[ ] ( l) ( ) Option 2. Option 3. Option 4. Correct Answer 1. Explanation n. Q. No to n terms = ( 10-1 ) 3
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1 Q. No. The fist d lst tem of A. P. e d l espetively. If s be the sum of ll tems of the A. P., the ommo diffeee is Optio l - s- l+ Optio Optio Optio 4 Coet Aswe ( ) l - s- - ( l ) l + s+ + ( l ) l + s- + ( l ) Expltio l + ( -) d s [ + l] s +l s l - + d l l - d s - + ( l) Q. No to tems If 7, the to 0 tems Optio 5 Optio 6 Optio 7 Optio 4 40 Coet Aswe Expltio 6+ ( -) ( 0- ) [ 4+] [ ] [ + ] 5 (7) 5
2 Q. No. If,,,... e i A.P. whee > 0V, the Optio - Optio + Optio - Optio Coet Aswe Expltio d -d -d d - - -d - d ( ) Q. No. 4 If the sum of the 0 tems of A.P. is 4 times to the sum of its 5 tems, the the tio of fist tems d ommo diffeee is Optio : Optio : Optio : Optio 4 : Coet Aswe Expltio [ ] +9 d 4 [ +4d] 5 + 9d 4 + 8d d d
3 Q. No. 5 A seies whose th tems is + y, the sum of tems will be x Optio ( +) + y, x Optio ( -) x Optio ( -)- y, x Optio 4 ( +) - x y Coet Aswe Expltio s + y x ( + ) + y x Q. No. 6 If α, β be oots of x - x + 0 d γ, δe δ the oots of x - x + b 0 d α, β, γ, δ (i ode) fom iesig G.P., the Optio, b Optio, b Optio, b Optio 4 4, b 6 Coet Aswe Expltio α A, β A, γ A, δ A A + A, A. A A + A A.A b (A + A) ± A b Q. No. 7 If, b, e thee uequl umbes suh tht, b, e i A.P. d b -, - b, e i G.P., the : b : Optio : : 5 Optio : : 4 Optio : : 5 Optio 4 : : Coet Aswe 4 Expltio (- b) (b- ) () d d ( ) d b + d + d tio is : :
4 Q. No. 8 The sum of tems of the followig seies +(+ x)+(+ x+ x )+... will be Optio -x -x Optio x(- x ) -x Optio (- x)- x(- x ) (- x) Optio 4 oe Coet Aswe Expltio (- x)+(- x)(+ x)+(- x) ( + x+ x ) +... (- x) (- ) x x x x ( -x ) -x (- x) -x Q. No. 9 If the thid tem of G.P. is 4, the the podut of its fist 5 tems is Optio 4 Optio 4 4 Optio 4 5 Optio 4 Noe Coet Aswe Expltio Let the 5 tems be,,,, 4 podut Q. No. 0 If, b, e thee uequl umbes suh tht, b, e i A.P. d b -, - b, 4 - e i G.P. the : b : is Optio : : Optio : : 4 Optio : : 4 Optio 4 : : 4 Coet Aswe Expltio (- b) (b- ) (4- ) d d(4-( + d ) d - d d d b + d + d Rtio : :
5 Q. No. The lest vlue of fo whih the sum to tems is gete th Is Optio 7 Optio 9 Optio Optio 4 Coet Aswe Expltio > - > 4000 > 400 Q. No. If, 4, b e i A.P;,, b e i G.P., the,, b e i Optio HP Optio AP Optio GP Optio 4 oe of these Coet Aswe Expltio + b 8 b 4 b + b HP Q. No. If p, q, e i A.P., the pth, qth d th tems of y G.P. e i Optio A.P. Optio G.P. Optio H.P. Optio 4 A.G.P. Coet Aswe Expltio q p + T p R p - T q R q- T R - T q R q- T p. T R p + - R q - T q T p.t Q. No. 4 I G.P. if the(m + ) th tem be p d (m- ) th tems be q the the m th tems is Optio pq Optio p/ q Optio q/ p Optio 4 p/ q Coet Aswe Expltio T m + p m+- T m - p m-- pq m- m - pq Tm
6 Q. No. 5 The fouth, seveth d teth tems of G.P. e p, q, espetively the, Optio p q + Optio q p Optio p q Optio 4 pq + pq + 0 Coet Aswe Expltio T y p R T 7 q R 6 T 0 R q q p Q. No. 6 The sum of ifiite G.P. seies is. A seies whih is fomed by sques of its tems hve the sum lso. Fist seies will be Optio,,, Optio,,, Optio,,, Optio 4,-,,-,... Coet Aswe Expltio S - S Seies,,,
7 Q. No. 7 Optio A.P. Optio G.P. Optio H.P. Optio 4 Noe Coet Aswe 4 Expltio b + + b If, b, e i A.P., the,. e i b b 4,, ( + ) ( + ) 4 ) + ( + ) ( + ) would be +4 ( + ) ( + ) ( + + ) ( + ) ( + ) + ( + ) Not ( ) 8 b) ( + ) Not (b) ) ( + ) + 4 ( + ) +( + ) 4 ( + )(+ ) 4 Not ( ) Q. No. 8 If, b, e i H.P., the whih oe of the followig is tue Optio + b- b- b Optio b + Optio b+ b+ + b- b- Optio 4 Noe Coet Aswe 4 Expltio ) + S b + + b b- b-( + ) b ( b- )( b- ) b - b ( + )+ b - b b - +
8 ( b - ) b( b - ) b Flse b) Flse ( b+ )( b- )+( b+ )( b- ) ) ( b- )( b- ) b + b - b - + b - b + b b - b( + )+ ( b - ) b( b - ) Flse Q. No to tems Optio + - Optio Optio - - Optio 4 Noe Coet Aswe Expltio - T - - S - ( -) Q. No. 0 If x,, z e i A.P. d x,, z e i G.P., the x, 4,z will be i Optio A.P. Optio G.P. Optio H.P. Optio 4 Noe Coet Aswe Expltio x + z xz 4 xz 8 4 x+ z HP
9 Q. No. If the sum of tems of G.P., is S, podut is P d sum of thei iveses is R, the P Optio R S Optio S R Optio R S Optio 4 S R Coet Aswe 4 Expltio - S - P ( (- ) - R. - - R - S R - S R - ( -) P Q. No. If, b, e positive umbe i A.P. d, b, e i H.P., the Optio b Optio b + Optio / b 8 Optio 4 Noe Coet Aswe Expltio b + b ( + + ) ( + ) 8 ( + ) + ( + ) ( + ) ( + )- 4 ( - ) + ( + - ) 0 ( + ) (- ) + (- ) 0 (- ) [( + ) + ] 0 b
10 Q. No. If, b, e i H.P., the the vlue of Optio 0 Optio Optio Optio 4 Coet Aswe Expltio b + b ( b+ )( b- )+( b+ )( b- ) ( b- )( b- ) b + b - b - + b - b + b - ( b -) b - b( + )+ b - b + b ( b ) ( b -) - b b b- b- is Q. No. 4 The sum of thee oseutive tems i G.P. is 4. If is dded to the fist d the seod tems d subtted fom the thid, the esultig ew tems e i A.P. The the lowest of the oigil tems is Optio Optio Optio 4 Optio 4 8 Coet Aswe Expltio ( + ) + ( - ) ( + ) (- 8) (- ) 0, 8, Lowest tem is
11 Q. No. 5 If l(x + z) + l(x - y + z) l(x - z), the x, y, z e i Optio A.P Optio GP Optio H.P. Optio 4 oe of these Coet Aswe Expltio ( + )( - ) l ( x z x y+ z) l( x+ z) (x + z)(x - y + z) (x- z) x - xy+ xz+ xz- yz+ z x + z -xz 4xz xy + yz xz xy + yz Divided by xzy + HP y x Q. No. 6 The vlue of will be Optio Optio Optio Optio 4 Coet Aswe Expltio Q. No. 7 If x y b z d, b, i G.P., the x, y, z e i Optio A.P. Optio G.P. Optio H.P. Optio 4 oe Coet Aswe Expltio x y b z t t x t z b t y t x+z y x + z
12 Q. No. 8 The hmoi me of oots of the equtio (5+ ) x -(4+ 5) x+(8+ 5)0is Optio Optio 4 Optio 6 Optio 4 8 Coet Aswe Expltio αβ H.M. α + β Q. No. 9 The hmoi me betwee two umbes is getest umbe betwee them is : Optio 7 Optio 6 Optio 8 Optio 4 60 Coet Aswe Expltio 7 HM4 5 5 GM 4 b 7 b4 + b 5 (576) 7 + b 5 + b , 8 Lgest No d the geometi me is 4. The Q. No. 0 If, b d e positive el umbes, the the lest vlue of ( + b+ ) + + is b Optio 9 Optio Optio 0/ Optio 4 oe of these Coet Aswe Expltio + b+ + + b ( + b+ ) b Lest vlue 9
13 Q. No. b If, b d e positive el umbes the + + b Optio Optio 6 Optio 7 Optio 4 oe of these Coet Aswe Expltio b + + b b.. b b + + b is gete th o equl to Q. No. Let x be the ithmeti me d y, z be the two geometi mes betwee y two y + z positive umbe. The vlue of is xyz Optio Optio Optio Optio 4 Coet Aswe Expltio + b x b b b y. b b z b y + z b+ b xyz + b ( b) b( + b) + b ( b)
14 Q. No. Six ithmeti mes e iseted betwee d 9, the 4th ithmeti me is Optio Optio Optio Optio 4 4 Coet Aswe Expltio 9,,,, 4, 5, 6, 9 +7 d 7 7 d d 4 + 4d +4. Q. No. 4 The sum of seies upto 0 tems is Optio Optio Optio Optio 4 Noe Coet Aswe Expltio S T 0 T ( + ) ( ) ( +) ( +)( +) ( +) S S 4(0) + 4(870) Q. No. 5 The sum (!) + (!) + (!)+ +(!) equls to Optio (!) + - Optio ( +)!-(- )! Optio ( +)!- Optio 4 (!)- - Coet Aswe Expltio T (!) (( + ) - )! T ( + )! -! S ( +)!-
15 Q. No. 6 If th tem of seies is ( +)( +)' the sum of ifiite tems of the seies Optio Optio Optio 5 Optio 4 5 Coet Aswe 4 Expltio T T - 4 T - 5 T 4-6 T S + Q. No. 7 The sum of the tems of the seies + ( + ) + ( + + 5).. Optio Optio ( +) Optio ( +)( +) 6 Optio 4 oe of these Coet Aswe Expltio T ( +)( +) S 6 Q. No. 8 The sum of the seies is Optio Optio Optio Optio Coet Aswe 4 Expltio It is GP with S
16 Q. No. 9 If the sum to tems of seies be 5 +,the seod tems is Optio 5 Optio 7 Optio 0 Optio 4 5 Coet Aswe Expltio S Q. No. 40 Let 4 f( ), the ( -) Optio f() - 6f() Optio f() - 7f() Optio f( - ) - 8f() Optio 4 oe of these Coet Aswe Expltio f( ) 4 4 ( -) (- ) ( ) f() - 6f() 4 is equl to Q. No. 4 Coeffiiet of x 99 i the polyomil (x- )(x - ) (x- 00) is Optio 00! Optio Optio 5050 Optio 4-00 Coet Aswe Expltio Coeffiiet of x 99 ( ) Q. No Optio A miimum Optio A mximum Optio A miimum 50 Optio 4 A mximum 50 Coet Aswe Expltio x + x x 50 If x + x + x x 50d ( x +... x50 ) x +... x 50 x x... x Athe
17 m Mx vlueof x... x50 A miimum Q. No. 4 Give p A.P. s, eh of whih osists of tems. If thei fist tems e,,,, p d ommo diffeees e,, 5,, p - espetively, the sum of the tems of ll the pogessios is Optio p( p +) Optio ( +) p Optio p( + ) Optio 4 oe of these Coet Aswe Expltio S [ +( -)] ( +) S [ 4+( -)] [ +] S [ 6+( -)5] [5 +] S [( -) +] Sum of ll tems is. ( p )( p +) - p +P p + p p [ p +] Q. No. 44 If oe G.M. g d two A.M. s p d q e iseted betwee two umbe d b, the (p- q)( p- q) g Optio Optio - Optio Optio 4 - Coet Aswe Expltio g b, p, q, b e i Ap b + d b- d + b- + b p
18 b- +b q-- (p- q)( p- q) 4 + b -- b g + b--4b b (- b) - b Q. No. 45 Optio Optio Optio Optio 4 Coet Aswe Expltio x (0 digits) If x (0 digits), y (0 digits) d z (0 digits,) the x 9 9 y ( )(0 digits) ( ) y y 0 0 ( 0 -) ( 0 -) x y z 0 ( ) ( 0 -)( 0 +) -( 0 -) 0 ( 0 -) x-y z Q. No. 46 The sum of the fist 0 tems of is Optio Optio Optio - -0 Optio 4 oe of these Coet Aswe Expltio S S
19 Q. No. 47 The sum of the seies to 0 tems is Optio 9600 Optio 760 Optio Optio Coet Aswe Expltio tems [ ].. [ ] 0 40 (4) 0 0 () -8 (0) - 8 (0) Q. No to tems Optio 0 Optio Optio -0 Optio 4 - Coet Aswe Expltio (- )( + ) + (- 4)( + 4) + (9-0)(9 + 0) + -[ ] Q. No. 49 Coside the sequee,,,,, whee ous times. The umbe tht ous s 007 th tem is Optio 6 Optio 6 Optio 6 Optio 4 64 Coet Aswe Expltio ( +) 007 ( +) 404 6
20 Q. No. 50 If the A.M. of two umbes d b, > b > 0is twie thei G.M., the b Optio + Optio 7+ Optio 4+ Optio Coet Aswe 4 Expltio AM GM A+ A -G No A- A -G G+ G G- G Q. No. 5 A G.P. osist of eve umbe of tems. If the sum of the tems oupyig the odd ples is S d tht of the tems i the eve ples is S, the the ommo tio of the G.P. is Optio S S Optio Optio Optio 4 S S S S S S Coet Aswe Expltio,,, S - S - S S ( ) - ( ) -
21 Q. No. 5 If 5x - y, x + y, x + y e i A.P. d (x- ),(xy + ),(y + ) e i G.P., x 0, the x + y Optio 4 Optio Optio -5 Optio 4 oe of these Coet Aswe Expltio (x + y) 5x - y + x + y 4x + y 6x + y y x (xy + ) (x- ) (y + ) xy+ ± ( xy+ x-y-) xy + xy - y+ x - -x x - y -4 xy + -xy- x+ y + x 4 y x+ y -6 o 4 Q. No. 5 If the sum of m oseutive odd iteges is m 4, the the fist itege is Optio m + m + Optio m + m - Optio m - m - Optio 4 m - m + Coet Aswe 4 Expltio. +(m+) +m m m - m + Q. No. 54 The lgest positive tem of the H.P., whose fist two tems e d 5 is Optio Optio 6 Optio 5 Optio 4 8 Coet Aswe Expltio 5, e i A. P 5-7 d - Lst positive tem : 0
22 5-7 +( -) ( -) 0 ( -) Lgest tem of H.P 6 Q. No. 55 Fou distit iteges, b,, d e i A.P. If + b + d, the + b+ +d Optio Optio 0 Optio - Optio 4 oe of these Coet Aswe 4 Expltio + b + d -, b 0,, d + b + + d Q. No. 56 Assetio : If ll tems of seies with positive tems e smlle th 0-5, the the sum of the seies upto ifiity will be fiite. Reso : If S < the lim S is fiite. Optio Optio 5 0 If ASSERTION is tue, REASON is tue, REASON is oet expltio fo ASSERTION. If ASSERTION is tue, REASON is tue, REASON is ot oet expltio fo ASSERTION. If ASSERTION is tue, REASON is flse If ASSERTION is flse, REASON is tue ASSERTION d REASON Both e flse Optio Optio 4 Optio 5 Coet Aswe 5 Expltio ASSERTION d REASON Both e flse
23 Q. No. 57 Assetio : If thee positive umbe i G.P. epeset sides of tigle, the the 5-5+ ommo tio of the G.P. must lie betwee d Reso : Thee positive el umbes fom sides of tigle if sum of y two is gete th the thid. Optio If ASSERTION is tue, REASON is tue, REASON is oet expltio fo ASSERTION. Optio If ASSERTION is tue, REASON is tue, REASON is ot oet expltio fo ASSERTION. Optio If ASSERTION is tue, REASON is flse Optio 4 If ASSERTION is flse, REASON IS tue Optio 5 ASSERTION d REASON Both e flse Coet Aswe Expltio + > + - > 0 - ± ,, () - > -- > 0 ± , () Tkig itesetio 5-5+, Q. No. 58 Assetio : Thee exists A.P. whose thee tems e,, 5. Reso : Thee exists distit el umbes p, q, stisfyig Optio Optio Optio Optio 4 Optio 5 Coet Aswe Expltio A+( p-) d A+( p-) d, A+( q-) d, 5A+( -) d. If ASSERTION is tue, REASON is tue, REASON is oet expltio fo ASSERTION. If ASSERTION is tue, REASON is tue, REASON is ot oet expltio fo ASSERTION. If ASSERTION is tue, REASON is flse If ASSERTION is flse, REASON IS tue ASSERTION d REASON Both e flse A+( q-) d 5A+( -) d - p-q - 5 q- Itiol Rtiol
24 No suh t exists Assetio d eso both e flse. Q. No. 59 Assetio : The mximum umbe of ute gles i ovex polygo of sides is Reso : The sum of itel gles of y ovex polygo is (- ) 80 0 Optio If ASSERTION is tue, REASON is tue, REASON is oet expltio fo ASSERTION. Optio If ASSERTION is tue, REASON is tue, REASON is ot oet expltio fo ASSERTION. Optio If ASSERTION is tue, REASON is flse Optio 4 If ASSERTION is flse, REASON IS tue Optio 5 ASSERTION d REASON Both e flse Coet Aswe Expltio Let us ssume thee exist 4 ute gles Thei sum is less th 60. So, sum of est (- 4) gles should be gete th (- ) (- 4)80 Some gles must be gete 80 Hee ot ovex polygo. Q. No. 60 Assetio : The sum of ifiite A.G.P. +( + d) x+( + d) x +( + d) x +..., whee x < lwys exist. Reso : The sum of the ifiite seies oveges if < Optio If ASSERTION is tue, REASON is tue, REASON is oet expltio fo ASSERTION. Optio If ASSERTION is tue, REASON is tue, REASON is ot oet expltio fo ASSERTION. Optio If ASSERTION is tue, REASON is flse Optio 4 If ASSERTION is flse, REASON IS tue Optio 5 ASSERTION d REASON Both e flse Coet Aswe Expltio If ASSERTION is tue, REASON is tue, REASON is oet expltio fo ASSERTION. Q. No. 6 If the fist two tems of pogessio e log 56 d log 8 espetively, the whih of the followig sttemets e tue : Optio If thid tem is log 4 6, the the tems e i G.P. Optio If thid tem is log 6, the the tems e i A.P. Optio If thid tem is log 6, the the tems e i H.P. Optio 4 If the thid tem is log 8, the tems e i A.P. Expltio log 8 8 log 8 4 ) log 4 4 Coet b) 0, Coet 4 8 ) log the Coet d) 0 Flse
25 Q. No The vlue of fo whih , is Expltio S S ( +) +. -S ( ) - -S - - -S S (- ) ( -) + ( -).. 9 (- ) Q. No. 6 5 If S - + the S, the the umeil qutity k must be + ( +) k Expltio S ( +) 5 S S S S S S k544
26 Q. No. 64 Expltio The oly itege solutio of the equtio ( x-) +( x-) +( x-) +( x-007) 0is This is zeo t middle tem Q. No. 65 Two oseutive umbes fom,, e emoved. The ithmeti me of emiig - umbes is 05. The must be 4 Expltio + - x - y 05-4 ( + ) - 4( + y) 05( - ) x + y x x 8 x should be the itege should be eve x x 4 (- ) +0 x 4 5, 9,, 7,, 5, fo 5 x 7 50 Fo > 5, x > must be 50
27 Q. No. 66 The sum of the sques of thee distit el umbes, whih e i G.P., is S. If thei sum is αs, show tht α (,). ' Expltio,, ( -) αs - ( ) 6 ( -) αs + + S - S ( ) S ( + + )( - + ) ( + +) S α α ( + +) α t - + (t- ) -(t- ) + (t- ) 0 0, t ( t-) 4( t-)( t-) 0 ( t-) -(t -) 0 ( t++t-)( t+- t+) 0 (t -)(- t+) 0 (t -)( t+) 0 t, -{,} Fo t whih is ot possible α (,) ' ( )( ) Q. No ( ) Show tht (-)( +) 48 6( +) Expltio 4 T ( -)( +) ( )( ) 6( 4 -) T + 6( -)( +) T ( -)( +) S S
28 ( +)( +) S ( ) Fo S T T - T - 5 T S - + S 6( +) ( ) S ( +) Q. No. 68 Fid the sum of the seies tems Expltio T..5...( -)(+) T ( -)..5...( +) T -. T T T ( -)..5...( +) ( +)
29 Q. No. 69 A sequee of el umbes,,, is suh tht Expltio 0 0, +, +, pove tht - i. i ( - ) + ( ) ( ) ( -) ( ) - Poved. Q. No. 70 Fo positive el umbes x, y, z pove tht Expltio ( x+ y+ z) ( x+ y+ z) x + y + z x y z x+ y+ z X Y Z x+ y+ z x+ x+... x+ y+ y+... y+ z+ z... z x y z ( x y z ) x+ y+ z x+ y+ z x + y + z x y z x + y + z x+ y+ z x y z x + y + z x+ y+ z x + y + z x+ y+ z x+ y+ z x + y + z x+ y+ z x+ y+ z x+ y+ z x+ y+ z
30 Q. No. 7 If the equtio x 4-4x + x + bx + 0 hs fou positive oots, the fid d b. Expltio x 4-4x + x + bx + 0 Let the oot be α, β, γ, δ α + β + γ + δ 4 AM 4 4 GM( αβ γ δ ) 4 () 4 s AM GM α β γ δ 4 α 4 α + + αβ + αγ + αδ + βγ + βδ + γδ +6 b αβγ + αγδ + αβγ + βγδ b -4 Q. No If, b, e diffeet positive umbes pove tht + b + > b ( + b+ ). Expltio b + + b+ > b + + b+ > b + b+ > b + + b+ > b 4 Q. No. 7 If x,y, z e positive el umbes stisfyig the equtio x + 9y + 5z xy + 5yz + 5zx the fid the pogessio of x, y, d z. Expltio x + 9y + 5z xy + 5yz + 5zx x + 8y + 50z - 6xy - 0yz - 0zx 0 (x + 9y - 6xy) + (9y + 5z - 0yz) + (x +5z - 0zx) (x- y) + (y- 5z) + (x- 5z) y y, y 5z, x 5z y t x t, z t 5 + y x y x, y, z e i H.P
31 Q. No. 74 Mth the oditios fo the equtio x + bx + x + d 0 hvig oots i No. Colum A Colum B Colum C Id of Additiol Aswe AP (P) b d R GP (Q) 7 d 9bd - d P HP (R) b - 9b + 7 d 0 Q Expltio x + bx + x + d 0 () α -D, α, α +D - α -D+ α + α +D+ b -b α -b b b d0 7 9 b b - + d0 7 b - 9b + 7 d 0 (R) (b) α, α, α α -.. d α α -d α -d d d + b - + d0 bd d bd b d (P) (),, e oots of x + bx + x + d α -D α α +D α -D, α, α +De oots of x + x + bx α d - α d - b d d 9d d b d d - 9bd + 7d 0 7d 9bd - d (Q)
32 Q. No. 75 Mth the followig. If, b, e i HP, the No. Colum A Colum B Colum C Id of Additiol Aswe, b, (P) HP P b b + b- (Q) GP R,, b- b b- b b b (R) AP Q -,, - 4, b, P b+ + + b Expltio (A),, i A.P b Multiple by + b + + b+ + b+ + b+ i AP () b Subtt b b + b-,, i AP b, b, i HP b b + b- (A) (P) I () subtt b+ + + b,, i AP b, b, i HP b+ + + b (D) (P) (B) b + +,, ,, ( - ) ( - ) Addig st d d tem ( - ) ( - ) + ( - ) ( - ) + + (B) (R) (C) -,, ,, Multiplyig st d d tem ( tem) ( + ) (C) (Q) d
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