nkhwl;Lit gy;fiyf;fof nghwpapaw;gPl jkpo; khztu;fs;

elhj;Jk; f.ngh.j cau;ju khztu;fSf;fhd

gapw;rpg; guPl;ir - 012020

,ize;jfzpjk;tpilfs;

,ize;j fzpjk; - I

1. 1y x x

1 ; 0

2 1 ; 0 1

1 ; 1

x

x x

x

2y x x

2 ; 0

2 2 ; 0 2

-2 ; 2

x

x x

x

2 1y x ----(1)

2 2y x ----(2)

(1)+(2) 2 1y

1

2y

3

4x

2 1 1x x x

1 2 2 1x x x x x

1 2x x x x

3

4x

3 1

4x

1

4x

gFjp A

2. 𝑐𝑜𝑠𝐴 =𝑏 2+𝑐2−𝑎2

2𝑏𝑐

=3𝑎2−𝑎2

2𝑏𝑐

=𝑎2

𝑏𝑐

𝑠𝑖𝑛𝐴

𝑎=

𝑠𝑖𝑛𝐵

𝑏=

𝑠𝑖𝑛𝐶

𝑐= 𝑘 vd;f

𝑠𝑖𝑛𝐴 = 𝑎𝑘

𝑐𝑜𝑡𝐴 =𝑐𝑜𝑠𝐴

𝑠𝑖𝑛𝐴=

𝑎2

𝑏𝑐⁄

𝑎𝑘=

𝑎

𝑏𝑐𝑘→ (1)

𝑐𝑜𝑡𝐵 + 𝑐𝑜𝑡𝐶 =𝑐𝑜𝑠𝐵

𝑠𝑖𝑛𝐵+

𝑐𝑜𝑠𝐶

𝑠𝑖𝑛𝐶

=sin(𝑏+𝑐)

𝑠𝑖𝑛𝐵𝑠𝑖𝑛𝐶

=𝑠𝑖𝑛𝐴

𝑠𝑖𝑛𝐵𝑠𝑖𝑛𝐶

=𝑎𝑘

𝑏𝑘×𝑐𝑘

=𝑎

𝑏𝑐𝑘→ (2)

(1), (2) ,ypUe;J>

3. lim𝑥→0

(1−cos 4𝑥

√𝑥2+9−3) = lim

𝑥→0(

2 sin2 2𝑥 (√𝑥2+9+3)

(√𝑥2+9−3)(√𝑥2+9+3))

=4 x 2x [ lim2𝑥→0

(sin 2𝑥

2𝑥)

2] lim

𝑥→0(√𝑥2 + 9 + 3)

=8x 1x 6

=48

4. 𝑑

𝑑𝑥{𝑥√𝑎2 − 𝑥2 + 𝑎2𝑠𝑖𝑛−1 𝑥

𝑎} = (1)√𝑎2 − 𝑥2 + 𝑥

𝑑

𝑑𝑥√𝑎2 − 𝑥2 + 𝑎2 𝑑

𝑑𝑥𝑠𝑖𝑛−1 𝑥

𝑎

= √𝑎2 − 𝑥2 + 𝑥 ×1

2√𝑎2−𝑥2 × (−2𝑥) + 𝑎2 ×1

√1−𝑥2

𝑎2

×1

𝑎

= √𝑎2 − 𝑥2 −𝑥2

√𝑎2−𝑥2+

𝑎2

√𝑎2−𝑥2(𝑎 > 0)

=√𝑎2 − 𝑥2 +𝑎2−𝑥2

√𝑎2−𝑥2

=2√𝑎2 − 𝑥2

𝑎 = 3 →𝑑

𝑑𝑥{𝑥√9 − 𝑥2 + 9𝑠𝑖𝑛−1 𝑥

3} = 2√9 − 𝑥2

∫ √9 − 𝑥2𝑑𝑥 = 𝑥

2√9 − 𝑥2 +

9

2𝑠𝑖𝑛−1 𝑥

3+ 𝐶

1) a) sin8 𝜃 + cos8 𝜃 =17

32

(sin4 𝜃 + cos4 𝜃)2 − 2 sin4 𝜃 cos4 𝜃 =17

32

[(sin2 𝜃 + cos2 𝜃)2 − 2 sin2 𝜃 cos2 𝜃]2 −1

2(2 sin2 𝜃 cos2 𝜃 )2 =

17

32

[1 −1

2(2 sin 𝜃 cos 𝜃)2]

2

−1

2[

1

2(2 sin 𝜃 cos 𝜃)2]

2

=17

32

(1 −1

2sin2 2𝜃)2 −

1

8sin4 2𝜃 =

17

32

sin2 2𝜃 = 𝑥 vd;f.

(1 −𝑥

2)

2

− 𝑥2

8=

17

32

1 +𝑥2

4− 𝑥 −

𝑥2

8=

17

32

𝑥2

8− 𝑥 +

15

32= 0

4𝑥2 − 32𝑥 + 15 = 0

(2𝑥 − 1)(2𝑥 − 15) = 0

𝑥 =1

2 𝑜𝑟 𝑥 =

15

2

Mdhy;> 𝑥 = sin2 2𝜃 ≤ 1

/ 𝑥 =1

2

sin2 2𝜃 =1

2

cos 4𝜃 = 1 − 2 sin2 2𝜃

cos 4𝜃 = 1 − 2 ×1

2

cos 4𝜃 = 0

cos 4𝜃 = cos𝜋

2

4𝜃 = 2𝑛𝜋 ± 𝜋

2 ; 𝑛 ∈ 𝑧

𝜃 =1

4(2𝑛𝜋 ±

𝜋

2 )

Mdhy;> 0 ≤ 𝜃 ≤ 𝜋

∴ 𝜃 =𝜋

8,3𝜋

8,5𝜋

8,7𝜋

8

gFjp B

b) 2𝑎2 + 4𝑏2 + 𝑐2 = 4𝑎𝑏 + 2𝑎𝑐

(𝑎2 − 2𝑎𝑐 + 𝑐2) + (𝑎2 − 4𝑎𝑏 + 4𝑏2) = 0

(𝑎 − 𝑐)2 + (𝑎 − 2𝑏)2 = 0

𝑎 − 𝑐 = 0 , 𝑎 − 2𝑏 = 0 [(𝑎 − 𝑐)2, (𝑎 − 2𝑏)2 ≥ 0]

∴ 𝑎 = 𝑐 = 2𝑏

cos 𝐴 =𝑏2 + 𝑐2 − 𝑎2

2𝑏𝑐=

𝑏2 + (2𝑏)2 − (2𝑏)2

2𝑏 × 2𝑏=

1

4

cos 𝐵 =𝑎2 + 𝑐2 − 𝑏2

2𝑎𝑐=

(2𝑏)2 + (2𝑏)2 − 𝑏2

2 × 2𝑏 × 2𝑏=

7

8

cos 𝐴

cos 𝐵=

14⁄

78⁄

=2

7

cos 𝐴 ∶ cos 𝐵 = 2 ∶ 7

c)

(i) 4√3 = ∆𝐴𝐵𝐶 + ∆𝐴𝐷𝐶

4√3 = 12⁄ × 2 × 5 × sin 60° + 1

2⁄ 𝑥𝑦 sin 120°

= 5.√3

2+

𝑥𝑦

2×

√3

2

𝑥𝑦 = 6

∴ 𝐶𝐷 × 𝐷𝐴 = 6𝑐𝑚2

(ii) Nfhird; tpjpg;gb>

∆𝐴𝐵𝐶 ∶= 𝐴𝐶2 = 22 + 52 − 2 × 2 × 5 × cos 60° = 19

∆𝐴𝐷𝐶 ∶= 𝐴𝐶2 = 𝑥2 + 𝑦2 − 2𝑥𝑦 cos 120°

= 𝑥2 + 𝑦2 − 2 × 6 × (− 12⁄ )

19 = 𝑥2 + 𝑦2 + 6

𝑥2 + 𝑦2 = 13

(𝑥 + 𝑦)2 − 2𝑥𝑦 = 13

(𝑥 + 𝑦)2 = 13 + 2 × 6 = 25

𝑥 + 𝑦 = 5

𝑥 + 6𝑥⁄ = 5

𝑥2 − 5𝑥 + 6 = 0

(𝑥 − 2)(𝑥 − 3) = 0

𝑥 = 2 𝑜𝑟 𝑥 = 3

𝑦 = 3 𝑜𝑟 𝑦 = 2

Vida gf;f ePsq;fs; - 2𝑐𝑚, 3𝑐𝑚

d) tan−1 1

3= 𝛼 ⇒ tan 𝛼 =

1

3 ; 0 < 𝛼 <

𝜋

6

tan−1 1

4= 𝛽 ⇒ tan 𝛽 =

1

4 ; 0 < 𝛽 <

𝜋

6

tan−1 2

9= 𝛾 ⇒ tan 𝛾 =

2

9 ; 0 < 𝛾 <

𝜋

6

tan(𝛼 + 𝛽) =tan 𝛼+tan 𝛽

1+tan 𝛼 tan 𝛽

= 1

3+

1

4

1−1

3×

1

4

= 7

11→ (1)

tan (𝜋

4− 𝛾) =

tan𝜋

4−tan 𝛾

1+tan𝜋

4tan 𝛾

=1−2

9⁄

1+29⁄

= 7

11→ (2)

(1), (2) ⇒ 𝛼 + 𝛽 =𝜋

4− 𝛾

𝛼 + 𝛽 + 𝛾 =𝜋

4

tan−1 1

3+ tan−1 1

4+ tan−1 2

9=

𝜋

4

,ize;j fzpjk; - II

1.

njhFjpf;F I mv ,l

0 2 3 0 3 4 2 3mv m m u m u

0 2 6mv mu

3 0v u

MfNt B ,d; ,af;fj;jpir khWfpd;wJ.

3BV u

2.

(𝑖)𝑉𝐵,𝐴 = (3√2 cos 45)𝒊 + (3√2 sin 45)𝒋

= 3𝒊 + 3𝒋

(𝑖𝑖)𝑉𝐵,𝐴 = 𝑉𝐵,𝐸 + 𝑉𝐸,𝐴

= 𝑥𝒊 + 𝑦𝒋 + (−3𝒊 − 3𝒋)

= (𝑥 − 3)𝒊 + (𝑦 − 3)𝒋

(𝑖𝑖𝑖)𝐵, 𝐴 Ir; re;jpg;gjhy;>

𝑥 − 3 = 0, 𝑦 − 3 < 0

𝑥 = 3, 𝑦 < 3

5 = √𝑥2 + 𝑦2

𝑦2 = 16

𝑦 = −4 (𝑦 < 3)

II (2 )B m(3 )A m

4u 3u

0 v

20𝑚

5𝑚𝑠−1

3√2𝑚𝑠−1

𝐵

𝐴 450

G+kpapd; rl;lk;

20𝑚

𝑦 − 3

𝑥 − 3

𝐴

A apd; rl;lk;

gFjp A

3.

P=FV

2H=(R+Mg sin ) u

P=FV

2H=(R- Mg sin ) 3u

(R+Mg sin ) u=(R- Mg sin ) 3u

R+Mg sin = 3R- 3Mg sin

2R=4Mg sin

R=2Mg sin

R= 2𝑀𝑔

𝑝

4. 𝜃 = tan−1(13⁄ )

rkepiyf;F>

Gs;sp 𝑋 gw;wp tyQ;Ropj; jpUg;gk; 0

0 =𝑤

2∙ 𝑂𝑌𝑐𝑜𝑠𝜃 − 2𝑤 ∙ 𝑂𝐺𝑠𝑖𝑛𝜃

𝑂𝑌𝑐𝑜𝑠𝜃 = 4 ∙3𝑎

8𝑠𝑖𝑛𝜃

𝑂𝑌 =3𝑎

2𝑡𝑎𝑛𝜃

𝑂𝑌 =𝑎

2

sin =1

𝑝

2W

2

W

Y

O

G XG

90

R

a

1) a)c

AB=AE +EB

AO+OB =AAD-pBC - - - -

-g_+q_ = )...(AO+ OD)-p(BO+OC)

= A(-g_ + (1 + /J)OB)-p(-Q + (1 + a )OA) = J. ( -g_ + (1 + /J)Q) - p( -Q + (1 + a )g_)

(1-J.-p(l+ a))Q +(J.(l+ /J)+ p-1)'2_ = 0

Q.Q � 0.

f!i'2.1- Jc -p(l + a) = O G) --t(l+/J)+p-1=0 @

@x(J+a)+ G) =>1-A + --t(l+ /J)(l+a)-(1+ a)= 0

a

E

AE: ED= a : ( a/J + /J)

BE= /J ·BCa/J+a+ fJ

J3 B

BE: EC= /J: (a/J + /J)

A(a/J+a+ /J) = a a A=---

a/J+a+ /J µ = 1-).(1 + /J)

= 1-a

(l+ /J) afJ+a+ /J

l' =/J

a/J+a+ /J

AE= a ·ADa/J+a+ /J

a[J + [3 D

a[J+a c

gFjp B

(b)

( i ) 1 3 2 2 cos45 2 2 cos45X

2

2 4 2 2 sin 45 2 2 sin 45Y

2

3N

2N

1Nx

4N

y

A

CD

B

y

xA

(0, )k

2

2 2

2 2 2 2R --------------------------- (1)

1 2tan 45

2

R ∥ DB

( ii ) A tpirfspd; jpUg;gk; A tpisAspd; jpUg;gk;

2 2 2 3 22

aa k

2 5

2

ak

gbj;jpwd; tan135 1

2 5

2

ay x

( iii ) AapD}L nry;tjhy; 0k 2 5 0

5

2

( iv ) (1) 5 2R N ,

( v )

0 2G a

5 ,G a Nm tyQ;Rop

2

R

45