SCIENCE ENTRANCE ACADEMY PREPARATORY EXAMINATION-3 (II P.U.C) SCHEME 0F EVALUATION Marks:150 Date: duration:4hours MATHEMATICS-35

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1 JNANASUDHA SCIENCE ENTRANCE ACADEMY PREPARATORY EXAMINATION- (II P.U.C) SCHEME F EVALUATION Mrks:5 Dte:5-9-8 durtion:4hours MATHEMATICS-5 Q.NO Answer Description Mrk(s)!6 Zero mtri. + sin 4det A, 4 R 4 5 sin m d cos 6 7 Writing the definition 8. b b 9 Writing the definition. n(s)6,let E be the event of getting n even prime number on ech die, then n( E) n(e), hence P n s 6 sin 8 f nd g gof g f fog f g 8 Let y sin sin y cos y cos y sin + cos Put sec sec, 4 cot cot cot ( cot) sec sec tn ( y) Let P, be ny point on the line joining the points A(,) nd B(9,) Then 9 y y +

2 5 d ( cos ) nd sin d d sin cot d cos d ( ) 6 cos y log log y cos.log(log ) ; cos sin log log y d log cos cos (log ) sin log ( log ) d log 4 dv We hve V r 4r dr dv dv. r dv 47 (.).9m dr Put tn t d dt + t t tn I e dt e + c e + c e d e d e d e e e e d e d + Order -, degree - not defined + The unit vector in the direction of the given vector is i j 5 5 The vector hving mgnitude equl to 7 nd in the direction of is i j i j AB iˆ + ˆj + kˆ nd AC 4 ˆj + kˆ AB AC 6iˆ ˆj + 4kˆ AB AC 6 6 re of the tringle is AB AC sq units.

3 4 5 Let P(,y,z ) be the foot of the perpendiculr direction cosines of OP re,y,z 4 nd direction rtios re,, getting the point,, P A B P A B ( ) ( ) -P ( A B) Let A,,,4,5,6 (, ),(, ),(,4 ),( 4,5 ),( 5,6) ( 6,6 ) R, Ris not refleive. Rbut R nd R s Note tht, R but, R R is not symmetric. Now (,) nd (,), R is not trnsitive. 6 8 Let sin nd sin y sin nd sin y cos nd cos y 5 7 Nowcos( y) cos cos y + sin sin y cos( y) y cos sin sin cos Writing A IA A R R R A 5 R R 5 A 5 5 R R + R m m m

4 getting 5 5 A f is continuous on,4 ( f is polynomil function ) f 4. f is differentil on (, 4) f ( b) f By MVT f (c) b c 4 c (, ) 4 MVT is verified. 9 Given ysec.put cos. Then ysec y sec ( sec ) cos. y cos d f f 4 Now f. And showing f ( ) is strictly incresing in,, nd strictly decresing in,. I cos. Put cos cos sin d t t tdt d I-t sin t costdt sin t tdt tcos t + sin t+ c 4 (( ) cos ) + c Put + t dt 5 d ( 5 ) d tdt t ( ) d nd getting 5 + d

5 4 Drwing the figure Are A y y 8 8(4 ) Eqution of fmily of circles with centre (,) nd rdius ( y ) y y is + + diff () with respect to + y + y or d d d d + y sub ( ) in ( ) d we get + y y d y d y 5 We hve AB iˆ ˆj 6kˆ BC ˆ i ˆj + kˆ ˆ ˆ,, CA i + j + 5k Note tht AB BC + CA,hence tringle is right + 6 ngled tringle, b, c d +. b ( c + d ).( ) b c + b d. b c +. b d, b, c +, b, d 7 Writing the eqution of the plne y + z 4 + ( + y + z ) 8. Applying condition to the plne which psses through (,, ), getting Getting eqution of the plne 7-5y+4z-8 Writing P(E ), P(E ), P(E ), 6 P( A / E )., P( A / E )., P( A / E ).5 P(E )P(A/E ) writing P(A/E ) P(E )P(A/E )+P(E )P(A/E ) + P(E )P(A/E )

6 Getting the nswer.9 9. ) ( ) Let f:r 5, be function defined by f() y + 6 Writing y ( + ) 6 nd getting R ) ( ) defining function g: 5, R by g( y) getting gof()g + -6) nd stting gofi Getting fog ( y ) f - f is invertible nd f () + y+6- y nd stting fogi +6- y A+B -5, (A+B)C, AC, BC 8, AC+BC (A+B)C Getting A Getting cofctor of A Writing Adj A Writing X A B Getting, y, z m n y Ae + Be d m Ame + d y Bne n Am e + Bn e d d y ( m + n) + mny d d m n R+ -5, ) m n m n m n Am e + Bn e m + n Ame + Bne + mn Ae + Be 4. Let r be the rdius, h be the height nd V be the volume of the snd cone t time t m m

7 44 dv r Given cm / s, h r 6h dt 6 (6 ) dv dh V r h h h h, getting 6 h dt dt dh Getting cm / s dt 48 Height of the snd cone is incresing t the rte of / 48 cm s I d put sec sec, d sec tn d sec tnd Writing I sec Getting log sec + tn + K Getting log + + C + log d + + C 45 Solving the circle +y 8 nd prbol y 4,, 4 46 Are yd + yd d + 4 ( 4) d 4 4 / ( 4) sin sq.units y ycos + () d y cos Showing () is homogeneous D.E Writing y v dv v+...(), d d d substituting () in () nd getting cos vdv Integrting both sides nd getting sin v log + log C or sin v log C y y Replcing v by, we get sin log C 47 Drwing figure Writing AP b Getting r + b 8 4 +

8 48 iˆ + yj ˆ + zkˆ ( + ) iˆ + ( y + b) ˆj + (z + c) kˆ y y z z b c n n P( ) Cq p where n,p P( ) C 5 ( i) getting P(6) 5 9 ( ii) getting P( 6) 5 5 ( iii) getting P( 6) 64 m 49.() (b) Writing + f d f d f I f d, put t d dt, when, t when, t Getting ( ) I f d f d I f d f d f d Writing ( ) + Cse : If f is even; Getting Cse : If f is odd; f d f d Getting f d getting I R R + br 7 sin d LHS + + b -b + + b b - + b b b -b + + b b - + b b b +

9 5.() (b) R R R LHS + + b + + b (+ + b ) (+ + b ) -b b b -b b b R we get LHS + + b Epnding long Writing M Z45+5y, +y 5, 5+8y 4,,y Plotting the grph of +y5, 5+8y4 Finding the fesible region Finding corner points, Corner points Z45+5y (5,) 5 (,5) 5 (,75) 875 k Getting LHL lim f Getting RHL k lim f + Given f, since f is continuous t Getting k6. m m m m

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