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 +
5 d ( cos ) nd sin d d sin cot d cos d ( ) 6 cos 7 8 9. 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 7 4 7 7 i j i j 5 5 5 5 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.
4 5 Let P(,y,z ) be the foot of the perpendiculr direction cosines of OP re,y,z 4 nd direction rtios re,, 9 9 9 8 4 getting the point,, 9 9 9 P A B P A B ( ) ( ) -P ( A B) 5-8 8 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 5 7 8 sin nd sin y 5 7 4 5 cos nd cos y 5 7 Nowcos( y) cos cos y + sin sin y 4 5 8 cos( y) + 5 7 5 7 84 8 84 y cos sin sin cos 85 5 7 85 7. Writing A IA A R R R A 5 R R 5 A 5 5 R R + R m m m
getting 5 5 A 5 5 8 f is continuous on,4 ( f is polynomil function ) f 4. f is differentil on (, 4) f ( b) f By MVT f (c) b + 6 5 c 4 c (, ) 4 MVT is verified. 9 Given ysec.put cos. Then ysec y sec ( sec ) cos. y cos d f 4 + 6 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 4 5 4 Put + t dt 5 d 4 5 5 ( 5 ) d tdt t ( ) + + 5 + d + 4 5 5 4 5 4 nd getting 5 + d
4 Drwing the figure 4 4 4 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 4 6 + 5 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 )
Getting the nswer.9 9. ) ( ) Let f:r 5, be function defined by f() + 6 + 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 + 6 4 7 8 A+B -5, 8 6 9 (A+B)C, AC, BC 8, AC+BC (A+B)C 8-8 4. Getting A 7 4 8 Getting cofctor of A 5 6 9 7 5 Writing Adj A 8 6 9 7 5 8 Writing X A B 8 6 9 7 7 4 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
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 4 5 4 5 d + + C 45 Solving the circle +y 8 nd prbol y 4,, 4 46 Are 4 8 4 8 yd + yd d + 4 ( 4) d 4 4 / 4 4 4 4. 4 ( 4) sin + + 4 + 4 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 +
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 +
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