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XII STD 1. Matrix and Determinantsand 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 that3

( ) ( )A adjA adjA A A I

6.

3 3 4

2 3 4

0 1 1

If A

then ,Find inverse matrix and verify3 1A A

7.

2 2 11

2 1 23

1 2 2

If A

then ,prove that1 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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1. Matrix and Determinantsand 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 ofk, the system of equations1, 1, 1kx y z x ky z x y kz have

i. unique solutionii. more than one solutioniii .no solution

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

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 z bethe no 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 21, 5, 0

x y z x y z x y z

6. Solve

:

, , . .1, .2 .5 .

30

2 5 100

Data

Let x y z bethe no 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

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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 lines1 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 is2

.(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 , then2 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.

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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 13 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 b d 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

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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.If1 1

2cos 2cosx and yx y

then Prove that

. 2cos( )

. 2 sin( ).

m n

n m

m n

n m

x yi m ny 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. 2 cos( )

. 2 cos 2( )

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 deduct9 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

arg

3 2

z

z

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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]

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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 of3

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

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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 1tan x

uy

then Prove that2 2u u

x y y x

6. Using Eulers theorem, prove that1

tan2

u ux y u

x y

if

1

sin

x y

u x y

7. Using Eulers theorem, prove that sin2u u

x y ux y

if

3 31tan x y

ux y

8. Ifax 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 for2 2

1( , )f x y

x y

10. i. If log(tan tan tan )u x y z then prove that sin2 2u

xx

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

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7. INTEGRAL CALCULUS 10 10 100

Answer any 10 questions

1. Find the area between the curves

2

2y x x ,x-axis and the linesx = 2 andx = 4.

2. Find the area between the liney=x + 1 and the curvey =x21

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

the lines 0x and x

4. Find the area of the curve2( 5) ( 6)y x x

i. betweenx = 5 andx = 6 (ii) betweenx = 6 andx = 7

5. Find the area of the loop of the curve2 23 ( ) .ay x x a

6. Find the area bounded byx-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

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

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

11. Find the length of the curvex = 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 aboutx-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).

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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 where a b and ab

Define a b a b ab a b Gii G Q where a 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

aTo prove G isabelian group

v G n z

To prove G isabelian group

4. ( , )n n

Z .

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), .

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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.X6, 5.

( ) (0 8) ( ) ( 6 10)i P X ii P X .

4.?5.?6.?7.,.

3

3(2 ),0 2,

04( ) ( ) ( ) ( )0

0

1, 12 12 3 0

( ) ( ) ( ) ( )240

0

0,( ) ( )

0

x

x

x

x x x xe xi f x iv f xelsewhere

elsewhere

x e xii f x v f x

elsewhereelsewhere

e xiii f x

elsewhere

8.x

1 , , 0( )

0

xkx e x

f xelsewhere

( )i k. ( ) ( 10)i i 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

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THIRU TUITION CENTRE XII STD

FIRST MID-TERM MODELTEST-2011 MATHS

Section-A 6 1 6

1. Find the rank of the matrix 7 12 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 matrix3 1 5 1

1 2 1 5

1 5 7 2

8. If1 2 2

4 3 4

4 4 5

A

then prove that 1A A .

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

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

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: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 1

A...

2 17.

3 4A

( )adjA A ...

8.A n adjA ...

0 09.

0 5A

,

12A ...

10. A 3 det( )kA ...

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

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

a

b

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

. ()

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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 ab d 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

Email id: [email protected]

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