w'fmd'i' ^W'fm<& Wmldrl iïuka;%Kh - 2016 ixhqla; .Ks;h - I m;%h ms
+
1.
f(n) = 4 + 15n − 1; n ∈ Z hehs .ksuq'
n = 1 úg f(1) = 4 + 15 − 1 = 18 = 9 × 2
n
∴ f(1), 9 ka fnfoa'
5
∴ n = 1 g m%ldYh i;H fõ'
n = p " p ∈ Z i|yd m%ldYkh 9 ka fnfoa hehs Wml,amkh lruq'
tkï" f(p) = 4 + 15p − 1 = 9k ; k ∈ Z fõ' 5
f(p + 1) = 4
+
p
+ 15(p + 1) − 1
= 4. 4 + 15p + 15 − 1
= 4 [9k − 15p + 1] + 15p + 15 − 1
.co
p+ 1
= 4 × 9k − 45p + 18
= 9 [4k − 5p + 2] = 9 λ ; λ = 4k − 5p + 2 ∈ Z
+
∴ f(p + 1), 9 kA fnfoa'
ha
5
wa
p
m
+
vib
∴ n = p + 1 jk úg m%ldYh i;H fõ' 5
∴ .Ks; wNHqyk uQ,Or®ufhka" ishÆ Ok ksÅ, n i|yd ° we;s m%ldYkh 9kA
(√ 2 + 7
1 5 10
(
=
25
w.
2.
(
10
Σ
r=0
11 − r
(
10
(
1 10 − r 2
Cr 2 r −1
(
fnfoa' 5
(7
1 r 5
(
ww
10 5 2 Tr = C ; fuys 1 ≤ r ≤ 11 fõ' 5 2 7 r− 1
(
(
2 yd 7 m%:ul neúka" m˙fïh mo i|yd 11 − r = 2p iy r − 1 = 5q úh hq;h = ' p, q ∈ Z
5
5
tkï r ∈ {1, 3, 5, 7, 9, 11} ∩ {1, 6, 11 } ∴ r = 1 fyda 11 fõ' 5
m˙fïh moj, ftlHhh =
+
C 0 2 + C 10 7
10
5
10
2
5 = 32 + 49 = 81
25
[2 jk mssgqj n,kak
-23. lsisu iSudjlska f;drj 5 fofkl=f.ka hq;a lKavdhula f;dard.; = yels wdldr .Kk
14
C
5
= 2002
ms˙ñ
= C
= 56
.eyeKq
= C
∴ fojr®.uh ksfhdackh jk m˙† 5 fofkl=f.ka iukaú; lKavdhula f;dard .; yels wdldr .Kk
5 = 6 14 8 6 = C − C + C 5 5 5
= 2002 − 56 − 6
4.
y ^w;d;a;aúl& B(0, 6)
θ
(0, 3)
5
6
5
5
(
= 1940
5
5
25
x ^;d;a;aúl&
vib
A(3, 0)
Arg Z = π iy Arg (Z − 3) = π jk m˙† jQ Z = Z0 ixlSr®K ixLHdjg wkqrEm ,laIHh rEmfha 4 2 m˙† P fõ' rEmhg wkqj θ = π fõ' 5 5 4 tu ksid Arg (Z0 − 6i) = 7π fõ' 5 25 4
w.
8
5
ha
π/4
lim (1 + kx)2 − (1 − kx)2 x 0 √ 1 + k2x − √ 1 − k2x
5.
=
ww
P(Z)
5
5
θ
0
m
.co
wa
(
lim 1 + 2kx + k x − 1 + 2kx − k x x 0 (1 + k 2x) − (1 − k2 x) 2 2
=
lim 4kx x 0 2k2x
=
2 lim k x
4 k
=
2 2
(√ 1 + k x + √ 2
(√ 1 + k x + √ 2
0 1
∴ k = 4
×
1 − k2x
1 − k2x
)
(√ 1 + k x + √ 2
) =
1 − k2x
)
10
; k, x = 0
( 2k ) × 2 =
4 k
5
5 5
25 [3 jk mssgqj n,kak
-3-
y
6.
y = (x − 2)
2
5
4
=
jr®.M,h
∫{ ∫
0
2
=
=
}
(4 − 2x) − (x −2)2 dx
(4 − 2x) dx −
0
0
[
2 4x − 2x
2
5
]
∫
[
2
3 − (x −2)
0
5
(x −2)2 dx
3
= (8 − 4) −
2
]
2 0
5
[ 0 + 83 ]
.co
2
25
5
vib
7. t úIhfhka wjl,kfhka" dx
ha
8 = 4− 3 4 = 3
m
x y = 4 − 2x
2
wa
0
dy 5 = 2t = 3at − 2t dt dt dy dy dt dy dx
=
dt
.
dx
w.
dx
= (3at2 − 2t) . 1 2t
2
=
3at − 2 ; t 0 ≠ 2 dy 3a − 2 dx t= −1 = − 2
5
ww
(dxdy ) = 3a − 2 ( ) 5 2 , t=1
iamr®Yl tlsfklg
( 3a2− 2 ) (− 3a2− 2 )
ksid"
= −1
5
9a − 4 = 4 ⇒ a = 8 /9 2
a > 0 ksid a =
2
2 √2
5 3
25
[4 jk mssgqj n,kak
-4 C (3t, −t)
8.
B (4, −3)
E
A(2, −1)
D
5 4
= −1
(
∴ C = 15 , − 5 4 4
.co
⇒ t =
)
5 5
)
wa
(
−1 −2 + t 3 − 3t
m
E = (3, −2) AB, CE g ,ïnl neúka mAB . mCE = −1 fõ'
5
D = ( x, y ) f,i .ksu'q
ha
(
)
vib
5
25
w.
x = 2 × 3 − 3t = 6 − 3 × 5 = 9 4 4 5 y = 2 × −2 + t = − 4 + = − 11 4 4 ∴ D = 9 , − 11 5 4 4
+1
2=
0
9.
ww
3x
−
4y
y
C
(h,h)
0
x
[5 jk mssgqj n,kak
-5-
wjYH jD;a;h S ≡ x + y + 2gx + 2fy + c = 0 hehs .ksu'q 2
2
jD;a;h x yd y wlaI iamr®Y lrk neúkA" C = (h,h) fõ'
5
;jo 3x − 4y +12 = 0 fr®Ldj jD;a;h iamr®Y lrk neúka
√ 32 + 42
=h
5
=5 h
−h +12
(−h +12) = ±5h
5
∴ h = −3 fyda h = 2 fõ'
5
⇔
∴ jD;a;j, iólrK 2
2
2
2
2
2
(x − 2) + (y − 2) = 2
=
1 − tan α tan α
=
1 − tan2 α tan α
=
2 (1 − tan2 α) 2 tan α
=
2 cot 2α
=
2 tan 2α
=
cot2α − tan 2α = 2 cot4 α
2
5
cot4α − tan4α = 2 cot8 α
3
5
+ 2 ×
1
2 + 4 × 3
cot α = tanα + 2 tan 2 α + 4 tan 4α + 8 cot8α
25
u.ska"
cot α − tan α − 2 tan 2 α − 4 tan 4α = 8 cot8α
5 1
ww
∴
2 cot2 α
w.
cotα − tan α
wa
cot α − tan α
ha
10.
vib
.co
5
(x + 3) + (y + 3) = 3
m
3h − 4h +12
10
25
[6 jk mssgqj n,kak
-611. (a)
ax2 + bx + c = 0
a x2 + bx + c a a
[ ]=0 a [(x + b ) − b + c ] = 0 a 2a 4a 2
2 2
)
]
2
a x+ b 2a
iumd; uQ, ;sîu i|yd b2 − 4ac = 0 úh hq;=hs'
k a + b x+c x − c = 2x
a(x − c) + b(x + c) = k 2 2 2x x −c
x2 [k − 2a − 2b] − 2(bc − ac) x − kc2 = 0 iumd; uQ, ;sîug kï
4(bc − ac)2 − 4(k − 2a − 2b) (− kc2 ) = 0 úh hq;=hs'
tkï k2 − 2(a + b) k + (b − a )2 = 0
fuys uQ, k1 yd k2 kï"
k1 + k2 = 2(a + b)
(k1 − k2)
k1 k2 = (b − a)2
= (k1 + k2)2 − 4 k1 k2
= 4(a + b)2 − 4(b − a)
2
10
∴ k1 − k2 = 4 √ ab
ww
(b) f(x) = (λ+ 1)x2 + (6 − 3λ)x +(20 − 12λ)
(i) λ = − 1 úg f(x) talc fõ'
5
(ii) uQ, fol α iy − α f,i .ksuq'
5
5
10
= 16ab
túg α + (−α) = − (6 − 3λ) (λ+ 1)
10
5
vib
5
w.
10
ha
2
20
10
wa
10
.co
2
m
[(
) = 0 − (b − 4ac 4a2
55
5
∴ 0 = 6 − 3λ. ⇒ λ = 2. 5
[7 jk mssgqj n,kak
-7
f(x) = (λ + 1)x2 + (6 − 3λ)x + (20 − 12λ)
5
ix.=Kl iei|Sfuka" − b = λ + 1 ⇒ b = −(λ + 1) 3 (2 − λ) 2ab = 6 − 3λ ⇒ a = − 2 (λ + 1)
1
2
2
9 (2 + 10) h = 4( 5 + 30) − 4 (−10 + 1) f(x) ys Wm˙u w.h
10 10
= 176
5
75
wa
10
3
2 ⇒ 4( λ + 1) = −( 6 − 3λ) ⇒ 4 λ + 4 = − 6 + 3λ ⇒ λ = − 10
5
5
x = 2 jk úg f(x) Wm˙uhla jk neúka a = 2 fõ'
5
(2 − λ)2 h − ba2 = 20 − 12λ ⇒ h = 4( 5 − 3λ) − 9 4 (λ + 1)
= h − b (x − a)2 = h − b (x2 − 2ax + a2) = − bx2 + 2abx + (h − ba2)
m
(iii) f(x)
.co
−2
y =
l − 5x
y =
kx −3
0
A ,laIHh i|yd l + 10 = 2k + 3 l − 2k = −7
B
l/5
4/9
y
=
5x
− l
−3
5
5 5
(ii) = 4
10
x
3/k
(i)
k
kx = y
5
− 9l + 4k = 7
(i) iy (ii) u.sk a l = 1 " 5
= −k . 4 + 3 9
B ,laIHh i|yd − l + 5 . 4 9
y = − = -5 kx + 3 x+ l
ww
1 2
y
w.
vib
ha
12. (a) l − 5x < kx −3 ys úi∫ï l=,lh {x | −2 < x < 4/9} neúka m%ia;dr fol my;ska ±lafjk m˙† msysghs' 1 y 2 10 A
5
50
[8 jk mssgqj n,kak
-8-
3n 2n + 1
=
5
lim S = 3 n ∞ n 2
5 5
m˙ñ; fõ'
5
∴ tu fY%aKsh wNsid¯ fõ'
= Sr − Sr−1
Ur
=
3r 2r + 1
=
3 4r2 − 1
'
Sn =
Σ r=1
5
n
r Σ r=1
3 f,i .ksuq' 4r2 − 1
2
3 (4r2 − 1) + 3 4 4 n
=
3 3 + 1 2 4 r = 1 4r − 1 r=1 4
=
3n + 1 S n 4 4
Σ
3n + 4
{
1 3n 1 + 4 2n + 1
= 3n (n + 1) 2(2n + 1) ∴ nlim ∞
5
n
}
5
r U Σ r=1 2
r
5
{
1 = nlim ∞ 3n 1 + 4 2n + 1 =
m˙ñ; fkdfõ'
10
5
3n 1 4 (2n + 1)
ww
=
'
Σ
w.
=
5
5
(4r2 − 1)
n
5
5
vib
=
n
− 3(r − 1) 2r − 1
wa
5
ha
Ur
m
Sn
.co
(b)
∞
}
5
5
5
∴ tu fY%aKsh wNsid¯ fkdfõ' 5
100
[9 jk mssgqj n,kak
-913. (a)
3
det A =
p
= − 9 + 2p
− 2 − 3
5
A−1 mej;Sug det A ≠ 0.
[ ]
tkï" p ≠ 9/2 úh hq;=h'
− 3 −p
(2p − 9)
2
5
3
[ ] [ ]
A−1 = A 1
− 3 −p
(2p − 9)
2
3
=
3
p
5
− 2 − 3
m
1
.co
A−1 =
5
wkqrEm wjhjhka iei|Sfuka" −p
= 3
2p − 9 2
3
= −2 ,
2p − 9
= p
2p − 9
= −3
2p − 9
5
5
5
wa
3
5
ha
−
⇒ 2p − 9 = −1 and p [1 + 2p − 9] = 0 p ≠ 0 neúka p = 4 úh hq;=h'
⇒ AA
−1
∴ I
4
− 2 − 3
= A
w.
A−1
vib
[ ] 3
túg, A =
5
= A . A = A
2
= A
2
5
⇒ 0 = A − I
ww
2
5
⇒ 0 = (A − I) (A + I) ; I = I 2
[ ] [ ] [ ] [ ]
tkï 0 = BC wdldrh .kS' fuys B
= A− I =
5
=
3
4
− 2 − 3 2
4
− 2 − 4
−
= 2
1
0
0
1
1
2
−1 −2
5
[10 jk mssgqj n,kak
- 10 -
iy C= A + I =
5 =
(b) (i) Let Z
4
+
− 2 − 3
4
4
− 2 − 2
1
0
0
1
2
= 2
2
= x + iy, hehs .ksuq' x, y ∈ R
(
2
fõ'
.co
Z
(ii) Let Z1 = x1 + iy1 o Z2 = x2 + iy2 o hehs .ksuq' x1, x2, y1, y2 ∈ R fõ'
= (x1 + iy1 ) (x2 + iy2 )
Z1 Z2
= x1 x2 + i x1 y2 + i y1 x2 + i 2y1y2
= (x1 x2 − y1 y2 ) + i (x1 y2 + y1 x2 )
ha
=
=
Z1 Z2 = Z1 Z2 fõ'
ww
=
(
)(
⇒
Z 1 − 2 Z2
2
⇒
(Z
1
− 2 Z2
) (Z
(
)
1
− 2 Z2
)(
5
)
2 2 Z Z Z Z − − ( ( ) ) ) =
1
2
1
2
10
5
2 − Z1 Z2
⇒ Z − 2 Z 2= 2 − Z Z 1 2 1 2 Z 1 − 2 Z2 Z 1 − 2 Z2 ⇒ = 2 − Z1 Z2 2 − Z1 Z2
10
5
(x1 − iy1 ) (x2 − i y2 )
(iii) Z1 − 2 Z2 = 1 2 − Z1 Z2
5
x1 (x2 − i y2 ) − iy1 (−i y2 + x2 )
w.
75
∴ Z1 Z2 = (x1 x2 − y1 y2 ) − i (x1 y2 + y1 x2 )
vib
wa
∴ Z Z =
5
)
5
= (x + iy) (x − iy)
ZZ
= x2 + y2 2 2 = √ x2 + y2 = Z
5
−1 −1
m
[ ] [ ] [ ] [ ] 3
5 5
[11 jk mssgqj n,kak
- 11 Z1 Z1 − 2 Z1 Z2 − 2 Z2 Z1 + 4 Z2 Z2 Z1
2
+ 4 Z2
2
Z1
2
+ 4 Z2
2
Z1
2
(1 −
(1 −
Z2
2
Z2
= 4 + Z1
2
Z2
2
2
− 4
− Z1 2
) (Z
)
1
(
Z2
.
2
)
−4
)=0
Z2 ≠ 1" ksid Z1 − 4
= 0
2
5
Z1 > 0, ksid Z1 = 2
Z−3 <2
Arg (Z − 3) = π
3
(5, 0) ;d;a;aúl wlaIh
0
vib
(1, 0)
(3, 0)
π/3
(1, 0)
(3, 0)
w.
Z − 3 < 2 iy Arg (Z − 3) = π ; hk folu ;Dma; lrk P ys m:h
3
ww
w;d;a;aúl wlaIh
ha
w;d;a;aúl wlaIh
0
35
wa
(c)
.co
∴ Z1 = 4
= 0
5
= 0
2
2
− 4 1 − Z2
2
5
m
= 4 − 2 Z1 Z2 − 2 Z1 Z2 + Z1 Z1 Z2 Z2
(5, 0) ;d;a;aúl wlaIh
w;d;a;aúl wlaIh
0
5
A (3, 0)
5 2
π/3
5
B
5 ;d;a;aúl wlaIh
20
[12 jk mssgqj n,kak
- 12 14. (a)
y = (sin x)x 0 ≤ x ≤ π
2
ln y = x ln sin x dy 1 = ln sin x + x cot x y dx
10
dy = [x cot x + ln (sin x) ] (sin x)x dx
(b) gexlsfha m˙udj = πx2 y + 2 πx3
5
= x2 (y + 2 x)
3
y = 452 − 2 x 3 x
gexlsfha mDIaG jr®.M,h
5
A = 2πx + πx + 2πxy 2
2
A = 3πx + 2πx
2 2 A = 3πx + 90π − 4π x x 3
A=
dA 90π = 10πx − 2 dx 3 x
( 45x − 23 x )
ha
2
5π x2 + 90π 3 x
vib
dA = 0 fõ' dx
x
dA dx
5
5
0
3
5
> 0
∴ x = 3 úg mDIaG jr®.M,h wju fõ'
5
5
w. x = 3 jk úg
2
3 10π (x − 27) 2 3x 2 = 10π2 (x − 3) (x + 3x + 9) 3x
ww
10
A = 3πx + 2πxy 2
=
.co
∴ 45
5
wa
3
m
∴ πx2 y + 2 πx3 = 45π
25
3
10
5
6 y = 45 − 9
= 3
3
5
55
[13 jk mssgqj n,kak
- 13 =
a b 2 + (x − 1) (x + 1)
= 2 neúka"
f(0)
a + b = 2
f (x)
f (0)
2a − b = 0
1
1 4 / f (x) = − 4 (x − 1)3 − 3(x + 1)2 3
/
/
=
2
5
4 2 2 ka a = 3 , b = 3
5
(x + 1) + (x − 1) 3 2 (x − 1) (x + 1) 2
3
2
3
2
2
3
2
ha
2
x /
−∞ < x < −1
−1 < x < 0
< 0 f wvqfõ'
< 0 f wvqfõ'
> 0
f jeäfõ'
1
< 0 f wvqfõ'
10
ww
túg f(0) = 2 fõ' x
± ∞ úg f(x)
0
−1 , f (x)
−∞
x
−1 , f (x)
+ ∞
x
1 , f (x)
+ ∞
x
1 , f (x)
+ ∞
5
x = 0 ° f Y%s;hg ia:dkSh wjuhla mj;S' 5
x
0
w.
f (x)
vib
2
/
ishÆ x ∈ R i|yd (x − 1) + 4 > 0 neúka" x = 0 kïu muKla f (x) = 0 fõ'
3
5
wa
2
3
5
} { x − 2x + 5x =− 4 [ 3 (x − 1) (x + 1) ] x − 2x + 5 = − 4x [ 3 (x − 1) (x + 1) ] (x − 1) + 4 = − 4x [ 3 (x − 1) (x + 1) ]
5
= 0 neúka"
yd
2a b 3 − 2 (x − 1) (x + 1)
−
= − 4 3
1
m
f(x)
.co
(c)
−
+
−
+
10
[14 jk mssgqj n,kak
- 14 -
y
2
0
x
1 f
m%ia;drh 10
m
−1
(2 + x)
1/ 2
1
=
(2 − x)
3/ 2
x = 2 sin θ wdfoaYfhka"
dx = 2 cos θ dθ
x = 0, sin θ = 0
θ = 0
θ = π 6
I
∫
=
0
1
=
0
∫
2 cos θ
ww
dθ
(4 − 4sin2 θ) / 2 (2 − 2sin θ)
/6
π
1 = 2
0
∫
1 + sin θ cos2 θ
[
π
π
dθ = 1 2
0
5
[
[
[
π
/6
∫
5
sec dθ 2
1 + 2
/6
0
∫
secθ tanθ dθ
5
π
/6 /6 1 1 sec θ = + tan θ 2 2 0 0
5
dθ
2 cos θ 2(1− sin θ) /6
(2 − x)
5
2 cos θ
π
5
w.
(4 − x )
vib
x = 1, sin θ = 1 2
/6
0
dx
2 1/2
5
π
∫
wa
0
∫
dx
70
ha
1
15. (a)
.co
f m%ia;drh 5
5
[15 jk mssgqj n,kak
=
[
1 2
1 + 2 − 1 √3 √3
= √3 − 1 2
5
(b)
[
G(x) =
A Bx + C + 2 (x + 2) (x + 8)
5 5 5
1
x2 ix.=Kl ( 0
= A + B
⇒ A = −B
x ix.=Kl ( 0
= 2B + C
⇒ C = −2B
ksh;h (
1
= 8A + 2C
1
= −8B − 4B ⇒ 12B = −1
12
5
∫
1
(x + 2) (x2 + 8)
dx
x 1 1 1 dx dx + 1 dx − 1 g(x) = 2 (x + 2) 12 (x + 8) 6 (x2+ 8) 12 1 12
vib
=
∫
5
( (
x ln |x + 2| − 1 ln (x2+ 8) + 1 tan−1 +C 6 24 2 √2 5
[
w.
∫
∫
g(x) =
5
⇒ B =− 1
ha
5
6
5
5
wa
12
= A (x2+ 8) + (Bx + C) (x + 2)
A = 1 " C = 1
50
m
.co
- 15 -
[
2 1 1 = 24 ln (x + 2) + 1 tan−1 2 6 2 √2 x+8
ww
5
[
[
5
( (
x +C 2 √2
( (
2 1 1 x = 24 ln (x + 2) + tan−1 +C 2 x + 8 12√ 2 2 √2
5
60
[16 jk mssgqj n,kak
- 16 In
=
∫
x sin x dx n
∫
n = − x d (cos x) dx n−1 n = [− x cos x] + (cos x) nx dx n
{x
= − x cos x + n
= − x cos x + nx
In + n(n − 1) In − 2
n
n−1
x
n−1
n−1
n
5
∫
sin x − sin x(n − 1) xn − 2 dx
sin x − n(n − 1) In − 2
[n sin x − x cos x]
= x
n−1
d (sin x) dx
10 5
.co
= − x cos x + n
10
{
∫ ∫
5
m
(c)
5
40
0
ha
P ( x, y ) kï
=
16. (a) ´kEu fldaK iuÉf√olhla u; msysá ,laIHhla
wa
N
5
x,
x
y
a2x + b2y + c2
√a2 + b2 2
2
w.
∴ a1x + b1y + c1 = ± √a12 + b22
5
2
c
+ 2
by
+
P ( x, y )
2
ax
vib
5 PL = PN a1x + b1y + c1 a2x + b2y + c2 5 = 2 2 2 2 √a1 + b2 √a2 + b2
a1x + b1y + c1 = 0
L
y f,i m%;sia:dmkh lsrSfuka
5
ww
fldaK iuÉf√ol fr®Ldj, iólrK
a2x + b2y + c2 = ± 5 √a1 + b2 √a22 + b22 a1x + b1y + c1 2
2
fldaK iuÉf√olj, iólrKh 4x + y + 3
x + 4y − 3
=
±
+ : 3x − 3y + 6
=
0 ⇒
x−y+2 =0
− : 5x + 5y
=
0 ⇒
x+y =0
√4 + 1 2
2
√ 42 + 12
5 5
[17 jk mssgqj n,kak
- 17 x + y = 0 iy x − y + 2 = 0 úi£fuka x = −1,
y =1
5
A = (−1, 1) hehs .ksuq' B = (0, 2), x − y + 2 = 0 u; msysghs'
5
P = (x, y) hkq x + y = 0 u; msysá ,laIHhlaa f,i .ksuq' PB neúka
( yx +− 11) × 1
= −1
5
x+ 1 1
y −1 = −1
∴ x = −1 + t,
5
= t ; t hkq mrdñ;shls
y = 1−t
x + y = 0 u; AD = AB jk m˙†
5
túg D = (−1 + T, 1 − T)
T
2
⇒ T + T 2
5
= ±1
2
= 1 + 1 2
2
=
2
5
vib
2
ha
D ,laIhg wkqrEm mrdñ;sh T f,i .ksuq'
AD = AB
m
.co
wa
PA
∴ D = (0, 0) fyda (−2 , +2)
5
5
w.
D ≡ (0, 0) úg CD iólrKh
/
=0
ww
x − y
/
D
C
5
/
D ≡ (−2, +2) úg C D iólrKh
5
x − y +4 = 0
A (−1, 1)
B
(0, 2)
/
BC iy BC fr®Ldjkays iólrKh x + y −2 = 0
5
D
/
/
C
100
[18 jk mssgqj n,kak
- 18 (b) S = x2 + y2 − 2x + 4y − 3 = 0 1
S = x2 + y2 + 2gx + 2fy + c = 0 hehs .ksuq' fuys g, f, c ksh; fõ'
S − S = 0 fr®Ldj u; S = 0 ys flakaøh msysghs'
− 2x(g + 1) − 2y (f − 2) − 3 − c = 0
∴ 2(g) (g + 1) + 2(f) (f − 2) − c − 3 = 0
S = 0 jD;a;h (1, 1) yryd hk neúka 2 2 1 + 1 + 2g + 2f + c = 0
1
5
1
1 ka yd 2
2
5
ka
2g + 2g + 2f − 4f − (− 2g − 2f − 2) − 3 = 0
2g + 2f + 4g − 2f − 1 = 0
2
2
2
wa
2
5
1
m
∴ c = − 2g − 2f − 2
5
.co
S = 0 ka S = 0 iuÉf√o jk ksid"
5
5
2(−g) + 2(−f ) − 4(−g) + 2(−f ) − 1 = 0
∴ (−g, − f) ,laIHh 2x2 + 2y2 − 4x + 2y − 1 = 0 jD;a;h u; msysghs' 5
fuys flakaøh(1, − 1 ) 2
wrh =
=
√7
5
√
=
7 4
a = BC = BD + DC a = c cos B + b cos C c b
5
2 A
ww
17. (a)
B
D
ha
√
2 1 + 1 + 1 4 2
5
w.
2
vib
2
50
C
a
A
a = b cos C + 0 = b cos C + c cos 90° b = b cos C + c cos B c B
a
5
C [19 jk mssgqj n,kak
- 19 -
A
c
b
π−B
D
B
a = BC = CD − BD = b cos C − c cos (π − B) = b cos C + c cos B
a
C
tm˙†u b = a cos C + c cos A
a cos C = b − c cos A
a2 cos2 C = b2 − 2bc cos A + c2 cos2 A
m
a2 + c2 sin2 A − a2 sin2 C = b2 + c2 − 2bc cos A ;
∴
10
a2 − a2 sin2 C = b2 + c2 − 2bc cos A − c2 sin2 A
c a = sin C ksid sin A
∴ a2 = b2 + c2 − 2bc cos A 2 2 2 cos A = b + c − a 2bc
a, b, c iudka;r fY%aVßhl kï
a + c = 2b
5
b cos C + c cos B + a cos B + b cos A = 2b
cos A + cos C + 2 cos B
(
)
(
(
)
(
= 2
)
(
)
0 < x, y < π 2
ww
(b)
)
B A−C 2 cos π − cos 2 2 2 A−C cos 2
w.
∴ 0 < π −y< π 2 2
)
5
= 4 sin2 B 2
= 2 sin B 2
5
50
5
(
)
sin x > cos y = sin π − y 2
(
= 2 (1 − cos B)
vib
A−C 2 cos A + C cos 2 2
5
ha
wa
.co
5 = 0
5
5
sin x > sin π − y 2
(0, π2 ) jiu ;=< fldaKh jeä jk úg ihska w.h jeä jk ksid"
10
∴ x > π −y 5 2 x + y > π 2
25
[20 jk mssgqj n,kak
- 20 (c)
f(x)
=
=
3 cos2 x + 8 sin x cos x − 3 sin2 x 3 cos2x + 4 sin2x
5
5
= 5( 3 cos2x + 4 sin2x) 5 5
5
= 5(sinα cos2x + cosα sin 2x)
= 5sin(2x+ α)
= A sin(2x + α)
fuys A = 5, α hkq tan α = 3 jk m˙† jQ iqΩ fldaKhhs' 5 4 f(x) =
m
5 2
.co
5 2
5sin(2x + α) =
1 = sin π 2 6
sin(2x + α) =
5
wa
5
n 2x + α = n π + (−1) π 5 6 nπ α n x = − + (−1) π , fuys n ∈ Z 2 12 2 5
f(x) = 5 sin(2x + α)
ha
vib
f(x) Wm˙u = 5 ; x = π − α 4 2 5
w.
f(x) wju = − 5 ; x = − π − α 4 2 5
ww
5 (α < π ksid) 4
5
y
5
π α − − 4 2 x π 0 π α π − − 2 4 2 2
15
−3 −5
75
w'fmd'i' ^W'fm<& Wmldrl iïuka;%Kh - 2016 ixhqla; .Ks;h - II m;%h ms
1.
2
2
v = u + 2as fh°fuka m g 2 v = 2gh ∴ v = √ 2gh 5
m
Δ (mv) fh°u ;
=
I
u=0 h
P yd m g ∴ −J = 3mv1 − mv
1
Q g
2 ka
yd
v1 = v
=
5
2m
Q
P
v1
5
5
= 2m
5 √ 2gh 5
25
vib
J
√ 2gh
P
v
wa
1
5
2
= 2mv1 − 0
Q 2m
J
J
ha
J
5
v1
.co
= (2m + m) v1 − mv − 2m × 0
−J
m
2. ;;amrhl° msglrk c, m˙udj = 8 (0.005) m
w.
3
5
3 = 0.040 m
ww
;;amrhl° msglrk c, ialkaOh = 10 × 0. 040 kg 3
= 40 kg
;;amrhl° fmdïmh u.ska flfrk ldr®hh
∴ fmdïmfha Èu;djh
5
= mgh +
1 mv2 2
2 1 = (40 × 10 × 4) + × 40 × 8 2 5 5 −1 = 2880 js
= 2880 W
5
25
[2 jk mssgqj n,kak
-23.
y
5 (t = T)
u
0
θ
x
t
= T úg"
v
= u +gT
5
m
AC = AB + BC
A
gT sinθ = u
gT
∴
θ
10 v
5
u
T = g sinθ
wa
u
.co
B
u cosecθ
= g
θ
C
u
m
km
5u 2
u 2
vib
5u
ha
4.
moaO;shg .uH;d ixia:; ß h s kshuh fh°fuka 5mu − kmu = kmu − 5mu 2 2 10 − 2k = k − 5
5
∴ k = 5
5
w.
25
ww
ksõgkaf.a m¯laIdKd;aul kshufhka u 5u + = e(u +5u) 2 2
5
5
3u
= 6ue
1 2
= e
I −I I
= Δ(mv) = −m. 5u − m. 5u 2 =
15mu 2
5
25
[3 jk mssgqj n,kak
-35.
b ksid a ' b = 0
a
5
∴ (2i + 3j) ' (λi + μj) = 0 2λ + 3μ b =1
2
μ = ±
μ > 0 ksid μ =
2
ksid λ2 + μ2 = 1
1 yd
5
1
=0
5
2
√ 13
2
5
√ 13
m
3 λ =− 5 √ 13 6. jia;=j iSudld¯ wjia:dfõ mj;sk úg fõ'
P
5
s
5
w sin (λ + α) cos (θ − λ)
α
w
ha
5
tkï θ = λ
θ
λ
π −λ 2 π −α 2
P wvq;u ùug cos (θ − λ) Wm˙u úh hq;=h'
5
wa
,dï m%fïhfhka" P w sin[π − (α+λ)] = sin[ π − (θ − λ)] 2 P =
(A)
4
1 (m
(B)
st
2
nd
|
(fojk)
|
|
w.
X = (A ∩ B ) ∪ (A ∩ B)
(i)
|
kuq;a (A ∩ B ) ∩ (A ∩ B) = φ
∴ P (X)
|
|
ww
= P (A ∩ B ) + P (A ∩ B) |
|
= P(A) P(B ) + P(A ) P(B)
= 1 ×
= 1
= P (A ∩ X) P (X)
= P (A) P(B ) P (X)
1 × 3 3 4 = 5 12
=
(ii) P (A | X)
25
5
3 4
( 1 − 14 ) + ( 1 − 13 ) . 14 5 + 2 × 1 = (1 × 5 ) = 12 3 4 4 3 |
3 5
(
5
5
∴
vib
∴ P ^wvq;u& = w sin (λ + α) 5 7. P (A) = 1 , P (B) = 1 hehs .ksuq'
3
25
.co
1 ka"
III jk m%;HÈh) (iajdh;a; neúka)
5 5
25
[4 jk mssgqj n,kak
-4-
|
P (A ∪ B)
= 0.6
1 − P (A ∪ B)
= 0.6
P (A ∪ B)
= 0.4
P (A ∪ B) − P (A ∩ B)
= 0.2 + 0.1
∴ P (A ∩ B)
= 0.4 − 0.3 = 0.1
P (A ∩ B)
|
|
P (A ∩ B ) =
|
5
= 0.1
5 5
= P(B) − P (A ∩ B)
0.1 + 0.1 = P(B) P (A ∩ B) 0.1 ∴ P (A | B) = = 0.2 P(B) = 1
2
5
m
|
P (A ∩ B ) = 0.2, P (A ∩ B)
.co
|
8.
5
25
x = 5 iy
sx = 2
(i) yi ∈ {12, 13, 14, 15, 16, 17, 18}
yi = xi + 10 hehs .ksuq'
fuys xi ∈ {2, 3, 4, 5, 6, 7, 8}
∴ y
= x + 10 = 5 + 10 = 15
iy sy = sx = 2
5
vib
ha
9.
wa
yi = 10xi hehs .ksuq'
fuys xi ∈ {2, 3, 4, 5, 6, 7, 8}
ww
w.
(ii) yi ∈ {20, 30, 40, 50, 60, 70, 80}
∴ y
= 10 x
= 10 × 5 = 50
iy sy = 10 sx = 10 × 2 = 20
(iii) yi = axi + b hehs .ksuq'
túg y = ax + b 2 sy = a2 s 2 x sy = a s
= 2a
= 5a + b
x
5
5 5
5
25
[5 jk mssgqj n,kak
-5 10. ui fi
−3
−2
−1
0
1
2
5
10
25
30
20
10
fu −15 −20 −25 0 20 Σf u u
i
i
=
i
Σfi
= − 20 100
5
xi − 35 a
∴ x
= a u + 35
33
= − a + 35
a
= 10
5
0 − 10
10 - 20
20 - 30
5
wa
5
m%dka;r
.co
5
ui
5
= −1
=
20
m
i
30 - 40
40 - 50
50 - 60
5
11. (a)
ha
fi 5 10 25 30 20 10 ai + 20j
(2, 8)
vib
B
5
i + 25j (−2, , −2)
ww
w.
A
25
;sria m%fõ.h
issria m%fõ.h 25
5
1
P
a
Q
0
T
20
5
ld,h
5 0
yuqùu i|yd
P ys isria úia:dmkh = Q ys isria úia:dmkh + 10
P ys isria úia:dmkh − Q ys isria úia:dmkh = 10
T
5
Q
P
ld,h
10 [6 jk mssgqj n,kak
-6
5T = 10
T =
5
2
P ys ;sria úia:dmkh = Q ys ;sria úia:dmkh + 4
P ys ;ssria úia:dmkh − Q ys isria úia:dmkh = 4
(1 − a) 2 =
1 − a
=
a
= − 1
5
.co
u α
5
v
=
taug v (P, E)
u ⇒ BC + AB = AC 1 1
+
α
v
=
ha
hdug v (P, E)
wa
v (P, E) = v (P, S) + v (S, E)
u ⇒ BC + AB = AC 2 2
+ α
v
vib
C2
w.
v
A
ww
u
∧
C1AC2
=
60
m
(b) v (S, E) = v (P, S) =
5
4 2
10
5 5
15
α
α
B v C1
π 2
∴ C1C2 úYalïNh jk m˙† jQ jD;a;h A yryd .uka lrhs'
10
C1C2 ys uOH ,laIHh B neúka" B C1 = B C2 =
BA =
u
5
v = u
∧
BAC1 =
π − α iy 2 4
5
∧ BAC2 = π + α 4 2 5
[7 jk mssgqj n,kak
-7-
S1
π− α 4 2
.ukg .;jk uqΩ ld,h t kï"
I
AC2
10
π + 4 + AC2 sin ( π + 4 d 2d = v cos α u cos α
π α S1I sin ( − ) 4 2 AC1 sin ( π − α ) 2 4
=
S2I
+
d v cos α
=
+
S2I sin (
^¥m;&
α ) 2 α) 2 (
10
m
AC1
5
v =
u)
.co
S1I
=
d
∴
t
S2
π+α 4 2
5 5
90
v − u + 2ga (1 + cos θ) = 0 mg
vib
F = ma fh°fuka
2 R + mgcos θ = mv a 1 ka 2 g wdfoaYfhka
2 R = mu a
2
[
10
O
θ
v
P
R
mg
5 P.E = 0
a
m
u
5
]
m = a u2 − 2ga (1 + cosθ)
w.
R + mg cos θ
5
1
2
θ
15
ha
2
wa
12. (a) m g Yla;s ixia:S;s kshufhka" 1 mu2 − mga = mga cos θ + 1 mv2 2 2
− mg (2 + 3 cos θ)
ww
OA Wvq isri iu. idok fldaKh α jk úg wxY=j
mDIaGfhka bj;a fõ kï" túg R = 0 fõ' ∴ u2 − 2ga − 3ga cos α = 0
5 ∴
2 cos α = u − 2ga > 0 ( 3ga
5
u2 > 2ga)
∴ α iqΩ fldaKhla fõ' ;jo 0 < α < π jk ksid 0 < cos α < 1 fõ' 2 u2 − 2ga < 1 3ga
5
u2 < 5ga [8 jk mssgqj n,kak
-8
m ialkaOh mDIAGfhka bj;a jk úg cos α = 1
√3
=
u2 − 2ga 3ga
1
√3
fõ'
u2 − 2ga = √ 3 ga
5
u2 = (2 + √ 3 ) ga túg m%fõ.h v = u2 − 2ga (1+ 2
1
) = 2ga + √ 3 ga − 2ga −
√3
2ga √3
=
ga √3
5
m
m ialkaOh f.da, mDIAGfhka bj;a jQ miq m%laIsma;hl wdldrhg .uka lrhs' wk;=rej is≥jk p,s;fhaoS" a sinα ;sria ≥rla hdug .;jk ld,h t0 kï
y = =
=
vsinα × a sinα v cos α
2a ga2 3 − 1 2ga √3 √3 2a − √3a √3 a
=
−
=
− a cos α
2 y = (v sinα ) t0 − 1 g t 0 2
− 2 3 1 3
1 2
ga2 sin2α v2 cos2α
5
wa
túg by
ha
vib
.co
a sinα = (v cos α) t0 5
√3
5
ww
w.
m ialkaOh O yryd hk isria fr®Ldj miqlr hkúg my
T
.
θ
x
.
P
5
+y v (Q, O) = π − θ
x
.
2
x
O
θ A
T
T
mg
R
T
Q
R2
mg
5mg
B
moaO;sh i|yd .uH;d ixia:ß;s kshuh fh°fuka
.
.
.
.
.
5m x + m(x − y cos θ ) + m (x − y sinθ ) = 0
.
10
.
7m x = m y (cos θ + sinθ ) [9 jk mssgqj n,kak
-9
.
( 35
= y.
7x
.
.
5x
+4 5
)
5 1
= y
20
moaO;sh i|yd Yla;s ixia:ß;s kshuh fh°fuka"
. . . 1 5m x. 2 + 1 m {(x − y cos θ )2 + (y sinθ )2} 2 2
1 m {(x. − y. sinθ )2 + (y. cosθ )2} −mgy sinθ − mg(l − y) cos θ = ksh;hla 2
{x.
2
.2
} + {x. + y. − 2 x. y. sinθ }
. .
2
+ y − 2 x y cosθ
2
− 2gy sin θ + 2gy cosθ = ksh;hla
.2
. .
.2
.2
. .
)
.2
7x + 2y − 2x y
4 3 + 5 5
5
) − 2gy 45
3 + 2gy 5 = ksh;hla
35 x + 10 y − 14 x y − 2gy = ksh;hla
25
wa
2
1 yd 2 ka .2 .2 .2 35 x + 250 x − 70 x − 2gy = ksh;hla
.2
m
. 2 5x +
.co
+
20
t úIfhka wjl,kh ls¯fuka
ha
215 x − 2gy = ksh;hla
vib
. .. . 5 430 x . x − 2gy = 0 . .. . . (∴ x ≠ 0) 430 x . x − 2g . 5x = 0 ∴ .. = g x 43 P i|yd F = ma fh°fuka ..
..
= mg sinθ − m (5 x − x cosθ )
ww
T
..
.. x : mg sinθ − T = m (y − cosθ )
w.
θ
(
.. 3 = mg 4 − m x 5 − 5 5
5
10 5
5
)
= 4mg − m. 1 g . 22 5 5 43
{
= 2mg 2 − 11 43 5
}
= 2mg × 75 5 43
= 30mg 43 5
15
[10 jk mssgqj n,kak
- 10 13. ;ka;=j iajdNdúl †f.a isg x ≥rla we|S we;s úg ;ka;=fõ wd;;sh T kï" R B 2mgx T λx T = = 5 a P x a
O a
30° mg
wxY=fõ p,s;h i|yd F = ma
..
= mx
.. 2mgx = mx a
2
.. x
2g a = − a (x − 4 )
x
=
.
x
a
5
1
+ A cos ωt + B sin ωt
4
2
= − Aω sin ωt + Bω cos ωt
..
2
3
2
x
= −Aω cos ωt − Bω sin ωt
..
= −ω (A cos ωt + Bω sin ωt) 2 a = − ω (x − ) 5
4
2
2 ka
4
1 yd 5 ie,lSfuka ω2 = 2g a
5
5
t = 0 úg x = a fõ'
5
ww
ω ≠ 0 ksid B = 0 fõ'
2
ka" a −
∴ x = x −
a
4
3a
4
=
3a
a
a
4
5
4 = A ⇒ A = 4
cos ωt + 3a
4
40
a
4
cos ωt
wxY=fõ f∞a,k flakaøh x − tkï x =
5
a
w.
ka" 0 = Bω
fõ'
2g
5
vib
.
t = 0 úg x = 0 fõ'
3
√
ω=
5
ha
x
5
20
.co
−
wa
mg × 1
10
m
mg sin 30° − T
a
4 = 0 ° ,efí'
f∞a,k flakaøh fõ'
5
5
10
[11 jk mssgqj n,kak
- 11 -
úia;drfha° x = 0 fõ' túg t = t1 hehs .ksuq'
0 = −Aω sin ωt1
sin ωt1 = 0 =
x −
x −
a
3a 4 = 4 cos ωt1
a
± 3a 4 = 4
5
∴ wxY=fõ ir, wkqjr®;S p,s;fha úia;drh =
3a
4
5
.co
5
wxY=j m
4
cos ωt
=− a
4
= − 1
V = −Aω sin ωt 2g = − 3a 4 a = − 3a 4
√
.
√ √ 2g
a
= − √ag
x = 0 úg m%fõ.h
5
3
√1
− cos2 ωt
5
8 9
= − 3a
4
√
√ag
fõ'
5
20
w.
wxY=j m
−
a
=
3a
5
cos ωt
ww
5
2g . 2 √2 a 3
vib
cos ωt
20
wa
+
nπ ; n ∈ Z0
m
ωt1
5
ha
.
0 4 cos ωt0 = − 1 5 3 ωt0 = π − cos−1 ( 1 ) 3 t0 = 1 π − cos−1 ( 1 ) = 3 ω
4
[
]
√
a 2g
[π − cos
−1
( 1 )
3
]
5
wxY=j O olajd .=re;ajh hgf;a p,kh fõ' B isg O olajd hdug ld,h t2 kï
[12 jk mssgqj n,kak
- 12 = ut + 1 at2
S
= a, u =
√ ag , a
a
=
1 g t 2 2 2 2
2
√ ag t2 −
g t2 − √ ag t2 + a 2
√ ag
t = 2 2
±
√ ag −
4
g
√
5
= 0
4
t2 =
= − g sin 30°
g a
4
2
a g
m
S
5
.co
∴ O olajd hdug ld,h t0 + t2
√
a −1 1 2 2g (π − cos ( 3 )) +
√
a 2g
5
[π −
√
a g
wa
=
]
=
cos ( 1 ) + 2 √2
3
ha
−1
;ka;j = Wm˙u †f.a we;af;a A ,laIHfha°h' tkï x
= a úg"
vib
λa 5 a TA =
= λ TA = 2mg
5
w.
ww
14. (a) (i) a = b = c = 1 ( a + 2 b ) (5a − 4 b)
( a + 2 b ) . (5a − 4 b)
kï
30
10
5
5 = 0 5 a . a + 10 b . a − 4 a . b − 8 b . b = 0 2 2 5 a + 10 a . b − 4 a . b − 8 b = 0 5 + 6a . b − 8 = 0 6a . b = 3
a.b = 1
2
a b cos θ = 1
2
5 5
1 × 1 cos θ = 1 ⇒ θ = 60° 5 2
25
[13 jk mssgqj n,kak
- 13 2
2
a−b + b−c + c −a
2
= (a − b ) . (a − b ) + (b − c ).(b − c ) + (c − a ).(c − a ) = a 2+ b
2
− 2 a . b + b 2 + c 2 − 2b. c + c
= 6 − 2(a . b + b . c + c . a )
2
5
+ a
2
−2 c . a
5
(
2
∴ 2(a . b + b . c + c . a ) = 6 − a − b 2 + b − c + c − a 2
≥ 0
a 2+ a . b + a . c + b . a + b
5 2
+b .c + c .a + c .b + c
3 + 2 (a .b + b . c + c . a ) ≥ 0
2
)
1
5
∴ (a + b + c ) . (a + b + c ) ≥ 0
1 yd
5
2
2
2
2
wa
5
D
4
C
vib
(b)
5
2 5√2 2√2 8 F α E
w.
2m
x
B
ww
A
3
(i)
50
ha
2 2 2 ∴ a− b + b− c + c − a ≤ 9 5
5
≥ 0
5
ka
3+6 −( a − b + b − c + c − a )≥ 0
2
.co
a +b + c
2
5
m
(ii)
X = 4 − 3 + 5√2 cos 45° − 2√2 cos 45° = 4N Y = 2 − 8 + 5√2 cos 45° + 2√2 cos 45° = 1N iïm%hqla;h R kï"
5
5
R = √ X2+ Y 2 = √ 42 + 12 = √17 N
5
iïm%hqla;h ;sri;a iuÛ idok fldaKh α kï" 1 tan α = 4
()
−1 1 α = tan 5 4
25
[14 jk mssgqj n,kak
- 14 -
iïm%hqla;fha l%shd fr®Ldj AB lmk ,laIHh E kï AE = x hehs .ksuq' A 1 × x = 2 × 2 − 4 × 2 + 2√2 . 2 cos 45° D
5
5
x = 0
A ≡ E ^iumd; fõ&
F
iïm%hqla;fha l%shd fr®Ldj A yryd hhs' ∴ n, moaO;sh iu;=,s; ùu i|yd √17 N n,hla FA †Ydjg
A
C
R
α
B
20
√17 × AG sinα
= 39
√17 × AG × 1 = 39 √17
5
G
α y
A
α
wa
ABC
.co
m
5 A ys° fh†h hq;=hs' 5 C (ii) ABC w;g 39Nm hq.auhlg W!kkh ls¯u i|yd fh†h hq;= D n,h †lal< BA u; Aisg y ≥rlska AF g iudka;r FA †Ydjg jQ √17 N n,hla l%shd lrkafka hehs .ksuq' F 5 R R B
AG = 39m 5
=
√17 × 2 ×
1 √17
5
5
C
D
= 2Nm
5
7N
√17 × BA sinα
vib
=
ha
(iii) B ys° ;ks n,hlg W!kkh ls¯u i|yd tla l< hq;= >Qr®Kh
F
w.
√1
Aliter α α tla l< hq;= >Qr®Kh M kï A B √17 N = 0 M − √17 × 2 sinα 1 = 2 Nm M = √17 × 2 × √ 17 Y1 15. (a) AB = BC = 2a f,i .ksuq' X
ww
15
15
1
A α
G1
Y
10
X G2 X B θ W Y W C 5 B (i) BC g W W asin θ = 2Wa cos θ 5 tan θ = 2
10
10 [15 jk mssgqj n,kak
- 15 BC fldgfia iu;=,s;;dj ie,lSfuka" X = W
5
Y = W
5
√ W2 + W2
∴ RB =
√2
=
5
W
20
π RB ys †Ydj ;sri iuÛ tan −1 1 = 4 fldaKhla idohs'
5
A
10
X. 2 a cos α = Wa sin α + y. 2a sin α
m
AB g
=
α =
tan α tan −1
(23 )
5
AB ys iu;=,s;;dj ie,lSfuka"
Y1 = 2W
5
5
ha
X1 = X = W
√ X12 + Y12 =
√5
5
W
vib
∴ RA =
RA ys †Ydj ;sri iuÛ idok fldaKh
(b)
10N
d
ww
B
X A Y
e
a
20
5
tan −1 (2)
w.
=
f E R
15
wa
2 3
.co
W 2 cos α = W sin α + W. 2 sin α
c
C
30° 30°
30°
g
D 5N
b
oKavl os. 2a hehs .ksuq' ^DE yer& moaO;sh i,ld" A (i) R 2a − 10 × 3a − 5 × 5a R
= 55 N 2
5
∴ E ys fhfok isria n,h
= 0
5
= 55 N 2 [16 jk mssgqj n,kak
- 16 − Y + R − 10 − 5 = 15 − 55 2 25 N ∴ Y = 2
−Y
= 0 = − 25 N 2
= 0
X
A wiõfõ m%;sl%shdfõ isria ixrplh
;sria ixrplh = 0
g
60° 60°
b
úYd,;ajh
c
AE
25 √3 N 6
10
AB
25 √3 N 3
wd;;sh
BE
25 √3 N 3
f;rmqu
BC
25 √3 N 3
wd;;sh
CE
20 √3 N 3
f;rmqu
CD
5 √3
wd;;sh
ED
10 N
f;rmqu
5
25 2 60°
a
vib
e
5
oKa~
d
20
20
m%;Hdn,h
35
w.
16. iuñ;sfhka ialkaO flakaøh x wlaIh u; fõ' y y
a
10
∫ πρ (a2 − x2) dx
5
ww
∫ πρx (a2− x2) dx
x =
0
a 0
[ [
3 a2x − x 3
= 3a
[ [
=
a2x2 − x4 2 4
a
x O G(x, y)
x
a
0
a
5
0
8 5 ∴ G ≡ 3a , 0 8
(
55
dx
5
= 0
f;rmqu
wa
60°
5
ha
(iii)
f
25 N 2
=
m
(ii)
.co
)
30
[17 jk mssgqj n,kak
- 17 y
(a)
ka O
x
a
iuñ;sfhka ialkaO flakaøh Ox u; fõ'
bj;a l< wr®O f.da,h b;s˙ fldgi
3a 8
2 π(ka)3 ρ 3
3ka 8
2 π a3 ρ (1 − k3) 3
x
3a (1 − k4) 8 (1 − k3)
=
5
15
(1 + k2) (1 − k) (1 + k) = 3a 8 (1 − k) (1 + k + k2)
vib
10
ha
2 π a3 ρ 3a 3 3 − 2 πk a ρ 3ka 3 3 8 8 2 π a3 ρ (1 − k3) 3
=
m
2 π a3 ρ 3
wa
wr®O f.da,h
x
O isg ialkaO flakaøhg ≥r
ialkaOh
.co
jia;=j
= 3a 8
(1 + k2) (1 + k) (1 + k + k2)
10
40
w.
(b) ialkaO flakaøh G1(x1, y1) f,i .ksuq' bj;a l< fldgi rEmfha whqre y iïnkaO l< úg o Ox jgd iuñ;sl ksid y1 = 0 fõ' 5
ww
O
G1
x
(i) bj;a l< fldgfia ialkaOh m o wrh a o jQ wr®O f.da,fha ialkaOh M hehso .ksuq' m = M
5 2π k3 a3 ρ 3 = k3 2π a3 ρ 3 5
5
m = Mk 3
20 [18 jk mssgqj n,kak
- 18 (ii) ixhqla; jia;=fõ ialkaO flakaøhg O isg ≥r x1 fõ' (M − m) x + m − 3 ka 15 8 x1 = ( M − m) + m
(
)
( ) − m ( 38 ka) neúka M ( 3a ) − m ( 3 ka ) − m ( 3 ka ) 8 8 8
;jo (M − m) x = M 3a 8 x1 =
5
3a (M − 2mk) 8 M
=
3a 8
k (1 − 2m M )
10
.co
=
m
M
5
2k4 = 1
1 √2
5
G1 O
10
vib
k2 = ±
ha
3a (1 − 2k4) = 0 8
P k2 > 0 ksid k2 = 1 √2
(i) P (A ∪ B) = P(A) + P(B) − P(A) P(B)
(A iy B iajdh;a; ksid)
ww
0.37
= 0.1 + P(B) − 0.1 P(B)
0.3
= P(B)
0.37 − 0.1 = 0.9 P(B) |
|
P( B ∩ A ) (ii) P(B' | A') = P(A| )
fuys P(B ∩ A ) = P[(B ∪ A) ] = 1 − P (A ∪ B) |
|
30
P(A) = 0.1, P(A∪B) = 0.37 iy P(C) = 0.2
w.
17. (a)
30
wa
ixhqla; jia;=fõ ialkaO 4 = 3a (1 − 2k ) flakaøhg O isg ≥r 8 (iii) G1 ialkaO flakaøh O iumd; úh hq;=h' 5 R tkï" x1 = 0 úh hq;=hs' 5
|
|
=
1 − 0.37
= 0.63
=
0.63 0.9
= 0.7
P(A ) =
|
|
∴ P(B | A )
1 − P(A) = 1 − 0.1 = 0.9
5 5 5
15
5 5 5 5
20
[19 jk mssgqj n,kak
- 19 -
|
|
(iii) P(A ∩ B ∩ C)
|
=
5
|
P(A ) P(B ) P(C)
=
0.9 × 0.7 × 0.2
5
= 0.126 |
|
|
|
|
5
|
(iv) X : (A∩B ∩C ) ∪ (A ∩ B ∩ C ) ∪ (A ∩B ∩ C) |
|
|
|
|
|
P(X) = P(A∩B ∩C ) + P(A ∩ B ∩ C ) + P(A ∩B ∩ C) |
|
|
|
|
|
= P(A) P(B ) P(C ) + P(A ) P(B) P(C ) + P(A ) P(B )P(C)
= 0.1 × 0.7 × 0.8 + 0.9 × 0.3 × 0.8 + 0.9 × 0.7 × 0.2
5
10
m
10
⇒ P(A | X) = P (A ∩ X) P (X)
' ' = P(A∩B ∩C ) P (X)
= 0.1 × 0.7 × 0.8 0.398
=
5
56 398 28
vib
5
ha
= 199 n
=
10
r=1 r
n
5
,l=Kj q , uOHkHh x = 28 + 56 + 23 + 94 + 8 + 5 + 13 + 846
ww
Σx
w.
(b) (i) (α) uOHkHh
5
28
1073
= 28 = 38.32
20
wa
.co
= 0.398
5
15
(β) 94 isg ,l=Kq 49 olajd wvq ù we;s neúka fjki − 45 la fõ'
05 isg ,l=Kq 50 olajd jeä ù we;s neúka fjki + 45 la fõ'
∴ uOHkHh fjkia fkdfõ' 5
5
10
[20 jk mssgqj n,kak
- 20 -
n 2 5 Σi = 1(xi − x ) iïu; wm.ukh = n
√
n 2 Σi = 1(xi − x ) 2 sx = úp,;djh = n n 2 Σi = 1(xi − 2xix + x 2 ) = n =
2
r=1 i
n
n
Σx
Σx 2 −2 x n i + x
5
2
i 2 2 −2 x + x = r=1 n
n
Σ
m
n
Σx
5
x 2 = i=1 i − x n X = {x1, x2, ..., x20} iy Y = {y1, y2, ..., y10} f,i .ksu'q 20 20 x = 320 iy xi2 = 5840
.co
Σ
i=1 i
5
i=1
10
10
y 2 = 2380 y = 130 iy i=1 i i=1 i
Σ
Σ
20
Σ
20
Σx i=1
2
i
20
= 292 − 256 = 36 ∴ s = 6 x
10
Σ
ww
yi = i = 1 = 130 = 13 y 10 10 10
Σy
5
2 2 − 16 = 5840 − 16 20
w.
2 sx =
5
vib
x i=1 i = x = 320 = 16 20 20
ha
iy
wa
Σ
2
5 5
2
2 − 13 = 2380 − 169 = 69 100 10 5 ∴ sy = 8.30 2
sy =
i=1 i
Z = X ∪ Y f,i .ksu'q 20
z
=
Σx
i=1 i
+
10
Σy i=1
i
30 = 320 + 130 = 15 30
5
[21 jk mssgqj n,kak
- 21 20
2
sz =
10
Σx + Σ i=1 i
2
30
i=1
= 274 − 225
yi2
−z
2
5
= 49
5
sz = 7
ww
w.
vib
ha
wa
.co
m
60
