Memory Based Questions & Solutions

Transcription

1 PAPER- (B.E./B. TECH.) JEE (Main) 09 COMPUTER BASED TEST (CBT) Memory Based Questions & Solutions Date: 09 April, 09 (SHIFT-) TIME : (9.0 a.m. to.0 p.m) Duration: Hours Ma. Marks: 60 SUBJECT : MATHEMATICS Registered & Corporate Office: CG Tower, A-6 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.)-005 Tel.No.: , 000, 0, Toll Free: Fa: Website: contact@resonance.ac.in CIN: U800RJ007PLC009

2 Registered & Corporate Office: CG Tower, A-6 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.)-005 Tel.No.: , 000, 0, Toll Free: Fa: Website: CIN: U800RJ007PLC009

3 JEE MAIN-09 DATE : (SHIFT-) PAPER- MEMORY BASED MATHEMATICS PART : MATHEMATICS Straight Objective Type This section contains 0 multiple choice questions. Each question has choices (), (), () and () for its answer, out of which Only One is correct.. There are 5 girls and 7 boys. A team of boys and girls is to be formed such that no two specific boys are in the same team. Number of way to do so () 00 () 50 () 00 () 00 Ans. () Sol. Without any restriction, total number of ways of forming team is 7 C 5 C 50 If two specific boys B, B are in same team then total number of ways of forming team equals to 5 C 5 C 50 ways total ways ways. The equation has roots and. Then value of is () 5 () 56 () 5 () 56 Ans. () Sol. roots are + i and i Let + i and b i then ( + i) 5 + ( i) 5 ( i) ( i) i i ( i i) i 6 8( ) cos d is equal to () () Ans. () Sol. / cos d cos d 0 0 / cos d 0. (by walli's formula) () 0 () 8. If n +, n N then sin( ) sin( ) sin( ) sin( ) d is equal to () lncos + c () lncos + c () ln sec + c () lnsec + c Ans. () CORPORATE OFFICE : CG Tower, A-6 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Reg. Office : J-, Jawahar Nagar, Main Road, Kota (Raj.)-005 Ph. No.: FAX No. : Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC009 This solution was download from Resonance JEE MAIN 09 Solution portal PAGE #

4 JEE MAIN-09 DATE : (SHIFT-) PAPER- MEMORY BASED MATHEMATICS Sol. ( cos( )) ( cos( )) sin cos.tan t d d d d dt dt tant nsect + c nsec c 5. If A i j, B i + j + k are two vectors and C is another vector such that A C + B 0 and A. C 0 then C () 9 () 8 () 9 () 7 Ans. () Sol. a (a c ) + a b 0 ( a c ) a (a. a ) c + a b 0 a c + a b 0 ˆi ˆj kˆ a b 0 î ( ) ĵ () + k() î ĵ + ˆk c ( ˆ i ˆ j ) + ( î ĵ + ˆk ) c î 5 ĵ + ˆk c a 5 c 8 c 8 c 9/ 5 ; a b ; 6. f() then b 5 ; 5 0 ; 5 () f()is discontinuous a R, b R () f() is continuous if a 0 & b 5 () f() is continuous if a 5 & b 0 () f() is continuous if a 5 & b 0 Ans. () CORPORATE OFFICE : CG Tower, A-6 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Reg. Office : J-, Jawahar Nagar, Main Road, Kota (Raj.)-005 Ph. No.: FAX No. : Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC009 This solution was download from Resonance JEE MAIN 09 Solution portal PAGE #

5 JEE MAIN-09 DATE : (SHIFT-) PAPER- MEMORY BASED MATHEMATICS 5 ; a b ; Sol. f() b 5 ; 5 0 ; 5 for function to be continuous at f() lim f( + h) lim f( h) h0 h0 f() 5 lim f( + h) lim a + b( + h) a + b h 0 h0 lim f( h) lim 5 5 h 0 h0 hence a + b 5.. () for function to be continuous at f() lim f( + h) lim f( h) h0 h0 f() b + 5 lim f( + h) lim b + 5( + h) b + 5 h 0 h0 lim f( h) lim a + b( h) a + b h 0 h0 hence b + 5 a + b a + b 5 () for function to be continuous at 5 f(5) lim f(5 + h) lim f(5 h) h0 h0 f(5) 0 lim f(5 + h) 0 h 0 lim f(5 h) lim b + 5(5 h) b + 5 h 0 h0 hence b b 5..() from equation () a 5 from equation () a 0 hence f() is discontinuous for a, br s 7. Average height & variance of 5 students in a class is 50 and 8 respectively. A new student whose height is 56 cm is added to the group. Find new variance. () 0 () () 6 () Ans. () Sol. Let 5 students are,,,, 5 5 Given i 50 i () 5 i () 8 5 i i i ( ) 5 (50) 8 5 i 590 () i Height of new student 56 (Let 6) now new 5..() 6 6 CORPORATE OFFICE : CG Tower, A-6 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Reg. Office : J-, Jawahar Nagar, Main Road, Kota (Raj.)-005 Ph. No.: FAX No. : Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC009 This solution was download from Resonance JEE MAIN 09 Solution portal PAGE #

6 Variance (new) JEE MAIN-09 DATE : (SHIFT-) PAPER- MEMORY BASED MATHEMATICS 6 i () 6 from equation () and () (56) var (new) 6 (5) (5) a, b, c are in G.P. a + b + c b, can not be () () () () Ans. () Sol. a + ar + ar ar since a 0 so r r ; + r r r r + r (, ] [, ) (, ] [, ) 0 9. k then find k. 5 5 () () 8 () () Ans. () Sol ( + 5) 50 8( + 5) hence remainder is lim y0 y y () Ans. () Sol. Using rationalization () () ( ) () does not eist lim y0 y y y y lim y y y y y y0 lim y 0 y y by putting value of limit CORPORATE OFFICE : CG Tower, A-6 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Reg. Office : J-, Jawahar Nagar, Main Road, Kota (Raj.)-005 Ph. No.: FAX No. : Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC009 This solution was download from Resonance JEE MAIN 09 Solution portal PAGE #

7 JEE MAIN-09 DATE : (SHIFT-) PAPER- MEMORY BASED MATHEMATICS. There is a parabola having ais as -ais, verte is at a distance of unit from origin & focus is at (, 0). Which of the following point does not lie on the parabola. () (6, 8) () (5, 6 ) () (8, ) () (, ) Ans. () Sol. As per the given data diagram of parabola must be as below (, 0) (, 0) then equation of parabola will be (y 0).( ) y 8( ) now check all options (C) isin. Find sum of all possible values of in the interval, for which is purely imaginary isin () Ans. () Sol. Let z isin, isin for z to be purely imaginary z + z 0 isin isin 0 isin isin () () 6isin isin sin 6isin isin sin sin () sin 0 sin n ± Given, hence,, sum of all possible value of CORPORATE OFFICE : CG Tower, A-6 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Reg. Office : J-, Jawahar Nagar, Main Road, Kota (Raj.)-005 Ph. No.: FAX No. : Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC009 This solution was download from Resonance JEE MAIN 09 Solution portal PAGE # 5

8 . Let A JEE MAIN-09 DATE : (SHIFT-) PAPER- MEMORY BASED MATHEMATICS cos sin sin cos Find the value of A 50 at () () Ans. () cos sin Sol. A sin cos A 50 cos( 50 ) sin( 50 ) sin( 50 ) cos( 50 ) 5 5 A 50 cos sin 6 6 at is 5 5 sin cos 6 6 cos sin 6 6 sin cos 6 6 () (). If (A B) ( A B) A B what should be proper symbol in place of and to hold the equation () and () and () and () and Ans. () Sol. Check all option repeatedly (i) (A B) (~A B) A(B (~ A B)) A (B) A B (i) is correct (ii) (A B) (~A B) (A~A) B fb f (iii) (A B) (~A B) B (iv) (A B) (~A B) B (A ~A) B f f only () is correct CORPORATE OFFICE : CG Tower, A-6 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Reg. Office : J-, Jawahar Nagar, Main Road, Kota (Raj.)-005 Ph. No.: FAX No. : Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC009 This solution was download from Resonance JEE MAIN 09 Solution portal PAGE # 6

9 JEE MAIN-09 DATE : (SHIFT-) PAPER- MEMORY BASED MATHEMATICS 5. If y() is solution of dy d + y, y() then value of y () 9 6 Ans. () Sol. dy +y d dy d + y () 9 6 This is linear differential equation d I.F. e n e Solution of differential equation is y.. d + C y + C C y given y () () 5 8 () C C Hence y() + y 6 + y From a well shuffled deck of cards, cards are drawn with replacement. If represent numbers of times ace coming, then value of P( ) + P ( ) is () 5 () () 9 () Ans. () Sol. P( ) P( ) P( ) + P ( ) If eccentricity of the hyperbola cos y sin is more than when 0, Find the possible values of length of latus rectum () (, ) () (, /) () (, ) () (, ) Ans. () CORPORATE OFFICE : CG Tower, A-6 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Reg. Office : J-, Jawahar Nagar, Main Road, Kota (Raj.)-005 Ph. No.: FAX No. : Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC009 This solution was download from Resonance JEE MAIN 09 Solution portal PAGE # 7

10 Sol. JEE MAIN-09 DATE : (SHIFT-) PAPER- MEMORY BASED MATHEMATICS cos y sin e > (given) e sin > + cos > + tan > tan > 0, hence, sin Latus rectum tan sin cos for,, tan sin is An increasing function Hence latus rectum (, ) Ans() 8. If slant height of a right circular cone is cm then the maimum volume of cone is () cm () cm () ( + ) cm () ( ) cm Ans. () Sol. V r h, r + h 9 V h (9 h ) dv dh (9 h ) 0 9 h 0 h, h V () (6) 9. If cos + cos, > then () 5 () Ans. () Sol. cos cos 5 () 6 0 () 6 cos CORPORATE OFFICE : CG Tower, A-6 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Reg. Office : J-, Jawahar Nagar, Main Road, Kota (Raj.)-005 Ph. No.: FAX No. : Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC009 This solution was download from Resonance JEE MAIN 09 Solution portal PAGE # 8

11 JEE MAIN-09 DATE : (SHIFT-) PAPER- MEMORY BASED MATHEMATICS square both side ± > hence 5 5, 0 0. If p + q + r 0 represent a family of straight lines such that p + q + r 0 then () All lines are parallel () All lines are inconsistence () All lines are concurrent at, Ans. () Sol. Given relation is p + q + r 0 p + q + r 0...(ii) comparing () and () we get, y hence we can say that these lines are concurrent at, () All lines are concurrent at,. Consider the system of equations + y + z, + y + z, + y + (a ) z a + then () system has a unique solution for a () system is inconsistence for a () system is inconsistence for a () system is inconsistence for a Ans. () Sol. D a ( a ) 6 (a ) + a a If a if a system has unique solution y z y z Hence system is inconsistent for a y z. The value of (cos sin) + 6(sin + cos) + sin 6 is where, () cos () cos 6 () cos 6 + sin cos () cos + sin cos Ans. () CORPORATE OFFICE : CG Tower, A-6 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Reg. Office : J-, Jawahar Nagar, Main Road, Kota (Raj.)-005 Ph. No.: FAX No. : Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC009 This solution was download from Resonance JEE MAIN 09 Solution portal PAGE # 9

12 Sol. JEE MAIN-09 DATE : (SHIFT-) PAPER- MEMORY BASED MATHEMATICS (sin cos) + 6(sin + cos) + sin 6 ( sin) + 6( + sin) + ( cos ) ( sin + sin ) + 6( + sin) + ( cos + cos cos 6 ) + sin + cos (cos ) cos 6 + sin cos sin cos cos 6 cos 6. circles of radii a, b, c (a < b < c) touch each other eternally and have -ais as a common tangent then () a, b, c are in A.P. () b a + c Ans. () () a, b, c are in A.P. () c + b a Sol. b a c (a b) (a b) + (a c) (a c) (b c) (b c) ab + ac bc c + b a. If f(), f (), f () then find J() such that f o J o f () f () () f () () f () () f () () f() Ans. () Sol. f (Jo(f())) f() J (f()) f () J() J J J() f () CORPORATE OFFICE : CG Tower, A-6 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Reg. Office : J-, Jawahar Nagar, Main Road, Kota (Raj.)-005 Ph. No.: FAX No. : Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC009 This solution was download from Resonance JEE MAIN 09 Solution portal PAGE # 0

13 JEE MAIN-09 DATE : (SHIFT-) PAPER- MEMORY BASED MATHEMATICS 5. Find the equation of line through (,, ) & parallel to the plane + y + z while the line intersects another line whose equation is + y z + y z + 5 () y z () y z y z () () y z Ans. () Sol. Family of plan containing the line of intersection of planes of + y z 0 + y z + 5 is ( + y z) + ( + y z + 5) 0 If this plane passes through (,, ), we get ( + ) + ( ) ( 6) Hence equation of this plane is ( + y z) ( + y z + 5) 0 now required line lies in this plane and is parallel to + y + z 5 ˆ Hence vector, v (say), along the line is v i ˆj kˆ i ˆ j ˆ kˆ so required line is y z 0 6. Consider the curves y + and y 0. Let be the angle between both the curves at point of intersection, then find tan () 8 () 5 () () Ans. () Sol. y + & y 0 meet at (±, 6) dy d for first curve and dy for second. d Hence slope are and so tan ( ) 8 ( ) 5 7. A plane parallel to y-ais passing through line of intersection of planes + y + z & + y z 0 which of the point lie on the plane () (,, ) () (, 0, ) () (,, ) () (,, ) Ans. () Sol. Equation of required plane is ( + y + z ) + ( + y z ) 0 ( + ) + ( + )y +( ) 0 since given plane is parallel to y-ais + 0 Hence equation of plane is z 0 + z + 0 CORPORATE OFFICE : CG Tower, A-6 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Reg. Office : J-, Jawahar Nagar, Main Road, Kota (Raj.)-005 Ph. No.: FAX No. : Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC009 This solution was download from Resonance JEE MAIN 09 Solution portal PAGE #

14 JEE MAIN-09 DATE : (SHIFT-) PAPER- MEMORY BASED MATHEMATICS 8. Find common tangent of the two curve y and + y 6 0 () y () y () y () y Ans. () Sol. Tangent to y is y m + m If it is tangent to given circle it's distance from (, 0) is equal to Hence m m m m + m m squaring both sides, we get 9m + 6m + 9m + 9m m Hence common tangents are y + or y 9. If the area bounded by the curve y tangent to it at (, ) and y-ais is () () () 8 () Ans. () Sol. (, ) 5 area dy dy 5 y 5 y dy 5 y 5y / (y ) CORPORATE OFFICE : CG Tower, A-6 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) Reg. Office : J-, Jawahar Nagar, Main Road, Kota (Raj.)-005 Ph. No.: FAX No. : Ph.No. : , To Know more : sms RESO at Website : contact@resonance.ac.in CIN : U800RJ007PLC009 This solution was download from Resonance JEE MAIN 09 Solution portal PAGE #

15 Registered & Corporate Office: CG Tower, A-6 & 5, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.)-005 Tel.No.: , 000, 0, Toll Free: Fa: Website: CIN: U800RJ007PLC009