: 3 "Efut Time : 3 Hours PAPER-2 Code [I] 3l<P : 180 Max. Marks : 180 pq<rl f.'r<!vif em tliR I 5 fi:Ac fclit"!if "@l Cf)ll{ 1""[1) I Please read the instructions carefully. You are allotted 5 minutes specifically for this purpose. "f.1"irn /INSTRUCTIONS: A. A. General: 1. <:15 -3lfC!Cl5T I "(!"<[ This booklet is your Question Paper. Do not break ([Cl) "'1 ent "\if<! ([Cl) clo "i<RJ "ii"'<!Cl5T f.fcf'!T the seals of this booklet before being instructed to do so by the invigilators. 2. CJ5T cp);s (CODE) '):"'D cj) BJ"1it The question paper CODE is printed on the right C!5T""r dtx "ii"'" cj) 'l:"'D ('J:"'D "ffis;m hand top corner of this sheet and on the back page 44) "'l"X t9Cll I (Page No. 44) of this booklet. 3. CJ5T'1 cj) fc:rit <SilC'!t 'J:"'D dtx \if1l"5 Blank spaces and blank pages are provided in the it t\ I CJ5T'1 clo question paper for your rough work. No additional CJ5T'l\il "'lt\ "\ill"irrr I sheets will be provided for rough work. 4. CJ5T'l\il. fuc;rq <ITt (CLIP BOARD). C'ii'T Blank papers, clipboards, log tables, slide rules, ([JR1CJ5T, calculators, cameras, cellular phones, pagers, and -q\iR dtx J<CJ5Tx cj) \l""Cj"Cj)XUT electronic gadgets are NOT allowed inside the examination hall. 11k'im1 CJ5l1 it "'lt\ I 5. Write you name and roll number in the space -3l'1"'ll "'ll'1 -3ftx xTC'f provided on the back cover of this booklet. 6. cj) dtx -3l'T"it l?CJ) Answers to the questions and personal details are cD-"lH'T xftcl CJ5T'l\il, \iiT -3lC'f'T "fr to be filled on a two-part carbon-less paper, which \iflirrr, "'l"X "lffi \iflirf\ I "1i\ mT \1'1Tl1D cj) <IT?; is provided separately. These parts should only be separated at the end of the examination when f.'R\mCJ) cj) f.fcf'!T "CJ"X tt <:15 "lH'T -3JC'f'T 1 instructed by the invigilator. The upper sheet is a i3l11k't '!Ttc machine-gradable Objective Response Sheet (.:ffix.-3llx.1?1'[., ORS) \iiT f.'R\mCJ) "i;TXT i'JT'1\1 c'f (ORS) which will be retained by the invigilator. You C'\\ \iflirf\ I f.'ri1c;rr '):"'D -3ll11 11k'im1 cj) <!R will be allowed to take away the bottom sheet at the end of the examination. 7. i3l11k't '):"'D clo (BUBBLES) C!5T CJ5Tc'r Using a black ball point pen, darken the <rfu Cj)C'f1'f "fr Cj)jC'fj 1 "ii"(f'l] Ci<ITCJ bubbles on the upper original sheet. Apply fcl;" '):"'D "'l"X f.'mR <1"'1 \iflit I sufficient pressure so that the impression is created on the bottom sheet. 8. (ORS) <TT it DO NOT TAMPER WITH/MUTILATE THE ORS ,g-x-cilx;fclcpfcl "" 1 OR THE BOOKLET. 9. "ii"'" ctt cj) "Cj"'fqR( cp:l'lT On breaking the seals of the booklet check that it c'r fcl;" 44 'l:"'D dtx \1lft 60 dtx contains 44 pages and all the 60 questions and fclq;-c;q cl\q;" "fr \iiT I \1lft corresponding answer choices are legible. Read carefully the Instructions printed at the beginning of xsiit cj) it 31? f"rc{'W C!5T UlT'l "fr 1 each section. B. (ORS) cf; "llT'1 q;) 1N"11 B. Filling the Right Part of the ORS 10. cj) dtx <rr"(f "lH'T it "l1'\ cp);s The ORS has CODE printed on its left and right sl? 1 parts. 11. \i'[]'i'[ c'r fcl;" '):"'01) "CJ"X t9ClT cp);s Check that the same CODE is printed on the ORS "ii"'" "CJ"X C!5T;s cj) m tt dtx (on both sheets) is the same as that on this booklet mfur \if1l"5 it !,:X([]i\TX "[["5 and put your signature affirming that you have verified this. 12. C!5T;s Pi"'"'l en C!5T ctt IF THE CODES DO NOT MATCH, ASK FOR A +fi'T CHANGE OF THE BOOKLET. www.examrace.com
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
~=p=rlf : 3 "Efut
Time : 3 Hours ~-2
PAPER-2 Code [I] l'f~Tl'f 3l<P : 180
Max. Marks : 180
pq<rl ~ f.'r<!vif em tliR ~ ~ I ~ 5 fi:Ac fclit"!if "@l ~ ~ Cf)ll{ ~ ~ ~ 1""[1) ~ I Please read the instructions carefully. You are allotted 5 minutes specifically for this purpose.
"f.1"irn /INSTRUCTIONS: A. ~= A. General:
1. <:15 ~ -3lfC!Cl5T 1<~?1" ~ I ~ ~ "(!"<[ This booklet is your Question Paper. Do not break ([Cl) "'1 ent "\if<! ([Cl) f.'Rt~ clo "i<RJ "ii"'<!Cl5T f.fcf'!T the seals of this booklet before being instructed to
"'1~~1 do so by the invigilators.
2. 1<~?1" CJ5T cp);s (CODE) ~ '):"'D cj) BJ"1it ~ The question paper CODE is printed on the right
C!5T""r dtx "ii"'" 3~ cj) fG~ 'l:"'D ('J:"'D "ffis;m hand top corner of this sheet and on the back page
44) "'l"X t9Cll ~ I (Page No. 44) of this booklet.
3. ~ CJ5T'1 cj) fc:rit <SilC'!t 'J:"'D dtx <S~TC'\\ \if1l"5 ~ Blank spaces and blank pages are provided in the
3~ it t\ ~ I ~ CJ5T'1 clo ~ cp)~ ~ question paper for your rough work. No additional
CJ5T'l\il "'lt\ ~ "\ill"irrr I sheets will be provided for rough work.
-q\iR dtx ~ J<CJ5Tx cj) ~~Tf.1q;- \l""Cj"Cj)XUT electronic gadgets are NOT allowed inside the examination hall.
11k'im1 CJ5l1 it ~ "'lt\ ~ I 5. "ii"\'l~clofG~'):"'D"C!"X~"l"J1?~it Write you name and roll number in the space
-3l'1"'ll "'ll'1 -3ftx xTC'f ~x ~I provided on the back cover of this booklet.
6. 1<~ cj) ~ dtx -3l'T"it ~ ~ l?CJ) Answers to the questions and personal details are
cD-"lH'T ~"'1 xftcl CJ5T'l\il, \iiT -3lC'f'T "fr ~ to be filled on a two-part carbon-less paper, which
\iflirrr, "'l"X "lffi \iflirf\ I "1i\ mT \1'1Tl1D cj) <IT?; is provided separately. These parts should only be separated at the end of the examination when
f.'R\mCJ) cj) f.fcf'!T "CJ"X tt <:15 "lH'T -3JC'f'T ~ ~ 1 instructed by the invigilator. The upper sheet is a
3. A steady current I flows along an infinitely long hollow cylindrical conductor of radius R. This cylinder is placed
coaxially inside an infinite solenoid of radius 2R. The solenoid has n turns per unit length and carries a steady
current I. Consider a point Pat a distance r from the common axis. The correct statement(s) is (are) :
(A) In the region 0 < r < R, the magnetic field is non-zero.
(B) In the region R < r < 2R, the magnetic field is along the common axis.
(C) In the region R < r < 2R, the magnetic field is tangential to the circle of radius r, centered on the axis.
(D) In the region r > 2R, the magnetic field is non-zero.
100 R ~ "ct ~ ~ ~ ~ m ctl ~ ii 100 ~ trRT I qg W ~ I ~ ~c;r-=r coT 2R ~ ctl ~ q R'll FC1 CJ) 1 ct ~ "fll11~ xm Tf<Tf ~ I ~ q R'll fC1 CJ) 1 ii n "WI~ >lfCl ~ ~ ii ~ cr 100 ~ trRT I ~ I 100 ~
P ct ~ \Jl'r "fll11~ "fr r ~~ LR t CJ)f"'lX1f ("fr) >rcP~ ~ ~ (~) :
(A) 0 < r < R ii, ~~ ~?f ~L~ "'l!ft ~ I (B) R < r < 2R ii, ~~ ~?f ctl ~TT "fll11~ ctl ~TT ii ~ I (C) R < r < 2R ii, ~~ ~?f r ~ "ct CJ:~ "fr ~f~m ~ ~ "ct~ "fll11~ LR ~ I
(D) r > 2R ii, ~~ ~?f ~L~ "'l!ft ~ I Ans. (A,D)
www.examrace.com
PHYSICS
Sol.
H
(A) For 0 < r < R => B * 0
(D) For r > 2R => B * 0
Hindi.
H
(A) 0 < r < R "ct ~ => B * 0
(D) r > 2R "ct ~ => B * 0
www.examrace.com
PHYSICS
4. Two vehicles, each moving with speed u on the same horizontal straight road, are approaching each other.
Sol.
Wind blows along the road with velocity w. One of these vehicles blows a whistle of frequency f1
• An observer
in the other vehicle hears the frequency of the whistle to be f2
. The speed of sound in still air is V. The correct
statement(s) is (are) :
(A) If the wind blows from the observer to the source, f2
> f1
.
(B) If the wind blows from the source to the observer, f2
> f1
.
(C) If the wind blows from the observer to the source, f2
< f1
.
(D) If the wind blows from the source to the observer, f2
< f1
.
efT ~. ~ ~cp CJ5T l"lfCl u ~. 100 Bt m~ 1.\lm ~ LR 100 ~-m: CJ5T 311~ 311 \% ~ 1 ~ ~ CJ5T ~rr ii w
R (A) Reaction I : P and Reaction II : P (B) Reaction I : U, acetone and Reaction II : Q, acetone (C) Reaction I: T, U, acetone and Reaction II: P (D) Reaction I : R, acetone and Reaction II : S, acetone
(A)~I: P3fix~ll: P
(B)~I: U, ~-;::r 3fix~ll: Q, ~-;::r
(C)~I:T,U, ~-;::r3fix~II:P
(D)~I:R,~-;::r3fix~II:S,~-;::r Ans. (C)
Sol.
Sol.
0
Reaction I : C~H 3 3
1.0 mol (Unreacted)
(In basic medium complete haloform reaction takes place since the rate of reaction increases with each a-halogenation)
0
Reaction II : c~H 3 3
1.0 mol
Br2 (1 .0 mol)
(In acidic medium monohalogenation takes place with 1-mol of halogen)
This section contains 4 paragraphs each describing theory, experiment, data etc. Eight questions relate to four paragraphs with two questions on each paragraph. Each question of a paragraph has only one correct answer among the four choices (A), (B), (C) and (D).
An aqueous solution of a mixture of two inorganic salts, when treated with dilute HCI, gave a precipitate (P) and a filtrate (Q). The precipitate P was found to dissolve in hot water. The filtrate (Q) remained unchanged, when treated with H2S in a dilute mineral acid medium. However, it gave a precipitate (R) with H2S in an ammoniacal medium. The precipitate R gave a coloured solution (S), when treated with Hp2 in an aqueous NaOH medium.
8RT P 3fix Q31(Y[T[-31(Y[T[ ~ CJ5\ ~ 31~ ~ ~ ~ <lTfTrcp S, T 31~ U q;:n ~~I
~t~~ COOH
s
H~*~HH COOH
T
31. Compounds formed form P and Q are, respectively (A) Optically activeS and optically active pair (T, U) (B) Optically inactiveS and optically inactive pair (T, U) (C) Optically active pair (T, U) and optically activeS
(D) Optically inactive pair (T, U) and optically inactiveS
P cr~ o 8RT ~ -mm ~T: ~
H~~: COOH
u
(A) ~CfUT ~uf"CJ) S 10i ~CfUT ~uf"CJ) ~TI'f (T, U) (B) ~CfUT ~ S 10i ~CfUT ~ ~TI'f (T, U)
(C) ~CfUT ~uf"CJ) ~TI'f (T, U) 10i ~CfUT ~uf"CJ) S (D) ~CfUT ~ ~TI'f (T, U) 10i ~CfUT ~ S Ans. (B)
32. In the following reaction sequences V and W are respectively:
Pi 9 ft1 Ftl C1 ~ 31JW1'11 ii, v 3fi-x w ~T: ~ :
Q v
O•v AICI3 (anhydrous)
0
(A)QOand/~cy vO wo
0
(C) Qo and/~co vO w
Ans. (A)
1. Zn-Hg I HCI
2. H3P04 w
v w
www.examrace.com
Sol. (31 & 32)
HOOC"'-. /COOH H/c = c"'--H
'P' (Cis)
H, ....-COOH "C = C"
Hooc/ "'--H
'Q' (trans)
HOOC-CH=CH-COOH
Q
0
Cold alk.
KMn04
(Syn addition)
Cold alk.
KMn04
(Syn addition)
H/Ni
Anhydrous AICI3
Sol.
+
0 v
(31 & 32)
HOOC"'-. /COOH H/c = c"'--H
'P' (Cis)
H"'-. /COOH C=C
Hooc/ "'--H
'Q' (trans)
Cl"J~ KMn04
(ftR <Trl)
Cl"J~ KMn04
(ftR <Trl)
COOH
H+OH
HIOH
COOH Meso (S)
COOH
o=+~H COOH
T
COOH
+ H~+~H COOH
u (Racemic Mixture)
0
0
v
COOH
H*OH H OH
COOH f11fl'r (S)
COOH
H*OH OH H
COOH
T
+
Zn- Hg
HCI
COOH
HO*H H OH
COOH
u (~~ fii>J.TUT)
CHEMISTRY
OH
0
l HP04
www.examrace.com
CHEMISTRY
HOOC-CH=CH-COOH
Q
0
H/Ni
0
0
v
OH
0 Zn- Hg
HCI ~ 0
v 1 HP04
@:0 Paragraph for Questions 33 to 34
"Sl~ 33 ~ 34 ~ fi1il' 3TJ~"G
0
A fixed mass 'm' of a gas is subjected to transormation of states from K to L to M to Nand back to K as shown
in the figure
100 frx:r cfi ~ ~~ 'm' CJ5T ~ ~-;or K -fr L -fr M -fr Ncr~ ~ K ii ~ 8RT ~ ~ ~
Pressure Gl6f
K
N
L
M
Volume~
33. The succeeding operations that enable this transformation of states are (A) Heating, cooling, heating, cooling (B) Cooling, heating, cooling, heating
(A) ll+l, i~, ll+l, i~ ~ LR (B) i~, ll+l, i~ ll+l ~ LR
(C) ll+l, i~, i~, ll+l ~ LR (D) i~, ll+l, ll+l, i~ ~ LR
Ans. (C)
www.examrace.com
K L
Sol. Pressure
N M
Volume
K ~ L => V t at constant P
Hence T t (Heating)
L ~ M => P ~ at constant V
Hence T ~ (Cooling)
M ~ N => V ~ at constant P
Hence T ~ (Cooling)
N ~ K => P t at constant V
Hence T t (Heating)
K L
Sol. ~
N M
K ~ L => f.n:rcl P -qx V t , 31Cl: T t ('Tl"f)
L ~ M => f.n:rcl V -qx P ~ , 31Cf: T ~ (i~)
M ~ N => f.n:rcl P -qx V ~ , 31Cf: T ~ (i~)
N ~ K => f.n:rcl V -qx P t , 31Cl: T t ('Tl"f)
34. The pair of isochoric processes among the transormation of states is (A) K to L and L to M (B) L to M and N to K (C) L to M and M to N (D) M to N and N to K
~ ~-;:ff CJ5T ~ ii x:r=r311G!C1PI¢ >rWll ~l11 ~
(A) K -fr L 3fix L -fr M (B) L -fr M 3fix N -fr K
(C) L -fr M 3fix M -fr N (D) M -fr N 3fix N -fr K Ans. (B) Sol. L ~ M
M~K
Both are having constant volume therefore these processes are isochoric. cfr-;:ff >l"Wll f.n:rcl ~ -qx -g'r \% ~ 31Cl: <:ffi >l"Wll x=r=!311 Gl C1 Pi¢ >l"Wll ~ I
CHEMISTRY
www.examrace.com
CHEMISTRY
Paragraph for Questions 35 to 36
"Sl~ 35 ~ 36 ~ fi1il' 3TJ~"G
The reactions of Cl2 gas with cold-dilute and hot-concentrated NaOH in water give sodium salts to two (different) oxoacids of chlorine, P and Q, respectively. The Cl2 gas reacts with S02 gas, in presence of charcoal, to give a product R. R reacts with white phosphours to give a compoundS. On hydrolysis, S gives an oxoacid of
This section contains 4 multiple choice questions. Each questions has matching lists. The codes for the lists have choices (A), (B), (C) and (D) out of which ONLY ONE is correct.
~ ~ ii 4 CSI§fcl45(>q ~ ~ I ~CJ) ~-;:or ii ~c;r-=r ~ ~ I ~ cfi ~ co'r~ cfi fctcp-c;q (A), (B), (C) 3fix (D) l ~-fl-~~~~1
The unbalanced chemical reactions given in List I show missing reagent or condition (?)which are provided in List II. Match List I with List II and select the correct answer using the code given below the lists :
38. Match the chemical conversions in List I with the appropriate reagents in List II and select the correct answer using the code given below the lists :
~ Iii ~ ll<'t \1*11Glf.'145 mCRUtf co'r ~ II ii ~ ll<'t ~~col cfi "fiT~ ~ ~ cr~ ~ cfi m ~ ll<'J cp'r~ COT >r<Tf'T ~ ~ \3"m ~f.n'r :
List 1/ ~Tft I List II/ ~Tft II
P. -j-cl )= 1. (i) Hg(OAc)2
; (ii) NaBH4
a. -j-oNa -j-oEt 2. NaOEt
R.(J O<OH 3. Et-Br
s.(J 0: ~OH
4. (i) BH3 ; (ii) HP/NaOH www.examrace.com
CHEMISTRY
Codes: p Q R s
(A) 2 3 1 4 (B) 3 2 1 4 (C) 2 3 4 1 (D) 3 2 4 1
Ans. (A)
-)-c1 NaOEt Sol. P.
E2 )= -)-oNa EtBr
Q. SN2 -)-oEt
rr (i) Hg(OAc)2
R. (ii) NaBH4
Q<oH (Markovnikov addition) ('"11¢lP1¢IL/) <Tr"r)
S. a (Anlimarkovnikov addition) M'ce l<t>1f.'l '""' il'l) ''''oH
39. An aqueous solution ox X is added slowly to an aqueous solution of Y as shown in list I. The variation in conductivity of these reactions is given in List II. Match List I with List II and select the correct answer using the code given below the lists :
Ans.
List I List II P. (C
2HJ
3N + CH
3COOH 1. Conductivity decreases and then
X Y increases Q. Kl (0.1 M) + AgN0
3(0.01 M) 2. Conductivity decreases and then
X Y does not change much R. CH
3COOH + KOH 3. Conductivity increases and then X Y does not change much
S. NaOH + HI 4. Conductivity does not change much X Y and then increases
X cfi ~ ~ ii ~T: Y "CJ)T ~ ~ ~-~ ~ \JfTCfT t ~"fr ~ Iii~ ll<Tf ~I ~ ~311
"fr ~ iliC'1¢C11 CJ5T ~ ~ II ii eft~~ I~ I co'r ~II "fr "WJ~ ~ cr~ ~ cfi ~ ~ ll<'t co'r~ "CJ)T >r<TITf ~ ~ ~ ~f.n'r :
As CH3COOH is a weak acid, its conductivity is already less. On addition
of weak base, acid-base reaction takes place and new ions are created. So conductivity increases.
(Q) KI (0.1 M) + AgN03
(0.01 M) ~ Agi ~ (ppt) + KN03
(aq).
As the only reaction taking place is precipitation of Agi and in place of Ag+, K+ is coming in the solution, conductivity remain nearly constant and then increases.
(R) CH3COOH + KOH ~ CH
3COOK (aq) + Hp OH- (aq) is getting replaced by CH
3COO-, which has poorer conductivity.
So conductivity dereases and then after the end point, due to common ion effect, no further creation of ions take place. So, conductivity remain nearly same.
(S) NaOH + HI ~ Nai (aq) + Hp AsH+ is getting replaced by Na+ conductivity dereases and after end point, due to OH-, it increases.
So answer of 39 is: (P)- (3) ; (Q)- (4); (R)- (2); (S)- (1). Answer is (D).
40. The standard reduction potential data at 2SDC is given below.
1=fT'"iCP ~ fct'l'fCI 2SDC -qx Pi 9 R1 Ftl C1 ~ I
ED (Fe3+.Fe2+) = + 0.77 V; ED (Fe2+.Fe) =- 0.44 V; ED (Cu2+.Cu) = + 0.34 V; ED (Cu+.Cu) = + 0.52 V; ED (0
2(g) + 4H+ + 4e- ~ 2Hp) = + 1.23 V ;
ED (02(g) + 2Hp + 4e- ~ 40H) = + 0.40 V
ED (Cr3+.Cr) =- 0.74 V; ED (Cr2+.Cr) =- 0.91 V Match ED of the rebox pair in List I with the values given in List II and select the correct answer using the code given below the lists :
www.examrace.com
Ans.
Sol.
CHEMISTRY
~ I ii ~ ll<'t ~ <jn:IT co'r ~ II ii ~ ll<'t ~ fcr:rcr -fr "WJ~ ~ cr~ ~ cfi ~ ~ ll<'t co'r~ COT >r<TfTf ~ ~ ~ ~f.n'r :
P. Q.
R. S.
Codes:
(A) (B) (C) (D) (D)
(P)
List I I ~Tft I ED (Fe3+· Fe) ED (4Hp ~4H++ 40H-) ED (Cu2++ Cu ~ 2Cu+) ED(Cr3+, Cr2)
p Q R s 4 1 2 3 2 3 4 1 1 2 3 4 3 4 1 2
ED +0.77V Fe2• -0.44V Fe
Fe3+, Fe ._I __ n_=_1 ____ n _= -2 _ _.t xV n = 3
1. 2. 3. 4.
::::> 1 X 0.77 + 2 X(- 0.44) = 3 X X
0.11 => X = - -
3- V::- 0.4 V.
(Q) 4Hp ~ 4H+ + 40H-
(R)
(s)
2Hp ~ 02
+ 4H+ + 4e+
-1.23 v
02
+ 2Hp + 4e-~ 40H- + 0.4 v
4Hp ~ 4H+ + 40H- -0.83 v
Cu2+ xV Cu + -0.52V Cu n = 1 f I
n = 1
+ 0.34V, n=2 X X 1 + 0.52 X 1 = 0.34 X 2 x=0.16V.
Cu2+ + e-~ Cu+ 0.16 v +
Cu ~cu++e- -0.52 v
Cu2+ + Cu ~ 2Cu+ -0.36 v However, in the given option,- 0.18 Vis printed.
~ ll<'t co'r~ (option) -0.18 V ~ s311 ~ I
Cr3' __ x --7 Cr'+ -0.91V Cr I n=1 n=2 f
-0.74V, n=3 X X 1 + 2 X (- 0.91) = 3 X (- 0.74) X - 1 .82 = - 2.22 X=- 0.4 V Hence, most appropriate is (D).
SECTION -1 :(One or more options correct Type) ~-1:(~m~~~~)
(MATHEMATICS)
This section contains 8 multiple coice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.
r W+2a+ .... +na) 1 For a E R (the set of all real numbers), a* -1, 1m (
1)a-1[(
1) (
2) ( )] 60 . Then a=
n--*oo n + na + + na + + ... + na + n
. . r w +2a + .... +na) 1 a E R (wlt qi«JfCICJ) ~311 COT Xllj~), a* -1 "ct ~ lm ( 1)a-1[( 1) ( 2) ( )] 60 Cf6l
n--*oo n + na + + na + + ... + na + n
a=
(A) 5 (B) 7 -15
(C)-2
-17 (D)-
2
Sol. (8, D)
2a+1
2
(2a + 1)(a + 1) 60
120 = (2a + 1) (a + 1) a= 7, -17/2
42. Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length 2J7 on y-axis is (are)
X-31~ coT~~ "fr 3 ~~ LR ~f m qrc;ff (~) Cf~ y-31~ LR 2 j7 ~:~ ~ qrc;ff (~) CJ:~ ~ (~)
(A) x2 + y2 - 6x + 8y + 9 = 0
(C) x2 + y2 - 6x- 8y + 9 = 0
Sol. (AC) Let 1=f"l-lT x2 + y2 + 2gx + 2fy + c = 0 g2 - c = 0 g2 = c ... (i)
2Jf2- c = 2-fi
r- c = 7 ... (ii) 9 + 0 + 6g + 0 + c = 0 9 + 6g + g2 = 0 (g + 3)2 = 0 g = -3 r= 16
c=9 f= ± 4
:. x2 + y2- 6x ± 8y + 9 = 0
(B) x2 + y2 - 6x + 7y + 9 = 0
(D) x2 + y2 - 6x- 7y + 9 = 0
www.examrace.com
43.
Sol.
(MATHEMATICS)
y z y z Two lines L1 : x = S,
3 _a = _
2 and L2 : x =a, -=-1 =
2 _a are coplanar. Then a can take value(s)
...,',.-',. . y z y z ~ ""' .,rl. ""' ctl x~ L
1: x = S, -
3-= -
2 CfQJT L
2: x= a, -
1 = -
2_ fP""lC1C1N c> I C'fq a Cf>Tl=f"l-l t>l "flCf)C1T c>-
-a - - a
(A) 1 (A, D)
(B) 2 (C) 3
x-S =~=_.!:.__ 0 a-3 -2
x-a =.1_=_z_ 0 -1 2-a
S-a 0 0
0 3-a -2 0 -1 2-a
=0
(S - a) ((3 - a) (2 -a) - 2) = 0 (a2 - Sa + 6- 2) = 0 (a- S)(a2 - Sa+ 4) = 0 a= 1, 4, S
1
(D) 4
44. In a triangle PQR, Pis the largest angle and cosP = 3. Further the incircle of the triangle touches the sides
PQ, QR and RP at N, Land M respectively, such that the lengths of PN, QL and RM are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)
f:f:!"l PQR ii, P ~ co'ruT ~ CfQJT cosP = ~ I~~ f:f:!"l Cf>T3f'C"l":C[Cl" ~\JIT31'f PQ, QR CfQJT RP co'r ~: N, L CfQJT M -qx ~ ~ ~f ~ ~ f¢ PN, QL CfQJT RM CJ5T ~<:IT W1""lT'TC1" "fl1""l 'lul ~~I C'fq f?r~\Jf CJ5T ~\JIT
Graph of y = f(x) is minima at x = -2, 0 ; maxima at x = -2/3
y = f(x) COT 31R~ ~
x= -2, O-qx ~ ; x= -2/3 -qx ~
(C) 2 2
(D) 3
-4x
2x+4
4x
2
www.examrace.com
SECTION- 2: (Paragraph Type)
~ - 2 : (3TJ~"G "S!Cf)"R)
(MATHEMATICS)
This section contains 4 paragraphs each describing theory, experiment, data etc. Eight questions relate to four paragraphs with two questions on each paragraph. Each question of a paragraph has only one correct answer among the four choices (A), (B), (C) and (D).
Paragraph for Question Nos. 49 to 50 >r~ 49 "fi 50 ~ ~ 3fj~"G
Let f: [0, 1] ~ R (the set of all real numbers) be a function. Suppose the function f is twice differentiable, f(O) = f(1) = 0 and satisfies f"(x)- 2f1(X) + f(x) ~ex, x E [0, 1].
~ f: [0, 1] ~ R (wlt qi«JRI¢ ~311 CJ5T x=Pj~) 100 ~~I 1=f"lrf ~ ~ f efT 6fR 31q¢C1'1lll t f(O) = f(1) = 0 Cf~ f"(x)- 2f1(X) + f(x) ~ex, X E [0, 1] coT ~te: ~~I
49. Which of the following is true for 0 < x < 1 ?
"F-19 ii "fr ctf-;::r O<x< 1 cfi ~ ~ ~?
1 1 (A) 0 < f(x) < oo (B) - - < f(x) < -
2 2 Sol. (D)
f" (x) - 2f1 (x) + f(x) ~ ex f" (x) ·e-x- f 1 (x)e-x- f 1 (x)e-x + f(x)e-x ~ 1
d d dx (f1(X)e-x)- dx (f(x) ·e-x)~ 1
d dx (f1(X) e-x_ f(x) e-x)~ 1
d2 ::::> - 2 (e-xf(X)) ~ 1 \;f X E [0, 1]
dx
Let 1=f"lriT ~(x) = e-x f(x)
=> ~(x) is concave upward f(O) = f(1) = 0
1 (C) - - < f(x) < 1
4
=> ~(0) = 0 = ~(1) => f(x) < 0
=> ~(x) < 0
(D) - oo < f(x) < 0
1 50. If the function e-x f(x) assumes its minimum in the interval [0, 1] at x = 4 , which of the following is true ?
1 3 (A) f 1 (x) < f(x), 4 < x < 4
1 (B) f 1 (x) > f(x), 0 < x < 4
1 3 (C) f 1(X) < f(x), 0 < x < 4 (D) f 1(X) < f(x), 4 < x < 1
Sol. (C) ~1 (X) < 0, X E (0, 1/4)
and~ ~~(x) > 0, x E (1/4, 1) ::::>e-x f 1(X)- e-xf(x) < 0, x E (0, 1/4) f 1(X) < f(x), 0 < x < 1/4 www.examrace.com
Paragraph for Question Nos. 51 to 52 >r~ 51 "fi 52 ~ ~ 3fJ~"G
(MATHEMATICS)
Let PQ be a focal chord of the parabola y2 = 4ax. The tangents to the parabola at P and Q meet at a point lying on the line y = 2x + a, a> 0.
~ f¢ PQ ~ y2 = 4ax CJ5T 10P -;:n~ ~ ~ I ~~ P Cf~ Q LR ~ CJ5T ~f ~~ 10P ~ LR ~
~ l:il'r fcO ~m y = 2x + a, a > 0 LR ~ ~ I
51. Length of chord PQ is
~ PQCJ5T~~: (A) 7a (B) Sa
Sol. (B) R lies on y = 2x + a
R, y = 2x + a ~m LR ~ ~
1 t- - = -1
t
(t+~r =1+4=5
(C) 2a
52. If chord PQ subtends an angle 8 at the vertex of y2 = 4ax, then tan 8 =
~ ~ PQ, y2 = 4ax ct ~\Ttf LR co'rur 8 3fcrf'«f ~ ~ C'f6l tan 8 =
( _!_ X ~X ~J + (~ X ~X _i_J + (~ X i X ~J 6 9 12 6 9 12 6 9 12
6 +36 +40 6x9x12
(MATHEMATICS)
82 648
56. If 2 balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these 2 balls are drawn from box B
This section contains 4 multiple choice questions. Each questions has matching lists. The codes for the lists have choices (A), (B), (C) and (D) out of which ONLY ONE is correct.
~ ~ ii 4 C51§Fcl¢(>q ~ ~ I ~"CJ) ~-;:or ii ~c;r-=r ~ ~ I ~ cfi ~ -cp'r~ cfi fclcp-c;q (A), (B), (C) 3fix (D) l ~-fl-~100~~1
www.examrace.com
(MATHEMATICS)
57. Match List I with List II and select the correct answer using the code given below the lists:
p
Q.
R.
S.
List -I List -II
1 cos (tan-1 y) + y sin(tan-1 y) 4
[ 2 ]1/2
;[ cot(sin-1 y) + tan(sin-1 y) J + y takes value
If cos x + cos y + cox z = 0 = sin x + sin y + sin z then
x-y possible value of cos -
2- is
If cos ( ~-x) cos 2x +sin x sin 2x sec x =cos xsin 2x sec x +
cos ( ~ + x) cos 2x then possible value
of sec xis
If cot ( sin-1 J1- X
2) =sin (tan-1(xJ5)), x * 0,
then possible value of xis
1.
2.
3.
4.
_]_[ 2~3
2
~ I coT ~ II "fr "WJ~ ~ Cf~ ~ cfi m ~ ~ coT~ COT wTI'T ~ ~ \3"m ~f.n'r :
p
Q.
R.
S.
Codes:
(A) (B) (C) (D)
~Tft -I ~-II [ , r _1 [ cos(tan-1 y) + ysin(tan-
1 y) J + Y4
y2 cot(sin-1 y) + tan(sin-1 y) COT 1=f"lrf ~ 1.
~ cos x + cos y + cos z = 0 = sin x + sin y + sin z C'f6l 2.
x-y cos -
2- COT ~ 1=f"lrf ~
~cos ( ~-x) cos 2x +sin x sin 2x sec x =cos xsin 2x sec x + 3.
cos ( ~ + x) cos 2x C'f6l sec x COT ~ 1=f"lrf ~-
~ cot ( sin-1 J1- X
2) = sin (tan-1(xJ5)), x * 0,
C'f6l X COT x+l~ 1=f"lrf ~-
p Q R s 4 3 1 2 4 3 2 1 3 4 2 1 3 4 1 2
4.
_]_# 2 3
J2
2
www.examrace.com
Sol. (P)- 4 ; (Q)-3 ; (R)- 2 or 4 ; (S) - 1
(P) [ 2 ]1/2
_1 [cos(tan-1 y) + ysin(tan-
1 y)l + Y4
y2 cot(sin-1 y) + tan(sin-1 y)
= (v; .y'(1-v•l+v•J"' = 1 Ans.4
1/2
(MATHEMATICS)
(Q) COS X + COS y = - COS Z
sin x + sin y = -sin z square and add cr1 ~ l:il'r~ -qx
2 + 2 cos (x - y) = 1 cos (x- y) = -1/2
2cos2 (x;y)-1 =-1/2, :::>cos (x;y) = 1/2
Ans.3
(R) cos 2x (cos(~- x)- cos(~+ x) J + 2 sin2 x = 2 sin x cos x
cos 2x (.fi sin x) + 2 sin2 x = 2 sin x cos x
.fi sin x [cos 2x + .fi sin x- .fi cos x] = 0
<:rr ill Either sin x = 0 OR <:rr cos2x- sin2 x = .fi (cos x- sin x)
I secx = 11 OR <:rr cos x = sin x
=> I sec x = .fi I
Ans. 2 OR m4
(S) cot (sin-1 J1_x2) =sin (tan-1(x .j6))
X
x xJ6 J1-x2 = J1+6x2
=> 1 + 6x2 = 6 - 6x2
=> 12x2 = 5
Ans.1
www.examrace.com
(MATHEMATICS)
58. A line L : y = mx + 3 meets y- axis at E(O, 3) and the arc of the parabola y2 = 16x, 0 :c:; y :c:; 6 at the point F(x
0, y
0). The tangent to the parabola at F(x
0, y
0) intersects they-axis at G(O, yJ The slope m of the line
L is chosen such that the area of the triangle EFG has a local maximum
Sol.
Match List I with List II and select the correct answer using the code given below the lists: List -I List -II
P. m= 1. 2
Q. Maximum area of ~EFG is 2. 4 R. Yo= 3. 2 S. y1 = 4. 1
~ - I co'r ~ - II~ "WJ~ ~ Cf~ ~ "ct m ~ ~ co'r~ COT >r<:fr<T ~ ~ "\3"m ~f.n'r : List- I I ~Tft I List- II I ~ II P. a= 1. 13 Q. b = 2. -3 R. c= 3. S. d= 4. Codes: