10TH MATHS MATERIAL CHAPTERS 4,7,8,12_doozy study.pdf ...

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DY

mur¨d® nkše¨iyg¦ gŸs¨, gŸs¨, ehaf‹ng£il. ehaf‹ng£il.

ST U

Àò¾¡õ ÅÌôÒ

DO OZ

Y

fz¡F Á¡½Å÷ ¨¸§ÂÎ

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i. ec, B.Sc., B.Ed., M.Com., rcr - . ac!"#$%n'(, *+#, ,cp$.. 603601. PHONE : 9842558987

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4. «½¢¸û

(2). A = -2 ÁüÚõ B = ( 1 3 -6 ) 4 5 ±ýÈ «½¢¸ÙìÌ( AB ) T = B T A T ±ýÀ¨¾ º¡¢À¡÷ì¸×õ.

(1). A = 5 2 ; B = 2 -1 7 3 -1 1 T T T ±É¢ø, ( AB ) = B A ±ýÀ¨¾ º¡¢À¡÷ì¸×õ.

¾£÷× : ;

5 2 7 3

AB AB

2 -1

-1 1

5(2)+2(5(2)+2(-1) 7(2)+3(7(2)+3(-1)

=

10 - 2 14 - 3

5(5(-1)+2 (1) 7(7(-1)+3(1)

-5 + 2 -7 + 3

=

8 11

-3 -4

(AB) T =

8 -3

11 -4

AT

B A

-1 1

=

BTAT T

2 -1

T

B TA T

5 7 2 3 = 2(5)2(5)-1(2) -1(5)+1(2) =

10 - 2 -5 + 2

=

8 -3

=

-2 4 5

-2 -6 12

-6 12 15

12 -24 -30

4 12 -24

5 15 -30

4

5 )

R. H. S. : B T A T

DO OZ

=

AB

( AB ) T =

R. H. S. : B T A T BT

-2 ; B = ( 1 3 -6 ) 4 5 L .H. S. : ( AB ) T AB = -2(1) -2(3) -2(2(-6) 4(1) 4(3) 4(--6) 4( 5(1) 5(3) 5(5(-6)

Y

AB =

A=

ST U

=

¾£÷× :

-1 1

( AB ) T

L .H. S. : AB

B= 2 -1

DY

A= 5 2 7 3

2(7)2(7)-1(3) -1(7)+1(3)

14 - 3 -7 + 3

11 -4

L. H. S. = R. H. S. S. T T ( AB ) = B AT º¡¢À¡÷ì¸ôÀð¼Ð

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BT

=

1 3 -6 = ( -2

AT T

T

=

BTAT

=

B A

1(1(-2) 1(4) 1(5) 3(3(-2) 3(4) 3(5) -6(6(-2) -6(4) -6(5) -2 -6 12

4 12 -24

5 15 -30

L. H. S. = R. H. S. ( AB ) T = B T AT º¡¢À¡÷ì¸ôÀð¼Ð

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-2+1 6-5

=

-1 1

2

(3). A = 1 -1 ±É¢ø A - 4A + 5I 5I2=0 2 3 ±É ¿¢Ú×¸.

¾£÷× : 1 2

A (B+C)= 3 2 -1 4

- 4A + 5I2 = 0 -1 3

1 -1 2 3

=

- 4 1 -1 + 5 1 0 = 0 2 3 0 1

-4 7

-

4 8

-4 + 5 12 0

0 =0 5

-4 + 12

5 0

0 =0 5

-1-4 -4+4 + 8 -8 77-12

5 0

0 =0 5

5 0

0 5

-5 0

0 -5

+

0 0

=0

A 2 - 4 A + 5 I 2 =0 ¿¢ÚÅôÀð¼Ð.

(4). A = 3 2 B = -2 5 ÁüÚõ -1 4 6 7 ÁüÚõ C = 1 1 ±É¢ø, -5 3 A (B+C) = AB +AC ±ýÀ¨¾ º¡¢À¡÷ì¸×õ.

¾£÷× : A ( B + C ) = AB + AC L. H. S. : A ( B + C ) B + C = -2 6

5 7

+ 1 -5

-1 5

1 3

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6 10 3(6)+2(10) -1(6)+4(10)

18+20 -6+40 38 34

R. H. S. : AB + AC AB = 3 2 -2 5 -1 4 6 7 AB = 3(3(-2)+2(6) 3(5)+2(7) -1(-1(5)+4(7) 1(-2)+4(6)

AB

=

AB

=

=0

DO OZ

0 0

A (B+C)=

ST U

-1 8

- 4 8

Y

-1-3 -2+9

-1 1

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

= -3+2 1+4

1(1)1(--1)1(1)-1(2) 1( 1)-1(3) - 4 -4 + 5 0 =0 2(1)+3(2) 2(2(-1)+3(3) 8 12 0 5 1 -2 2+6

6 10

DY

A

2

5+1 7+3

AC

-6+12 2+24

15+14 -5+28

6 29 26 23 1 1 = 3 2 -1 4 -5 3

AC

=

AC

=

AC

=

3(1)+2(3(1)+2(-5) -1(1)+4(1(1)+4(-5)

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

3-10 3+6 -1-20 -1+12 -7 9 -21 11

AB + AC = 6 29 + -7 9 26 23 -21 11 AB + AC = 6 - 7 29 + 9 26 - 21 23 + 11 AB + AC =

-1 5

38 34

L. H. S. = R. H. S. A (B + C ) = AB + AC º¡¢À¡÷ì¸ôÀð¼Ð.

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2 1 ; 2 3

B = 0 ÁüÚõ 1 2 C = ( 2 1 ) ±É¢ø, ( AB ) C = A ( BC ) ±ýÀ¨¾ º¡¢À¡÷ì¸×õ.

¾£÷× :

(6). A = 3 3 ; B = 8 7 ÁüÚõ 7 6 0 9 C = 2 -3 , ±É¢ø (A+B) (A+B) C ÁüÚõ 4 6 AC +BC «½¢¸¨Çì ¸¡ñ¸. §ÁÖõ, (A+B) C = AC +BC ±ýÀÐ ¦ÁöÂ¡ÌÁ¡ ?

¾£÷× :

L. H. S. : ( AB ) C AB = -1 2 1 1 2 3

(A+B)

AB

=

( A +B )

AB

= 0+2+2 0+2+6 = 4 8 = 4 (2 1) 8

( AB ) C ( AB ) C

= =

4(2) 8(2) 8 16

(A+B)C=

DO OZ

1 ) =

0 0 2 1 4 2

0 0 2 1 4 2

A (BC) = -1(0 1(0)+2(2)+ )+2(2)+1(4 1(4) -1(0 1(0)+2(1) +2(1)+ (1)+1(2 1(2) 1(0 1(0)+2(2)+3(4) 1(0 1(0)+2(1)+3(2) )+2(1)+3(2) A ( BC ) A ( BC )

= 0+4+4 0+4+12

=

8 16

8 0

0+2+2 0+2+6

4 8

L. H. S. = R. H. S. ( AB ) C = A ( BC ) º¡¢À¡÷ì¸ôÀð¼Ð.

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7 9

3+8 7+0

3+7 = 6+9

11 7

10 15

( A + B ) C = 22+40 14+60

4 8

A ( BC ) = -1 2 1 1 2 3

+

( A + B ) C = 11(2) 11(2)+ +10(4) 7(2)+15(4)

4(1) 8(1)

R. H. S. : A (BC ) BC ( 2 = 0 1 2 BC = 0(2) 0(2) 0(1) 1(2) 1(1) 2(2) 2(1)

=

3 6

ST U

( AB ) C

= 3 7

(A+B)C =

Y

AB

0 1 2 -1(0)+2(1)+1(2) 1(0)+2(1)+3(2)

DY

(5). A = -1 1

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62 74

11 7 2 4

10 15 -3 6

11(11(-3)+10 )+10(6) 7(7(-3)+ 3)+15(6) 15(6)

-33+60 -21+90 27 69

AC

= 3 7

AC

= 3(2)+3(4)3( 3(2)+3(4)3(--3)+3(6) 7(2)+ 7(2)+6(4)6(6(4)6(-3)+6(6)

AC

=

6+12 14+ 24

AC

=

BC

= 8 0

BC

=

BC

=

BC AC + BC

3 6

1

18 38 7 9

2 4

-9+18 -21+36 9 15 2 4

8(2)+7(4) 0(2)+ (2)+9(4) 16+28 0 + 36

=

44 36

=

18 38

-3 6

-3 6 8(8(-3)+7(6) 0(-3)+ 3)+9(6)

-24+42 0 +54 18 54 9 15

+

44 36

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18 54

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= 18+44 38+36

AC + BC

1

=

9+18 15+54

62 74

ÁüÚõ

1

X2 => 4x + 6y =

=> 9x + 6y = 2 X3 =>

27 69

2

-5x

Ä¢ÕóÐ

2

3 0

1

3x + 2y =

2 -1

-2 5

2

9 0

=> 6x + 4y = 2 X2 =>

4 -2

-4 10

+ 5y =

6-4 12+2

+ 5y =

2 14

y=

9+4 0-10

=

2 5

-12 5

-11 5

15 5

X

6 12

6 -3

-6 15

-2 11

6+6 0-15 12 -15

-2 -5

12 -5

11 -5

-15 -5

¾£÷× :

=

Y

X3 => => 6x + 9y =

X

=

ST U

2 4

DO OZ

1

2x + 3y =

6 0

= 4-6 8+3

-5x

(A+ B) C = AC + BC ±ýÀÐ ¦ÁöÂ¡Ìõ.

(7). 2x + 3y = 2 3 ÁüÚõ 4 0 3x +2y= 2 -2 ±É¢ø, x ÁüÚõ y -1 5 ¬¸¢Â «½¢¸¨Çì ¸¡ñ¸. ¾£÷× :

4 8

DY

AC + BC

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y =

2 5

13 5

14 5

-10 5

13 -10

2 5

13 5

14 5

-10 5

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http://doozystudy.blogspot.in/ (4).¸¡Ã½¢ôÀÎòÐ¸. 2 x 3 - 9 x 2 + 7 x + 6.

þÂü¸½¢¾õ (1). ¸¡Ã½¢ôÀÎòÐ¸. 2 x 3 - 3 x 2 - 3 x + 2.

¾£÷× :

2 - 3 = -1 -3 + 2 = -1

2

¾£÷× :(x+1) (x+1) µ÷ ¸¡Ã½¢ 2 0

-3 -3 +2 -2 +5 -2

2

-5

+2

®× : 2 x 2 - 5 x + 2

1-10= -9 -3+24=21

-3 2

-10 -2

+24 -24

1

-1

-12

0

Y

1 0

(x(x-2) ´÷ ¸¡Ã½¢ ¬Ìõ. ®× : x 2 - 1 x - 12

DO OZ

-12

¸¡Ã½¢¸û : (x(x-4) (x+3) 3 2 x - 3 x - 10x + 24 ý ¸¡Ã½¢¸û (x(x-2) (x(x-4) (x+3)

-4

(3). ¸¡Ã½¢ôÀÎòÐ¸. x 3 - 5 x 2 - 2 x + 24

1-2= -1 -5+24= 19

3

-5 - 2 +24 -2 +14 -24

2

-5x-3

(5). ¸¡Ã½¢ôÀÎòÐ¸. 2 x 3 + 11 x 2 - 7 x - 6.

®× : x - 7 x + 12 ¸¡Ã½¢¸û : (x(x-4) (x(x-3) 3 2 x - 5 x - 2x + 24 ý ¸¡Ã½¢¸û (x+2) (x(x-4) (x(x-3)

+12 -4 -3

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-6 -6 1 2 2

2 - 7 = -5 11 - 6 = +5

¾£÷× :(x(x-1) 2 0

+11 -7 -6 +2 +13 +6

2

+13

+6

0

(x(x-1) ´÷ ¸¡Ã½¢ ¬Ìõ. ®× : 2 x

2

+ 13 x + 6

+12

¸¡Ã½¢¸û ¸¡Ã½¢¸û : (x+6) (2x+1)

12 1 2 2 3 2 2 x + 11 x - 7 x - 6 ý ¸¡Ã½¢¸û (x(x-1) (x+6) (2x+1) (6). ¸¡Ã½¢ôÀÎòÐ¸. x3- 2 x 2 - 5 x + 6

1 - 5 ==-4 -2 + 6 = +4

¾£÷× : 1

(x-1) 1 0

-2 - 5 +1 -1

1

-1

+6 -6

-6

0

(x+2) ´÷ ¸¡Ã½¢ ¬Ìõ. 2

0

2 x 3 - 9 x 2 + 7 x + 6 ý ¸¡Ã½¢¸û (x(x-2) (x(x-3) (2x+1)

1

-7 +12

-3

¸¡Ã½¢¸û : (x(x-3) (2x+1)

-1 -4 2 2

¾£÷× :

1

-5

ST U

(2). ¸¡Ã½¢ôÀÎòÐ¸. x 3 - 3 x 2 - 10 x + 24

-2 1 0

2 ®× : 2 x

4

¸¡Ã½¢¸û : (2x(2x-1) (x(x-2) 3 2 2 x - 3 x - 3 x + 2 ý ¸¡Ã½¢¸û (x+1) (2x(2x-1) (x(x-2)

¾£÷× :

-9 +7 +6 4 -10 -6 (x(x-2) ´÷ ¸¡Ã½¢ ¬Ìõ.

0 (x+1) ´÷ ¸¡Ã½¢ ¬Ìõ.

2

2 0

DY

-1

2+7= 9 -9+6= -3

0

(x(x-1) ´÷ ¸¡Ã½¢ ¬Ìõ. ®× : x 2 - x - 6 ¸¡Ã½¢¸û : (x(x-3) (x+2) (x+2) 3 2 x - 2 x - 5x + 6 ý ¸¡Ã½¢¸û (x(x-1) (x(x-3) (x+2)

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-6 -3 +2

http://doozystudy.blogspot.in/ (7). (7). ¸¡Ã½¢ôÀÎòÐ¸. x 3 - 10 x 2 - x + 10

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1-1=0 -10+10 = 0

¾£÷× :

(x(x-1)

Å÷ì¸ãÄõ ¸¡½ø (1). Å÷ì¸ãÄõ ¸¡ñ¸: x4 - 10 x 3 + 37 x 2 - 60 x + 36

1 0

-10 -1 +10 +1 - 9 -10

1

-9

-10

1x 2 - 5 x + 6 1x2

0

1x 4

(x(x- 1) ´÷ ¸¡Ã½¢ ¬Ìõ.

(8). (8). ¸¡Ã½¢ôÀÎòÐ¸. x 3 - 23 x 2+142 x -120

-10 +1

1+142=143 -2323-120 = -143

¾£÷× : 1

-10

(x(x-1)

1 0

-23 1

+142 -120 - 22 +120

1

-22

+120 +120

2x2-5x

- 10 x 3 + 37 x2 -10 x 3 + 25 x2 +12 x 2 - 60 x +36 + 12x2 - 60x +36

2x2-10x+6

ST U

®× : x 2 - 9 x - 10 ¸¡Ã½¢¸û : (x(x-10) (x+1) (x+1) 3 2 x - 10 x - x + 10 ý ¸¡Ã½¢¸û (x(x- 1) (x(x- 10) 10) (x + 1) 1)

1x 4 - 10 x 3 + 37 x 2 - 60 x + 36

DY

1

0

x4 - 10 x 3 + 37 x 2 - 60 x + 36 ý Å÷ì¸ãÄõ : ± x2 -5x + 6 (2). Å÷ì¸ãÄõ ¸¡ñ¸:

x4 - 6 x 3 + 19 x 2 - 30 x + 25

0

(9). (9). ¸¡Ã½¢ôÀÎòÐ¸. 4x 3 - 5 x 2+7 x -6

¾£÷× : 1

+120 +120

-12 -10

DO OZ

®× : x 2 -22 x + 120 120 ¸¡Ã½¢¸û : (x(x-12) 12) (x(x-10) 10) 3 2 x - 5 x - 2x + 24 ý ¸¡Ã½¢¸û (x(x- 1) (x(x- 12) 12) (x(x- 10) 10)

Y

(x(x-1) ´÷ ¸¡Ã½¢ ¬Ìõ.

4+7= +11 -5 - 6 = - 11 (x(x-1)

4 0

-5 +4

+7 -6 - 6 +6

4

-1

+6

0

(x(x-1) ´÷ ¸¡Ã½¢ ¬Ìõ.

®× : 4x 2 - x + 6 ¸¡Ã½¢ôÀÎò¾ þÂÄ¡Ð. x 3 - 5 x 2 - 2x + 24 ý ¸¡Ã½¢¸û

1x 2 - 3 x + 5 1x2

1x 4 - 6 x 3 + 19 x 2 - 30 x + 25 1x 4

2x2-3x - 6 x 3 + 19 x2 -6x3 + 9x 2x2-6x+5 +10 x 2 - 30 x +25 + 10x2 - 30x +25 0 x4 - 6 x 3 + 19 x 2 - 30 x + 25 ý Å÷ì¸ãÄõ : ± x2 -3x + 5

+24

(x(x- 1) (4 (4x 2 - x + 6 )

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http://doozystudy.blogspot.in/ (5). Å÷ì¸ãÄõ ¸¡ñ¸:

(3). Å÷ì¸ãÄõ ¸¡ñ¸:

9x 4 - 6 x 3 + 7 x 2 - 2 x + 1

x4 - 4 x 3 + 10 x 2 - 12 x + 9

3x2 -1x + 1

1x 2 - 2 x + 3 1x 4 - 4 x 3 + 10 x 2 - 12 x + 9

9x 4

1x 4

6x2-1x

- 4 x 3 + 10 x2 - 4 x 3 + 4 x2

2

2x -4x+3

+ 6 x - 12 x +9 + 6 x2 - 12x +9 0

x4 - 4 x 3 + 10 x 2 - 12 x + 9 ý Å÷ì¸ãÄõ : ± x 2 - 2 x + 3 (4). Å÷ì¸ãÄõ ¸¡ñ¸:

Y

DO OZ

2x 2 + 2 x + 1

2x24x 4 + 8 x 3 + 8 x 2 + 4 x + 1 + 8 x 3 + 8 x2 4x2+2x + 8 x 3 + 4 x2 4x2+4x+1

+ 6 x 2 -2 x +1 + 6 x2 -2 x +1 0

9x 4 - 6 x 3 + 7 x 2 - 2 x + 1 ý Å÷ì¸ãÄõ : ± 3x 2 - x + 1 (6). 4x 4 - 12 x 3 + 37 x 2 + a x + b ¬ÉÐ ÓØÅ÷ì¸¦ÁÉ¢ø a, b ý Á¾¢ôÒ ¸¡ñ¸.

4x 4 + 8 x 3 + 8 x 2 + 4 x + 1

4x 4

6x2-2x+1

2

- 6 x 3 + 7 x2 - 6 x 3+1x2

ST U

2x2-2x

9x 4 - 6 x 3 + 7 x 2 - 2 x + 1

DY

1x2

3x2

2x2 -3x + 7 2x2

4x 4 - 12 x 3 + 37 x 2 + a x + b 4x 4

4x2-3x

- 12 x 3 + 37 x2 - 12 x 3 + 9x2

4x2- 6x+ 7

+ 4 x 2 +4 x +1

+ 4 x2 +4 x +1

0

4x 4 - 8 x 3 + 8 x 2 + 4 x + 1 ý Å÷ì¸ãÄõ : ± 2x 2 + 2 x + 1

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+ 28 x 2 +a x + b + 28 x2 -42 x +49 0

¾£÷× : a= - 42 b = 49

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¬Âò ¬Âò¦¾¡¨Ä ÅÊÅ¢Âø

(7). x4 - 4 x 3 + 10 x 2 - a x + b ¬ÉÐ ÓØÅ÷ì¸¦ÁÉ¢ø a, b ý Á¾¢ôÒ ¸¡ñ¸.

¿¡ü¸Ãò¾¢ý ÀÃôÒ ¸¡½ø : (1). ((-4,4,-2), ((-3,3,-5), (3,(3,-2), (2,3) ¬¸¢Â ÒûÇ¢¸¨Ç Ó¨É¸Ç¡¸ì ¦¸¡ñ¼ ¿¡ü¸Ãò¾¢ý ÀÃôÀÇ× ¸¡ñ¸.

x2 - 2 x + 3 x2 x 4 - 4 x 3 + 10 x 2 - a x + b

¾£÷× :

4

2x2-2x

- 4 x 3 + 10 x2 - 4 x 3 + 4 x2

2x2- 4x+ 3

+ 6 x 2 -a x + b + 6 x2 - 12 x + 9

=½

=½ = ½x = 28

a = 12 b = 9

Y

(8). m - n x + 28 x2 + 12 x3 + 9 x4 ¬ÉÐ ÓØÅ÷ì¸¦ÁÉ¢ø m, n ý Á¾¢ôÒ ¸¡ñ¸. 3x2 +2x + 4

DO OZ

9x 4 + 12 x 3 + 28 x 2 - n x + m

9x

6x2+2x 2

6x +4x+4

4

+ 12 x 3 + 28 x2 + 12 x 3 + 4 x2

2

+ 24 x -n x + m + 24 x2 +16 x +16 0

¾£÷× :

-2

-3

-5

3

-2

x4

x1 º.«.

y4 2

y1 -4 º.«.

3

-2

=½ (20 + 6 + 9 - 4) - (6 -15 - 4 -12) º«.

¾£÷× :

3x2

-4

ST U

0

DY

¿¡ü¸Ãò¾¢ý ÀÃôÀÇ× x2 x3 x1 =½ Y1 y2 y3

x

m = 16 n = -16

31 + 25 56 º.«. º.«.

º.«.

(2). (6,9), (7,4), (4,2), (3,7) ¬¸¢Â ÒûÇ¢¸¨Ç Ó¨É¸Ç¡¸ì ¦¸¡ñ¼ ¿¡ü¸Ãò¾¢ý ÀÃôÀÇ× ¸¡ñ¸.

¾£÷× : ¿¡ü¸Ãò¾¢ý ÀÃôÀÇ× x1 x2 x3 =½ Y1 y2 y3 =½

6

7

4

x4

x1 º.«.

y4

y1

3

6 º.«.

9

4

2

7

9

=½ (24+14+28+27)(24+14+28+27)-(63+16+6+42) º«. = ½ 93 - 127 º.«. = ½x ( -34 ) º.«. = 17 º.«. [ ¿¡ü¸Ãò¾¢ý ÀÃôÒ Ì¨È ±ñ½¡¸ þÕì¸ þÂÄ¡Ð ]

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(3). ((-3,4), 3,4), ((-5,5,-6), (4,(4,-1), (1,2) ¬¸¢Â ÒûÇ¢¸¨Ç Ó¨É¸Ç¡¸ì ¦¸¡ñ¼ ¿¡ü¸Ãò¾¢ý ÀÃôÀÇ× ¸¡ñ¸.

Sn =7 + 77 + 777 + ....+ n ¯ÚôÒ¸û Å¨Ã = 7[ 1+11+111+....+ n ¯ÚôÒ¸û] ¯ÚôÒ¸û] =7 [ 9+99+999+.....+ n ¯ÚôÒ¸û ]

¿¡ü¸Ãò¾¢ý ÀÃôÀÇ× x2 x3 x1 =½ Y1 y2 y3 -3

=½

-5

4

x4

x1 º.«.

y4 1

y1

9 = 7 [(10[(10-1)+(100)+(100-1)+(10001)+(1000-1)+.....+n 9 = 7[(10+100+1000+....+ n ¯ÚôÒ¸û ] 9 [ 1+1+1+.....+ n ¯ÚôÒ¸û

-6

-1

2

9

Y x1

º.«.

5

y4

-4

10 n - 1 10 - 1

10

a(rn-1) (r-1)

- n

( 10n - 1 ) - n

(2). (2). 6 + 66 + 666 + ....... ±ýÈ ¦¾¡¼¡¢ý Ó¾ø n ¯ÚôÒ¸Ç¢ý ÜÎ¾ø ¸¡ñ¸.

DO OZ

=½

0

7 9

Sn =

S n = 70 ( 10 n - 1 ) - 7n 81 9

(4). ((-4,5), (0,7), (5,(5,-5), ((-4,4,-2) ¬¸¢Â ÒûÇ¢¸¨Ç Ó¨É¸Ç¡¸ì ¦¸¡ñ¼ ¿¡ü¸Ãò¾¢ý ÀÃôÀÇ× ¸¡ñ¸.

-4

=

=

x4

]

ST U

=½ 35 + 51 º.«. = ½x 86 º.«. = 43 º.«.

¿¡ü¸Ãò¾¢ý ÀÃôÀÇ× x2 x3 x1 =½ y2 y3 Y1

a = 10 , r = 10 ;

4

=½ (18+5+ 8 + 4) - (-20 -24 - 1 -6) º«.

¾£÷× :

]

= 7[10+102+103+.... n ¯ÚôÒ¸û - n ]

-3 º.«.

4

DY

¾£÷× :

y1

-4

º.«.

5 7 -5 -2 5 =½ (-28+0 -10 -20) - (0 +35+20+8) º«. = ½ -58 - 63 º.«. = ½x -121 º.«. = 60.5 º.«.

¦Áö¦Âñ¸Ç¢ý ¦Áö¦Âñ¸Ç¢ý ¦¾¡¼÷¸Ùõ ¦¾¡¼÷Å¡¢¨º¸Ùõ

(1). 7 + 77 + 777 + ....... ±ýÈ ¦¾¡¼¡¢ý n ¯ÚôÒ¸Ç¢ý ÜÎ¾ø ¸¡ñ¸.

¾£÷× : http://doozystudy.blogspot.in/

¾£÷× : Sn =6 + 66 + 666 + ....+ n ¯ÚôÒ¸û Å¨Ã = 6 [ 1+11+111+....+ n ¯ÚôÒ¸û ] =

6 [ 9+99+999+.....+ n ¯ÚôÒ¸û ] 9 = 2 [(10[(10-1)+(1001)+(100-1) +(1000+(1000-1)+.....+n 3 = 2[(10+100+1000+....+ n ¯ÚôÒ¸û ] 3 [ 1+1+1+.....+ n ¯ÚôÒ¸û ]

= 2[10+102+103+.... n ¯ÚôÒ¸û - n ] 3 a = 10 , r = 10 ;

=

2 3

= 2 3

10

10 n - 1 10 - 1

Sn =

a(rn-1) (r-1)

- n

10 ( 10 n - 1 ) - n 9

S n =20 ( 10 n - 1 ) - 2n 27 3

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]

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http://doozystudy.blogspot.in/

(3). À¢ýÅÕõ ¦¾¡¼÷¸Ç¢ý ÜÎ¾¨Äì ¸¡ñ¸.

(i) 26 + 27 + 28 +.....+ 60

¾£÷× :

= 5525

- 1240

=

26 + 27 + 28 +.....+ 60 (iv) 163 + 17 3 + 18 3 +.....+ 35 3

= ( 1+2+3+.....+60 ) - ( 1+2+3+....+25 1+2+3+....+25 )

¾£÷× :

= +

16 3 + 17 3 + 18 3 +.....+ 35 3 = ( 13 +2 3 +....+353 ) - (13 +2 3 +....+ 153 )

= + − +

=

= 35 (35 + 1) 2

=

ST U

= 35 x 36 2

(ii) 12 2 + 13 2 + 14 2 +.....+ 35 2

¾£÷× :

= + +

- 15(15+1)2 2

-

15 x 16 2

2

= ( 35 x 18 ) 2 - ( 15 x 8 ) 2

= ( 630 ) 2 - ( 120 ) 2 = 396900 - 14400

=

=

=

¾£÷× :

= 28 (28 + 1) 2 = 28 x 29 2

16 2 + 17 2 + 18 2 +.....+ 25 2 = ( 12 +2 2 +....+252 ) - (12 +2 2 +....+ 152 )

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2

2

- 10(10+1) 2

-

10 x 11 2

= ( 14 x 29 ) 2 - ( 5 x 11 ) 2

= + +

= 25(25+1)(50+1) - 15(15+1)(30+1) 6 6 = 25 x 26 x 51 - 15x16x31 6 6

n ( n+1 )2 2

(iii) 16 2 + 17 2 + 18 2 +.....+ 25 2

2

2

= 35(35+1)(70+1) - 11(11+1)(22+1) (v) 11 3 + 12 3 + 13 3 +.....+ 28 3 6 6 ¾£÷× : = 35 x 36 x 71 - 11x12x23 6 6 11 3 + 12 3 + 13 3 +.....+ 28 3 = 14910 = ( 13 +2 3 +....+283 ) - (13 +2 3 +....+ 103 )

DO OZ

5

Y

12 2 + 13 2 + 14 2 +.....+ 35 2 =( 12 +2 2 +....+352 ) - (12 +2 2 +....+ 112 )

n ( n+1 )2 2

= ( 30 X 61 ) - ( 25 X 13 ) = 1830 -325

DY

= ( 406 ) 2 - ( 55 ) 2

= 164836 - 3025

=

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2

2

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(vi)11 ¦º.Á£., 12 ¦º.Á£, 13 ¦º.Á£,.....24 ¦º.Á£. ¬¸¢ÂÅü¨È Ó¨È§Â Àì¸ «Ç×¸Ç¡¸ì ¦¸¡ñ¼ 14 ºÐÃí¸Ç¢ý ¦Á¡ò¾ô ÀÃôÒ ¸¡ñ¸.

=

¾£÷× :

= 3 30 0 x 31 2

- 15(15+1) 2

-

15 x 16 2

DY

= ( 465 ) 2 - ( 120 ) 2 = 216225 - 14400

=

ST U

= 24(24+1)(48+1) - 10(10+1)(20+1) 6 6 = 24 x 25 x 49 - 10x11x21 6 6 = 4900 - 385

2

2

= ( 15 15 x 31 ) 2 - ( 15 x 8 ) 2

= + +

2 = 30 30 (30 (30 + 1) 2

14 ºÐÃí¸Ç¢ýÀÃôÒ¸Ç¢ý ÜÎ¾ø 11 2 + 12 2 + 13 2 +.....+ 24 2 = ( 12 +2 2 +....+242 ) - (12 +2 2 +....+ 102 )

n ( n+1 )2

=

(vii) 12 ¦º.Á£, 13 ¦º.Á£, 14 ¦º.Á£,.....23 ¦º.Á£. ¬¸¢ÂÅü¨È Ó¨È§Â Àì¸ «Ç×¸Ç¡¸ì ¦¸¡ñ¼ 12 ºÐÃí¸Ç¢ý ºÐÃí¸Ç¢ý ¦Á¡ò¾ô ÀÃôÒ ¸¡ñ¸.

¾£÷× :

= + +

DO OZ

Y

12 ºÐÃí¸Ç¢ýÀÃôÒ¸Ç¢ý ÜÎ¾ø 12 2 + 13 2 + 14 2 +.....+ 23 2 = ( 12 +2 2 +....+232 ) - (12 +2 2 +....+ 112 )

= 23(23+1)(46+1) - 11(11+1)(22+1) 6 6 = 23 x 24 x 47 - 11x12x23 6 6 = 4324 - 506

=

(viii) 16 ¦º.Á£, 17 ¦º.Á£, 18 ¦º.Á£,.....3 .....30 ¦º.Á£. ¬¸¢ÂÅü¨È Ó¨È§Â Àì¸ «Ç×¸Ç¡¸ì ¦¸¡ñ¼ 15 15 ¸ÉîºÐÃí¸Ç¢ý ¸ÉîºÐÃí¸Ç¢ý¸É«Ç×¸Ç¢ý ºÐÃí¸Ç¢ý¸É«Ç×¸Ç¢ý ÜÎ¾ø ¸¡ñ¸.

¾£÷× : 15 ¸ÉîºÐÃí¸Ç¢ý ¸É«Ç×¸Ç¢ý ÜÎ¾ø 16 3 + 17 3 + 18 3 +.....+ 30 3 = ( 13 +2 3 +....+303 ) - (13 +2 3 +....+ 153 )

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2

2

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http://doozystudy.blogspot.in/ 2. 200 «Ê ¯ÂÃÓûÇ ¸Äí¸¨Ã ¸Äí¸¨Ã Å¢Çì¸ò¾¢ý

1. 700 Á£ ¯ÂÃò¾¢ø ÀÈóÐì ¦¸¡ñÊÕìÌõ ´Õ ¦†Ä¢¸¡ô¼¡¢Ä¢ÕóÐ ´ÕÅ÷ µ÷ ¬üÈ¢ýþÕ ¸¨Ã¸Ç¢ø §¿§Ã¾¢Ã¡¸ ¯ûÇþÕ ¦À¡Õð¸¨Ç 30° 30°, 45° 45°þÈì¸ì §¸¡½í¸Ç¢ø ¸¡ñ¸¢È¡÷ ±É¢ø, ¬üÈ¢ý «¸Äò¨¾ì ¸¡ñ¸.

¾£÷× : AB= ¦†Ä¢¸¡ô¼÷ ÀÈìÌõ ¯ÂÃõ 700 Á£ CD = ¬üÈ¢ý «¸Äõ = ? ²üÈì§¸¡½í¸û = 45°, = 45°, 30 °, 30° 30° ¦ºí§¸¡½ ∆ ABC Â¢ø,

1

=

√

"#$ ° =

!

=

!

"#$ ( =

1

)

) )

DO OZ

C, D = §¾¡½¢ ÁüÚõ À¼Ì ²üÈì §¸¡½í¸û = 45° 45°, 30° 30° ¦ºí§¸¡½ ∆ ABC ABCÂ¢ø

!

BC BC BC = 700 = 700 Á£ ¦ºí§¸¡½ ∆ ABD ABDÂ¢ø, "#$ ° =

AB = ¸Äí¸¨Ã Å¢Äì¸ò¾¢ý ¯ÂÃõ 200 «Ê

Y

"#$ ° =

¾£÷× :

ST U

=

B BD = 700 = 700√ = 700 Á£

¬üÈ¢ý «¸Äõ = BC BC + + B + BD

= 700 + 700 √ 700 + 700

= 700[ 1 + √ ] = 700 [ 1 + 1.732 ] 700 [ 1 + 1.732 ] = 700 x 2.732 700 x 2.732

¯îº¢Â¢Ä¢ÕóÐ, «¾ý «¾ý ¸¡ôÀ¡Ç÷ §¾¡½¢ ÁüÚõ À¼Ì ¬¸¢ÂÉ ´§Ã ¾¢¨ºÂ¢ø ´§Ã §¿÷ì§¸¡ðÊø «¨Á¸¢ýÈÉ. §¾¡½¢, À¼Ì ¬¸¢ÂÅüÈ¢ýþÈì¸ì§¸¡½í¸û Ó¨È§Â 45° 45°ÁüÚõ 30° 30° ±ý¸. þùÅ¢ÃñÎõ À¡Ð¸¡ôÀ¡¸ þÕì¸§ÅñÎ¦ÁÉ¢ø, «¨Å¸ÙìÌ þ¨¼ôÀð¼ àÃõ Ì¨Èó¾Ð 300 «ÊÂ¡¸ þÕì¸§ÅñÎõ. þÕì¸§ÅñÎõ. þ¨¼¦ÅÇ¢Ì¨Èó¾¡ø ¸¡ôÀ¡Ç÷ ±îº¡¢ì¨¸ ´Ä¢ ±ØôÀ §ÅñÎõ. «Å÷ ±îº¡¢ì¨¸ ´Ä¢ ±ØôÀ §ÅñÎÁ¡ ?

DY

7. Óì§¸¡½Å¢Âø Óì§¸¡½Å¢Âø

¬üÈ¢ý «¸Äõ = 1912.4Á£ 1912.4Á£.

=

=

!

! !

BC BC BC = 200 = 200 «Ê = 200 «Ê ¦ºí§¸¡½ ∆ ABD Â¢ø "#$ ( = ) "#$ ° =

=

√

)

)

B BD = 200√«Ê D = 200 «Ê §¾¡½¢, À¼Ì þ¨¼ôÀð¼ àÃõ = BD D - B BC = 200 √3 √3 - 200 200 √3 200 200 = 200 [ √3 = 200 [ √3 √3 - 1 ] 1 ] 1 ] = 200 [ 1.732 = 200 [ 1.732 - 1 ] 1 ] = 200 X 0.732 = 200 X 0.732 = 146.4«Ê = 146.4«Ê «Ê

§¾¡½¢, À¼Ì þ¨¼ôÀð¼ àÃõ146.4 àÃõ146.4 «Ê ±É§Å ¸¡ôÀ¡Ç÷ ´Ä¢ ±ØôÀ§ÅñÎõ. ±ØôÀ§ÅñÎõ.

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3.

60 Á£. ¯ÂÃÓûÇ ´Õ §¸¡ÒÃò¾¢Ä¢ÕóÐ ´Õ ¸ðÊ¼ò¾¢ý ¯îº¢ ÁüÚõ «Ê ¬¸¢ÂÅüÈ¢ýþÈì¸ì §¸¡½í¸û Ó¨È§Â 30° 30°, ÁüÚõ 60° 60°. ±É¢ø, ¸ð¼¼ò¾¢ý ¯ÂÃò¨¾ì ¸¡ñ¸.

"#$ ( =

¾£÷× : AE = ¸ð¼¼ò¾¢ý ¯ÂÃõh ¯ÂÃõh = ?

√3 =

¦ºí§¸¡½ ∆ ABD Â¢ø,

√3 =

AB B=

A= √

Á£

¦ºí§¸¡½ ∆ECD ECDÂ¢ø, !) "#$ ( = 8!

Á£

√

AB = EC =

"#$ ° =

√

¦ºí§¸¡½ ∆ ECD Â¢ø

9: √

DO OZ

√

=

√

Y

"#$ ° =

9:

A=

= 60 60 - h

√; √

= = h = 60 – 20

¸ð¼¼ò¾¢ý ¯ÂÃõ = 40 Á£.

Á£. ¯ÂÃÓûÇ ´Õ §¸¡ÒÃò¾¢ý ¯îº¢ ÁüÚõ ÁüÚõ «Ê ¬¸¢ÂÅüÈ¢Ä¢ÕóÐ ´Õ ¸Äí¸¨Ã Å¢Çì¸ò¾¢ý ¯îº¢Â¢ý ²üÈì§¸¡½í¸û Ó¨È§Â 30° 30° ÁüÚõ 60° 60° ±É¢ø, ¸Äí¸¨Ã Å¢Çì¸ò¾¢ý ¯ÂÃò¨¾ì ¸¡ñ¸. ¸Äí¸¨Ã Å¢Çì¸ò¾¢ý ¯îº¢Â¢Ä¢ÕóÐ §¸¡ÒÃò¾¢ý «ÊìÌ ¯ûÇ àÃò¨¾ì ¸¡ñ¸.

¾£÷× :

http://doozystudy.blogspot.in/

√; √

A:

:

B= √

=

√

!) "#$ ( = 8!

4. 40

Á£

√

AB = EC =

A=

ST U

"#$ ° =

A:

AB B=

)

DY

þÈì¸§¸¡½í¸û= ²üÈì§¸¡½í¸û = 30° 30°, 60° 60°

?@ A:

"#$ ° =

BD = §¸¡ÒÃò¾¢ý ¯ÂÃõ = 60 Á£.

"#$ ( =

AE= §¸¡ÒÃò¾¢ý ¯ÂÃõ = 40 Á£. BD = ¸Äí¸¨Ã Å¢Çì¸ò¾¢ý ¯ÂÃõ = ? ²üÈì§¸¡½í¸û = 30° 30°, 60° 60° ¦ºí§¸¡½ ∆ABD Â¢ø,

:

B= √

= h

= h

40+ 40+ h = 3 h 40 = 3h – h 2h = 40 ;

h =

h = 20 Á£. BD =B =BC + CD =40 + 20 = 60 Á£ ¸Äí¸¨Ã Å¢Çì¸ò¾¢ý Å¢Çì¸ò¾¢ý ¯ÂÃõ = 60 Á£. ¦ºí§¸¡½ ∆ADB Â¢ø,

= DE$ ° = √

=

)

)

F@

F@ ; ; ; ; = √ √ √ AD = √

AD= 40√

§¸¡ÒÃò¾¢ý «ÊÂ¢Ä¢ÕóÐ ¸Äí¸¨Ã ¸Äí¸¨Ã Å¢Çì¸ò¾¢ý ¯îº¢Å¨Ã ¯ûÇàÃõ40√Á£ 40 Á£.

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5.

§¿÷ìÌò¾¡É ´Õ ÁÃò¾¢ý §ÁøÀ¡¸õ ¸¡üÈ¢É¡ø ÓÈ¢óÐ, «õÓÈ¢ó¾ ÀÌ¾¢ ¸£§Æ Å¢ØóÐ Å¢¼¡Áø, ÁÃò¾¢ý ¯îº¢ ¾¨ÃÔ¼ý 30° 30° §¸¡½ò¨¾ ²üÀÎòÐ¸¢ÈÐ. ÁÃò¾¢ý ¯îº¢ «¾ý «ÊÂ¢Ä¢ÕóÐ 30 Á£. ¦¾¡¨ÄÅ¢ø ¾¨Ã¨Âò ¦¾¡Î¸¢ÈÐ ±É¢ø, ÁÃò¾¢ý ÓØ ¯ÂÃò¨¾ì ¸¡ñ¸.

¾£÷× :

?G

"#$ ° = =

√

BC =

; √ ; √

=

√

BC = 10 √ Á£.

¦ºí§¸¡½ ∆ABC Â¢ø, Â¢ø,

!

!

DO OZ

=

DE$ ° =

=

√

FG √ FG

AC = √

=

√

AB = 50 √Á£ Á£. ¦ºí§¸¡½ ∆ABCÂ¢ø, ABCÂ¢ø, ! "#$ ( = ?G

"#$ ° =

√ ?G

√

BC= = 50 X 3 BC= 50√ ; √= ÌýÈ¢ý ¯ÂÃõ = 150 Á£.

√

√3=

?G

Y

"#$ ° =

ST U

AC = ÁÃò¾¢ý ÓÈ¢ó¾ ÀÌ¾¢ BC = ÁÃò¾¢ý ÓÈ¢Â¡¾ ÀÌ¾¢ AB = ÁÃò¾¢ý «ÊÂ¢Ä¢ÕóÐÁÃõ ÓÈ¢ó¾ ÀÌ¾¢ Å¨Ã ¯ûÇàÃõ 30 Á£. ²üÈì§¸¡½õ = 30° 30° ¦ºí§¸¡½ ∆ ABC Â¢ ! "#$ ( =

AD = §¸¡ÒÃò¾¢ý ¯ÂÃõ = 50 Á£. BC = ÌýÈ¢ý ¯ÂÃõ = ? ²üÈì§¸¡½í¸û = 60° 60°, 30° 30° ¦ºí§¸¡½ ∆DAB Â¢ø, ) "#$ ( =

DY

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x 2 =20√

AC = 20 √ Á£.

BC+ BC+ AC AC = 10 √ + 20 √

Á£. ÁÃò¾¢ý ¦Á¡ò¾ ¯ÂÃõ= ¯ÂÃõ= 30 √Á£

6.

´Õ §¸¡ÒÃò¾¢ý «ÊÂ¢Ä¢ÕóÐ ´Õ ÌýÈ¢ý ¯îº¢Â¢ý ²üÈì§¸¡½õ 60° 60° ±ý¸. ÌýÈ¢ý «ÊÂ¢Ä¢ÕóÐ §¸¡ÒÃò¾¢ý ¯îº¢Â¢ý ²üÈì§¸¡½õ 30° 30°ÁüÚõ §¸¡ÒÃò¾¢ý ¯ÂÃõ 50 Á£ ±É¢ø, ÌýÈ¢ý ¯ÂÃò¨¾ì ¸¡ñ¸.

¾£÷× :

http://doozystudy.blogspot.in/

7. ´Õ ¦ºíÌò¾¡É ÍÅÕõ, ´Õ §¸¡ÒÃÓõ ´Õ ÌÈ¢ôÀ¢ð¼ þ¨¼¦ÅÇ¢Â¢ø ¯ûÇÉ. §¸¡ÒÃò¾¢ý ¯îº¢Â¢Ä¢ÕóÐ À¡÷ìÌõ§À¡Ð, ÍÅüÈ¢ý ¯îº¢ ÁüÚõ«Ê ÁüÚõ«Ê ¬¸¢ÂÅüÈ¢ýþÈì¸ì§¸¡½í¸û Ó¨È§Â 45° 45°ÁüÚõ 60° 60° ¬Ìõ. §¸¡ÒÃò¾¢ý ¯ÂÃõ 90 Á£. ±É¢ø, ÍÅüÈ¢ý ¯ÂÃò¨¾ì ¸¡ñ¸.

¾£÷× : BD= BD= §¸¡ÒÃò¾¢ý ¯ÂÃõ = 90 Á£ AE = ÍÅ¡¢ý ¯ÂÃõ = ?

þÈì¸§¸¡½í¸û= ²üÈì§¸¡½í¸û = 45° 45°, 60° 60°

¦ºí§¸¡½ ∆ ABD Â¢ø, ) "#$ ( = "#$ ° = √=

AB =

√

=

; √ ; √ √

= 30√

AB =51.96Á£.

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http://doozystudy.blogspot.in/

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AB = EC = 51.96 Á£ ¦ºí§¸¡½∆ECD ECDÂ¢ø, !) "#$ ( = 8! 1 =

2h = 300√3 300√3 ; h =

√

h = 150 √3 Á£.

9:

¸¼ø Áð¼ò¾¢Ä¢ÕóÐ ¸Äí¸¨Ã

. 9:

Å¢Çì¸ò¾¢ý ¯ÂÃõ150 ¯ÂÃõ150 √3 Á£.

.

9. ´Õ ¸ð¼¼ò¾¢ý ¸ð¼¼ò¾¢ý §Áø ´Õ ¦¸¡Êì¸õÀõ ¿¢ü¸¢ÈÐ. ¾¨ÃÂ¢ÖûÇ ´Õ ÒûÇ¢Â¢Ä¢ÕóÐ, ÒûÇ¢Â¢Ä¢ÕóÐ, ¦¸¡Êì¸õÀò¾¢ý ¯îº¢ ÁüÚõ «Ê ¬¸¢ÂÅüÈ¢ý ²üÈì§¸¡½í¸û Ó¨È§Â 60° 60°ÁüÚõ 45° 45° ±ý¸. §ÁÖõ ¦¸¡Êì¸õÀò¾¢ý ¦¸¡Êì¸õÀò¾¢ý ¯ÂÃõ 10 Á£ ±É¢ø, ¸ð¼¼ò¾¢ý ¸ð¼¼ò¾¢ý ¯ÂÃò¨¾ì ¸¡ñ¸.

51.96 = 90 90 - h h = 90 – 51.96

ÍÅüÈ¢ý ¯ÂÃõ = 38.04 Á£.

¾£÷× :

ST U

8. ¸¼ü¸¨ÃÂ¢ø ¸¼ü¸¨ÃÂ¢ø ¯ûÇ ¦ºíÌò¾¡Éô ¦ºíÌò¾¡Éô À¡¨È ´ýÈ¢ý

DY

"#$ ° =

300 √3 + h= h= 3h

Á£Ð ¸ð¼ôÀðÎûÇ ´Õ ¸Äí¸¨È Å¢Äì¸ò¾¢ý Å¢Äì¸ò¾¢ý ¯îº¢Â¢ø ¿¢ýÚ¦¸¡ýÊÕìÌõ ´Õ º¢ÚÁ¢, ¸¢ÆìÌ ¾¢¨ºÂ¢øþÕ À¼Ì¸¨Çô À¡÷ì¸¢È¡û. «ôÀ¼Ì¸Ç¢ý þÈì¸§¸¡½í¸û Ó¨È§Â 30° 30°, 60° 60°ÁüÚõþÕ À¼Ì¸Ùì¸¢¨¼§Â ¯ûÇàÃõ 300 Á£. ±É¢ø, ¸¼ø Áð¼ò¾¢Ä¢ÕóÐ ¸Äí¸¨Ã Å¢Çì¸ò¾¢ý ¯îº¢Â¢ý àÃò¨¾ì ¸¡ñ¸.

DC = ¦¸¡Êì¸õÀò¾¢ý ¯ÂÃõ = 10 Á£ BC = ¸ð¼¼ò¾¢ý ¯ÂÃõ = h Á£ ²üÈì§¸¡½í¸û = 60° 60°, 45° 45°

¾£÷× :

"#$ ° =

Y

AD = ¸¼ø Áð¼ò¾¢Ä¢ÕóÐ ¸Äí¸¨Ã Å¢Çì¸ò¾¢ý ¯ÂÃõ = h Á£. BC = þÕ À¼Ì¸ÙìÌ þ¨¼ôÀð¼ àÃõ = 300 Á£ þÈì¸§¸¡½í¸û= ²üÈì§¸¡½í¸û = 30° 30°, 60° 60°

DO OZ

¦ºí§¸¡½ ∆DAB Â¢ø, "#$ ( =

"#$ ° = √= =

:

X=

I

)

: I

:

√

Á£

¦ºí§¸¡½ ∆ DACÂ¢ø, DACÂ¢ø, ) "#$ ( = ! "#$ ° =

√

=

:

AI :

: A √

;

√

=

300 √3 + h = h √3 x √3

: √ + = √

http://doozystudy.blogspot.in/

¦ºí§¸¡½ ∆aBCÂ¢ø,

1

=

"#$ ( =

:

!

J : J

x= h ¦ºí§¸¡½ ∆ ABDÂ¢ø, BDÂ¢ø, ) "#$ ( = "#$ ° = √= =

A :

I A : =

h√ = 10 + h=h√ = = h =

√9

x

√A √A

=

h = 13.66 Á£

,.A . 9

¸ð¼¼ò¾¢ý ¯ÂÃõ13.66 ¯ÂÃõ13.66 Á£

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8. «ÇÅ¢Âø 1. ´Õ Å¢¨ÇÂ¡ðÎÀõÀÃÁ¡ÉÐÜõÀ¢ý Á£Ð «¨Ãì§¸¡Çõ þ¨½ó¾ ÅÊÅ¢ø ¯ûÇÐ. «¨Ãì§¸¡Çò¾¢ý «¨Ãì§¸¡Çò¾¢ý Å¢ð¼õ 3.6 ¦º.Á£. ÁüÚõ ÀõÀÃò¾¢ý ¦Á¡ò¾ ¯ÂÃõ 4.2 ¦º.Á£. ±É¢ø, «¾ý ¦Á¡ò¾ô ÒÈôÀÃô¨Àì ¸¡ñ¸.

¾£÷×: ×: ÜõÒ :«¨Ãì§¸¡Çõ : ¬Ãõr ¬Ãõr = 1.8¦º.Á£ 1.8¦º.Á£.

DY

Å¢ð¼õ d= 3.6 ¦º.Á£. ¬Ãõr ¬Ãõr = 1.8 ¦º.Á£. ¯ÂÃõh ¯ÂÃõh =4.2 - 1.8 = 2.4 ¦º.Á£.

ST U

º¡ÔÂÃõl= √ h2 + r2 = √ 2.42 + 1.82 = √ 5.76 + 3.24 = √ 9.00 º¡ÔÂÃõl= 3 ¦º.Á£.

ÀõÀÃò¾¢ý ¦Á¡ò¾ô ÒÈôÀÃôÒ = ÜõÀ¢ýÅ¨ÇÀÃôÒ + «¨Ãì§¸¡Çò¾¢ý Å¨ÇÀÃôÒ 2 = Mr l+ 2M r

= r [l + 2 r ] = M x 1.8 [ 3 + 2 x 1.8 ]

ÀõÀÃò¾¢ý ¦Á¡ò¾ô ÒÈôÀÃôÒ = 11.88M 11.88M º.¦º.Á£.

= M x 1.8 x 6.6

¾£÷× :

DO OZ

Y

2. ´Õ ¾¢ñÁ ÁÃô¦À¡õ¨ÁÂ¡ÉÐ «¨Ãì§¸¡Çò¾¢ý §ÁøÜõÒ þ¨½ó¾ ÅÊÅ¢ø ¯ûÇÐ. «¨Ãì§¸¡ÇõÁüÚõÜõÒ ¬¸¢ÂÅüÈ¢ý ¬Ãõ 3.5 ¦º.Á£. §ÁÖõ ¦À¡õ¨ÁÂ¢ý ¦Á¡ò¾ ¯ÂÃõ 17.5 ¦º.Á£. ±É¢ø, «ô¦À¡õ¨Á ¾Â¡¡¢ì¸ô ÀÂýÀÎò¾ôÀð¼ ÁÃò¾¢ý ¸É «Ç¨Åì ¸¡ñ¸.

ÜõÒ :«¨Ãì§¸¡Çõ :

¬Ãõr ¬Ãõr = 3.5 ¦º.Á£. ¯ÂÃõh ¯ÂÃõh =17.5– 17.5–3.5 = 14 ¦º.Á£.

¬Ãõr ¬Ãõr = 3. 3.5 ¦º.Á£.

§¾¨ÅÂ¡É ÁÃò¾¢ý ¸É «Ç× = ÜõÀ¢ý ¸É «Ç× + «¨Ãì§¸¡Çò¾¢ý ¸É «Ç× =

=

=

=

Mr 2 h+

M r3

Mr 2 [ h +2 r ]

X

X 3.50.5 X 3.5 [ 14 + 2 X 3.5 ]

X 22 X 0.5 X 3.5 X 21 7

§¾¨ÅÂ¡É ÁÃò¾¢ý ¸É «Ç× = 269.5 ¸.¦º.Á£.

http://doozystudy.blogspot.in/

http://doozystudy.blogspot.in/

http://doozystudy.blogspot.in/

http://doozystudy.blogspot.in/

3. ´Õ ÁÕóÐìÌôÀ¢Â¡ÉÐ ´Õ ¯Õ¨ÇÂ¢ýþÕÒÈÓõ «¨Ãì§¸¡Çí¸û þ¨½ó¾ ÅÊÅ¢ø ¯ûÇÐ. ÁÕóÐìÌôÀ¢Â¢ý ¦Á¡ò¾ ¿£Çõ 14 Á¢.Á£. ÁüÚõ Å¢ð¼õ 5 Á¢.Á£. ±É¢ø, «õÁÕóÐìÌôÀ¢Â¢ýÒÈôÀÃô¨Àì ¸¡ñ¸. ¾£÷× : «¨Ãì§¸¡Çõ :

¯Õ¨Ç :

Å¢ð¼õ d =5 Á¢.Á£.

¬Ãõ r = Á¢.Á£

=

2 x

x

, 2x + 9 .

ST U

=

DY

¬Ãõ r = Á¢.Á£ ¯ÂÃõ h = 14 - 5 = 9 Á¢.Á£. ÁÕóÐìÌôÀ¢Â¢ýÒÈôÀÃôÒ = 2 «¨Ãì§¸¡Çí¸Ç¢ýÅ¨ÇÀÃôÒ + ¯Õ¨ÇÂ¢ýÅ¨ÇÀÃôÒ = 2 x 2 M r2 + 2Mrh = 2Mr , 2r + h .

x 5 x 14

ÁÕóÐìÌôÀ¢Â¢ýÒÈôÀÃôÒ= ÁÕóÐìÌôÀ¢Â¢ýÒÈôÀÃôÒ= 220 º.Á¢.Á£.

¾£÷× :

Y

4. ´Õ Ü¼¡ÃÁ¡ÉÐ ¯Õ¨ÇÂ¢ý Á£Ð ÜõÒ þ¨½ó¾ ÅÊÅ¢ø ¯ûÇÐ. Ü¼¡Ãò¾¢ý ¦Á¡ò¾ ¯ÂÃõ 13.5 Á£. ÁüÚõ Å¢ð¼õ 28 Á£. §ÁÖõ ¯Õ¨Çô À¡¸ò¾¢ý ¯ÂÃõ 3 Á£. ±É¢ø, Ü¼¡Ãò¾¢ý ¦Á¡ò¾ô ÒÈôÀÃô¨Àì ¸¡ñ¸. ÜõÒ :

Å¢ð¼õ d= 28 Á£. ¬Ãõ r = 14 Á£.

¬Ãõ r= 14 Á£. ¯ÂÃõh ¯ÂÃõh= 13.5 - 3 = 10.5 Á£.

DO OZ

¯Õ¨Ç :

¯ÂÃõ ¯ÂÃõ h = 3 Á£.

º¡ÔÂÃõ l =

√ h2 + r2

= √ 10.52 + 142

l= 17.5 Á£.

Ü¼¡Ãò¾¢ý ¦Á¡ò¾ ÒÈôÀÃôÒ = ÜõÀ¢ýÅ¨ÇÀÃôÒ + ¯Õ¨ÇÂ¢ýÅ¨ÇÀÃôÒ ¯Õ¨ÇÂ¢ýÅ¨ÇÀÃôÒ = rl + 2 M r h = Mr [ l + 2 x h ]

=

x

14 , 17.5 + 2 x 3 .

Ü¼¡Ãò¾¢ý ¦Á¡ò¾ ÒÈôÀÃôÒ= ÒÈôÀÃôÒ= 1034 º.Á£.

http://doozystudy.blogspot.in/

http://doozystudy.blogspot.in/

http://doozystudy.blogspot.in/

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5. ´Õ º÷ì¸Š Ü¼¡ÃÁ¡ÉÐ ¯Õ¨ÇÂ¢ý Á£Ð ÜõÒ þ¨½ó¾ ÅÊÅ¢ø ¯ûÇÐ. Ü¼¡Ãò¾¢ý ¦Á¡ò¾ ¯ÂÃõ 49 Á£. «¾ý «ÊôÀì¸ Å¢ð¼õ 42 Á£. §ÁÖõ ¯Õ¨Çô À¡¸ò¾¢ý ¯ÂÃõ 21 Á£. §ÁÖõ 1 º.Á£. ¸¢ò¾¡ý Ð½¢Â¢ý Å¢¨Ä r.. 12.50 ±É¢ø, Ü¼¡Ãõ «¨Áì¸ò §¾¨ÅÂ¡É ¸¢ò¾¡ý Ð½¢Â¢ý Å¢¨Ä¨Âì ¸¡ñ¸. ¯Õ¨Ç :

ÜõÒ :

Å¢ð¼õ d= 42 Á£. ¬Ãõr =21 Á£. ¬Ãõr

¬Ãõr ¬Ãõr = 21 Á£. ¯ÂÃõh ¯ÂÃõh = 49 - 21 = 28 Á£.

¯ÂÃõh ¯ÂÃõh = 21 Á£.

º¡ÔÂÃõ l = =

DY

¾£÷× :

√ h2 + r2

√28 2 + 21 2

l = 35 Á£.

ST U

Ü¼¡Ãò¾¢ý ¦Á¡ò¾ ÒÈôÀÃôÒ = ÜõÀ¢ýÅ¨ÇÀÃôÒ + ¯Õ¨ÇÂ¢ýÅ¨ÇÀÃôÒ = rl + 2 M r h = Mr [ l + 2 x h ] =

x 21 , 35 + 2 x 21 .

Y

Ü¼¡Ãò¾¢ý ¦Á¡ò¾ ÒÈôÀÃôÒ= ÒÈôÀÃôÒ= 5082 º.Á£. 1 º.Á£ ¸¢ò¾¡ý Ð½¢Â¢ý Å¢¨Ä = r..12.50 ±É§Å §¾¨ÅÂ¡É ¸¢ò¾¡ý Ð½¢Â¢ý¦Á¡ò¾ Å¢¨Ä = 5082 X 12.50 = r..63525

¾£÷× :

DO OZ

6. 7 ¦º.Á£. Å¢ð¼ÓûÇ ¯Õ¨Ç ÅÊÅ Ó¸¨ÅÂ¢ø º¢È¢¾Ç× ¾ñ½£÷ ¯ûÇÐ. «¾¢ø ´ù¦Å¡ýÚõ 1.4 ¦º.Á£. Å¢ð¼ÓûÇ º¢Ä §¸¡Ç ÅÊÅ ÀÇ¢íÌì ¸ü¸û §À¡¼ôÀÎ¸¢ÈÐ. ¯Õ¨ÇÂ¢ÖûÇ ¿£¡¢ý Áð¼õ 5.6 ¦º.Á£. ¯ÂÃ ±ò¾¨É ÀÇ¢íÌ ¸ü¸¨Ç Ó¸¨ÅÂ¢Ûû §À¡¼§ÅñÎõ ±Éì ¸¡ñ¸. ¯Õ¨Ç : Å¢ð¼õ d= 7 ¦º.Á£ ¬Ãõr1 = 3.5¦º.Á£ 3.5¦º.Á£. ¯ÂÃõh ¯ÂÃõh= 5.6 ¦º.Á£.

§¸¡Çõ : Å¢ð¼õ d = 1.4 ¦º.Á£. ¬Ãõr2 = 0.7 ¦º.Á£.

ÀÇ¢íÌ ¸ü¸Ç¢ý ±ñ½¢ì¨¸

=

¯Â÷ó¾ ¿£¡¢ý ¸É «Ç×

§¸¡Çò¾¢ý ¸É «Ç×

=

=

Mr12 h

M r2

3

N . N . N . N . N . N .

ÀÇ¢íÌ ¸ü¸Ç¢ý ±ñ½¢ì¨¸ = 150

http://doozystudy.blogspot.in/

http://doozystudy.blogspot.in/

http://doozystudy.blogspot.in/

http://doozystudy.blogspot.in/

7. 8 ¦º.Á£. Å¢ð¼Óõ 12 ¦º.Á£. ¯ÂÃÓõ ¦¸¡ñ¼ ´Õ §¿÷ Åð¼ ¾¢ñÁ þÕõÒìÜõÀ¡ÉÐ ¯Õì¸ôÀðÎ 4 Á¢.Á£. ¬ÃÓûÇ ¾¢ñÁì §¸¡Ç ÅÊÅ ÌñÎ¸Ç¡¸ Å¡÷ì¸ôÀð¼¡ø ¸¢¨¼ìÌõ §¸¡Ç ÅÊÅ ÌñÎ¸Ç¢ý ±ñ½¢ì¨¸¨Âì ¸¡ñ¸. ¾£÷× : ÜõÒ :

§¸¡Çõ :

M O =

N N

=

ST U

=

§¾¨ÅÂ¡É §¸¡Çí¸Ç¢ý ±ñ½¢ì¨¸

DY

Å¢ð¼õd ¬Ãõr2 = 4 Á¢.Á£ Å¢ð¼õd = 8 ¦º.Á£ ¬Ãõr1= 4 ¦º.Á£ = 40 Á¢.Á£. ¯ÂÃõh ¯ÂÃõh= 12 ¦º.Á£ = 120 Á¢.Á£. §¾¨ÅÂ¡É §¸¡Çí¸Ç¢ý ±ñ½¢ì¨¸ = ÜÁÀ¢ý ¸É «Ç× §¸¡Çò¾¢ý ¸É «Ç×

MO

N N N

= 750

Y

8. 12 ¦º.Á£. Å¢ð¼Óõ 15 ¦º.Á£. ¯ÂÃÓõ ¦¸¡ñ¼ §¿÷Åð¼ ¯Õ¨Ç ÓØÅÐõÀÉ¢ìÜÆ¢É¡ø ¿¢ÃôÀôÀðÎûÇÐ. þôÀÉ¢ìÜÆ¡ÉÐ 6 ¦º.Á£. Å¢ð¼Óõ, 12 ¦º.Á£. ¯ÂÃÓõ ¦¸¡ñ¼ §ÁüÒÈõ «¨Ãì§¸¡Çõ þ¨½ó¾ ÅÊÅ¢Ä¨Áó¾ ÜõÀ¢ø ¿¢ÃôÀôÀÎ¸¢ÈÐ. ±ò¾¨É ÜõÒ¸Ç¢øÀÉ¢ìÜÆ¢¨É ÓØÅÐÁ¡¸ ¿¢ÃôÀÄ¡õ ±Éì ¸¡ñ¸. ¾£÷× :

DO OZ

¯Õ¨Ç : ÜõÒ : «¨Ã§¸¡Çõ : Å¢ð¼õ d=12 ¦º.Á£ Å¢ð¼õ d= 6 ¦º.Á£ ¬Ãõr2 = 3 ¦º.Á£ ¬Ãõr1 =6 ¦º.Á£ ¬Ãõr2= 3 ¦º.Á£ ¯ÂÃõ= ¯ÂÃõ= ¯ÂÃõ =5 ¦º.Á£ ÂÃõ =12 ¦º.Á£ ¦º.Á£ ÀÉ¢ìÜú ¿¢ÃôÀò §¾¨ÅÂ¡É ÜõÒ¸Ç¢ý ±ñ½¢ì¨¸

=

¯Õ¨ÇÂ¢ý ¸É «Ç× ÜõÀ¢ý ¸É «Ç× + «¨Ãì§¸¡Çò¾¢ý ¸É «Ç× = M O = = 3 x M X 6 x 6 x 15

=

M O = +

M O [ = + 2r 2r2 .

M O

3 x 6 x 6 x 15 3 x 3 x , 12 + (2 x 3) .

ÀÉ¢ìÜú ¿¢ÃôÀò §¾¨ÅÂ¡É ÜõÒ¸Ç¢ý ±ñ½¢ì¨¸

http://doozystudy.blogspot.in/

= 2 x 6 x 15 18

= 10

http://doozystudy.blogspot.in/

http://doozystudy.blogspot.in/

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9. ¸Ç¢Áñ¨½ô ÀÂýÀÎò¾¢ ´Õ Á¡½Åý 48 ¦º.Á£. ¯ÂÃÓõ 12 ¦º.Á£. ¬ÃÓõ ¦¸¡ñ¼ §¿÷Åð¼ ¾¢ñÁì Üõ¨Àî ¦ºö¾¡÷. «ìÜõ¨ÀÁü¦È¡Õ Á¡½Å÷ ´Õ ¾¢ñÁì §¸¡ÇÁ¡¸Á¡üÈ¢É¡÷. §¸¡ÇÁ¡¸Á¡üÈ¢É¡÷. «ùÅ¡Ú Á¡üÈôÀð¼ Ò¾¢Â §¸¡Çò¾¢ý ¬Ãò¨¾ì ¸¡ñ¸. ÜõÒ : ¬Ãõr ¬Ãõr1 = 12 ¦º.Á£.

§¸¡Çõ : ¬Ãõr ¬Ãõr = ?

§¸¡Çò¾¢ý ¸É «Ç×

=

Mr 3

ÜõÀ¢ý ¸É «Ç×

M P h

=

I I

r3 =

ST U

¯ÂÃõh = 48 ¦º.Á£

DY

¾£÷× :

r = ∛12 x 12 x 12 Ò¾¢Â §¸¡Çò¾¢ý ¬Ãõ

= 12 ¦º.Á£.

Y

10. 18 ¦º.Á£. ¬ÃÓûÇ ¾¢ñÁ ¯§Ä¡¸ §¸¡ÇÁ¡ÉÐ ¯Õì¸ôÀðÎãýÚ º¢È¢Â ¦Åù§ÅÚ «Ç×ûÇ §¸¡Çí¸Ç¡¸ Å¡÷ì¸ôÀÎ¸¢ÈÐ. «ùÅ¡Ú Å¡÷ì¸ôÀð¼ þÃñÎ ¾¢ñÁì §¸¡Çí¸Ç¢ý ¬Ãí¸û Ó¨È§Â 2 ¦º.Á£. ÁüÚõ 12 ¦º.Á£. ±É¢ø, ãýÈ¡ÅÐ §¸¡Çò¾¢ý ¬Ãò¨¾ì ¸¡ñ¸. ¾£÷× :

º¢È¢Â §¸¡Çí¸û :

¬ÃõR ¬ÃõR= 18 ¦º.Á£ P = 2 ¦º.Á£ P = ?

¬Ãí¸û P = 12 ¦º.Á£

DO OZ

¦À¡¢Â §¸¡Çõ :

º¢È¢Â §¸¡Çí¸Ç¢ý ¸É «Ç×

M O +

M

O

M O + M O + O + O

12 3 + 2 3 +O

= ¦À¡¢Â §¸¡Çò¾¢ý ¸É «Ç×

=

=

=

O

=

P

MR3

MR3

18 3

5832 - 1736

= ∛ 4096

ãýÈ¡ÅÐ §¸¡Çò¾¢ý ¬ÃõP ¬Ãõ = 16 ¦º.Á£.

http://doozystudy.blogspot.in/

=

= 4096

∛ 16 x 16 x 16

http://doozystudy.blogspot.in/

http://doozystudy.blogspot.in/

http://doozystudy.blogspot.in/

11. Á½Ä¡ø ¿¢ÃôÀôÀð¼ ´Õ ¯Õ¨Ç ÅÊÅ Å¡Ç¢Â¢ý ¯ÂÃõ 32 ¦º.Á£. ÁüÚõ ¬Ãõ 18 ¦º.Á£. «õÁ½ø ÓØÅÐõ ¾¨ÃÂ¢ø ´Õ §¿÷Åð¼ì ÜõÒÅÊÅ¢ø ¦¸¡ð¼ôÀÎ¸¢ÈÐ. «ùÅ¡Ú ¦¸¡ð¼ôÀð¼ Á½üÜõÀ¢ý ¯ÂÃõ 24 ¦º.Á£. ±É¢ø, «ìÜõÀ¢ý ¬ÃõÁüÚõ º¡ÔÂÃò¨¾ì ¸¡ñ¸. ¾£÷× : ÜõÒ :

¯Õ¨Ç : ¬ÃõP Ãõ = 18 ¦º.Á£ ¯ÂÃõ= ¯ÂÃõ = 32 ¦º.Á£

DY

¯ÂÃõ= ¯ÂÃõ = 24 ¦º.Á£ ¬ÃõP Ãõ = ? º¡ÔÂÃõl = ?

ÜõÀ¢ý ¸É «Ç× = ¯Õ¨ÇÂ¢ý ¸É «Ç×

M O =

º¡ÔÂÃõ

=

M O =

O

=

I I I

P

= √9 x 2 x 9 x 2 x 2 x 2

P = 36 ¦º.Á£

ST U

l= √ h2 + r2 = √ 24 2 + 36 2= √ 576 + 1296 = √1872 12 √13 ¦º.Á£.

Y

º¡ÔÂÃõ l =

DO OZ

12. 14 Á£. Å¢ð¼õ ÁüÚõ 20 Á£. ¬ÆÓûÇ ´Õ ¸¢½Ú ¯Õ¨Ç ÅÊÅ¢ø ¦Åð¼ôÀÎ¸¢ÈÐ. «ùÅ¡Ú ¦ÅðÎõ§À¡Ð §¾¡ñÊ¦ÂÎì¸ôÀð¼ Áñ º£Ã¡¸ ÀÃôÀôÀðÎ 20 Á£. X 14 Á£. «Ç×¸Ç¢ø «ÊôÀì¸Á¡¸ì ¦¸¡ñ¼ ´Õ §Á¨¼Â¡¸ «¨Áì¸ôÀð¼¡ø, «õ§Á¨¼Â¢ý ¯ÂÃõ ±ýÉ ? ¾£÷× : ¯Õ¨Ç : Å¢ð¼õ Å¢ ð¼õ d = 14 Á£. ¬Ãõr ¬Ãõr = 7 Á£. ¯ÂÃõh ¯ÂÃõh = 20 Á£.

¸Éî¦ºùÅ¸õ : ¿£Çõ l= 20 Á£. «¸Äõ b = 14 Á£. ¯ÂÃõh1 = ?

¸Éî¦ºùÅ¸ §Á¨¼Â¢ý ¸É «Ç× = §¾¡ñÊ¦ÂÎì¸ôÀð¼ Áñ½¢ý ¸É «Ç×

l b h1

=

h1 =

Mr 2 h

; ; ; N N

Ò¾¢¾¡¸ «¨Áì¸ôÀÎõ §Á¨¼Â¢ý ¯ÂÃõ = 11 Á£.

http://doozystudy.blogspot.in/

http://doozystudy.blogspot.in/

http://doozystudy.blogspot.in/

http://doozystudy.blogspot.in/

12. ¿¢¸ú¾¸× 1. ãýÚ ¿¡½Âí¸û ´§Ã §¿Ãò¾¢ø Íñ¼ôÀÎ¸¢ýÈÉ. À¢ýÅÕõ ¿¢¸úîº¢¸ÙìÌ ¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸. (i) Ì¨Èó¾Ð ´Õ ¾¨Ä ¸¢¨¼ôÀÐ. (ii) þÕâì¸ûÁðÎõ ¸¢¨¼ôÀÐ. (iii) Ì¨Èó¾Ð þÕ ¾¨Ä¸û ¸¢¨¼ôÀÐ.

¾£÷× :

H

(F)

P(A) = (U) B B n(B)

= = =

P(B) = P(B) =

T

H

T

H

H T

T

H T H

T

ST U

P(A) =

H

DY

ÜÚ¦ÅÇ¢ S = { HHH,HHT,HTH,HTT, HHH,HHT,HTH,HTT, THH, THT, TTH, TTH, TTT } n(S) = 8 A = Ì¨Èó¾Ð ´Õ ¾¨Ä ¸¢¨¼ò¾ø. A= {HHH, HHT, HTH, HTT, THH, THT, TTH } n(A) = 7

T

þÕâì¸ûÁðÎõ ¸¢¨¼ò¾ø. { HTT, THT, TTH } 3 (?)

(U)

(G)

(U)

=

DO OZ

P(C) =

Y

C = Ì¨Èó¾Ð þÕ ¾¨Ä¸û ¸¢¨¼ò¾ø. C = { HHH, HHT, HTH, THH } n(C) = 4

P(C) =

2. ãýÚ ¿¡½Âí¸û ´§Ã §¿Ãò¾¢ø Íñ¼ôÀÎ¸¢ýÈÉ. ¿¢¸ú¾¸Å¢ý Üð¼ø §¾üÈò¨¾ô ÀÂýÀÎò¾¢, º¡¢Â¡¸ þÕâì¸û «øÄÐ Ì¨Èó¾Àðºõ ´Õ ¾¨ÄÂ¡ÅÐ ¸¢¨¼ìÌõ ¿¢¸úîº¢Â¢ý ¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸.

¾£÷× :

H

ÜÚ¦ÅÇ¢ S = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTH, TTT } n(S) = 8 A = º¡¢Â¡¸ þÕâì¸û ¸¢¨¼ò¾ø. A = { HTT, THT, TTH } n(A) = 3

P(A) =

P(A) =

H H

T

T

T H

H T

H T H

(F)

(U)

http://doozystudy.blogspot.in/

http://doozystudy.blogspot.in/

T T

http://doozystudy.blogspot.in/

http://doozystudy.blogspot.in/

B = Ì¨Èó¾Ð ´Õ ¾¨Ä ¸¢¨¼ò¾ø. B = { HHH, HHT, HTH, HTT, THH, THT, TTH } n(B) = 7 P(B) = P(B) =

(?)

(U)

A ∩ B = {HTT, {HTT, THT, TTH }

n (A ∩ B) = 3 ;

P (A ∩ B) =

P ( A∪ A∪ B ) = P (A) + P (B) - P ( A∩ A∩B ) P ( A∪ A∪ B ) =

+

-

ST U

=

DY

AÁüÚõB ¬¸¢ÂÉ ´ý¨È¦Â¡ýÚÅ¢Äì¸¡¾ ¿¢¸úîº¢¸û. ±É§Å,

3. þÕ À¸¨¼¸û ´§Ã §¿Ãò¾¢ø ¯Õð¼ôÀÎõ§À¡Ð ¸¢¨¼ìÌõ Ó¸ ±ñ¸Ç¢ý ¦ÀÕì¸üÀÄý ´Õ À¸¡ ±ñ½¡¸ þÕôÀ¾ü¸¡É ¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸.

¾£÷× :

DO OZ

Y

ÜÚ¦ÅÇ¢ S = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1),(2,2),(2,3),(2,4),(2,5),(2,6), (3,1),(3,2),(3,3),(3,4),(3,5),(3,6), (4,1),(4,2),(4,3),(4,4),(4,5),(4,6), (5,1),(5,2),(5,3),(5,4),(5,5),(5,6), (6,1),(6,2),(6,3),(6,4),(6,5),(6,6) } n(S) = 36 A = Ó¸ ±ñ¸Ç¢ý ¦ÀÕì¸üÀÄý ´Õ À¸¡ ±ñ½¡¸ þÕò¾ø. A = {(1,2), {(1,2), (1,3), (1,5), (2,1), (3,1), (5,1) } n(A) = 6 P(A) =

P(A) =

(F)

(U)

=

4. ´Õ À¸¨¼ þÕÓ¨È ¯Õð¼ôÀÎ¸¢ÈÐ. Ì¨Èó¾Ð ´Õ ¯Õð¼Ä¢Ä¡ÅÐ ±ñ 5 ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸.

¾£÷× :

ÜÚ¦ÅÇ¢ S = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) } n(S) = 36

http://doozystudy.blogspot.in/

http://doozystudy.blogspot.in/

http://doozystudy.blogspot.in/

http://doozystudy.blogspot.in/

A = Ó¾ø ¯Õð¼Ä¢ø 5 ¸¢¨¼ò¾ø. A = { (5,1), (5,2), (5,3), (5,4), (5,5), (5,6) (5,6) } n(A) = 6 P(A) = P(A) =

(F)

(U)

P(B) = P(B) =

DY

B = þÃñ¼¡ÅÐ ¯Õð¼Ä¢ø 5 ¸¢¨¼ò¾ø. B = { (1,5), (2,5), (3,5), (4,5), (5,5), (6,5), } n(B) = 6 (?)

(U)

A ∩ B = { ( 5,5 ) }

n (A ∩ B) = 1 ;

ST U

A ÁüÚõB ¬¸¢ÂÉ ´ý¨È¦Â¡ýÚÅ¢Äì¸¡¾ ¿¢¸úîº¢¸û. ±É§Å, P (A ∩ B) =

P ( A∪ A∪ B ) = P (A) + P (B) - P ( A∩ A∩B ) =

+

-

Y

P ( A∪ A∪ B ) =

¾£÷× :

DO OZ

5. ´Õ À¸¨¼ þÕÓ¨È ¯Õð¼ôÀÎ¸¢ÈÐ. Ó¾Ä¡Å¾¡¸ ¯Õð¼ôÀÎõ§À¡Ð ´Õ þÃð¨¼ôÀ¨¼ ±ñ ¸¢¨¼ò¾ø «øÄÐ «ùÅ¢Õ ¯Õð¼Ä¢ø Ó¸ ±ñ¸Ç¢ý ÜÎ¾ø 8 ¬¸ þÕò¾ø ±Ûõ ¿¢¸úîº¢Â¢ý¿¢ ¿¢¸úîº¢Â¢ý¿¢¸ú¾¸Å¢¨Éì ¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸. ÜÚ¦ÅÇ¢ S = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) } n(S) n(S) = 36 A = Ó¾ø ¯Õð¼Ä¢ø þÃð¨¼ôÀ¨¼ ±ñ ¸¢¨¼ò¾ø. A = { (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) } n(A) = 18 P(A) = P(A) =

(F)

(U)

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B = Ó Ó¸ ¸ ±ñ¸Ç¢ý ÜÎ¾ø 8 ¬¸ þÕò¾ø. B = { (2,6 (2,6), (3,5), (4,4 (4,4), (5,3 (5,3), (6,2) (6,2) } n(B) = 5 P(B) = P(B) =

(?)

(U)

A ∩ B = { ( 2,6 ),( 4,4 ), ( 6,2 )} )}

n (A ∩ B) = 3 ;

P (A ∩ B) =

P ( A∪ A∪ B ) = P (A) + P (B) - P ( A∩ A∩B ) P ( A∪ A∪ B ) =

+

-

=

ST U

=

DY

A ÁüÚõB ¬¸¢ÂÉ ´ý¨È¦Â¡ýÚÅ¢Äì¸¡¾ ¿¢¸úîº¢¸û. ±É§Å,

6. þÕ À¸¨¼¸û ´§Ã §¿Ãò¾¢ø §ºÃ ¯Õð¼ôÀÎõ§À¡Ð ¸¢¨¼ìÌõ Ó¸ ±ñ¸Ç¢ý ÜÎ¾ø 3 ¬ø ÁüÚõ 4 ¬ø ÅÌÀ¼¡ÁÄ¢Õì¸ ¿¢¸ú¾¸× ¸¡ñ¸.

¾£÷× :

DO OZ

Y

ÜÚ¦ÅÇ¢ S = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) } n(S) = 36 A = Ó¸ ±ñ¸Ç¢ý ÜÎ¾ø 3 ¬ø ÅÌÀÎõ ±ñ½¡¸ þÕò¾ø. A = { (1,2), (1,5), (2,1), (2,4), (3,3), (3,6), (4,2), (4,5), (5,1), (5,4), (6,3), (6,6) } n(A) = 12 P(A) =

P(A) =

(F)

(U)

B = Ó¸ ±ñ¸Ç¢ý ÜÎ¾ø 4 ¬ø ÅÌÀÎõ ±ñ½¡¸ þÕò¾ø. B = { (1,3), (2,2),(2,6), (2,2),(2,6), (3,1),(3,5), (3,1),(3,5), (4,4), (5,3), (6,2), (6,2), (6,6) } n(B) = 9

P(B) = P(B) =

(?)

(U)

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A ÁüÚõB ¬¸¢ÂÉ ´ý¨È¦Â¡ýÚÅ¢Äì¸¡¾ ¿¢¸úîº¢¸û. ±É§Å, A ∩ B = { ( 6,6 )} )} n (A ∩ B) = 1 ;

P (A ∩ B) =

P ( A∪ A∪ B ) = P (A) + P (B) - P ( A∩ A∩B )

=

-

=

DY

P ( A∪ A∪ B ) =

+

P ( A∪ A∪ B ) = Ó¸ ±ñ¸Ç¢ý ÜÎ¾ø 3 ÁüÚõ 4 ¬ø ÅÌÀ¼¡Áø þÕò¾ø. P ( A∪ A∪ B ) = 1 - P ( A∪ A∪ B ) =

ST U

P ( A∪ A∪ B )

= 1 -

7. ¿ýÌ ¸¨ÄòÐ ¨Åì¸ôÀð¼ 52 º£ðÎ¸¨Çì ¦¸¡ñ¼ º£ðÎì ¸ðÊÄ¢ÕóÐ ºÁÅ¡öôÒÓ¨ÈÂ¢ø ´Õ º£ðÎ ±Îì¸ôÀÎ¸¢ÈÐ. «ó¾î º£ðÎŠ§À¼¡¸§Å¡ ÎŠ§À¼¡¸§Å¡ «øÄÐþÃ¡º¡Å¡¸§Å¡ «øÄÐþÃ¡º¡Å¡¸§Å¡ þÕôÀ¾ü¸¡É ¿¢¸ú¾¸Å¢¨É ¿¢¸ú¾¸Å¢¨Éì Å¢¨Éì ¸¡ñ¸.

¾£÷× :

P(A) =

52 Š§ÀÎ º£ð¼¡¸ þÕò¾ø 13 (F) (U)

DO OZ

P(A) =

Y

ÜÚ¦ÅÇ¢ n(S) = A = n(A) =

B = þÃ¡º¡Å¡¸ þÕò¾ø n(B) = 4

P(B) =

P(B) =

(?) (U)

A ÁüÚõB ¬¸¢ÂÉ ´ý¨È¦Â¡ýÚÅ¢Äì¸¡¾ ¿¢¸úîº¢¸û. ±É§Å, A ∩ B = 1 ; n (A ∩ B) = 1 ;

P (A ∩ B) =

P ( A∪ A∪ B ) = P (A) + P (B) - P ( A∩ A∩B ) =

P ( A∪ A∪ B ) =

+

-

=

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8. ¿ýÌ ¸¨ÄòÐ ¨Åì¸ôÀð¼ 52 º£ðÎ¸¨Çì ¦¸¡ñ¼ º£ðÎì ¸ðÊÄ¢ÕóÐ ºÁÅ¡öôÒÓ¨ÈÂ¢ø ´Õ º£ðÎ ±Îì¸ôÀÎ¸¢ÈÐ. «ó¾î º£ðÎ´Õ Î´Õ þÃ¡º¡ «øÄÐ ´Õ †¡÷ð «øÄÐ ´Õ º¢ÅôÒ ¿¢Èî º£ð¼¡¸ì ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸Å¢¨É ¿¢¸ú¾¸Å¢¨Éì Å¢¨Éì ¸¡ñ¸.

¾£÷× :

P(A) = P(A) =

DY

ÜÚ¦ÅÇ¢ n(S) = 52 A = þÃ¡º¡ º£ð¼¡¸ þÕò¾ø n(A) = 4 (F) (U)

P(B) = P(B) =

ST U

B = †¡÷ð º£ð¼¡¸ þÕò¾ø n(B) = 13 (?)

(U)

C= º¢ÅôÒ º£ð¼¡¸ þÕò¾ø n(C n(C) = 26

P(C) =

(G) (U)

Y

P(C) =

A , B ÁüÚõ C ¬¸¢ÂÉ ´ý¨È¦Â¡ýÚÅ¢Äì¸¡¾ ¿¢¸úîº¢¸û. ¿¢¸úîº¢¸û. ±É§Å, ;

P (A ∩ B) =

n (B ∩ C) = 13 ;

P (A ∩ B) =

n (A ∩ C) = 2

P (A ∩ C) =

DO OZ

n (A ∩ B) = 1

;

n(A∩ n(A∩B∩C) = 1 ;

P(A∩ P(A∩B∩C) =

P ( A∪ A∪ B∪C ) = P (A) + P (B) + P(C) - P ( A∩ A∩B ) - P(B∩ P(B∩C) - P( A∩ A∩C) + P(A∩ P(A∩B∩C) =

P ( A∪ A∪ B∪ C ) =

+

+

-

-

+

=

http://doozystudy.blogspot.in/

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9. ´Õ ¨ÀÂ¢ø 10 ¦Åû¨Ç, 5 ¸ÕôÒ, 3 Àî¨º ÁüÚõ 2 º¢ÅôÒÀóÐ¸û ¯ûÇÉ. ºÁÅ¡öôÒÓ¨ÈÂ¢ø §¾÷ó¦¾Îì¸ôÀÎõ ´Õ ÀóÐ, ¦Åû¨Ç «øÄÐ ¸ÕôÒ «øÄÐÀî¨º ¿¢ÈÁ¡¸ þÕôÀ¾ü¸¡É ¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸.

¾£÷× :

P(A) = P(A) =

(F) (U)

P(B) = P(B) =

ST U

B = ¸ÕôÒ ¿¢È Àó¾¡¸ þÕò¾ø þÕò¾ø n(B) = 5

DY

ÜÚ¦ÅÇ¢ n(S) = 10 + 5 + 3 + 2 = 20 A = ¦Åû¨Ç ¿¢È Àó¾¡¸ þÕò¾ø n(A) = 10

(?) (U)

C= Àî¨º ¿¢È Àó¾¡¸ þÕò¾ø n(C n(C) = 3

P(C) =

(G) (U)

Y

P(C) =

DO OZ

A , B ÁüÚõ C ¬¸¢ÂÉ ´ý¨È¦Â¡ýÚÅ¢ÄìÌõ ¿¢¸úîº¢¸û. P ( A∪ A∪ B ∪ C) = P (A) + P (B) + P ( C )

=

P ( A∪ A∪ B ) =

+

+

=

10. ´Õ ¨ÀÂ¢ø 10 ¦Åû¨Ç, 6 º¢ÅôÒÁüÚõ 10 ¸ÕôÒ ¿¢ÈôÀóÐ¸û ¯ûÇÉ. ºÁÅ¡öôÒÓ¨ÈÂ¢ø ´Õ Àó¾¢¨É ±ÎìÌõ§À¡Ð «Ð ¦Åû¨Ç «øÄÐ º¢ÅôÒ ¿¢Èô Àó¾¡¸ þÕôÀ¾ü¸¡É ¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸.

¾£÷× :

ÜÚ¦ÅÇ¢ n(S) = 10 + 6 + 10 = 26 A = ¦Åû¨Ç ¿¢È Àó¾¡¸ þÕò¾ø n(A) = 10 P(A) = P(A) =

(F) (U)

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B = º¢ÅôÒ ¿¢È Àó¾¡¸ þÕò¾ø þÕò¾ø n(B) = 6 (?)

P(B) = P(B) =

(U)

= P ( A∪ A∪ B ) =

+

=

DY

A ÁüÚõB ¬¸¢ÂÉ ´ý¨È¦Â¡ýÚÅ¢ÄìÌõ ¿¢¸úîº¢¸û. P ( A∪ A∪ B ) = P (A) + P (B)

11.

ST U

´Õ Á¡½Å¢ìÌ ÁÕòÐÅì ¸øæ¡¢Â¢ø §º÷ì¨¸ ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× 0.16 ±ý¸. ¦À¡È¢Â¢Âø ¸øæ¡¢Â¢ø ¸øæ¡¢Â¢ø §º÷ì¨¸ ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× 0.24 ÁüÚõþÕ ¸øæ¡¢¸Ç¢Öõ §º÷ì¨¸ ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× 0.11 ±É¢ø, (i) ÁÕòÐÅõÁüÚõ ¦À¡È¢Â¢Âø ¸øæ¡¢¸Ç¢ø ²§¾Ûõ ´Õ ¸øæ¡¢Â¢ø §º÷ì¨¸ ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× ¸¡ñ¸. (ii) ÁÕòÐÅì ¸øæ¡¢Â¢øÁðÎ§Á¡ «øÄÐ ¦À¡È¢Â¢Âø ¦À¡È¢Â¢Âø ¸øæ¡¢Â¢øÁðÎ§Á¡ §º÷ì¨¸ §º÷ì¨¸ ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× ¸¡ñ¸.

¾£÷× :

ÁÕòÐÅì ¸øæ¡¢ §º÷ì¨¸ ¿¢¸ú¾¸× P ( A)

= 0.16

¦À¡È¢Â¢Âø ¸øæ¡¢ §º¡¢ì¨¸ ¿¢¸ú¾¸× P (B)

= 0.24

Y

þÕ ¸øæ¡¢¸Ç¢Öõ §º÷ì¨¸ ¿¢¸ú¾¸× P ( A∩ A∩B ) = 0.11 ²¾¡ÅÐ ´Õ ¸øæ¡¢Â¢ø §º÷ì¨¸ ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× : P ( A∪ A∪ B ) = P (A) + P (B) P ( A∩ A∩B ) = 0.16 + 0.24 0.11 P ( A∪ A∪ B ) = 0.29

(ii)

²¾¡ÅÐ ´Õ¸øæ¡¢Â¢øÁðÎõ §º÷ì¨¸ ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× : P ( AÁðÎõ «øÄÐ B ÁðÎõ ) P ( A ∩ B) + P ( A ∩ B) = [ P ( A ) - P ( A∩ A∩B ) ] + [P ( B ) - P ( A∩ A∩B ) ] = [ 0.16 - 0.11 ] + [ 0.24 - 0.11 ] = 0.05 + 0.13

DO OZ

(i)

P ( A∩ A∩ B ) + P ( A ∩ B ) = 0.18

12.

´Õ Ò¾¢Â Á¸¢ú×óÐ «¾Û¨¼Â ÅÊÅ¨ÁôÀ¢ü¸¡É Å¢ÕÐ ¦ÀÚõ ¿¢¸ú¾¸× 0.25 ±ý¸. º¢Èó¾ Ó¨ÈÂ¢ø ±¡¢¦À¡ÕûÀÂýÀ¡ðÊü¸¡É Å¢ÕÐ ¦ÀÚõ ¿¢¸ú¾¸× 0.35 ÁüÚõþÕÅ¢ÕÐ¸Ùõ ¦ÀÚÅ¾ü¸¡É ¿¢¸ú¾¸× 0.15 ±É¢ø, «õÁ¸¢ú×óÐ (i) Ì¨Èó¾Ð ²¾¡ÅÐ ´Õ Å¢ÕÐ ¦ÀÚ¾ø. (ii) (ii) ´§Ã ´Õ Å¢ÕÐ ÁðÎõ ¦ÀÚ¾ø ¬¸¢Â ¿¢¸úîº¢¸Ùì¸¡É ¿¢¸ú¾¸Å¢¨Éì ¸¡ñ¸.

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¾£÷× : ÅÊÅ¨ÁôÀ¢ü¸¡É Å¢ÕÐ ¦ÀÚõ ¿¢¸ú¾¸× P ( A) = 0.25 ±¡¢¦À¡ÕûÀÂýÀ¡ðÊü¸¡É Å¢ÕÐ ¦ÀÚõ ¿¢¸ú¾¸× P (B) = 0.35 þÕÅ¢ÕÐ¸Ùõ ¦ÀÚÅ¾ü¸¡É ¿¢¸ú¾¸× P ( A∩ A∩B ) = 0.15 ²¾¡ÅÐ ´Õ Å¢ÕÐ ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× : P ( A∪ A∪ B ) = P (A) + P (B) P ( A∩ A∩B ) = 0.25 + 0.35 0.15 P ( A∪ A∪ B ) = 0.45

(ii)

²¾¡ÅÐ ´Õ Å¢ÕÐ ÁðÎõ ¸¢¨¼ôÀ¾ü¸¡É ¿¢¸ú¾¸× : P ( AÁðÎõ «øÄÐ B ÁðÎõ ) P ( A ∩ B) + P ( A ∩ B) = [ P ( A ) - P ( A∩ A∩B ) ] + [P ( B ) - P ( A∩ A∩B ) ] = [ 0.25 - 0.15 ] + [ 0.35 - 0.15 ] = 0.10 + 0.20

ST U

DY

(i)

P ( A∩ A∩ B ) + P ( A ∩ B ) = 0.30

,

13. A, B, C ¬¸¢§Â¡÷´Õ ¬¸¢§Â¡÷´Õ Å¢É¡Å¢üÌò ¾£÷× ¸¡ñÀ¾ü¸¡É ¿¢¸ú¾¸×¸û Ó¨È§Â

ÁüÚõ BþÕÅÕõ §º÷óÐ ¾£÷× ¸¡ñÀ¾ü¸¡É ¿¢¸ú¾¸×

¾£÷× :

P (A) =

P( B ∩ C ) =

, ãÅÕõ §º÷óÐ

, ±É¢ø Â¡§ÃÛõ ´ÕÅ÷ «ùÅ¢É¡Å¢ý ¾£÷× ¸¡ñÀ¾ü¸¡É¿¢¸ú¾¸Å¢¨Éì

DO OZ

¸¡ñ¸.

, ±ý¸. A

. B ÁüÚõCþÕÅÕõ ÁüÚõCþÕÅÕõ §º÷óÐ ¾£÷×

. A ÁüÚõ C þÕÅÕõ §º÷óÐ ¾£÷× ¸¡½ ¿¢¸ú¾¸×

¸¡ñÀ¾ü¸¡É ¿¢¸ú¾¸× ¾£÷× ¸¡½ ¿¢¸ú¾¸×

Y

,

P (B) =

,

P (A∩C) =

,

P(C) =

P( A ∩B) =

,

P (A∩B∩C) =

,

,

P ( A∪ A∪ B∪C ) = P (A) + P (B) + P(C) - P ( A∩ A∩B ) - P(B∩ P(B∩C) - P( A∩ A∩C) + P(A∩ P(A∩B∩C) =

=

= P ( A∪ A∪ B ∪ C ) =

+

+

-

-

-

+

( Á£.º¢.Á = 105)

N A N A N 9 N 9 N 9 N A N

A A 9 – 9 A

=

9

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