SOLUTION OF IIT JEE 2010 BY SUHAAG SIR &

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1 fo/u fokjr Hkh: tu] ugha vkjehks dke] foifr ns[k NksM+s rqjaar e/;e eu dj ';ke A iq:"k flag ladyi dj] lgrs foifr vusd] cuk* u NksM+s /;s; dks] j?kqcj jk[ks Vsd AA jfr% ekuo /kez iz.ksrk ln~q: Jh j.knksm+nklth egkjkt SOLUTION OF IIT JEE BY SUHAAG SIR & HIS STUDENTS OF CLASS MOVING FROM TH TO TH S.No. Student s Name School Syd. Almas Ali All Saints Public School Devashh Saena St. Mary s Sr. Sec. Mujahid Mohd. Khan All Saints Public School 4 Shahrukh Ahmed Peole s Public School 5 Anmol Rehani Vikram Hr. Sec. 6 Sarsh Mehta Peole s Public School 7 Geet Soni Jawahar Lal Nehru 8 Amit Sarathe K.V. 9 Ranjeet Singh Peragatheel School Rahul Jharwade Peole s Public School Anamika Singh K.V. Shailja Aouthanere Peole s Public School Yamini Jain Chavara Vidya Bhawan Mandidee 4 Kirti Choariya Bourbon School Results of year IIT & 7 AIEEE selections out of 7 Fresh students International Maths Olymiad Solution of IIT JEE also available on website : OR come to our Institute Same day at 5 m, we are dtributing, rinted coies, so very less time of Solution. Page of

2 MATHEMATICS PAPER Question sequence as er SECTION Paer CODE - Single Correct Choice Tye Th section contains 8 multile choice question. Each question has 4 choices A), B), C) and D) for its answer, out of which ONLY ONE correct. 9. Let f, g and h be real-valued function defined on the interval [, ] by f e e, g e e. If a, b and c denotes, resectively, the absolute maimum of f, g and h on [, ], then A) a b and c b B) a c and a b C) a b and c b D) a b c. Let and q be real numbers such that comle numbers satfying and as its roots, q and q q. If and are nonzero and q, then a quadratic equation having A) q q q B) q q q C) q 5 q q D) q 5 q q y z. Equation of the lane containing the straight line 4 and erendicular to the lane y z y z containing the straight lines and 4 4 A) y z B) y z C) y z D) 5 y 4z. If the angles A, B and C of a triangle are in an arithmetic rogression and if a, b and c denote the lengths of the sides oosite to A, B and C resectively, then the value of the eression a c sin C sin A c a A) B) C) D). Let be comle cube root of unity with. A fair die thrown three times. If r,r and r r are the numbers obtained on the die, then the robability that r r Same day at 5 m, we are dtributing, rinted coies, so very less time of Solution. Page of

3 A) 8 B) 9 C) 9 D) 6 4. Let P, Q, R and S be the oints on the lane with osition vectors i ˆ ˆj,4i,i ˆ ˆ j ˆ and i ˆ j ˆ resectively. The quadrilateral PQRS must be a A) arallelogram, which neither a rhombus nor a rectangle B) square C) rectangle, but not a square D) rhombus, but a square 5. The value of t n( t) lim dt t 4 4 A) B) C) 4 D) The number of matrices A whose entries are either or and for which the system A y z has eactly two dtinct solutions, A) B) 9 C) 68 D) 7. Let ABC be a triangle such that ACB 6 and let a,b and c denote the lengths of the sides oosite to A, B and C resectively. The value(s) of for which c (are) a,b and A) B) C) D) 4 8. Let A and B be two dtinct oints the arabola y 4. If the a of the arabola touches a circle of radius r having AB as its dimeter, then the sloe of the line joining A and B can be A) r B) r C) r D) r 9. The value(s) of 4 4 d (are) A) 7 B) 5 C) D) 7 5 Same day at 5 m, we are dtributing, rinted coies, so very less time of Solution. Page of

4 4. Let z and z be two dtinct comle number and let z t z tz for some real number t with t. If Arg (w) denotes the rincial argument of a nonzero comle number w, then A) z z z z z z B) Arg z z Arg z z C) z z z z z z z z D) Arg z z Arg z z 4. Let f be a real-valued function defined on the interval, by Then which of the following statement(s) (are) true? f () n sin tdt. A) f "() ets for all, B) f '() ets for all, and f ' continuous on,, but not differentiable on,, C) there ets such that f ' f () for all, D) there ets such that f() f '() for all, The circle y 8 and hyerbola y 9 4 intersect at the oints A and B. 4. Equation of a common tangent with ositive sloe to the circle as well as to the hyerbola A) 5y B) 5y C) 4y 8 D) 4 y 4 4. Equation of the circle with AB as its diameter A) C) y 4 B) y 4 D) y 4 y 4 Let be an odd rime number and T be the following set of matrices : a b T A : a,b,c,,,..., c a 44. The number of A in T such that A either symmetric or skew-symmetric or both, and det A) divible by Same day at 5 m, we are dtributing, rinted coies, so very less time of Solution. Page 4 of 4

5 A) B) D) D) 45. The number of A in T such that the trace of A not divible by but det A) divible by [NOTE- The trace of a matri the sum of its diagonal entries.] (a) B) C) D) 46. The number of A in T such that det A) not divible by A) B) 5 C) SECTION IV (Integer Tye) D) Th section contains TEN questions. The answer to each question a single-digit integer, ranging from to 9. The correct digit below the question number in the ORS to be bubbled. 47. Let f be a real-valued differentiable function on R (the set of all real numbers) such that f. If the y-intercet of the tangent at any oint P, y on the curve y f equal to the cube of the abscsa of P, then the value of f equal to 48. The number of values of in the interval, tan cot5 as well as sin cos4 such that n 5 for n,, and 49. The maimum value of the eression sin sin cos 5cos 5. If a and b are vectors in sace given by a b. a b a b a i j i j k and b, 5 4 then the value of y 5. The line y tangent to the hyerbola. If th line asses through the oint a b of intersection of the nearest directri and the -a, then the eccentricity of the hyerbola 5. If the dtance the lane A y z d and the lane containing the lines y z 4 and y z 4 6, then d 4 5 Same day at 5 m, we are dtributing, rinted coies, so very less time of Solution. Page 5 of 5

6 5. for any real number, let denote the largest integer less than or equal to. Let f be a real valued function defined on the interval, by f () if odd, even Then the value of f ()cos d 54. Let be the comle number z satfying z z z cos in. Then the number of dtinct comle numbers equal to 55. Let Sk, k,,...,, denote the sum of the infinite geometric series whose first term k and the common ratio. k! k Then the value of! k k k S k 56. The number of all ossible values of, where, for which the system of equations y z cos yz sin sin Have a solution,y,z with yz,. cos sin y z (yz)sin y z cos ysin SOLUTION IIT JEE (PAPER - ) 9. (D) a b c {all three functions are increasing in [, ]} Ans (D). (B), q equation a q q q q q. q q q. q- for new equation sum Proved that. (C) Plane contains y z 4 & y z 4 a by cz k. () Same day at 5 m, we are dtributing, rinted coies, so very less time of Solution. Page 6 of 6

7 then a 4b c, 4a b c a b c a b c 8 y z New lane contain & lane () 4 A By Cz k, A8 B C, A B 4C A B C 4 4 A B C A B C, y z Ans (C). (D) Angles A,B,C are in A.P. then let o o o A,B 6,c 9 BC a, o c 9, o B 6, o A, AB c AC b, a sin c c sin A, c a.sin 9.sin(.()) sin8.sin 6.,.. (C) 9 Ans (C) 4. (A) Make figure as, 4,,,., 5. (B) lim t n 4 t 4 t Aly L' hosital & Newton labrets. n 4 4, n lim 4 4 further L' hosital ut (A) a by cz..(), a by cz..() a by cz..() Same day at 5 m, we are dtributing, rinted coies, so very less time of Solution. Page 7 of 7

8 a b c a b c y a b c z hence eq. () & () satfy (,,) but eq. () will not ass throes origin. Hence only one solution ossible so we think are of th question should be (). 7. (B,C,D) 8. (C&D) Just draw the figure of arabola & circle both asses throes (,) it nearly radius (circle) 4 & line have solve nearly in first quadrant so it can also in fount quadrant. finally values case &. r r 9. (A&B) Due to vary less time, we aly check the otion here A&B having very near value. 4. (A,B,C,D) It shows the equation to straight line on argent lane. It condition & theorem. 4. (B,C,D) f ''() cos f () n sin tdt sin o Here due to less time we says. f '() sin A not Answer, B Answer, C&D just check. 4. According to conditions. y m c c should be ositive with ositive sloe only ossible in B&C otions only. Here according to figure answer C correct. 4. (A) y 8 y 9 4 Check center should (,) be lie between 8 so center will be (6,). 4 9y 6, 9 9y 7, 7 6, 6, y 4 utting = 6, y, , y,., 49. () Taking maimum value =. 5. (e = ) 54. Ans Same day at 5 m, we are dtributing, rinted coies, so very less time of Solution. Page 8 of 8