JEE(MAIN) 2018 TEST PAPER WITH SOLUTIONS (HELD ON SUNDAY 08 th APRIL, 2018) PART B MATHEMATICS ALLEN

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1 . The integal sin cos 5 5 (sin cos sin sin cos cos ) is equal to () ( tan ) C () cot C () cot C () ( tan ) C (whee C is a constant of integation) Ans. () Let I sin cos d [(sin cos )(sin cos )] sin cos (sin cos ) tan sec d ( tan ) d ut ( tan ) t tan sec d dt I dt t t C - Hence, I ( tan ) C. Tangents ae dawn to the hebola 6 at the oint P and Q. If these tangents intesect at the oint T(, ) then the aea (in sq. units) of DPTQ is - () 5 () 6 () 6 5 () 5 5 Equation PQ : chod of contact T Q JEE(MAIN) 8 TEST PAPER WITH SOLUTIONS (HELD ON SUNDAY 8 th APRIL, 8) PART B MATHEMATICS T (, ) R d P - 6 Aea : PQ TR TR 5, Point P ( 5, -) Þ PQ 6 5 CODE-C Aea of DPTQ sq. units. Tangent and nomal ae dawn at P(6, 6) on the aabola 6, which intesect the ais of the aabola at A and B, esectivel. If C is the cente of the cicle though the oints P, A and B and ÐCPB q, then a value of tanq is- () () () Ans. () Equation of tangent at P(6, 6) is 6 Tangent A ( 6, ) m PC m PB Hence, tan q tan q P(6, 6) q Nomal C(,) mpc -m m.m PC B (, ) PB PB (). Let u be a vecto colana with the vectos a i ˆ j ˆ-kˆ and b ˆj kˆ. If u is eendicula to a and u.b, then u is equal to- () 5 () 56 () 8 () 6

2 JEE(MAIN)-8 u l (a b) a l{ a.b (a.b)a} l { i ˆ 8j ˆ 6k ˆ } u l '{ i ˆ j ˆ kˆ} u.b Þ l' u i ˆ 8j ˆ 6kˆ u 6 5. If a, b Î C ae the distinct oots of the equation, then a b 7 is equal to- () () () () Ans. () a, b ae oots of \ a w and b w whee w is non-eal cube oot of unit so, a b 7 Þ ( w) ( w ) 7 Þ [w w] Þ [ ] (As w w & w ) 6. Let g() cos, f() and a, b (a < b) be the oots of the quadatic equation 8. Then the aea (in sq. units) bounded b the cuve (gof) () and the lines a, b and is- () ( ) () ( ) () ( - ) () ( -) 8 ; gof() cos ( ) (6 ) a 6, b A / /6 cos d - A 7. The sum of the co-efficients of all odd degee tems in the eansion of ( ) 5 - ( ) 5 - -, ( > ) is- () () () () Ans. () using ( a) 5 ( a) 5 [ 5 C 5 5 C a 5 C a ] ( ) 5 - ( ) [ 5 C 5 5 C ( ) 5 C ( ) ] Þ [ ] consideing odd degee tems, [ ] \ Sum of coefficients of odd tems is 8. Let a, a, a,..., a be in A.P. such that Ans. () ak 6 and a a 66. If k 7 a a... a m, then m is equal to- () 68 () () () 66 K a 6 k Þ [a 8d] 6 Þ a d... () a a 66 Þ a 5d () a 8d 6 b... () 7 7 d and a 8 Þ m a [ 8 ( -).] Þ m Þ m 7 ( 7) Þ m Þ m [ 5 - ] 6 Þ m 8.7 Þ m. If (i - 5) and i i (i - 5) 5, then the standad deviation of the items,,..., is- () () () () Ans. ()

3 Given Also, (i - 5) Þ S i 5... () i (i - 5) 5 i Þ S i.s i (5) 5 Þ S i 6 (using ())... () As, vaiance i S æsi -ç è ø 6 æ5 ç è ø 6 Hence standad deviation is (As standaed deviation va iance. PQR is a tiangula ak with PQ PR m. A T.V. towe stands at the mid-oint of QR. If the angles of elevation of the to of the towe at P,Q and R ae esectivel 5, and, then the height of the towe (in m) is- () 5 () () 5 () N P 5º º º h º Q M R Let height of towe MN is 'h' In D QMN MN tan º QM \ QM h MR...() Now in D MNP MN PM...() In D PMQ MP () - ( h) \ Fom () () - ( h) h Þ h m. Two sets A and B ae as unde A {(a, b) Î R R : a 5 < and b 5 < }; CODE-C B {(a, b) Î R R : (a 6) (b 5) 6}. Then :- () A Ì B () A Ç B f (an emt set) () neithe A Ì B no B Ì A () B Ì A Ans. () A {(a, b) Î R R : a 5 <, b 5 < } Let a 5, b 5 Set A contains all oints inside <, < B {(a, b) Î R R : (a 6) (b 5) 6} Set B contains all oints inside o on (-) (,) (,) (, ) (,) (, ) (,) (,) (, ) (±, ± ) lies inside the ellise Þ A Ì B. Fom 6 diffeent novels and diffeent dictionaies, novels and dictiona ae to be selected and aanged in a ow on a shelf so that the dictiona is alwas in the middle. The numbe of such aangements is- () less than 5 () at least 5 but less than 75 () at least 75 but less than () at least Numbe of was æ6 ç èø æ ç! èø 5 8

4 JEE(MAIN)-8. Let f() g() and, Î R {,, }. If h() f() g(), then the local minimum value of h() is : () () - () () Ans. () h() - æ ç - è ø - when < Þ - so will be local maimum value when > Þ ³ - so will be local minimum value. Fo each t Î R, let [t] be the geatest intege less than o equal to t. Then æé ù é ù é5ù lim ç... èê ëúû êëúû êë úûø () is equal to 5. () is equal to. () does not eist (in R). () is equal to. Ans. () æé ù é ù é5ù lim... ç ê ë ú û ê ë ú û ê ë ú è ûø { } { } { } æ... 5 æ 5 lim ç... - ç è ø è ø Q { } < { } < æ... 5 lim ç è ø The value of () Ans. () I / / sin d is : - () () sin d using oet I b f()d a... (i) / sin d... (ii) / adding (i) and (ii) I / / Þ I sin d /. sin d Þ I Þ I b a () 8 f(a b -)d 6. A bag contains ed and 6 black balls. A ball is dawn at andom fom the bag, its colou is obseved and this ball along with two additional balls of the same colou ae etuned to the bag. If now a ball is dawn at andom fom the bag, then the obabilit that this dawn ball is ed, is: () 5 Ans. () () 5 () () Let R i be the event of dawing ead ball in i th daw and B i be the event of dawing black ball in i th daw. Now, In bag thee ae R and 6B balls \ P(R ) and P(B ) 6 Now accoding to given infomation ær Pç R è ø 6 and P ær ç B è ø

5 CODE-C Requied obabiflit P(R ). ær Pç R P(B ). ær Pç è ø B è ø 6 6 Þ cos Þ cos \ n ±, n Î I Þ n ± 5 7. The length of the ojection of the line segment joining the oints (5,, ) and (,, ) on the lane, z 7 is : () Ans. () C uuu AC AB.AC ( ˆ ˆ ) () A(,, ) () A B (i ˆ ˆj k) ˆ i k. A B BC () B(5,, ) AB - AC - Length of ojection 8. If sum of all the solutions of the equation Ans. () æ æ æ 8 cos ç cosç.cosç -- è è6 ø è6 ø ø in [, ] is k, then k is equal to : () () 8 () æ 8cos ç cos -sin - è 6 ø æ ç - -cos è ø Þ 8cos ( ) æ Þ 8cosç cos - è ø () In Î [, ] :,, - onl sum. A staight line though a fied oint (, ) intesects the coodinate aes at distinct oints P and Q. If O is the oigin and the ectangle OPRQ is comleted, then the locus of R is : () () () 6 () 6 Ans. () Equation of PQ is h k (, k)q O (, ) asses though (, ) so P(h,) R(h,k) h k So locus 5. Let A be the sum of the fist tems and B be the sum of the fist tems of the seies If B A l, then l is equal to : () 8 () 6 () 6 () Ans. () B A T T T T 5

6 JEE(MAIN)-8 6 B A (..... ) (..... ) [ ] é( 6...6) ( ) ù ê ë tems tems ú û é ù ê ( 6) ( 6) ë úû [.8.8] [6 8].8 5. If the cuves 6, b 6 intesect each othe at ight angles, then the value of b is : () 7 () () () 6 Ans. () Let cuve intesect each othe at oint P(, ) 6 & P(, ) 6 b 6... (i) b 6... (ii) Now diffeentiate both cuve and get sloe of tangent to both cuve at oint P(, ) æd m \ ç èdø(,) & æd ç èd ø m b (,) 7 Q m m Þ b... (iii)... (iv) \ fom equation (i) b 5. Let the othocente and centoid of a tiangle be A(, 5) and B(, ) esectivel. If C is the cicumcente of this tiangle, then the adius of the cicle having line segment AC as diamete, is: () () 5 () 5 () Ans. () Othocente A(, 5) centoid B(, ) and AB A B C Centoid divides othocente and cicumcente in atio : \ AB : BC : Now AB AC AC.AB ( ) AC Radius of cicle with AC as diamete is 5 5. Let S {t Î R : f() (e ) sin is not diffeentiable at t}. Then the set S is equal to: () {} () {} () {, } () f (an emt set) f() (e ) sin we check diffeentiabilit at & at R.H.D L.H.D lim h lim h h h - (e -)sin h - h -h -h - (e -)sin -h - -h Q RHD LHD so function is diffeentiable at at R.H.D L.H.D lim h lim h h h - (e -)sin h - h -h -h- (e -)sin -h - -h Q RHD LHD so function is diffeentiable at set S is emt set, f

7 CODE-C 5. If (A B) ( A), then the odeed ai (A, B) is equal to : () (, ) () (, 5) () (, 5) () (, 5) Ans. () Put Þ Put Þ (A B) ( A) odeed ai (A, B) is (, 5) A Þ A (B ) ( ) æ æ ç B- ç è ø è ø B Þ B The Boolean eession ~ ( Ú q) Ú (~ Ù q) is equivalent to : () () q () ~q () ~ ~ ( Ú q) Ú (~ Ù q) (~ Ù ~q) Ú (~ Ù q) Þ ~ Ù (~q Ú q) Þ ~ Ù t º ~ 56. If the sstem of linea equations k z k z z has a non-zeo solution (,, z), then equal to : () () () () Ans. () Fo non zeo solution Þ k Now equations k k - - z...() z...() z...() z on equation () () we get 5 Þ 5 Now ut 5 in equation () we get 5 z Þ z z (-5)(-) 57. Let S { Î R : ³ and ( 6) 6 }. Then S : () contains eactl one element. () contains eactl two elements. () contains eactl fou elements. () is an emt set. Ans. () Case-I : Î [, ] ( ) 6 6 Þ 8 Þ, Case-II : Î [, ] 6, Þ ejected ( ) 6 6 Þ 6, So, 6 ejected is 7

8 JEE(MAIN) If the tangent at (, 7) to the cuve 6 touches the cicle 6 c then the value of c is : () 85 () 85 () 5 () 5 Ans. () Equation of tangent at (, 7) to 6 is 5. ( 8, 6) O 5 Now, eendicula fom cente O( 8, 6) to 5 should be equal to adius of the cicle C 6 6-C then, C 5 5. Let () be the solution of the diffeential equation sin d d cos, Î (, ). æ If ç è ø, then æ ç is equal to : è6 ø () -8 () () Ans. () sind cosd d Þ d(.sin) d Integate we get Þ.sin C () 8 æ Þ asses though ç, è ø Þ C Þ C Þ sin is the solution æ æ ç ç. - è6ø è 6 ø 8 6. If L is the line of intesection of the lanes z, z and L is the line of intesection of the lanes z, z, then the distance of the oigin fom the lane, containing the lines L and L is : () () () () Ans. () 6. Plane asses though line of intesectuion of fist two lanes is ( z ) l( z ) (l ) ( l) z(l ) (l )... () is having infinite numbe of solution with z and z then ( l ) ( l ) ( l ) Solving l z eendicula distance fom (,, ) is 6 8