Answers & Solutions. for MHT CET-2018 Paper-I (Mathematics) Instruction for Candidates

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Clemence Tyler
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1 DATE : /5/8 Qustion Booklt Vrsion Rgd. Offic : Aakash Towr, 8, Pusa Road, Nw Dlhi-5 Ph.: -75 Fa : -77 Tim : Hour Min. Total Marks : Answrs & Solutions for MHT CET-8 Papr-I (Mathmatics) Instruction for Candidats. This qustion ooklt contains 5 Ojctiv Tp Qustions (Singl Bst Rspons Tp) in th sujct of Mathmatics (5).. Th qustion papr and OMR (Optical Mark Radr) Answr Sht ar issud to amins sparatl at th ginning of th amination sssion.. Choic and squnc for attmpting qustions will as pr th convninc of th candidat.. Rad ach qustion carfull. 5. Dtrmin th corrct answr from out of th four availal options givn for ach qustion.. Each answr with corrct rspons shall awardd two () marks. Thr is no Ngativ Marking. If th amin has markd two or mor answrs or has don scratching and ovrwriting in th Answr Sht in rspons to an qustion, or has markd th circls inappropriatl.g., half circl, dot, tick mark, cross tc. mark/s shall NOT awardd for such answr/s, as ths ma not rad th scannr. Answr sht of ach candidat will valuatd computrizd scanning mthod onl (Optical Mark Radr) and thr will not an manual chcking during valuation or vrification. 7. Rough work should don onl on th lank spac providd in th Qustion Booklt. Rough work should not don on th Answr Sht. 8. Th rquird mathmatical tals (Log tc.) ar providd within th Qustion Booklt.

2 MHT-CET - 8 (Papr-I) Cod- MATHEMATICS K d. If, thn th valu of K is 8 Answr K d d 8 ( ) K tan K tan K hnc K tan. Th cartsian co-ordinats of th point on th paraola =, whos paramtr is, ar (, ) (, ) (, ) (, ) Answr (C, D)* *For th curv =, th paramtric form of coordinats of an point ar ( t, 8t). Hnc for t, point is (, ).. d sin.cos sc log sc tan c sc.tan c sc log sc tan c sc log cosc cot c Answr Th givn intgral can writtn as sin cos d sin cos = sc tan cosc d = sc log cosc cot c. If log Answr thn d = Taking antilog,. Appling componndo and dividndo, ( ( ) ( ) ) ( ) Simplifing, w hav Diffrntiating, d d d. 5. If f : R {} R is a function dfind f( ), thn its rang is R R {} R {} R {, } Answr Th function can simplifid as,. Th valu corrsponding to = is not in th rang. Hnc Rang is R {}. If f( ) for = for is continuous at = and f thn is

3 MHT-CET - 8 (Papr-I) Cod- Answr Sinc th function f() is continuous, w must hav lim f ( ) lim f ( ) f (). If A thn A 5A A = But and lim f ( ) lim, f () lim f( ) lim Hnc + =. f 7, If tan thn d d d...(i) Diffrntiating th givn function w gt tan d. Diffrntiating scond tim, w gt d tan. Rarranging th last d d quation w hav, tan. d ( ) Sustituting tan, w finall gt d d ( ) ( ) d d 8. Th lin 5 + = coincids with on of th lins givn 5 + k + = thn th valu of k is Answr If th lins rprsntd pair of lins a + + g + f + h + c = has slops m and m, h thn w hav m + m =, which in this cas is not dfind as =.Sinc slop of on of th lins is givn as m = 5, th slop of othr must infinit. Hnc th othr lin must of th form = k. Comparing trm trm (5 + )( k ) 5 + k +, w gt k = 5 = A A A A 5I Th quation of lin passing through (,, ) and prpndicular to th lins ˆ ˆ ˆ ˆ ˆ ˆ r i j k i j k and ˆ ˆ ˆ ˆ ˆ ˆ r i j k i j k is z z z z Answr Th dirction vctor of th lin is givn iˆ ˆj kˆ iˆ ˆj kˆ. Sinc lin passs through (,, ), its quation is z

4 MHT-CET - 8 (Papr-I) Cod-. Lttrs in th word HULULULU ar rarrangd. Th proailit of all thr L ing togthr is Answr siz of th sampl spac, n(s) = 8!!! numr of favoral cass, n(e) =!! P = n ( E)!!! n(s)! 8! = Th sum of th first trms of th sris... is [( ) + ( ) + ( ) + th trm] [( th trm) ] = = =. If A, B, C ar th angls of ABC thn cota.cotb + cotb.cotc + cotc.cota = Sinc tan(a + B + C) = tan =. Thus tan AtanBtan C tan A tanb tanc Dividing tan A tan B tan C and simplifing w gt cot A cot B + cot B cot C + cot C cot A = d A B C thn A + B =. If sin Answr 5. d d sin Comparing, w gt A =, B = Thrfor A + B = Answr sin d cos tan c tan c sin( ) d cos sin cos d cos = = sc tan d C tan c tan c = tan + c. A coin is tossd thr tims. If X dnots th asolut diffrnc twn th numr of hads and th numr of tails thn P(X = ) = tan AtanBtan C tan A tanb tanc tan A tan B tan B tan C tan C tan A

5 MHT-CET - 8 (Papr-I) Cod- Answr Thr ar two possil favoural cass: HHT or TTH. Whr H dnots Had and T dnots Tail. P(X = ) = C = 7. If sin cos thn tan= Answr sin cos sin cos cos sin sin cos cossin sin cos tan = 8. Th ara of th rgion oundd =, =, = and th -ais ling in th first quadrant is squar units. = 8 Rquird ara = = = = / 8 5 cos. If f( ), for is continuous at =, thn valu of f() is Answr Appl L'Hospital Rul, lim 5 cos sin = lim sin lim. Th maimum valu of + sujct to + 5 and 5 +,, is.5 7. Answr (, ), 5 (, ) 5,, 5 + = + 5 = Evaluating th function + at th cornr points of th shadd rgion, it can asil sn that th maimum occurs at th point (,) and is.. If a,, c ar mutuall prpndicular vctors having magnituds,, rspctivl, thn a c a c = 8 Answr a c a c = ( ac) i (( a) c) = ( ac) i ( c ac) = [ a c] [ a c] = [ ac] =

6 MHT-CET - 8 (Papr-I) Cod-. If points P(, 5, ), Q(,, ) and R (5, 8, ) ar collinar, thn th valu of + is 5 Answr = = = + = + =. If th slop of on of th lins givn a + h + = is two tims th othr thn 8h = a 8h = a 8h = a 8h = a Answr Lt th slops of lins m, m h sum of roots mm h m...(i) a product of roots m...(ii) liminating m from oth quations w gt 8h a. Th quation of th lin passing through th point (, ) and iscting th angl twn co-ordinat as is + + = + + = + = = Answr ** **Wrong Qustion Th lins iscting th angl twn coordinat as ar = and = which do not passs through th point (, ) 5. Th ngation of th statmnt : Gtting aov 5% marks is ncssar condition for Hma to gt th admission in good collg. Hma gts aov 5% marks ut sh dos not gt th admission in good collg Hma dos not gt aov 5% marks and sh gts admission in good collg If Hma dos not gt aov 5% marks thn sh will not gt th admission in good collg Hma dos not gt aov 5% marks or sh gts th admission in good collg Lt A dnot "Hma to gt th admission in good collg" B dnot "Gtting aov 5% marks" Thn givn statmnt is A B. Ngation of this statmnt is ~(AB) ~(~ A B) A ~ B Hnc th rquird ngation in words is "Hma dos not gt aov 5% marks and sh gts admission in good collg.. cos cos cos cos7 = Answr cos = 7. If plans c z =, c + az = and + a z = pass through a straight lin thn a + + c = ac ac ac ac Answr c c a a ( a ) c( c a) ( ac ) a c acac a c ac 8. Th point of intrsction of lins rprsntd + + = is : (, ) (, ),,

7 MHT-CET - 8 (Papr-I) Cod- Answr Lt s = + + ds d ds. A di is rolld. If X dnots th numr of positiv divisors of th outcom thn th rang of th random varial X is : {,, } {,,, } {,,,, 5, } {,, 5} Outcoms Divisors Numr of Divisors {} {,} {,} {,,} 5 {,5} {,,,} Hnc rang of X is {,,, }.. A di is thrown four tims. Th proailit of gtting prfct squar in at last on throw is : Rquird Proailit = P(No prfct squar in an throw) Prfct squar ar,. Proailit of prfct squar in an on throw = P(at last on prfct. squar) sc. d log log log Using intgration parts sc d= tan = log log tan d. In ABC, with usual notations, if a,, c ar in A.P. C thn a cos + c A cos = a Answr a c c cosc cosa = a c a cos C cos c A ac ac. If = (sin cos ), = (sin + cos ) thn d at = is Answr = (sin cos) = (sin + cos) cos sin sincos cossinsin cos d d

8 MHT-CET - 8 (Papr-I) Cod-. Th numr of solutions of sin + sin + sin 5 = in th intrval, is 5 sin + sin + sin5 = sin cos + sin = sin [cos + ] = ithr sin = or cos + = cas (), sin =, thn = n n =, n is an intgr If solution lis in th intrval n =, n =, Cas (), cos = Thn = =,, cos n, nz,, thn n =, n, which givs two solutions, Hnc total solutions ist in th intrval. 5. If tan + tan =, thn = Answr tan tan tan = + = ( + ) ( + ) = ( ) ( + ) = =, But = dos not satisf th givn quation. =. 5 Matri A = 7 thn th valu of a A + a A + a A is Answr It is amiguous that is dnotd a ij. Assuming a ij to lmnts of co-factor matri w hav a = ( ) + (5 ) = 7, a = ( ) + (5 ) =, a = ( ) + ( ) =. Hnc a A + a A + a A = 7 + ( ) + ( 7) =. 7. Th contrapositiv of th statmnt : If th wathr is fin thn m frinds will com and w go for a picnic. Th wathr is fin ut m frinds will not com or w do not go for a picnic If m frinds do not com or w do not go for picnic thn wathr will not fin If th wathr is not fin thn m frinds will not com or w do not go for a picnic Th wathr is not fin ut m frinds will com and w go for a picnic Lt A dnot "Wathr is fin" B dnot "m frinds will com" C dnot "w go for a picnic" Thn givn statmnt is AB C. To find its contrapositiv, w nd to valuat ~ (BC) ~ A i.., ( ~ B ~ C) ~ A. In words this coms "If m frinds do not com or w do not go for picnic thn wathr will not fin". 8

9 MHT-CET - 8 (Papr-I) Cod- 8. If f( ) is incrasing function thn th valu of lis in R (, ) (, ) (, ) Answr f() = ( ) ( ) f( ) > < (, ) n. If X ( n: n N) and Y {( n) : n N), thn X Y X Y {} Answr Considr st X, its lmnt ar of th form n n = ( + ) n n = n n C C n C C n p St Y consists of all non ngativ multipls of. Hnc vr lmnt of st X is a sust of Y. n n n =. Th statmnt pattrn p(~ p q) is A tautolog A contradiction Equivalnt to p q Equivalnt to p q Considr th truth tal of p~ pq p q ~ p ~ pq p(~ pq) T T F F F T F F F F F T T T F F F T F F Hnc a Contradiction.. If th lin = 5 touchs to th curv = a + at th point (, ) thn 7a + = Answr = a +...() Sinc (,) lis on th curv, w hav = 8a +...() Diffrntiating (), a a d d Sinc lin = 5 touchs th curv a a 7 7a + =. Th sids of a rctangl ar givn = ±a and = ±. Th quation of th circl passing through th vrtics of th rctangl is + = a + = a + + = a ( a) + ( ) = a + (a, ) and ( a, ) will nds of diamtr. Rquird quation of circl is ( + a) ( a) + ( + ) ( ) = + = a +. Th minimum valu of th function f() = log is Answr f() = log f'( ) log f'( ) and "( ) " f f minima ists. Thrfor minimum valu of f() will occur at f log

10 MHT-CET - 8 (Papr-I) Cod-. If X ~ B (n, p) with n =, p =. thn E(X ) =.. 8. Answr Var(X) = E(X ) (E(X)) E(X ) = Var(X) + (E(X)) = npq + (np) = (..) + (.) =. + = Th gnral solution of diffrntial quation d cos( ) is tan c tan c cot c cot c Answr d cos( ) Lt + = t d dt dt cost dt cost dt dt cos t cos t t dt sc tan t c tan c tan c. If plans r. pi j k and r. i pj k 5 includ angl thn th valu of p is,, Answr cos p p p 5 p 5 (p ) = p + 5 p p + = (p ) = p = 7. Th ordr of th diffrntial quation of all paraolas, whos latus rcturn is a and ais paralll to th -ais, is on four thr two Answr Rquird quation of paraola is ( m) = a ( n) Hr m and n ar two indpndnt aritrar constant. Ordr of rquird diffrntial quation is. z 8. If lins and intrsct thn th valu of k is 5 7 k z

11 MHT-CET - 8 (Papr-I) Cod- Answr Gnral point on ths lins ar ( +,, + ) and ( +, + k, ) For point of intrsction + = + and + = and = 5 and k k k k. If a lin maks angls and with th positiv dirctions of X and Z as rspctivl thn th angl mad th lin with positiv Y-ais is Answr 5 5 Lt th angls which th lin maks with positiv dirctions of X, Y and Z as, and Thn = = =? cos + cos + cos = cos cos cos or Hnc = 5 or 5 5. L and M ar two points with position vctors a and a rspctivl. Th position vctor of th point N which divids th lin sgmnt LM in th ratio : trnall is 5 a Answr Lt th position vctor of point N n. Thn n ( a ) ( a) = = 5

MATHEMATICS (B) 2 log (D) ( 1) = where z =

MATHEMATICS (B) 2 log (D) ( 1) = where z = MATHEMATICS SECTION- I STRAIGHT OBJECTIVE TYPE This sction contains 9 multipl choic qustions numbrd to 9. Each qustion has choic (A), (B), (C) and (D), out of which ONLY-ONE is corrct. Lt I d + +, J +