1

8 B

1

1

1 1

1

1 1 2

= 180

1=180

144=180144 = 4

5

2312

=2312180= 345

2 sin+ 5 - 2 cos+ 30= 1 2 x =+

5 = x -

5

2 sin x - 2 cos {x - 5 + 30 } = 1

2 sin x - 2 cosx - 6 = 1

2 cosx - 6 = 2cos x cos 6 + sin x sin 6 = 2 32 cos x + 12 sin x = 3 cos x + sin x

3601

1 1

180

1

2

2 sin x - ( 3 cos x + sin x) = 1

sin x - 3 cos x = 1

sin x - 3 cos x = 2sin x 12 - cos x 32 = 2sin x cos 3 - cos x sin 3 = 2 sinx - 3

sinx - 3 = 12

2

+5+

5 +

5

7

10 x 6

5

7

10-

3 x -

3 6

5-

3

1130

x -3 13

15

x

x -3

=56x = 7

6

+5

=76

= 76

-5

=2930

y

x

y =

1

1- 1

- 1

O

12

12

1130

1315

3

2

x log3 x xc 3 x 0c 03

log3 xlog3 x log3 xc 3

X 0 plog3 Xp = p log3 X

X 0Y 0log3

XY

= log3 X - log3 Y

(log3 x)(log3 x) 3 log3xc

(log3 x)2 3 (log3 x - log3 c)

t = log3 x

t2 3 (t - log3 c)

t 2 - 3t + 3 log3 c 0

c = 3 9 = 913 = (32)

13 = 3

23

t2 - 3t + 3 log3 323 0

t2 - 3t + 3 23

log3 3 0

log3 3 = 1

t2 - 3t + 2 0(t - 1) (t - 2) 0

t 1t 2

x ( 0)

log3 x 12 log3 x

3 log3 x 3132 3 log3 x

x 0

0 x 3x 9

4

x x 0

xtt = log3 x

t 2

x 0 c

t

f (t) = t2 - 3t + 3 log3 c

f (t) 0 c

f (t) =t - 32 2 - 94 + 3 log3 c f (t) f 32 = 3 log3 c - 94 c 3 log3 c -

94 0log3 c

34

3 log3 c 334 = (33)

14c 4 27

x

t

t = log3 x

1

31O

5

2

1

f (x) = px2 + qx + r

C A (1,1) ly = 2x - 1

2

f'(x) = 2px + qf'(1) = 2

2p + q = 2

q = - 2p + 2

C A (1,1)f (1) = 1

p + q + r = 1

p + (- 2p + 2) + r = 1r = p - 1

f (x) = px2 + (- 2p + 2) x + p - 1

S 1

S =v

1{px2 + (- 2p + 2)x + p - 1 - (2x - 1)} dx

=v

1(px2 - 2px + p) dx

=p3

x3 - px2 + pxv

1

= p3 v3 - pv2 + pv- p3 - p + p =

p3

v3 - pv2 + pv -p3

=p3

(v3 - 3v2 + 3v - 1)

T 1

2v - 1 0

T =1 + (2v - 1)

2(v - 1)

= v (v - 1)

= v 2 - v

U = S - T

=p3

(v3 - 3v2 + 3v - 1) - (v2 - v)

1

1 v

l

O

y

y = f(x)

x

1

1 vx

1

2v - 1

2

6

=p3

(v3 - 3v2 + 3v - 1) - v2 + v = g (v)

g'(v) =p3

(3v2 - 6v + 3) - 2v + 1 = p (v2 - 2v + 1) - 2v + 1

g (v) v = 2g'(2) = 0

p (22 - 22 + 1) - 22 + 1 = 0p = 3

g (v) = (v3 - 3v2 + 3v - 1) - v2 + v

= v3 - 4v2 + 4v - 1

g'(v) = 3v2 - 8v + 4

= (3v - 2) (v - 2)

g (v)

v 23 2

g'(v) + 0 - 0 +

g ( v)

v = 2

p = 3

U = 0g (v) = 0 v

v3 - 4v2 + 4v - 1 = 0

(v - 1) (v2 - 3v + 1) = 0

v = 13 5

2 v0 1

v0 =3 + 5

2

g (v) (v 1)

5 1 v0 = 3 + 52 3 + 12 = 2

U = g (v) 1 v v0 3

p = 3v 1 U = g (v)

g (2) = 23 - 422 + 42 - 1 = - 1

v (1) 2 v0 g'(v) - 0 + +

g ( v) (0) 0

v

g'(v)

12

+

23

-

7

2

F (x) f (x)

F'(x) = f (x) 7

W

W =t

1{0 - f (x)} dx

= -t

1f (x) dx

= - F (x)t

1

= - (F (t) - F (1))

= - F (t) + F (1) 4

W

h

h = (t2 + 1)2 - (t2 - 1)2

= (t4 + 2t2 + 1) - (t4 - 2t2 + 1)

= 4t2

= 2t t 1

W =12(2t2 - 2)2t

= 2t3 - 2t

- F (t) + F (1) = 2t3 - 2t

tF'(t) = f (t)

- f (t) = 6t2 - 2f (t) = - 6t 2 + 2

apqp q

p

q(x - a)2dx = 1

3(x - a)3

p

q

=13

(p - a)3 -13

(q - a)3

y = (x - a)2 x - a

x

W

t1

y =f(x)

t2 - 1

t2 + 1

2t2 - 2

h

y = x2 y = (x - a)2y

xq -a

-a

p -a pqaO

8

p

q(x - a)2dx

p - a

q - a x2dx

p - a

q - a x2dx = 1

3x3

p - a

q - a

= 13

(p - a)3 - 13

(q - a)3

S =v

1p (x2 - 2x + 1) dx

=v

1p (x - 1)2dx

=p3

(x - 1)3v

1

=p3

(v - 1)3

g (v) = p3

(v - 1)3 - v2 + v

g'(v) = p (v - 1)2 - 2v + 1 g'(2) = p12 - 22 + 1 = 0 p = 3

g (v) = (v - 1)3 - v (v - 1) = (v - 1) {(v - 1)2 - v } = (v - 1) (v2 - 3v + 1)

g (v) = 0 v = 1 v2 - 3v + 1 = 0

y = g (v) v

v = 1 3 52

1 v v0 g (v) 0

y

v1

O 3 + 523 - 5

2

y =g(v)

9

3

{an} d

a4 = a1 + 3d = 30

a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8

= a1 + (a1 + d) + (a1 + 2d) ++ (a1 + 7d)

= 8a1 + (1 + 2 ++ 7) d

= 8a1 + 28d

8a1 + 28d = 288

a1 = - 6d = 12

{an}

- 6 12

an = - 6 + 12 (n - 1) = 12n - 18

Sn =a1 + an

2n

=- 6 + (12n - 18)

2n

= (6n - 12)n

= 6n2 - 12n

{bn} r

b2 = b1 r = 36

b1 + b2 + b3 = b1 + b1 r + b1 r2

= b1 (1 + r + r2) = 156

r

1 + r + r2=

36156

=3

13

3 (1 + r + r2) = 13r

3r2 - 10r + 3 = 0

(3r - 1) (r - 3) = 0

r 1r = 3

10

b1 =1336 = 12

{bn}

12 3

bn = 123n - 1

Tn =n

k = 1123k - 1 = 12 3

n - 13 - 1

= 6 (3n - 1)

cn =n

k = 1(n - k + 1) (ak - bk)

cn + 1 =n + 1

k = 1{(n + 1) - k + 1} (ak - bk)

dn = cn + 1 - cn

=n + 1

k = 1{(n + 1) - k + 1} (ak - bk) -

n

k = 1(n - k + 1) (ak - bk)

= {(n + 1) - (n + 1) + 1} (an + 1 - bn + 1)

+n

k = 1{(n + 1) - k + 1} (ak - bk) -

n

k = 1(n - k + 1) (ak - bk)

= an + 1 - bn + 1 +n

k = 1{(n + 1) - k + 1 - (n - k + 1)} (ak - bk)

= an + 1 - bn + 1 +n

k = 1(ak - bk)

dn =n

k = 1ak + an + 1 - (

n

k = 1bk + bn + 1)

=n + 1

k = 1ak -

n + 1

k = 1bk

= Sn + 1 - Tn + 1 5

dn = 6 (n + 1)2 - 12 (n + 1) - 6(3n + 1 - 1)

= 6n2 - 63n + 1

= 6n2 - 23n + 2

cn + 1 - cn = dn

11

cn + 1 - cn = 6n2 - 23n + 2

c1 = a1 - b1 = - 6 - 12 = - 18

n 2

cn = c1 +n - 1

k = 1(6k2 - 23k + 2)

= - 18 + 6 (n - 1) n {2 (n - 1) + 1}6

- 2 33 (3n - 1 - 1)

3 - 1

= 2n3 - 3n2 + n + 9 - 3n + 2

n = 1

n 1

cn = 2n3 - 3n2 + n + 9 - 3n + 2

12

4

AB = FB - FA

= q - p 2

| AB |2 = | q - p |2

= ( q - p )( q - p )

= | q |2 - 2pq + | p |2

= | p |2 - 2pq + | q |2

D AB 13

FD =3 FA + 1 FB

3 + 1

=34

p +14

q

FD = srFE = tp

sr =34

p +14

q4sr = 3p + q

q = - 3p + 4sr

E BC a(1 - a)

FE = (1 - a) FB + a FCFE = (1 - a) q + ar

FE = tp

tp = (1 - a) q + ar(1 - a) q = tp - ar

0 a 11 - a0

q =t

1 - ap -

a

1 - ar

DE ABBCFABC

FA = pFC = r0

- 3 =

t1 - a

s =- a

4 (1 - a)

4s =- a

1 - at = - 3 (1 - a)

BAD

C

E

F

1 - a

a

qp

r

13

| p | = 1

| AB |2 = 1 - 2pq + | q |2

BE = FE - FB = tp - q = - 3 (1 - a) p - q

| BE |2 = {- 3 (1 - a)p - q}{- 3 (1 - a) p - q}

= 9 (1 - a)2 | p |2 + 6 (1 - a)pq + | q |2

| p | = 1

| BE |2 = 9 (1 - a)2 + 6 (1 - a)pq + | q |2

| AB | = | BE || AB |2 = | BE |2

1 - 2pq + | q |2 = 9 (1 - a)2 + 6 (1 - a)pq + | q |2

{6 (a - 1) - 2}pq = 9a2 - 18a + 8

2 (3a - 4) pq = (3a - 4) (3a - 2)

0 a 13a - 40

pq =3a - 2

2