fnYyh v/khuLFk lsok p;u vk;ksx izf'kf{kr Lukrd f'k{kd (efgyk oxZ) ijh{kk] 2018 gy iz'u&i= 1. lehdj.k x3 – 13x2 + 15x + 189 = 0 dks

gy

2. (C) fdUgha

nks ifjes; la[;kvksa ds chp ,d vkSj ifjes; la[;k gksrh gS] bls ?kuRo xq.k dgrs gSaA

djas] ;fn ,d ewy nwljs ls 2 vf/kd gSA (A) (3, –7, –9)

(B) (–3, 7, 9)

(C) (3, 7, 9)

(D) (–3, –7, –9)

1. (B) leh- x3 – 13x2 + 15x + 189 = 0

esa ewy a, a + 2, b gSA

— x 2 dk x.qkkad a x 3 dk xq.kkd 2a + 2 + b = 13

a + a + 2 + b =

b = 11 – 2a

...(i)

a(a + 2) + (a + 2) b + ab

x dk x.qkkad = 3 a x dk xq.kkd a2 + 2a + ab + 2b + ab = 15 a2 + 2a + 2b + 2ab = 15



0 1/8 1/4 3/8 1/2 3. fuEufyf[kr esa ls dkSu&ls osDVj (lfn'kksa) dk lewg R3 esa jSf[kd :i ls Lora= gSA

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

(B) 2 vkSj 3

(C) 1 vkSj 4

(D) 3 vkSj 4

3. (B) lHkh osDVj dk ;fn Determinant (lkj&

f.kd) ;fn 0 gks] rc osDVj jSf[kd :i ls Lora= gksaxsA

a + 2a + 2(11 – 2a) + 2a (11 – 2a) 2

a2 + 2a + 22 – 4a + 22a – 4a2

= 15

3a2 – 20a – 7 = 0

[(1,0,0), (0,1,0), (1,1,0)] [(1,0,0), (0,1,0), (0,0,1)] [(0,1,0), (1,0,1), (1,1,0)] [(0,0,1), (0,1,0), (0,1,1)]

(A) 1 vkSj 2

= 15

tSlsµ 0 vkSj 14 ds chp 18 la[;k gSA



;fn] 1 0 0



lewg (1) = 0 1 0 1 1 0

3a – 21a + a – 7 = 0

= 1 (0) – 0 – 0

3a (a – 7) + 1(a – 7) = 0

=0





2

(a – 7) (3a + 1) = 0



a–7=0

;k 3a + 1 = 0



a=7

;k



tc a = 7 rc leh- (i) ls



1 a = – 3

b = 11 – 2(7)

;g jSf[kd :i ls Lora= ugha gSA



= 1(1 – 0) – 0 + 0 =1

= 11 – 14

β=–3



\ a + 2 = 7 + 2 = 9



leh- ds ewy 7, 9, –3 gSaA

2. fdUgha

nks ifjes;] la[;kvksa ds chp ,d vkSj ifjesa;] la[;k gksrh gS] bls dgk tkrk gSA

(A) vkfdZfeMh;u (B) vlekurk (C) ?kuRo

xq.k

(fo"kerk)

xq.k

(D) e/;&eku

xq.k

lewg

1 0 0 (2) = 0 1 0 0 0 1

0 1 0

lewg (3) ÷ 1 0 1 1 1 0

= 0 – 1(0 – 1) + 0 =1 0 0 1

lewg (4) ÷ 0 1 0 0 1 1

= 0 – 0 + 1 (0) =0

vr% fodYi (2) rFkk (3) lR; gaS

ijh{kk frfFk % 22-11-2018 (f}rh; ikyh) 4. 143 vkSj 481 dk m, n cjkcj gSµ (A) – 9, 3

e-l- 143m + 481n gS] rc (B) – 8, 3

(C) – 10, 3

(D) – 7, 3

4. (C) 143 rFkk 481 dk

e-l-

481 = 143 × 3 + 52 143 = 52 × 2 + 39 52 = 39 × 1 + 13 39 = 13 × 3 + 0 143 481 3 429 52 143 2 104 39 52 1 39 13 39 3 39 ×



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

leh- (3) ls] 13 = 52 – 39 × 1 13 = 52 – (143 – 52 × 2) × 1 [leh- (2) ls] 13 = 52 – 143 × 1 + 52 × 2 = 52 × 3 – 143 × 1 = (481 – 143 × 3) × 3 – 143 × 1 [leh- (1) ls] = 481 × 3 – 143 × 9 – 143 × 1 = 481 – 3 – 143 × 10 = 143 × (–10) + 481 × 3 ...(5)

fn;k x;k] 13 = 143m + 48n dh rqyuk leh- (5) ls djus ij] m = – 10, n = 3 2 –6 5. b 3 l

dks fdl la[;k ls foHkkftr fd;k tk, –6 fd HkkxQy b 32 l ds cjkcj gSµ

2 6 (B) b 3 l 2 12 3 12 (C) b 3 l (D) b 2 l 5. (D) ekuk] la[;k = x 3 6 (A) b 2 l



iz'ukuqlkj]

b2l 3 x

–6

3 –6 = b 2 l

b2l 3 –6 = x b3l 2 –6

isij | 1



–6 b 2 × 2 l = x 3 3

2 b 3 l

2×– 6

–2+5+6–9=0

10. 360 ds

= x

2 –12 b 3 l = x

3 12 x = b 2 l V R SSS 1 1 1 WWW 6. vkO;wg SS a b c WW dk in gSµtgk¡ a = b SS 2 2 2WW Sa b c W X T ≠ c gSµ (A) 0 (B) 1 (C) 2 (D) 3 1 1 1 6. (C) a b c a2 b2 c2





0 0 0 1 1 1 a+b b+c c+a

gS %

(A) 7 (C) 0

cjkcj

8. lehdj.kksa 2x3 + 5x2 – 6x – 9 = 0 vkSj 3x3 + 7x2 – 11x – 15 = 0 ds nks mHk;fu"B ewy gS]

3 (B) b1, 2 l –3 (C) (1, –3) (D) b –1, 2 l 8. (A) lehdj.k 2x3 + 5x2 – 6x – 9 = 0 (A) (–1, –3)

2 |





x = – 1 – 3 + 7 + 11 – 15 = 0

x = – 1 j[kus

ij]

lHkh xq.ku[kaMksa dk ;ksx = 1170. Alternative Method : 360 = 23 × 32 × 51

\

xq.ku[k.Mksa dh la[;k



= (3 + 1) (2 + 1) ( 1 + 1) = 4 × 3 × 2 = 24



xq.ku[k.Mksa dk ;ksx



=



(x + 1) (3x2 + 4x – 15) (x + 1) (3x2 + 9x – 5x – 15) (x + 1) [3x(x + 3) –5 (3x – 1)] x = – 1, – 3, 5/3 nks u ka s lehdj.kks a ds mHk;fu"B (– 1, – 3) gSA

ew y

/kukRed iw.kk±d ds fy, 5

1 1 1 7. (A) 1 2 3 ! 0 1 4 k 1(2k – 12) + 1(3 – k) + 1(4 – 2) ≠0 2k – 12 + 3 – k + 2 k – 7 ≠ 0 k ≠ 7



45, 60, 72, 90, 120, 180 vkSj 360.

(x + 1) (2x2 + 3x – 9) (x + 1) (2x2 + 6x – 3x – 9) (x + 1) (x + 3) (2x – 3) 3 x = –1, –3, 2 leh- 3x3 + 7x2 – 11x – 15 = 0

9. n izR;sd

(B) 6 (D) 5

Kkr dhft,A

(D) 12, 810

2 3 + 1 - 1 3 2 + 1 - 1 51 + 1 - 1 2 -1 × 3 -1 × 5 -1 15 26 24 = 1 × 2 × 4 = 15 × 13 × 6 = 1170

ewyksa ds

?kuksa dk ;ksxQy Kkr dhft,A

0 0 0 = a–b b–c c–a ]a – bg]a + bg ]b – cg]b + cg ]c – ag]c + ag

vr% vkO;wg dk in 2 gSA 7. fuEufyf[kr ;qxir~ lehdj.k gS % x + y + z = 3 x + 2y + 3z = 4 x + 4y + kz = 6 k ds fy, ,d vf}rh; gy ugha gksxkA

(B) 24, 1080

(C) 24, 1170

11. lehdj.k x3 – 6x2 + 11x – 6 = 0 ds

0 0 0 a–b b–c c–a a2 – b2 b2 – c2 c2 – a2

= (a – b)(b – c)(c – a)

(A) 12, 540

10. (C) 360 ds xq.ku[kaM = 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40,

C 1 → C 1 – C 2, C 2 → C 2 – C 3, C 3 → C3 – C1 =

xq.ku[k.Mksa dh la[;k vkSj mudk ;ksx

gS %

3

n n7 + n5 + 23n – 105 gS % (A) fo"ke iw.kk±d (B) iw.kk±d (C) ½.kkRed okLrfod la[;k (D) ifjes; la[;k

n n 7 n 5 2n 3 9. (B) 7 + 5 + 3 – 105 tgk¡ n /kukRed iw.kk±d gSA



15n 7 + 21n5 + 70n3 – n 105 tc n = 1

...(i)

15 + 21 + 70 – 1 105 105 = 105 =1

tks ,d iw.kk±d gSA blh izdkj] tc n = 2 leh- (i) esa j[kus ij]

15 × 128 + 21 × 32 + 70 × 8 – 2 105 1920 + 672 + 560 – 2 = 105 3150 = 105 = 30 tks Hkh ,d iw.kk±d gSA

=

(A) 31 (B) 32 (C) 35 (D) 36 11. (D) fn;k x;k] leh- x3 – 6x2 + 11x – 6 = 0 ...(i) leh- (i) esa x = 1 j[kus ij ⇒ 1 – 6 + 11 – 6 = 0 vr% (x – 1), x3 – 6x2 + 11x – 6 = 0 dk ,d xq.ku[k.M gSA rc]



(x – 1) (x2 – 5x + 6) (x – 1) (x2 – 3x – 2x + 6) (x – 1) (x – 2) (x – 3) leh- ds rhuksa ewy 1, 2 rFkk 3 gaSA



iz'ukuqlkj] leh- ds rhuksa ewyksa ds ?kuksa dk ;ksx =

13 + 23 + 33 = 1 + 8 + 27 = 36 12. x esa f}?kkr cgqin dks tc (x – 1), (x – 2), (x – 3) ls foHkkftr fd;k tkrk gSA rc Øe'k% 4, 4, 0 'ks"kQy cprk gSA f}?kkr cgqin Kkr

dhft;sA

(A) –2x2 + 6x + 5 (B) –2x2 + 6x (C) –2x2 + 6x – 5 (D) –2x2 + 6x + 3

12. (B) ekuk] f (x) = ax2 + bx + c dksbZ

f}?kkr



tc cgqin f (x) dks x – 1 ls Hkkx nsrs gaSA



rc]

a2 = ^3 + 2 2 h = 9 + 8 + 12 2 2 a = 17 + 12 2

f (1) = 4

a + b + c = 4

...(1)





f (2)  4a + 2b + c = 4

...(2)



f (3)  9a + 3b + c = 0

...(3)





leh- (3) dks leh- (2) ls ?kVkus ij]

...(4) blh] izdkj leh- (2) dks leh- (1) esa ls ?kVkus ij] – 3a – b = 0 leh- (4) vkSj (5) ls]

...(5)



a dk



eku leh- (5) esa j[kus ij 6 – b = 0



b = 6

eku leh- (i) esa j[kus ij c = 0 \ f}?kkr cgqin] ax2 + bx + c a rFkk b dk

= –2x2 + 6x 13. ;fn a ,d iw.kZ la[;k vkSj p ,d vHkkT; la[;k gS] rks Fermat dh izes; ds vuqlkj% (A) 2p – 1 – 2, p ls foHkkT; gSA (B) 2p – 1, p ls foHkkT; gSA (C) 2p – 2, p ls foHkkT; ugha gSA (D) 2p – 2, p ls foHkkT; gSA 13. (D) Fermat dh

izes; vuqlkj]

ap ≡ a(mod p)



2p ≡ 2(mod p)

rc fodYi (D) lR; gksxkA

1 1 14. ;fn a = , b= 3–2 2 3+ 2 2 a2 + b2 dk eku gSµ (A) 36 (B) 37 (C) 34 (D) 35 14. (C) fn;k



gS] rks

x;k gS]

1 a = 3–2 2

rFkk

1 b= 3+ 2 2

3+ 2 2 1 × 3– 2 2 3+ 2 2 3+ 2 2 = 9–8 7]a + bg]a – bg = a2 – b2A a =

(C) iz R ;s d

a = 3 + 2 2

2

cgqin gSA

...(1)

blh izdkj] b=

3–2 2 1 × 3+ 2 2 3– 2 2

3–2 2 9–8 b = 3 – 2 2 b2 = 9 + 8 – 12 2 b2 = 17 – 12 2 ...(2) leh- (1) o (2) dks tksM+us ij] a2 + b2 = 17 + 12 2 + 17 – 12 2 = 34 15. ;fn ,d lfn'k lef"V V3 ls okLrfod la[;k R3 ds lHkh f=Hkqtksa dk leqPp; (x1, x2, x3) gS] rc ,d mèokZ/kj ry y = x ls fu:fir b =

mi&lef"V leqPp;ksa ds jSf[kd la;kstu }kjk izkIr dh tk ldrh gSµ (A) (1, 1, 0) vkSj (0, 0, 1) (B) (1, 0, 1) vkSj (0, 0, 1) (C) (1, 0, 0) vkSj (0, 1, 0) (D) (1, 1, 0) vkSj (1, 0, 0) 15. (A) iz'u esa fn;k x;k gS] fd R3 ,d lfn'k lef"V gS rFkk milef"V {(x1, x2, x3) ∈

R3 : y = x}

rc] lHkh f=xq.kksa esa ls fodYi (A) bldks lUrq"V djrk gSA 16. ;fn µ y ;wyj dk VksfV;aV Qyu gS] rc y f (92) gS%

(A) 44 (B) 46 (C) 48 (D) 42 16. (A) j(92) 92 = 23 × 2 × 2 j(23). j(4) ;wyj dks VksfV;aV Qyu ds vuqlkj] j(P) = P – 1 (tgk¡ P ,d vHkkT; la[;k gSA) = 22 × 2 = 44 17. 39312 dks fdrus rjhdkas ls nks xq.ku[kaMksa esa

foHkkftr fd;k tk ldrk gS] tks ,d&nwljs ls vHkkT; gSµ (A) 6 (B) 10 (C) 8 (D) 4 17. (C) 39312 = 24 × 33 × 7 × 13 ;s pkjksa i`Fkd la[;k gS] nh xbZ

la[;k ds xq.ku[kMa gSA rjhdksa dh la[;k = 23 = 8. 18. fuEufyf[kr esa ls dkSu&lk vlR; gSµ (A) 2 ,d ifjes; la[;k gSA (B) izR;sd vifjes; la[;k ,d okLrfod la[;k gSA

iw . kk± d ,d okLrfod la [ ;k gSA (D) izR;sd ifjes; la[;k ,d okLrfod la[;k gSA 18. (D) okLrfod l[;k,¡ ifjes; la[;k,¡ ugha gksrh] mUgsa vifjes; la[;k dgrs gSa] tSlsµ 2 , 8 , p vkfnA 19. pkbuht 'ks"kQy izes; }kjk X = 3(mod 5), x = 5(mod 7) dks gy djsaA mHk;fu"B gy gSµ (A) X = 33 (mod 35) (B) X = 31 (mod 5)

(C) X = 27 (mod 35) (D) X = 28 (mod 35) 19. (A) X = 3(mod 5) X = 5 (mod 7) M = m1 m 2 = 5 × 7 = 35 M 35 Z1 = m1 = 5 Z1 = 7

blh izdkj]

Z2 = 5 yi = (Zi)–1 (mod Mi) 1 y1 = 7 (mod 5) 7y1 = 1 (mod 5)

y1 = 3 1 y2 = 5 ]mod 7g 5y2 = 1 (mod 7)

y2 = 3

X = (a1 y1 Z1 + a2 y2 Z2) (mod M) = (3 × 3 × 7 + 5 × 3 × 5)

(mod 35)

= 63 + 75 (mod 35) = 138 (mod 35) X = 33 mod (35) 20. fuEufyf[kr

esa ls dkSu&lk leqPp; milef"V

W = )=

x y G, x + 2y + t = 0, y + t = 0 3 0 t

dk vk/kkj gSµ –1 1 G3 (A) )= 2 –1 1 –1 G3 (B) )= 0 1 1 0 0 1 0 0 (C) )= G, = G, = G3 0 0 0 0 0 1 2 1 1 –1 G, = G3 (D) )= 0 –1 0 1

isij | 3

20. (B) W

,d nks vk;keh milef"V gSA ge tkurs gSa] fd nks jSf[kd Lora= osDVj dsoy vkèkkj ij gSA vr% fodYi 1 ∉ W fodYi 3 ∉ W fodYi esa 2 ls T;knk vk;keh milfe"V gSA vr% fodYi (B) lR; gSA 21. lekUrj

Js.kh ds izFke 20 inksa dk ;ksxQy gS] ftldk izFke in 5 vksj lkoZvUrj 4 gSµ

(A) 830

(B) 850

(C) 820

(D) 860

(

dk xq.kkad ) x dk xq.kkd a

— x2

(a + b)2 = a2 + b2 + 2ab (12)2 = a2 + b2 + 2 × 32 a2 + b2 = 144 – 64 a2 + b2 = 80

α2 + β2 80 rc] α + β = 12 20 = 3 23. ;fn a1x + b1y + c1 = 0 vkSj a2x + b2y + c2 = 0 esa vuUr gy gS] rks fuEufyf[kr esa dkSu&lk c ! c12 c ! c12 c = c1 2 c1 =c 2



1 :0, 1 D (C) :– 2 , 1D (D) 2 24. (C) a2 + b2 + c2 = 1 (fn;k gS)

leh- (1) rFkk (2) dk xq.kk djus ij



⇒ 19 (a + b + c) b 1a + 1b + 1c l >

1 1 1 1/ 3 (abc)1/3 b a . b . c l



a2 + b2 – 2ab + b2 + c2 – 2bc + c2 + a2 – 2ca > 0 1 – (ab ≠ bc + ca) > 0



ab + bc + ca < 1



1 ab + bc + ca :– 2 , 1D

ds vUrjky esa

lehdj.k ax + bx + c = 0 ds nks ewy ]a – bg ]b – cg (a – b) vkSj (b – c) gS] rc c–a dk eku gSµ 2

b c (A) b (B) c bc ab (C) a (D) c 25. (A) leh- ax2 + bx + c ds nks ewy (a – b) rFkk (b – c) gSaA (a – b) + (b – c) = – ba b α + β = –b l a –b a – c = a ...(1) c (a – b) (b – c) = a b αβ = c l a ]a – bg]b – cg c a c – a = a × b c = b

1 1 1 26. ]a + b + cg b a + b + c l (A) 9 ls de vkSj cjkcj

gSµ

(B) 3 ls

1 (abc)1/3 ]abcg1/3 1 + 1 + 1l ^a + b + c h b a b c $9

⇒



;k 9 ls vf/kd vkSj cjkcj

27. ;fn Øfer ;qXe lehdj.k 2x – 3y = 18 vkSj 4x – y = 16 dks lUrq"V djrk gS] lehdj.k 5x – py – 23 = 0 dks Hkh lUrq"V djrk gSA rc p dk eku Kkr dhft,µ (A) –1

(B) 1

(C) 2

(D) –2

27. (C) fn;s

fLFkr gSA

25. ;fn

⇒ 19 (a + b + c) b 1a + 1b + 1c l >



2



x;s leh-

2x – 3y = 18

...(1)

4x – y = 16

...(2)



leh- (1) eas 2 dk xq.kk djds leh- (2) ls ?kVkus



–20 5 ,y=–4 y dk eku leh- (2) esa j[kus ij 4x – (– 4) = 16 4x + 4 = 16 4x = 16 – 4 4x = 12 x = 3 x rFkk y dk eku leh- 5x – py = 23 esa j[kus ij] 5 × 3 – p × (4) = 23 28. ;fn

26. (C) ge

(A) 2 1 (C) 2

tkurs gaS] lekUrj ek/; > xq.kksÙkj ek/; a + b + c > (abc)1/3 ...(1) vkSj

3

y =



de vkSj cjkcj (C) 9 ls vf/kd vkSj cjkcj (D) 3 ls vf/kd vkSj cjkcj

4 |

2

2(a2 + b2 + c2) – 2(ab + bc + ca) > 0

ab = 32

b ! b1 2 b1 =b 2 b1 !b 2 b1 =b 2

1 :–1, 1 D (A) : 2 , 1D (B) 2



nks



esa fLFkr gSµ

1 + 2 (ab + bc + ca) > 0 1 ab + bc + ca > – 2 (a – b)2 + (b – c)2 + (c – a)2 > 0

a + b = 12

a (A) a1 2 a1 (B) a 2 a1 (C) a 2 a1 (D) a 2

a1 b1 c1 a2 = b2 = c2 24. ;fn a2 + b2 + c2 = 1 rc ab + bc + ca vUrjky



ds nks ewy a rFkk b gSA

lR; gSµ



vuUr gy ds fy;s

(a + b + c ) + 2 (ab + bc + ca) > 0

= 860

a + b =



1 b 1 + 1 + 1 l b 1 1 1 l1/3 3 a b c > a b c ...(2)



a2x + b2y + c2 = 0

2

= 10 × 86



a1x + b1y + c1 = 0

(a + b + c)2 > 0

21. (D) lekUrj Js.kh ds n inksa dk ;ksxQy Sn = 2n 62a + ]n – 1g d @ izFke in a = 5 rFkk lkoZvUrj d = 4 20 = 2 62 × 5 + ]20 – 1g × 4@ = 10[10 + 19 × 4]

22. ;fn a vkSj b leh- x2 – 12x + 32 = 0 ds α2 + β2 ewy gS] rc α + β dk eku gSµ 8 8 (A) 3 (B) –3 20 20 (C) – 3 (D) 3 22. (D) fn;k x;k leh- x2 – 12x + 32 = 0

23. (D)

15 + 4p = 23 4p = 23 – 15 82 p = 4

p=2

lekÙkj Js.kh ds 16 inksa dk ;ksx 1624 gS] vkSj izFke in lkokZUrj dk 500 xquk gS] rc lkokZUrj Kkr dhft,A 1 (B) 5 (D) 5

28. (B) lkoZUrj = d

31. (D) fn;k





iz'ukuqlkj]

gSµ



Sn = 4n2 + 3n





Sn – 1 = 4(n – 1)2 + 3(n – 1)

203 = 1015d 203 d = 1015 1 d = 5 29. ;fn lehdj.k x2 + px + 12 = 0 dk ,d ewy 4 gS] tcfd lehdj.k x2 + px + q = 0 ds nksuksa ewy cjkcj gS] rc q dk eku gSµ (A) 4 (B) 12 49 (C) 3 (D) 4 29. (B) leh- x2 + px + 12 = 0

dk ,d ewy a = 4

(4)2 + p × 4 + 12 = 0

16 + 4p + 12 = 0



4p = – 28



p = – 7



nksuksa ewy cjkcj gSaA

x2 + px + q = 0 ds



rc] lkekU; in an = Sn – Sn – 1

c2 + b2 = 2a2

= 4n2 + 3n – [4(n2 + 1 – 2n) + 3n – 3]

35. eqds'k

= 4n + 3n – 4n – 4 + 8n – 3n + 3 2

2

= 8n – 1 32. ;fn

ifjes; xq.kkadkas okys f}?kkr lehdj.k dk ,d ewy 7 – 4 gS] rks f}?kkr lehdj.k gSµ (A) x2 – 2 7 x – 9 = 0 (B) x2 – 8x + 9 = 0 (C) x2 + 8x + 9 = 0 (D) x2 – 2 7 x + 9 = 0 32. (A) igyk

ewy] a = 7 – 4

nwljk

ewy] b = 7 + 4

a + b =

ab = ^ 7 – 4 h^ 7 + 4 h

= ^ 7 h – ]4 g 2

2

= 7 – 16



x2 – 2 7 x – 9 = 0

(A) x – 7 = 0 3

(B) x3 – 5x2 + 7x – 3 = 0 (C) x3 + 7x2 – 3 = 0 (D) x3 + 7x2 + 3 = 0 30. (B) fn;k

x;k gS]



a + b + g = 5



ab + bg + ga = 7



abg = 3



⇒ x3 – (a + b + g)x2 + (ab + bg + ga) x – abg = 0



x3 – 5x2 + 7x – 3 = 0

31. lekUrj

Js