Page 2 : OVFFERENTIAL CAL, , 352, , OETERMINATE FORMS, , ACL, , and c such that, 31. Find the valucs of a, b, , is equal to, , mxlogx, , lim, , c sin x, , ra+b-cOs.r), lim, , Agra 2001, Garhwal 1999,, , Kanpur 20M, , ANSWERS, , c), , Thevalue of lim, , 1, , 9, , 15. e, , 13. e., , 14. el/12, , 19., , 20. n/6., , loga, , 11., a=, , 16. a log a., , e/2, , 18. 1., , 1., , (a) 1, , (c), , 3, , (a), , (), , ii) -1., , (i) 2a/b., , e-e, , 9. Value of lim, iS, x0 log (l+bx), , ii) 0., , -, , (a), , log(1b), , 180, , of the following questions, four alternatives are given for the answer., , Choose the correct alternative., Which of the following is not an, , ., , Value of, , lim, , (c) o/o, , 2., , The formula of, , 3., , 8(a), , Consider () lim, (a) Both, ().i), , (d), , =1, (G) lim, are, , true, , (c) ) is true, (ii) is false, , 4. lim, I0, , (a) 0, , 12., , lim, a, , lim, , Lla), , ag(x), , (6) -1, , (c)0, , (), , s:, , sin6 0, , 1S, , (6)-2, , (c) 0, , (), , 13., , &'(), () 0, , (a)1, , 1,then:, , 4. lim O8x, , 6) () is false, (i) is true, )Both (), (i) are false, , x0 cotx, , (a), , x0, , log(aKanpur 20, 2001,, Agra, , a) 0, , (c) log (ab), , ()log(b/a), , iS, , (6), , 0, , 15, lim, , (e)log(alb), , (), , g'(x), , is equal to:, , (6), , c)0, , lim U - 8 c o s, , (a) 2, , ag(*), , 6) -1, lim s r c o s x, , 8+0, , c) l i m Q, , g{*), , (Agra 2001, Kanpur 20), , (b) lim, , g{z), , 11., , 1-cosx, , (a)0, , L' Hospital's rule is:, , (a) l i m ,, , Only one of them, , 0, , Og(+b), , x0, , x sin x, , indeterminate form?, (6) (d) 0x, , (a) +, , 9, , (Garhwal 2002), , 10., , is correct., , (d), , value of limcosx, 8. The, -0 32, , OBJECTIVE QUESTIONS, For each, , (Garhwal 2061, Avadh 2005), , is, , 0, , 30. Continuous at 0., , c=, , (c)1, , The value of lim, , 22. - el., , (i) -1., , 28. ()1., , 120, b =60,, , 12., , 17. 2., , 25. () 1., , 29. 0,0,0, 0,0,0,0,0., 31., , 11. b/2, , 10. -2, , 21. log (e/b) log be., , log(aie), , 24., , 26, , (b), , 6. -1, ., , 23. 1le/24., , -1, , (a) - 1, , 2.0, , 7. -2, , 353, , (b), , (a) 0, , S, , logx, , (6), , c), , (d)

Page 4 : DIFFERENTIAL, AL CACALC, , CESSIVE, , DIFFERENTIATION, , 189, , SUCCE, , ANSWERS, , (2-1)+m)., , (32+m), , 188, , ., , ., , ., , +m),, +2rV(1, , ( 2 n - 2 ), , (1, , (12+m), , m, , +m), , +m)(42, nm (22, n 1s, , -, , n, When, , n, , is odd, , O,, , ., , ., , 4., , -r5, , y, (U)=, I f n is even,, , ., , even, , 0; ifn, , 1s, , odd,, , y n (0)= n'3n - 5 ) 4, , when, , 17., , OBJECTIVE QUESTIONSs, , MISCELLANEOUS EXERCISES, , =, , -36y, -x))-(4-12r), , 1., , Show that, , ifr (1, , 3., , then, , -(4-), , x]y,1, , (12-2m), , [4-nr(1-x)y,+2-, , ofet, , sin, , (9-n)y,, , 6=, bx and, , tan", , Note:, , =0., , (b/a),, , prove, , Ify,, , denotes, , the nth, , Ifx=t-, , 1., , (a), Value, , 2., , (a), , Find the third, , of, , derivative, , of, , =, , , value of, , at, , dx, , (7, 2) will be, , (d), , (C), , (6), , 1, , (6), , n! a, , (Avadh 2002), , Value of Dax +b, n!a", (a)(ax + b)", , tan, , Va-, , n!, , (d), , (C)na b", , Find the value, , Ifu,, , denotes, , ofthe th, , derivative, , 4, , of, , (c)a sin (ax, , the nth derivative of (Lx+, , that, Mx*-2Bx +C), prove, , CU..1+2dU,1+U,, n+1, , nt---, , 6., , +132(, , x, , (d)bsin (ar +b+nm) (Manipur, , nr), 2, , nth, is used to find the, , differential coefficient, , difference, , (Rohilkhand 2002), , (d), (+1), , of D" (uv) is:, , D".uD v, , (c)"C, D"- u D'v, , (Manipur2, , -2, , +2) x( +1), , 7., , an +2 0)-4ny2, (0)+ (2n, sin (m, , +2n (3x' +1), +2n(n-1)(2n- 1)x*, , 2, , 8., , +m(n -1)?(n-, , Ify= sin (m sin'x),, , then, , =0, (c) (I -x*)y-xy -m*y, (x2+1)y, Ify= (tan' x), then, (6), , (a) 2, 9., , 1) 2ny2n-,(0) =0., tan' x), show that, , If y = [log {x -, , V, , +1D},, , -, , Y2n0)=0 and y2+1(0)=(-1", , m, , (m-1\(m, , (b), , (a), , 0, 2), , 2000), , of :, , oftwo, , (a) (1-x*)y-*y, + miy =0, , y=e""cosx, show that, , Ify=(1 +x*, , b sin ( +,nT), , (a), , 9. Ify=(tanx, then, , If, , (b), , b)} is, , functions, product of two, th term in the expression, "C+, , Deduce that, , (d), , (ar + b)"+I, , only, (a) trigonometric, function only, exponential, (6), functions, (c)sum and, , Bhopal, , +, , n!a", , (C), , functions, , =0., , D(a -2., , b, , +, , (ax + b", , nt), , +, , Leibnitz's theorem, , 5., , Ify=x*e", then, , +(4ndr, , b, , +, , +, , -1, , -1 n!a", , (b, , of D"{sin (ax, , Value, , (n+)(n + 2), , 11., , =I-cos, , is, D" (ar +b", , (u) d' sin (ax, , 2B, , 10., , Only one, , na", , V1+, , forx =0., , 8., , 1, y, , answer., , Ify= (a+br), , 4., , 7., , sin, , given for the, , are, , alternative., thecorect, , +n, , =, , 5. Provethat, 6., , Choose, , ofthem, , 'y-o, , 4., , For, , is correct., , that, , 8)., e " sin (bx, =0., (a c o s 8)", y,, tby,-1, + (a* prove that, that y,. 2ay,, Also show c o s kr + (c +dx) sin kx,, derivative, , 2., , 0,, , alternatives, f following questions, four, each ofthe, , Ifylm-y-l/m, (a) my, , =2x, then, , +2x (** 1»», , -2, , then value of(1, , D-/, , 6), , nC, , (d), , C,- D", , u, , D, , u, , Dv, , (6), (d), , (1-)y+*y-, , (c), , 2y, , None, , v, , y =0, , of these, (d)-2y, , equal, +*)y2+2, is, (c), , -, , 1, , (c), , tmy, , to, , (a), , 2, , (r 1)>2 *XY, , None of, , these, , these, , -, , (6)-my, , (Avadh 2005), , (a), , None of

Page 5 : DIFFERENTIAL CA, , ()y**-*y=0, 190, , y+x, , (o) - , + r y = 0, , +*y=0, , (c) r y - y , - r ' y = 0, , satisfies, , 1, , -, , 12. Function, , (a) (r+ 1)y'=ky, , 13. Ify-1(a-x),, , (a), , (d) +1)y"+xy'-Ry-g, , then y, , (-1(r+a)", 2a, , -(x, , (6)-15--a, , -a)"], , (d)None ofthese, , - 1 " [ r a)" + (x+ a)"], a, -, , 14., , (1+)y"+ ky' -y-0, , (), , Ifu= sin nr + cos nx, then u,=, , (a) [1 +(-1Y sin 2nx]/2, (c) [1 +(HYsin 2nx]l2, , (6), , [ 1 4+ sin 2nx]/2, , (d), , None of these, , 15. D'(sin r sin 3x) =, , ()None of these, 16., , Ify, , is, , a, , polynomial of degree n in x and first, coefficient is 2, then, , (a) 2 (n!), 17., , (b) 2(n) x, , Ify=xe and y, "C,y, +ay, +, ()2 (n-1)!x, then, (n-DC,y, a, (a) n (n-1), , D"-"y)=, (d) None ofta, , =, , =, , 18., , Ify=-' logx, then xy,=(6), , (a) n!, , 19., , Ify= e, (o) y,, , 20., , n(n-2), , (c) -nn- 1), , (6) (n-1)!, , , then (1, , fy=sin' x, then, , ()-n(n-, , (c) (n-2)!, , (d), , +*)y,+2+ [2 (n+ 1)x- 1]y,+1, , None ofts, , (6) n(n+ 1y,, , c), , (1-..-(2n+1)xy,,, (a) y, , -n, , (n+ 1y,, , (d)-, , (b) - *y, 1., ., , (a), , 9., 15. (d), , 2. (6), , 9. (6), 16. (), , 3. (c), , 10. (a), 17. (d), , ANSWERS, (c), , 11. (), 18. (b), , -(r+1)y,, , 5. (d), , 12. (e), , 19. (c), , a)Noneof, , 6. (c), , 13. (d), 20. (a), , 7, , 14. (C)

Page 6 : IFFERENTIA, , S54, , log, The, , 16., , lOg tan, , Value, , is, , cot x, , (, , ()0, , -, , 1, , (b), , (a), , 17, , f0, , valuc of, , CALCUL, , of 6, , is:, , (), , (c)-1, , log*, (6)1, , (a)o, , lim, , Vaueof, , 18., , 3, , (2+5), , is, , (), , (a), , tan5, 19., , >r/2, , is, , tan r, , (6), , (), , (c), , 5, , (a), , im, , 20., , (b), , (a) 0, tan, , -x), lim, 1L, , 21., , lim, , (b), , (a), , (a", , 1)x, , lim (sinx), , tan, , lim, , 0, , 2, , ()0, , lim (secx -, , -, , (a), , -1, , a2, , (c)a, , (6), , 0, , (c)-, , (d), , 1, , lim (cosecx -cotx) is, (a) 0, , 27., , (), , tan x) is:, , IR/2, 26., , log (1/a), , sin, , (6), , Value of, , (), , is:, , x, , (6), , (a), 25., , (c) log a, , rx/2, , Value of, , 2, , (), , 1, , (b), , (a) 0, , 24., , 1, , is:, , 1, , Value of, , 23., , -, , -, , s:, , -, , (a)0, , 22., , (a), , o, , Value of Ilim, 0 (1+x), (a)1, , (6), , 1, , (-1, , ()c, , is:, , (6)-1, , (c)e, , (a), (Agra 2003;, , lle, , 2003), Kanpur

Page 7 : E R N I E, , F O R M S, , q M I N A T EF O R M S, , 355, , is:, , (cosx)*%i, , he of lim, 0, , Mim, , (6) 1, , c)e, , ()1le, , C)e, , (d), , 1/e, , (c), , is, , Value oflim, , (6), , 1, , a) 0, , ANSWERS, , ., , (a), 7. (6), 13. (6), 19. a), , (6), , 2 (6), , & (c), 14. (), , 3. (a), 9. (a), , 20. («), , 15. (c), 21. (), , 26. (a), , 27. (), , (), , 5. (a), , 6., , 10. (), 16. (), , 11. (a), , 12., , (6), , 17. (6), 23. (a), , 18., , (6), , 24., , 29. (6), , 30., , 4., , 22., 28., , (a), , (b)