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Notes On Function -calculus Paper

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Published in: IIT JEE Mains | Indian National Mathematical Olympiad (INMO) | Mathematics
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This notes consist of questions based on what I teach for JEE, have a look.

Vikas K / Navi Mumbai

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Qualification: B.Tech/B.E. (Indian Institute of Technology (IITB), Mumbai - 2011)

Teaches: Algebra, Education, KVPY Exam, Logic, Mathematics, Statistics, NSTSE, NTSE, IISER Exam, NDA, BITSAT, CET, IIT JAM, IIT JEE Advanced, IIT JEE Mains

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  1. FUNCTION-TWT (45 Qs) Name &R011 no. MARKING SCHEME (200 Marks) 180mins. Batch Single correct QI-Q16 Multiple correct Q.17 to Q28 PSG (paragraph) Q29-Q37 Integer type Q.38-Q45 +4 for correct option,-l for wrong option, zero in all other cases +5 for all correct options, +1 for each correct option provided no wrong option is selected -1 if any wrong option is selected Zero in all other cases. +4 for correct option, zero in all other cases +5 for all option,-l for wrong option, zero in all other cases READ CAREFULLY AND SOLVE EASY QUESTIONS FIRST. SINGLE CORRECT (QI-Q16) 2tanlx-4 1. 2. 3. 4. The set of values of x satisfying < 0 is The domain of the real valued function f (x) for which 4 +4 (D) None of these 4x is 1] The range of function f (x) = cos x] + 4—12 +— where, [.] is GIF 3 (B) 2 (C) -,2+6 (D) -,2+6 Iff 8 (A) Only when m n n xy, then f (m, n) + f (n, m) 0 for (B) only when m n (D) for all m & n
  2. 5. 6. x —— is 7. 8. 9. 10. 11. If y logsmx then the possible set of values of x and y are (A) X e [2n1T, 2n1T + IT], y e {0} (C) U (2nn, 2n1T+ (D) U (2nn, — 2n1T+21T , y e {0} 2 Sum of the absolute values of all distinct real solution(s) of x 3 31+4 —+ 2 7 4 Let h(x) = kx+5 1,domain of f (x) is [ 5, 7] and domain of [ 6, l] and range of h(x) is the same as domain of f (x) ,then value of k is 1 3 4 5 Given y 21 x | +3 and y 31 x 21+5, where denotes the least integer value greater than or equal to x,then the value of px+3yl is (A) 15 (B) 17 (C) 25 (D) 37 let f (x) = x p + x —151+1 x p—15 f (x), Yr e [p, 15] is (B) 15 where p e (0, 15), then the minimum value of (C) 30 (D) does not exist Let f : R [0, T) defined by f (x) then complete set of values of is Let f R Ris defined as f (x) log 912 —121+1 +1 be a onto function, where 1 is parameter . If f (x) is injective then "a" belongs to
  3. 12. 13. 14. 15. 16. Let f (x) 2 5-1 Il (x 2 < x < 3, then number of solution of f (f (x)) — 2 are If (a, P) Ya e A lies on y f (x) ,where f : A + B but not ony f (x) (Wherea ß), then (A) y f (x) And y f (x) do not intersect (B) Either y f (x) and y f (x) do not intersect ory f (x) and y f I(x) intersect only on y = x. (C) y f (x) And y f I (x) intersect on y = x (D) y f (x) And y f I (x) cannot intersect on y = x The period of the function f (x) = cos(sin + coslxl) is (D) none of these 2 If f (x) and— < f (x) < 7 , then the value of cis If the equation 2 log (x + 3) = log (ux) has only one root then 01 e (C) oce (12m) u {1} MULTIPLE CORRECT (Q17-Q28) (B) 0) u {12} (D) (0, 12) u {24} 17. 18. Let g (x) be a function defined on [—1, l] . If the area ofthe equilateral triangle with two of its vertices at (0, 0) and (x, , then the function g (x) is 4 (D) I -k 12 11+1), Let f (x) = log then for x in valid domain, which of the following are correct? (A) f f (Il .12) Xl + 12 I -k Xl.X2 (B) f (x) is odd (D) f (x) is even
  4. 19. 20. 21. 22. In which of the following cases there does not exist any function (B) f 2 (12 x) —4f(2x 2) +12 +1—0 e R (C) f (x) + f (l/ x) 21 YxeR {0} (D) f (sin x) + f (cosx) x Yr e R Which of the following function(s) are odd function(s)? {0} cosx.cos —+1 3 cos x + cos —+1 3 sm x — sm 3 Yr e R Where [.]ls GIF 1 2 Function f (x) satisfies the relation f (xy) then 4 10000 4 (C) Ef 100k) 31 k=0 Which of the following options is/are true? f (x). f (y) for all positive number x & y and f (2) 4 (B) Ef(2k) 4 (D) Ef 100k) 32 10, (A) : IR IR e x Is many one & into function f (x) = 21+ | sin x Is one one and onto function Is many one and onto function 12 — 81+18 212 —x +5 Is many one and into function 712 -k -k 10
  5. 23. 24. 25. 26. If domain and range of f (x) are [0, 2] & [l, 3] respectively, then identify the correct options (A) Domain of 3 f (x) +5 is [0, 2] 3f(x-2) (B) Domain of f(x-2) 2 1 in its domain, is same as range of sec x in x e R (2n + l) (C) Range of 2)-2 (D) if f (x) is increasing function then range of 3 f (x) + sin (x —l) has 9 integer in it If the rational number—,q 0, p & q are relatively prime, is a root of the equation anxn +an I x n I + ...... +qx+ao = 0, where all coefficient ak are integers and leading coefficient an 0 . Then which of the following is necessarily true? (A) p Is divisor of ao (C) q is divisor of ao Let f (x) 21 3 (B) q is divisor of an (D) p, q both are divisor of an 2 < x < 3 , then true equations are - 2012 1004 times If f : [a, T) [a, T) be a function defined by f (x) f (x) f I (x) is 9, then the other solution may be — 2ax + a(a + l) . If one of the solution of I 2 (C) 10 0 0 (D) 11 Let f (x) = In(2x 12) + sing-E,then 2 (A) Graph of f (x) is symmetrical about line x (B) Graph of f (x) is symmetrical about line x (C) Maximum value of f (x) is 1 (D) Minimum value of f (x) is not finite.
  6. 28. 1 sin IT log 3 Consider the equation -27 0 Then which of the following statements is [are true? (A) There exists, infinitely many solutions for the equation in (0, 2m) (B) There exists, finitely many solutions for the equation in (0, 2m) (C) Sum of all the solutions of the given equation in (0, 9m) is 40.5 75 (D) Sum of all the solutions of the given equation in (0, 9m) is— PSG (29-31) If f : [0, 2] [0, 2] is a bijective function defined by f (x) numbers, then 2 ax +bx +c , where a, b, c are nonzero real 29. 30. 31. f (2) is equal to (B) a where a e (0, 2) Which of the following is one of the roots of f (x) 0 is 1 1 (C) 2 (D) cannot be determined 1 1 a 1 Which of the following is not a value of a 1 4 1 2 PSG (32-33) Let f (x) be a real valued function such that it satisfy functional equation f (x) + f (l —x) k, Yr e Q . Where k is constant quantity? Let m be a positive integer .Put x in the given equation, we get 111 -k I Ill -k I — r Ill -k I — r Ill -k I Ill -k I Let m +1 —r = t Ill -k I r=l mk 111 + I 111k mk 111 -k I 111 -k I 111 -k I Ill -k I r=l mk 2
  7. If f (x) If f (x) PSG (34-35) (525) 32. 33. I 3 a (55) Yr e Q, then the value of f —x -I-3X (B) 27 211-1 is equal to a (535) (C) 54 (D) 55 (2n — l)a (D) 2 Let f : R + R is a function satisfying f (2 x) f (2 + x) and f (20 34. 35. If f (0) — 5, then minimum possible number of values of x satisfying f (x) — 5 for x e [0, 170] ,is (A) 21 (B) 12 (C) 11 Graph of y f (x) is symmetrical about line (B) 1-5 0 0 (D) 22 (D)x 20 0 PSG (36-37) For a natural number n > 1, let g(n) denote the number of different prime factors of n. Thus g(500) 2Andg(37) l. Let N be the set of all natural numbers. A number theoretic function, called the MOBIUS function u: N + 0, l},is defined as : 1 "(n) 0, g(n) (-1) for n = I for n > I and n has sqaure divisor for n > I and n has no sqaure divisor 36. Let a e {0, l, 2, be {0, l, 2}. The number of values of n , such that n "(n) —l is (D)none of these The number of solutions of the equation +u(n+l) -kg(n+ 2) -kg(n+ 3) r 3b and 4 if ne{l, 2, 100 (D) none of these
  8. INTEGER TYPE (38-45) FOLLOW THESE CODES TO WRITE YOUR NUMERICAL ANSWER IN FORM OF ENGLISH ALPHABTES 3-1) 4-AB 5-AC 6-AD 7-BC 8-BD 9-CD IO-ABC O-A I-B 2-C Il-ABD 12-ACD13-BCD14-ABCD E.g if you have answer as 9 darken the bubble C&D. 38. 39. 40. 41. 42. 43. 44. 45. Let f (x) is a real function satisfying f : {I, 2, 3, 4} {I, 4, 9, 16} And : {I, 4, 9, 16} f (Xl) > f g(Xl) < g(X2) 1 1 1 111 YxeR {l}; then 6. f (0) is ? 1, are two bijective functions such that 2'3'4 Leu fl g -1 -1 , find value of n +3M , M (gof) 2 4 Sum of all value of x satisfying 3[x] + 2{x} is E . Where [.] is G.I.F. and {. } is F.P.F. 1 Find value of — kx•2 -k — 8 A function f : R {p, q} + R is defined by f (x) 2 , where x = p, q are roots of denominator for a given k . The number of integral values of k is? If fractional parts of —& for some x e are equal, then the value of x 3 4 is Let f (x) C Ax + B Ax + D E. If y f (x) and y g(x) intersect each other at points (4, 9)& (13, 2) find value of E C+ Let (x, Y) 12 3 Y 2 + 4 x 4 < 2 x -k I & B= y) 1 + Y2 < 4} , then find no of ordered pair (x, y) e AßlB such that x e W, y e W The number of function f : { 3, —2, 4, 5} {2, 5, 8, 20, 24} that can be defined such that f (i) < f (j) for i < j is L. what is the value of — 7 GO BACK AND RECHECK (1) FOR HALF SOLVED QUESTIONS. (2)Read more questions carefully to solve more. (3) Finish paper in two turns.

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