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1

XII STD 1. Matrix and Determinants and its Applications-1

MATHS THIRU TUITION CENTRE 8 10 80

1. Solve by matrix inverse method

2 7,3 5 13, 5x y z x y z x y z


3 8 10 0,3 4,2 5 6 13x y z x y x y z

3. Solve by Matrix inverse method

2 3 9, 6, 2x y z x y z x y z


9,2 5 7 52,2 0x y z x y z x y z

5. If

1 1 1

1 2 3

2 1 3

A

then, verify that 3( ) ( )A adjA adjA A A I

6.

3 3 4

2 3 4

0 1 1

If A

then ,Find inverse matrix and verify 3 1A A

7.

2 2 11

2 1 23

1 2 2

If A

then ,prove that 1 TA A

8.5 2 2 1

7 3 1 1A and B

then prove that

1 1 1. ( )

. ( )T T T

i AB B A

ii AB B A

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2

1. Matrix and Determinants and its Applications -2 7 10 70

1. Discuss the solutions of the system of equations for all values of

2, 2 2 2, 4 2x y z x y z x y z

1. For what values of k , the system of equations

1, 1, 1kx y z x ky z x y kz have

i. unique solution

ii. more than one solution

iii .no solution

2. Verify whether the given system of equations is consistent .if it is consistent , solve

them

2 5 7 52, 9,2 0x y z x y z x y z

4. Solve

: , , . , ,

100

240 260 300 25000

Data Let x y zbetheno of red blue and greenchairs

x y z

x y z

5. Solve by determinant method (Cramers method)

1 2 1 2 4 1 3 2 2

1, 5, 0x y z x y z x y z

6. Solve

:

, , . .1, .2 .5 .

30

2 5 100

Data

Let x y zbetheno of Rs Rs and Rs coins

x y z

x y z

7. Solve the non-homogeneous system of linear equations by determinant method

2 6, 3 2, 4 2 8x y z x y z x y z


3

THIRU TUITION CENTRE 10 6 10

2. VECTOR ALGEBRA -1 SECTION-B

1. If 0a b c

, 3, 5, 7,a b c

find the angle between .a and b

2. Show that the vectors 3 2 , 3 5 , 2 4i j k i j k and i j k

form a right angled

triangle.

3. Find the vectors whose length 5 and which are perpendicular to the vectors

3 4 , 6 5 2 .a i j k and b i j k

4. Prove by vector method 2

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

5. Show that the lines 1 1 2 1 1

1 1 3 1 2 1

x y z x y zand

intersect and find

their point of intersection.

6. Derive the equation of the plane in the intercept form.

7. Find the coordinates of the centre and the radius of the sphere whose vector equation is 2

.(8 6 10 ) 50 0.r r i j k

8. The volume of a paralleopiped whose edges are represented by

12 , 3 , 2 15 546.i k j k i j k is

Find the value of .

9. If ,a b

are any two vectors , then 2 2 2

2( . ) .a b a b a b

10. Find the area of the triangle whose vectors are (3, 1,2),(1, 1, 3), (4, 3,1).and

11. Find the magnitude and direction cosines of the moment about the point (1, 2, 3) of a

force 2 3 6i j k

whose line of action passes through the origin.

12. Prove by vector methodsin sin sin

a b c

A B C .

13. Diagonals of a rhombus are at right angles. Prove by vector methods

14. Angle in a semi-circle is a right angle. Prove by vector method.


4

THIRU TUITION CENTRE

2. VECTOR ALGEBRA-2 8 10 80

1. Find the vector equation and Cartesian equation

3 4 2

2 2

7

a i j k

b i j k

c i k


1,1, 1

2 2 1

2 3 2

A

x y zThe planecontaining theline


2 2 1

2 3 3

1 1 1

3 2 1

x y zThe planecontaining theline

x y zparallel totheline

4. If

2

2

2

a i j k

b i k

c i j k

d i j k

Then prove that ( ) ( )a b c d a bd c a b c d

5. Prove by vector method

. ( )

. ( )

. ( )

. ( )

i Cos A B CosACosB SinASinB

ii Cos A B CosACosB SinASinB

iii Sin A B SinACosB CosASinB

iv Sin A B SinACosB CosASinB


5

3. COMPLEX NUMBERS 8 10 80

1.Find all the values of

3

41 3

2 2i

and hence prove that the product of the values is 1.

2.Solve the equations

9 5 4

7 4 3

. 1 0

. 1 0

i x x x

ii x x x

3.If 1 1

2cos 2cosx and yx y

then Prove that

. 2cos( )

. 2 sin( ).

m n

n m

m n

n m

x yi m n

y x

x yii i m n

y x

4. If cos2 sin2 , cos2 sin2 , cos2 sin2a i b i c i then Prove that

2 2 2

1. 2cos( )

. 2cos2( )

i abcabc

a b cii

abc

5. If and are the roots of 2 2 4 0x x Prove that 2 sin3

n n ni

and deduct 9 9 .

6. If n is a positive integer, Prove that 13 3 2 cos6

n nn ni i

.

7. If and are the roots of 2 2 2 0 cot 1x x and y , ( ) ( ) sin

.sin

n n

n

y y nshow that

.

8.If P represents the variable complex number z .find the locus of P1

arg3 2

z

z


6

4. Analytical Geometry-1 11 10 110

1. Find the axis, vertex, focus, directrix, equation of the latus rectum, length of

the latus rectum for the following parabolas and hence draw their graphs.

2

2

2

2

. 8 6 9 0

. 8 6 1 0

. 6 12 3 0

. 2 8 17 0

i y x y

ii y x y

iii x x y

iv x x y

2. Find the eccentricity, centre, foci, and vertices of the following ellipses and

also trace the curve.

2 2

2 2

2 2

. 36 4 72 32 44 0

.16 9 32 36 92

. 4 8 16 68 0

i x y x y

ii x y x y

iii x y x y

3. Find the eccentricity, centre, foci and vertices of the following hyperbolas

and also trace the curve.

2 2

2 2

2 2

2 2

. 9 16 18 64 199 0

. 4 6 16 11 0

. 3 6 6 18 0

. 9 16 36 32 164 0

i x y x y

ii x y x y

iii x y x y

iv x y x y

Best wishes by M.THIRUPATHYSATHIYA M.Sc., M.Phil., CCA., KUNICHI MOTTUR(VILLAGE),KUNICHI(POST),TIRUPATTUR(TK)

Mobile no: +91 9790250740

Email id: [email protected]


7

4. ANALYTICAL GEOMETRY -2 10 10 100

1. Find the axis, vertex, focus, directrix, equation of the latus rectum, length of the latus

rectum for the parabola and draw the graph, 2 8 6 9 0y x y .

2. Find the eccentricity, centre, foci, and vertices of the ellipse

2 236 4 72 32 44 0x y x y

3. Find the eccentricity, centre, foci, and vertices of the Hyperbola

2 23 6 6 18 0x y x y

4. Find the equation of the asymptotes to the hyperbola

2 28 10 3 2 4 1 0x xy y x y

5. Find the equation of the asymptotes to the rectangular hyperbola

2 26 5 6 12 5 3 0x xy y x y

6. A comet is moving in a parabolic orbit around the sun which is at the focus of a parabola.

When the comet is 80 million kms from the sun, the line segment from the sun to the comet

makes an angle of 3

radians with axis of the orbit. Find

i. The equation of the comets orbit,

ii. How close does the comet come nearer to the sun? (Take the orbit as open right ward ).

7. A cable of a suspension bridge is in the form of a parabola whose span is 40mts.The road

way is 5mts below the lowest point of the cable. If an extra support is provided across the

cable 30 mts above the ground level. Find the length of the support if the height of the pillars

is 55mts.

8. Find the equations of the two tangents that can be drawn from the point (5,2) to the ellipse

2 22 7 14x y


8

6. Differential calculus and its applications

1. Trace the curve 3 1y x 10 10 100

2. Trace the curve 3y x

3. Trace the curve 2 32y x

4. Discuss the curve 2 2 2 2 2( ), 0a y x a x a for i. Existence ii. Symmetry

iii. Asymptote iv. Loops

5. If 1tanx

uy

then Prove that 2 2u u

x y y x

6. Using Eulers theorem, prove that 1

tan2

u ux y u

x y

if

1sinx y

ux y

7. Using Eulers theorem, prove that sin 2u u

x y ux y

if

3 31tan

x yu

x y

8. If ax byV ze and z is a homogeneous function of degree n in x and y Prove

that ( )V V

x y ax by n Vx y

9. Verify EULERS theorem for 2 2

1( , )f x y

x y

10. i. If log(tan tan tan )u x y z then prove that sin 2 2u

xx

ii. If ( )( )( )U x y y z z x then prove that 0x y ZU U U


9

7. INTEGRAL CALCULUS 10 10 100

Answer any 10 questions

1. Find the area between the curves2 2y x x , x-axis and the lines

x = 2 and x = 4.

2. Find the area between the line y=x + 1 and the curve y = x2 1

3. Compute the area between the curve siny x and cosy x and

the lines 0x and x

4. Find the area of the curve 2( 5) ( 6)y x x

i. between x = 5 and x = 6 (ii) between x = 6 and x = 7

5. Find the area of the loop of the curve 2 23 ( ) .ay x x a

6. Find the area bounded by x-axis and an arch of the cycloid

x = a (2t sin 2t), y = a (1 cos 2t)

7. Derive the formula for the volume of a right circular cone with radius r

and height h.

8.Find the length of the curve

2 2

3 3

1x y

a a

9. Find the surface area of the solid generated by revolving the

cycloid x = a(t + sin t), y = a(1 + cos t) about its base (x-axis).

10. Find the perimeter of the circle with radius a by using integral calculus method.

11. Find the length of the curve x = a(t sin t), y = a(1 cos t) between t = 0 and .

12. Find the surface area of the solid generated by revolving the arc of the

parabola 2 4y ax , bounded by its latus rectum about x-axis.

13. Prove that the curved surface area of a sphere of radius r intercepted

between two parallel planes at a distance a and b from the centre of the

Sphere is 2 ( )r b a and hence deduct the surface area of the sphere (b > a).


10

9. 10 10 100

1. ( , )Z .

2a b a b .

2. ; {0}x x

x Rx x

G

.

3. .

1, 2 3 4

1 2 3 4

) {1}, 1, 1, 1

, ,

) { 1}, 1, 1, 1

, ,

) { , , } {0}

1 1( ) , ( ) , ( ) , ( )

0) ,

0 0

i G Q wherea b and ab

Define a b a b ab a b G

ii G Q wherea b and ab

Define a b a b ab a b G

iii G f f f f or G C

Define f z z f z z f z f zz z

a aiv G

{0}

0

: ( ,.) .

) {2 ; }

: ( ,.) .

n

R

a

To prove G is abelian group

v G n z

To prove G is abelian group

4. ( , )n nZ .

5. 7 7( {[0]},. )Z .

6.11 {[1],[3],[4],[5],[9]}

.

7. 1- 4 .

8. 1- 3 .

9.1 0 1 0 1 0 1 0

, , ,0 1 0 1 0 1 0 1

. 10. i) (Cancellation law) , .


11

ii) (Reversal law) , .

10. 10 10 100

1.2 3( ) ,x xf x ce X

2, ,c .

2.22 4( ) ,x xf x ke X

2, ,k .

3. X 6 , 5 . ( ) (0 8) ( ) ( 6 10)i P X ii P X . 4. ? 5. ? 6. ? 7. , .

3

3(2 ),0 2, 0

4( ) ( ) ( ) ( )0

0

1, 12 12 3 0

( ) ( ) ( ) ( )240

0

0,( ) ( )

0

x

x

x

x x x xe xi f x iv f x

elsewhereelsewhere

x e xii f x v f x

elsewhereelsewhere

e xiii f x

elsewhere

8. x

1 , , 0( )

0

xkx e xf x

elsewhere

( )i k . ( ) ( 10)ii P X . 9.Refer :

Page Number Question Number

227 Exercise 10.1: 4,7,8,10

218 Examples: 10.3, 10.2

234 Exercise 10.2 : 1,2,6

238 Exercise 10.3: 5,6

240 and 242 Examples: 10.23, 10.24 and Exercise10.4: 3,4,5,6

250 Examples: 10.30,10.32

253 Exercise10.5: 4,5,6,8


12

THIRU TUITION CENTRE XII STD

FIRST MID-TERM MODEL TEST-2011 MATHS

Section-A 6 1 6

1. Find the rank of the matrix7 1

2 1

.

2. Solve by determinant method 2 3;2 4 8x y x y

3. The rank of an m n matrix A cannot exceed the minimum of m and n. that is...

4. Find the d.c.s of a vector whose direction ratios are 2,3, 6 .

5. If 13, 5 . 60a b and a b then find a b

6. Solve the fourth root of unity.

Section-B (Compulsory question-10) 9 6 54

7. Find the rank of the matrix

3 1 5 1

1 2 1 5

1 5 7 2

8. If

1 2 2

4 3 4

4 4 5

A

then prove that1A A .

9. Show that the points whose position vectors 4 3 ,2 4 5 ,i j k i j k i j

form a right

triangle.

10. 4. Find the magnitude and direction cosines of the moment about the point (1, 2, 3) of a

force 2 3 6i j k

whose line of action passes through the origin.

11. Prove by vector methodsin sin sin

a b c

A B C .

12. Find the square root of ( 7 24 )i .

13. If n is a positive integer .prove that 13 3 2 cos6

n nn ni i

.


13

14. Prove by vector method 2a b b c c a a bc

15. For any two complex numbers 1 2z and z then prove that

11

2 2

11 2

2

.

.arg arg( ) arg( )

zzi

z z

zii z z

z

OR

16. Solve the equation 9 5 4 1 0x x x

Section-C (Compulsory question-4) 4 10 40

1. 17..If cos2 sin2 , cos2 sin2 , cos2 sin2a i b i c i then

prove that

2 2 2

1. 2cos( )

. 2cos2( )

i abcabc

a b cii

abc

2. Find the vector and Cartesian equation of the plane containing the line

2 2 1

2 3 2

x y z

and passing through the point -1, 1 -1 .

3. 3. Prove by vector method Sin(A-B)=SinACosB-CosASinB

4. Verify whether the given system of equations is consistent. if it is consistent ,solve

them

5. 4x+3y+6z=25,x+5y+7z=13,2x+9y+z=1 OR

6. 5. Solve by Cramers rule 2 6,3 3 3,2 2 3x y z x y z x y z

Best wishes by M.THIRUPATHYSATHIYA M.Sc., M.Phil., CCA.,

Mobile no: +91 9790250740



14

: 100

12 :1.30hrs

SECTION-A 20 1 20

1 1 2

1. 2 2 4

4 4 8

.

1

2

2. 0

4

0

.

3. 2 0 1A , TAA .

1

4. 2

3

A

, TAA .

1 0

5. 0 1

1 0

2 .

6. 3 , 0k 1A ...

2 17.

3 4A

( )adjA A ...

8. A n adjA ...

0 09.

0 5A

,12A ...

10. A 3 det( )kA ...


15

11. ( ) ( ) ( )u a b c b c a c a b

12. 0, 3, 4, 5a b c a b c

ab

13. a b a b

14. 2 , 3 2PR i j k QS i j k

, PQRS ...

15. , , 64a b b c c a

, ,a b c

16. , ,i j j k k i

17. , , 8a b b c c a

, ,a b c

2 2 218. 6 8 10 1 0x y z x y z ...

19. r si t j

...

20. 2 3 4 ,i j k ai b j ck

...

SECTION-B 5 6 30

1 1 1

1. 1 2 3

2 1 3

A

.

1 2 1 3

2. 2 4 1 2

3 6 3 7

.

3. : 2 2 5, 1,3 2 4x y z x y z x y z

4. :sin sin sin

a b c

A B C .

5. 12 , 3 , 2 15i k j k i j k

546 .26. . (4 2 6 ) 11 0r r i j k

. ( )


16

1 1

1 1 3

x y z

2 1 1

1 2 1

x y z

. .

SECTION-C 5 10 50

1. .1 . 2 . 5 . 100 30 . .

2. k

1, 1, 1kx y z x ky z x y kz have

)i )ii )iii .

3. : ( )Cos A B CosACosB SinASinB

4. .

5. , 2 , 2 , 2a i j k b i k c i j k d i j k

( ) ( ) [ ] [ ]a b c d abd c a b c d

.

6. 2 3 , 2 5 , 3a i j k b i k c j k

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

.

7. (2,2, 1), (3,4,2), (7,0,6)and

. ( )

6 7 4

3 1 1

x y z

9 2

3 2 4

x y z

.

Best wishes by M.THIRUPATHYSATHIYA M.Sc., M.Phil. CCA.,

Mobile no: +91 9790250740



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