, 5, 7). Find the equation of plane.

25. Find the direction cosines of normal to the plane 3 x – 6 y + 2 z = 7.

26. Find the equation of plane having intercepts 1, 2 – 3 on the axes.

27. Find ‘p’ so that planes are perpendicular 013)32.( kjir

; 09)72.( kjipr

28. Find ‘p’ so that planes are parallel 010)323.( kpjipr

; 05).( kjir

29. Find the angle between the line kjikjir

22)432( and plane

010)236.( kjir

30. Find ‘m’ so that line kjikjir

22)2( is parallel plane 012)23.( kmjir

31. Show that line 1

3

12

1

zyx is lying in the plane x – y – z = 4.

32. Find the distance of the point kji

42 from the plane 9)1243.( kjir

.

33. Find the distance between the planes 2 x – y + 3 z + 4 = 0; 6 x – 3 y + 4 z – 3 = 0

34. Find p so that distance of point (1,2, p) from plane x + y + z – 10 = 0 is 32 .

35. Find the plane containing the lines jikjir

3)( ; kikir

32)4(

36. Show that lines are coplanar jiikjr

32)32( ; kjikjir

432)362(

37. (a) Find the equation of plane through the line of intersection of planes 2 x – 7 y + 4 z = 3 and

3 x – 5 y + 4 z + 11 = 0 and the point (-2, 1, 3)

(b) Find the equation of plane through the line of intersection of planes x + 2 y + 3 z = 4 and

2 x + y - z + 5 = 0 and perpendicular to the plane 5 x + 3 y – 6 z + 8 = 0.

(c) Find the equation of plane through intersection of planes 04).( jir

;

01)432.( kjir

and parallel to the line kjikjir

2)3(

38. Find the length and co-ordinates of foot of perpendicular from the point (7, 14, 5) to the plane

2 x + 4 y – z = 2


EXAMPLES: 4,7,14,16,17,19,21,22,25,26,27,28,30

Ex.11.: 1,3 Ex. 11.2: 2,3,4,6,8,10,11,12,15,17 Ex.11.3:2,4(a),5.(b),6(b),10,11,12,13(c),(e),14(b),(c)

MISC: 3,6,7,9,10,12,13,14,15,16,17,18,19,20,21,22

12

LINEAR PROGRAMMING Weightage 6 Marks

SYLLABUS

Introduction, related terminology such as constraints, objective function, optimization,

different types of linear programming (L.P.) problems, mathematical formulation of L.P.

problems, graphical method of solution for problems in two variables, feasible and infeasible

regions, feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial

constraints)



2009 0 0 1 6

2010 0 0 1 6

2011 0 0 1 6

2012 0 0 1 6

2013 0 0 1 6


2009

1. A diet is to contain at least 80 units of Vitamin A and 100 units of minerals. Two foods F1 and F2 are

available. Food F1 costs Rs. 4 per unit and food F2 costs Rs. 6 per unit. One unit of food F1 constrains 3

units of Vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of Vitamin A and 3 units

of minerals. Formulate this as a linear programming problem and find graphically the minimum cost for

the diet that consists of mixture of these foods and also meets the minimum nutritional requirements.

2010

1. One kind of cake requires 300 g of floor and 15 g of fat, another kind of cake requires 150 g of floor and

30 g of fat. Find the maximum number of cakes which can be made from 7.5 kg of floor and 600 g of fat,

assuming that there is no shortage of ingredients used in making the cakes. Make it LPP and solve ot

graphically.

2011 1. A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time

and 3 hours of craftsman’s time in its making while a cricket bat takes 3 hours of machine time

and 1 hour of craftsman’s time. In a day, the factory had the availability of not more than 42 hours

of machine time and 24 hours of craftsman’s time. If the profit on a racket and bat is Rs 20 and Rs 10

respectively, find the number of tennis rackets and cricket bats that factory must manufacture to

earn the maximum profit. Make it as an L.P.P. and solve it graphically.

2012

1. A small firm manufactures gold rings and chains. The total number of rings and chains manufactured

per day is at most 24. It takes 1 hour to make a ring and 30 minutes to make a chain. The maximum

number of hours available per day is 16. If the profit on a ring is Rs. 300 and that on a chain is

Rs. 190, find the number of rings and chains that should be manufactured per day, so as to earn the

maximum profit. Make it as an L.P.P. and solve it graphically.

2013

1. A cooperative society of farmers has 50 hectare of land to grow two crops X and Y. The profit from

crops X and Y per hectare are estimated as Rs 10,500 and Rs 9,000 respectively. To control weeds, a

liquid herbicide has to be used for crops X and Y at rates of 20 litres and 10 litres per hectare. Further,

no more than 800 litres of herbicide should be used in order to protect fish and wild life using a pond

13

which collects drainage from this land. How much land should be allocated to each crop so as to

maximise the total profit of the society?


Topic 2009 2010 2011 2012 2013

Manufacturing Problem ------------ ------------ 6 6 ------------

Diet Problem 6 ------------ ------------ ------------ ------------

Allocation Problems ------------ 6 ------------ ------------ 6


EXAMPLES: 3,6,7,10,11

Ex.12.1, 8

Ex. 12.2: 2,4,5,6,7,8,9,10

Ex.15.3: 2,3,4,6,7,8,9

MISC: 5,7,8,10

14

PROBABILITY Weightage 10 Marks

Conditional probability, multiplication theorem on probability. independent events, total

probability, Baye's theorem, Random variable and its probability distribution, mean and

variance of random variable. Repeated independent (Bernoulli) trials and Binomial

distribution.



2009 0 1 1 10

2010 0 1 1 10

2011 0 1 1 10

2012 0 1 1 10

2013 0 1 1 10

2009 1. A die is thrown again and again until three sixes are obtained. Find the probability of obtaining

the third six in the sixth throw of the die. 4

2. Three bags contains balls as shown in the table below

Bag Number of white balls Number of Black balls Number of Red balls

I 1 2 3

II 2 1 1

III 4 3 2

A bag is chosen at random and two balls are drawn from it. They happen to be white and red.

What is the probability that they are from third bag. 6

2010

1. On a multiple choice examination with three possible answers (out of which only one is correct)

for each of five questions, what is the probability that a candidate would get four or more correct

answers just by guessing? 4

2. One card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are

drawn at random and are found to be both clubs. Find the probability that lost card being of clubs. 6

OR

Out of a lot of 10 bulbs which includes 3 defectives, a sample of 2 bulbs is drawn at random.

Find the probability of the number of defective bulbs. 6

2011

1. Probabilities of solving a specific problem independently by A and B are 1

2and

1

3respectively.

If both try solve the problem independently, find the probability that (i) the problem is solved

(ii) exactly one of them solve the problem. 4

2. Suppose that 5% of men and 0.25% of women have grey hair. A grey haired person is selected at

random. What is the probability of this person being male? Assume that there are equal number of

males and females. 6

2012

1. A family has 2 children. Find the probability that both are boys, if it is known that

(i) at least one of the children is a boy (ii) the elder child is a boy. 4

2. A bag contains 4 balls. Two balls are drawn at random, and are found to be white.

What is the probability that all balls are white? 6

15

2013

1. A speaks truth in 60% of cases, while B in 90% of the cases. In what percentage of cases are they

likely to contradict each other in stating the same fact? In the cases of contradiction do you think,

the statement of B will carry more weight as he speaks truth in more number of cases than A? 4

2. Assume that the chances of a patient having a heart attack is 40%. It is also assumed that a

meditation and yoga course reduce the risk of heart attack by30% and prescription of certain drug

reduces its chances by 25%. At a time a patient can choose any one of the two options with equal

probabilities. It is given that after going through one of the two options the patient selected at

random suffers a heart attack. Find the probability that the patient followed a course of meditation

and yoga? 6


Topic 2009 2010 2011 2012 2013

Conditional Probability -------- -------- -------- 4 --------

Multiplication Th. , Independent Events -------- -------- 4 -------- 4

Bays’ Th. 6 6 6 6 6

Probability Distribution -------- -------- -------- -------- --------

Binomial distribution 4 4 -------- -------- --------

Important Results

Bayes’ theorem:- For the sample space S associated with a random experiment . Let nEEE ,,21 ,

are mutually exclusive and exhaustive events associated with a sample space S .If A is a

event of S then

n

i

iI

iIi

EAPEP

EAPEPAEP

1

)/().(

/.)/(

2. Conditional Probability: If A and B are two events of a sample space S associated with a random

experiment, then probability of occurrence of A provided B has already occurred is called conditional

probability and is denoted by P(A/B) and its value is given by )(

)(

BP

BAP .

3. Probability distribution:-If a random variable X takes values nxxx .,,........., 21 with respective

probabilities nppp .....,,........., 21 then following pattern is called probability distribution

X : nxxxx ............................321

P(X) : npppp .........................321

Mean E(X) =

n

i

ii xp1

. Variance V(X) = 22( ) ( )E X E X S.D. = Variance

4. Binomial distribution:- if an experiment is repeated finite number of times ,events associated are

independent and probability of success or failure is constant for all trials then probability of random

variable X having value r is given by rrn

r

n pqCrXP .)( .Here n: Total number of trials of

experiments.

.

16

Problems involving Dice (Up to 2 tosses) or Coins (Up to 3 tosses)

Form Sample Space and use the result ( )

( )( )

n EP E

n S

Q: A coin is tossed two times. Find the probability of getting at least one tail. Sol: S ={HH, HT, TH, TT} E: Getting at least one tail = {HT, TH, TT}

( ) 3

( )( ) 4

n EP E

n S

Problems involving Cards,

Marbles, Balls, Boys/ Girls etc

Only one object is selected

Use ( ) .

( )( ) .

n E no of favourable objectsP E

n S Total no of objects

Q: If one card is selected from a deck of playing cards, find the probability that it is a red king.

Sol. 2

( )52

noof red king cardsP red king

total noof cards

1

26

Note: Can use results of algebra of events.

1. ( ) ( ) ( ) ( )P E or F P E P F P E and F

2. /( ) 1 ( )P E P E

3. / / /( ) ( ) 1 ( )P E F P E F P E F

More than only one object is selected

Objects are selected one by one

with Replacement Events are independent. Use individual cases with rotation or use Binomial Distribution Q: If two balls are selected with replacement from a box having 2 red, 5 green and 3 black balls, find the probability of getting one black balls and one red balls.

Sol. ( ) ( ) ( )P BRorRB P BR P RB

( ). ( ) ( ). ( )P B P R P R P B

3 2 2 3

. .10 10 10 10

12 3

100 25

Objects are selected at random

without Replacement Events are dependent. Use combination(C) method or conditional probability Q: If two balls are selected from a box having 2 red, 5 green and 3 black balls, find the probability of getting one black ball and one red ball.

Sol. ( & )P onered oneblack ball

3 2

1 1

10

2

.C C

C =

3 .2 2

45 15

PROBABILITY

PROBLEM SOLVING ALGORITHM


EXAMPLES: 2,3,4,6,7,9,11,18,19,20,21,22,24,26,28,29,32,33,34,35,36

Ex.13.1: 3,5,7,8,9,10,13,14 Ex. 13.2:,3,5,7,8,9,11,13,14,16 Ex.13.3: 1,3,4,5,6,7,9,10,12,13

Ex.13.4: ,4,5,6,7,11,13,15 Ex.13.5: 2,3,5,6,7,8,12,13 MISC: 3,5,6,7,8,10,13,16

Class XII Mathematics Vector Algebra and 3D

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Transcript of Class XII Mathematics Vector Algebra and 3D