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4

4

-

. 7

GAT PAT O-NET

...

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(O-Net + PAT 1 + ) . , .

3

8 13

16

. ( We By The Brain)

23

35

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3

0, , , , , , 3 2

0 3 2

sin 0

1 2 3 1 0 - 1 0

cos 1

3 2 1 0 - 1 0 1

tan 0 1 1 3 0 0

cot 3 1 1 0 0

sec 1 2 2 2 - 1 1

csc 2 2 2 1 - 1

2

3

3

6 4 3 2 2

030 45 60 90 270180

3606 4 3 2

2

2

2

2

3

3

2

2

1. sin . csc = 1 5. cot = cos , sin 0 2. cos . sec = 1

3. tan . cot = 1 6. cos2+ sin2= 1 4. tan = sin , cos 0 7. 1 + tan

2= sec

2

8. cot2+ 1 = csc2cos

sin

. , . ,

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4

sin (A + B) = sin A cos B + cos A sin B sin (A - B) = sin A cos B - cos A sin B cos (A + B) = cos A cos B - sin A sin B cos (A - B) = cos A cos B + sin A sin B

tan (A + B) = tan A + tan B tan (A - B) = tan A - tan B

sin 2A = 2 sin A cos A cos 2A = cos2A - sin2A = 2 cos2A - 1 = 1 - 2 sin2A tan 2A = 2 tan A

sin 3A = 3 sin A - 4 sin3A

cos 3A = 4 cos3A - 3 cos A tan 3A = 3 tan A - tan

3A

sinA = 1 - cos A

cosA = 1 + cos A

tanA = 1 - cos A

2 sin A cos B = sin (A + B) + sin (A - B) 2 cos A sin B = sin (A + B) - sin (A - B) 2 cos A cos B = cos (A + B) + cos (A - B) 2 sin A sin B = cos (A - B) - cos (A + B) sin A + sin B = 2 sinA + B cosA - B

sin A - sin B = 2 cosA + B sinA - B

cos A + cos B = 2 cosA + B

cos

A - B

cos A - cos B = - 2 sinA + B sinA - B

1 - tan A tan B

1 - tan2A

1 - 3 tan2A

1 + tan A tan B

2

2

2

2

1 + cos A

2

2

2

2

2

2

2

2

2

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5

1. sin (arcsin x) = x, - 1 x 1 cos (arccos x) = x, - 1 x 1 tan (arctan x) = x, x R 2. arcsin (sin x) = x, -x

arccos (cos x) = x, 0 x arctan (tan x) = x, -< x <

3. arcsin (- x) = - arcsin x arccos (- x) = - arccos x arctan (- x) = - arctan x

ABC A + B + C = 180 A + B = 180- C

sin (A + B) = sin C A + B = 90-C

sinA + B = sin (90- C) = cosC

1. ABC = 1 ab sin C = 1 bc sin A = 1 ca sin B

2. : a = b = c

3. :a2 = b2+ c2- 2bc cos A

b

2

= c

2

+ a

2

- 2ca cos B c2 = a2+ b2- 2ab cos C 4. a = b cos C + c cos B b = c cos A + a cos C c = a cos B + b cos A

1. sin A - cos A = 0 csc A

2. 1

3. cos - sin = 5 sin 2

2

2

2

2 2

2

3

2

2 2

2

2

sin A sin B sin C

2

2

2

2

2

1 - xy

1 + xy

4. arcsin x + arccos x =

arctan x + arccot x =

arcsec x + arccsc x =

5. arctan x + arctan y = arctan x + y

arctan x - arctan y = arctan x - y

BA

C

2 sin x + 3 cos x

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6

1. ABC C sin B = 12 tan(B - A)

1. 1 2. 119 3. 13 4. 169

2. ABC B cosec (B - C) = 3 cot A + cot B + cot C

1. 5 2. 2 5 3. 9 5 4. 1

3. 1> 0,

2> 0

1+

2p sin

1= 3 cos

1= 5 sin (

1+

2)

1. 43 2. 54 3. 63 4. 64

4. tan 15+ tan 75 1. 1 2. 2 3. 3 4. 4

5. [sin (2 arccos 3 )] [cos (2 arctan 2)]

1. 96 2. 72 3. - 24 4. - 72

6. cos (arcsin (- 5))

1. 12 2. - 12 3. 5 4. - 5

13

12

59

2 5 13

65 6565 65

12

4. 1sin 10sin 50sin 70

5.

sin (arctan 2 + arctan 3)

6. cos (arctan 7 + arcsin 4 )

7. sec [ 1(arcsin 3+ arccos 3 )] + tan [ 1(arcsin 4+ arccos 4 )]

2 2

2

5

120 60

55

5

5

8

5

125 125 12525

13

13 13 13 13

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7. () arccos (- 1) - arcsin 1 = arctan (- 1)

() arctan 3 - arccos 3= arcsin 1

1. () () 2. () () 3. () () 4. () ()

8. arctan x = 2 arctan 1- arctan 1 cos (270- arctan x)

1. - 13 2. - 9 3. 9 4. 13

9. ABC a = 16 , b = 10 c = 14 a

1. 3 2. 3 3 3. 5 3 4. 7 3

10. ABC C tan B = 2.4 D E AC BC AB DE 13 12 DE AB AD

1. 1 2. 5 3. 5 4. 12

2

2

5 10 5 10 5 10 5 10

2

2 3

13 12 13

1. 2 2. 3 3. 3 4. 4 5. 4

6. 1 7. 3 8. 2 9. 3 10. 1

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8

{1, 2, 3, , n} a1, a

2, a

3,

1. n + 1 n n a

n + 1- a

n d

a

1 d

, a - 3d, a - d, a + d, a + 3d,

, a - 2d, a - d, a, a + d, a + 2d,

2. n + 1 n n an + 1 r

a1, a

1+ d, a

1+ 2d, a

1+ 3d, , a

1+ (n - 1) d,

n an= a

1+ (n - 1) d

a1, a

1r, a

1r2, a

1r3, , a

1rn - 1,

n an= a

1rn - 1

an

an b

n lim a

n= A lim b

n= B

1. lim c = c, c

2. lim kan= kA, k

3. lim (anb

n) = A B

4. lim (anb

n) = AB

5. bn0 n B 0 lim an = A

6. an0 n m > 1 lim a

n = A

7. k 7.1 lim nk 7.2 lim 1 = 0

n

n

n

n

n

n

n

n n

n

bn

nk

B

m m

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9

a

1, a

2, a

3, , a

k

an= a

1+ a

2+ a

3+ ... + a

kn= 1

k

a1, a

2, a

3, , a

k,

an= a

1+ a

2+ a

3+ ... + a

k+ ...

S

n= a

1+ a

2+ a

3+ ... + a

n n

n

Sn =n(a1+ an)

Sn = n[2a

1+ (n - 1) d]

n S

n =

a1- ra

n, r 1

Sn =

a1(1 - rn), r 1

a

1

, a2

, a3

, S

1 = a

1

S2 = a

1+ a

2

Sn = a

1+ a

2+ a

3+ ... + a

n

Sn n S

1, S

2, S

3,

1. c c = ck

2. (anb

n) = a

n b

n

3. (can) = c a

n

4. 1 + 2 + 3 + ... + n = n(n + 1)

5. 12+ 22+ 32+ ... + n2 = n(n + 1) (2n + 1)

6. 13+ 23+ 33+ ... + n3 = n2(n + 1)2 = (1 + 2 + 3 + ... + n)2

n= 1

n= 1

n= 1

n= 1 n= 1

n= 1 n= 1

k

k

k k

k k

2

6

4

2

2

1 - r

1 - r

.

.

.

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10

1. S

n lim S

n= S S

S

2. Sn lim S

n

1. : a

n= a

1+ (n - 1) d

a1= d = 0

2. : a

n= a

1rn - 1

|r| 1 |r| < 1 S = a1

3. 1 = 1 + 1 + 1 + ... p

p > 1 p 1

n

n

1. an lim a

n= 0

lim an= 0 a

n

2. lim an0 an

3. an b

n (a

n+ b

n)

4. an b

n (a

n+ b

n)

1 - r

n

n

n

n= 1

np 2p 3p

n= 1

n= 1

n= 1

n= 1

n= 1 n= 1 n= 1

n= 1 n= 1

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11

1. a1, a

2, a

3, ... a

2+ a

3+ ... + a

9= 100 S

10= a

1+ a

2+ ... + a

10

2. {100, 101, 102, ..., 600} 6 12

3. 1 + 6 + 11 + 16 + ... + 101

4. 10 2, 5, 10, 17, 26, ...

5. U = {1, 2, 3, ..., 500} 3

6. U = {200, 201, 202, ..., 2543} 5

7. 1 + 3 + 5 + 7 + ... + 209

8. 2 + 6 +18 + 54 + ... + 4347

9. 2 + 6 + 18 + 54 + 162 + 468

10. 1 + 1 + 1 + L + 1

8 .9 9 .10 10 .11 119 .120

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12

11. n 7 + 77 + 777 + 7777 + ...

12.

12

- 22+ 3

2

- 42

+ ... + (2n - 1)2

- (2n)2

1. an

an + 2= 2 n

an= 31 a

n

1. 21275- 1 2. 21276- 1 3. 22551- 1 4. 22552- 1

2. 4, a, 9, b, c 5 a > 0 a + 8c1. 156 2. 160 3. 164 4. 168

3. k x3- 3x2- kx + 3 = 0 1. 1 2. 2 3. 3 4. 4

4. x + 2, 9, x + 10 3 d xdn - 1

1. 410- 1 2. 49- 1 3. 3 (410) - 1 4. 3 (4

9) - 1

5. 2,000 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, ... 1. 10 2. 11 3. 12 4. 13

6. lim (3n + 1) - n2- 4n

7. an= n

3

- n2- 2

8. an=

1 + 2 + 3 + ... + n ( 3)

1. 121 2. 120 3. 111 4. 110

9. 4 + 1+ 2+ 3+ ... + n+ ...

1. 5 2. 6 3. 7 4. 8

an

3

n - 8n3+ 6n

2n2- 1

an

12+ 22+ 32+ ... + n2

2n + 1

3

n= 1

n= 1

n= 1

n= 1

10

10

10

2552

n 3

2 22 23 2n

10. n + 2 - n - 1n= 1

n2+ 3n + 2

1. 2 2. 4 3. 1 4. 1 5. 2 6. 2 7. 0.25 8. 2 9. 2 10. 2

2

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13

R

Q Q

I Q - I

I- {0} N

: p(x) = anxn+ a

n - 1xn - 1+ ... + a

1x + a

0

an, a

n - 1, , a

1, a

0 a

n0 n

p(x) x - c p(c)

: p(x) = anxn+ a

n - 1xn - 1+ ... + a

1x + a

0

an, a

n - 1, , a

1, a

0 a

n0 n

x - c p(x) p(c) = 0

a1

a2

an

an - 1

a1 a2 an - 2 anan - 1

+ +-

1 2 (x - a

1) (x - a

2) ... (x - a

n -1) (x - a

n) a

1< a

2< < a

n

3 x (x - a1) (x - a

2) ... (x - a

n - 1) (x - a

n) = 0

x = a1, a

2, , a

n - 1, a

n

4 x

5 +

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14

: a b 1. |a| 0 2. |a| = |- a| 3. |a2| = |a|2= a2 4. a2 = |a| 5. |ab| = |a||b| 6. a = |a|, b 0

7. |a - b| = |b - a| 8. |a + b| |a| + |b| 9. |a - b| ||a| - |b|| |a| - |b| 10. |a| = |b| a = b 11. a 12. a 11.1 |x| < a - a < x < a 12.1 |x| > a x < - a x > a

11.2 |x| a - a x a 12.2 |x| a x - a x a

6 < 0 > 0 =

a |a|

|a| = a, a 0

- a, a < 0

b |b|

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15

+

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16

+ >

1. 3 2. 3 3. 1 4. 6 5. 1

6. 4 7. 1 8. 9. 3 10. 193

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18

( ) ( )

( )

( )

( ) ( )

( ) ( )

+

+ +

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1. 720 2. 2. 3. 18 4. 135 5. 3 6. 6 7. 3 8. 5 9. 0.14 10. 4 11. 4 12. 2

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(O-Net + PAT 1 + )

.

By P'Golf We By The Brain

1. (a + b)3 = a3 + 3a2b+ 3ab2 + b3

2. (a b)3 = a3 3a2b+ 3ab2 b3

3. a3 + b3 = (a + b)(a2 ab + b2)

4. a3 b3 = (a b)(a2 +ab + b2)

a, b, am, an, bn a, b 0 m, n

1. am an = am + n

2. am

an = am n

3. (am)n = amn

4. (ab)n = anbn

5. a

b

n

= an

bn b 0

6. an = 1an

a 0

7. a1n = n a

8. a0 = 1 a 0

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24

Exponential Function

f = {(x, y) R R+ / y = ax ; a> 0 a 1}

1. 1

2. R

3. R+

4.

5. , a> 1 0< a< 1

Logarithmic Function

g = {(x, y) R+ R / y = logax ; a> 0 a 1}

1. 1

2. R+

3. R

4.

5. , a> 1 0 < a< 1

log

1. 6.loga1 = 0 logbna = 1

n logba

2. 7.logaa = 1 alogam = m

3. 8.loga(mn) = logam + logan alogbm = mlogba

4. 9. loga mn = logam logan logba =

log

m

a

logmb m> 0 m1

5. 10.logbam = m logba logba = 1

logab

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Problems

1. A = 7 3 5 , B = 5 3 7 , C = 3 5 7 D = 3 7 5

1. D > C > A > B 2. A > C > B > D

3. A > B > D > C 4. C > A > D > B

2. a = 248 , b = 336 c = 524

1. 2.1b> 1c>

1a

1a>

1

b> 1c

3. 4.1b> 1a>

1c

1a>

1c>

1

b

3. a = 7 + 4 3 , b = 2 2 2 2... c = 2 + 3

1. 2.1c> 1a> 1b

1c>

1

b> 1a

3. 4.1b> 1a>

1c

1

b> 1c>

1a

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4. x, y z x + y+ z = 16

xyz yx+ z = x2(x+ z) 3y = 3(9z)

5. R

C = xR (3x2 11x + 7)(3x2 + 4x+ 1) = 1

C

6. x2 + 5x +5 (x5) = 1

1. 2. 3. 0 4.5 52

5

2

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7. A 31+ 2x + 92x = 244

A

1. 2. 3. 4.(1,4) (2,0.5) (0,5) (3,0)

8. x

(2x 4)3 + (4x 2)3 = (4x + 2x 6)3

1. 2 2. 2.5 3. 3 4. 3.5

9. A

5(1+ x2 4x1) +5

( 5+ 4xx2

2+ x2 4x1)

= 126

A

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10. x, y, z 81x = 27

1y = 36

1z

x+yz

11. A x [0, 2)

2(1 + 3sinx) 5 22sinx + 2(2 + sinx) = 1 A

12. . (log2557)log2557

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13. (log2x)3 + 12

logx2 = 7(log 1

2

x)2

1. 8 2. 16 3. 24 4. 25

14. Suppose the number a satisfiesaloga +log (loga) log (log (log2) log (loga)) = 0

What is the value of aaaa

1. 1 2. 2 3. 4 4. 85. 16

15. x y x + y 5(x2A)2yA = (16)64 A = logy

logx

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16. log2x+6logx2 5 = 0

1. 8 2. 10 3. 12 4. 145. 16

17. If x and y are real numbers such that 2log (2y 3x) = logx+ log y , find xy

18. A log3(3(2x2 + 2x) +9) = x2 + x + 1

log3

B B = {x2 xA}

19. If and then the value ofx = 10 + 2

2 y = 10 2

2 , log2(x2

+ xy + y2

)is equal to1. 0 2. 2 3. 3 4. 4

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31

20. a b

log(1 + a2) loga 2log2 = 1 log(100+ b2) +log b 2ab

21. R A = {xR log

3(x 1) log 3

3(x 1) = 1}

B = {xR x +1 + x 1 = 2}

A B

22. A log( x+1 + 5) = logx B log2(3x) +log4(9x) + log8(27x) = 3 + 2log64(x)

A B

1. 2. 3. 4.129169

329

969

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23. 2x 2x+1 2x+ 2 = 4x + 4x+ 1 + 4x+ 2

1. 2. 3. 4.log221

10 log2

21

8 log2

21

6 log2

21

4

5. log2212

24. Positive number x, y and z satisfy xyz= 1081

and (log10x)(log10yz) + (log10y)(log10z) = 468

Find (log10x)2 + (log10y)2 + (log10z)2

25. A log6(3 4x +2 9x) = x + log65

B x + 1 x2 = 1+ 2x 1 x2

A B

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26. xy = 256 x + y logyx+ logxy = 10

3

27. R

A

3

5

(5x2 23x+ 3)>

5

3

(x + 5)

A

1. 2.{xR (5x 1)(x 3)

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29. 2log(y x) =log(2y) + log(y x 1)

1. 2.

3. 4.

5.

y

x

2

y

x

y

x

2

y

x

6

4

2

y

x

6

4

2

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3. A B det(A) = 23 3

x y B =

1 3 2

0 1 x0 2 y

AB + 3A = 2I I 3 3 x + y 1. 0 2. 3. 4.1 2 2.5

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4. x det A = 3A =

2x 1

x x

B I 2 2 BA + BA1 = 2I

det B 2 21. [1, 2] 2. 3. [0, 1] 4.[1, 0] [2, 1]

5. A B 2 2

2A B =

4 45 6

A 2B =

5 84 0

det(A4B1)

6. A =

3 1 0

2 4 35 4 2

B =

1 1 0

0 2 11 2 3

(A1B) C(BtA)

t = [det(2A)]I C1

1. 2.

1 12

3

2

1

2 9

2 2

3

2 2 5

28

1

8

3

8

1

8 9

8 4

8

3

8 4

8 10

8

3. 4.

1 12

32

12

9

2 2

32

2 5

2

8 1

8 3

8

18

9

8

4

8

38

4

8

10

8

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7. S x

4 2 7x 1 32 0 x

y S A=

y 1

1 y

det

A

t

1

t

1

8. a A =

a 1 a1 + a a

I =

1 0

0 1

det(A 2 I)(A 3 I)(A 5 I)(A 7 I)

1. 2.48 13a (a 2)(a 3)(a 5)(a 7)3. 17a 4. 17 5. 48

9. a, b, c, d, x y

A =

1 x

y 1

, B =

a b

c d

, C =

1 0

0 1

I =

1 0

0 1

AB = 2CA2 = I

det(B1)1. 0.25 2. 0.5 3. 2 4. 4

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10. 3 1 A =

1 2 4

3 8 01 2 1

A1

11. A =

1 20 1

, I =

1 0

0 1

B 2 2 x det(A2 + xI) = 0() det(A +xI) = 0

() det(A2 + xI B) = det(Bt)1. () () 2. () () 3. () () 4. () ()

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12. S = {3, 2, 1,1,2,3}

M=

a1 a2 a3

0 a4 a5

0 0 a6

ai S, 1i6

M 1 27 271. 2. 3. 4. 5.2

634

636

638

6310

63

13.

. A =

100 sin2a cos2a

200 2sin2b 2cos2b

300 3sin2c 3cos2c

det(A) = 0

. A = 2 2

3 2

det(3A4(A1)t(A At) ) = 72

1. . . 2. . . 3. . . 4. . .

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14. x A =

2 x 1

1 0 11 x 2 2 x

C22(A) = 14 det (adjA)

15. A 3 3 det (A) 0() (det(A))3 = det(adj(A))

() A2 = 2A det(A) = 21. () () 2. () () 3. () () 4. () ()

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17. A B det(A) 0 det(B) 0 det(A1 + B1) 0 det(A +B) 0

(A + B)1

1. 2.B1(A1 + B1)A1 B1(A1 + B1)1A1

3. 4.B(A1 +B1)A B(A1 +B1)1A

16. A i = 1, 2, 33 3 AXi = Bi

X1 =

1

0

5

, X2 =

1

2

5

, X3 =

1

3

1

B1 =

1

0

0

, B2 =

0

1

0

, B3 =

0

0

1

det(A)

1. 2. 3. 4. 1 5. 88 18

1

8

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18. a b A =

1 a

b 4 , I =

1 0

0 1

A ab 0 2(A I)1 = 4I A

() ab = 2() det(3A2AtA1) = 324

1. () () 2. () () 3. () () 4. () ()

19. x, y, z

= ax 2y +3z = bx 3y

= c2x 5y +5z

1 2 31 3 02 5 5

a

b

c

1 2 30 1 3

0 0 1

9

5

2

c

5/19/2018 Math Mama57

48/48

48

20. x, y, z 2x 2y z = 1

x 3y+ z = 7

x + y z = 5

1x+ 2y+ 3z1. 0 2. 2 3. 5 4. 8

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