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DIFFERENTIATION, , IIT JEE Syllabus, , 1. Differential Coefficient,  Differential Coefficients of some standard function, 3., , Theorems on Differentiation, , 4. Methods of Differentiation, , Total No. of questions in Differentiation are:, Solved examples……….......………………..…31, Level # 1 ….……………………………….… 133, Level # 2 …….……………………………….…34, Level # 3 …….……………………………….…28, Level # 4 ……………………………………..…22, Total No. of questions…..…………………….248, , ***, 1. Students are advised to solve the questions of exercises (Levels # 1, 2, 3, 4) in the, same sequence or as directed by the faculty members., PHOTON, , DIFFERENTIATION, , 1

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2. Level # 3 is not for foundation course students, it will be discussed in fresher and, target courses., , Index : Preparing your own list of Important/Difficult Questions, Instruction to fill, (A) Write down the Question Number you are unable to solve in column A below, by Pen., (B) After discussing the Questions written in column A with faculties, strike off them in the, manner so that you can see at the time of Revision also, to solve these questions again., (C) Write down the Question Number you feel are important or good in the column B., , EXERCISE, NO., , COLUMN :A, , COLUMN :B, , Questions I am unable, to solve in first attempt, , Good/Important questions, , Level # 1, , Level # 2, , Level # 3, , Level # 4, , Advantages, 1. It is advised to the students that they should prepare a question bank for the revision as it is, very difficult to solve all the questions at the time of revision., 2. Using above index you can prepare and maintain the questions for your revision., , PHOTON, , DIFFERENTIATION, , 96

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KEY CONCEPTS, 1. Introduction, The rate of change of one quantity with respect to, some another quantity has a great importance. For, example the rate of change of displacement of a, particle with respect to time is called its velocity, and the rate of change of velocity is called its, acceleration., The rate of change of a quantity 'y' with respect to, another quantity 'x' is called the derivative or, differential coefficient of y with respect to x., , 2. Differential Coefficient, Let y = f(x) be a continuous function of a variable, quantity x, where x is independent and y is, dependent variable quantity. Let x be an arbitrary, small change in the value of x and y be the, y, corresponding change in y then lim, if it exists,, x 0 x, is called the derivative or differential coefficient of, y with respect to x and it is denoted by, dy, . y', y1 or Dy., dx, , So,, , dy, y, = lim, dx x0 x, , f ( x  x ) – f ( x ), dy, = lim, , x, , 0, x, dx, The process of finding derivative of a function is, called differentiation., If we again differentiate (dy/dx) with respect to x, then the new derivative so obtained is called second, derivative of y with respect to x and it is denoted,  d2y , by  2  or y" or y2 or D2y. Similarly, we can find,  dx , successive derivatives of y which may be denoted by, , , , dn y, d3y d4 y, ,, , ……, n ……, 4, 3, dx, dx, dx, y, Note : (i), is a ratio of two quantities y and x, x, dy, where as, is not a ratio, it is a single quantity, dx, dy, i.e., dy ÷ dx, dx, PHOTON, , dy, d, is, (y) in which d/dx is simply a symbol, dx, dx, of operation and not 'd' divided by dx., , (ii), , DIFFERENTIATION, , 3. Differential Coefficient of Some Standard, Function, The following results can easily be established, using the above definition of the derivative–, d, (i), (constant) = 0, dx, d, (ii), (ax) = a, dx, d, (iii), (xn) = nxn–1, dx, d x x, (iv), e =e, dx, d, (v), (ax) = ax logea, dx, d, (vi), (logex) = 1/x, dx, d, 1, (vii), (logax) =, dx, x log a, d, (sin x) = cos x, dx, d, (ix), (cos x) = – sin x, dx, d, (x), (tan x) = sec2x, dx, d, (xi), (cot x) = – cosec2x, dx, d, (xii), (secx)= secx tan x, dx, d, (xiii), (cosec x) = – cosec x cot x, dx, d, 1, (xiv), (sin–1 x) =, , –1< x < 1, dx, 1 – x2, , (viii), , (xv), , d, (cos–1 x) = –, dx, , 1, , 1 – x2, d, 1, (xvi), (tan–1 x) =, dx, 1 x2, , ,–1 < x < 1, , 97

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f, f, and, are partial differential, x, y, coefficients of f(x,y) with respect to x and y, respectively., , Where, , (xiii), , Note : Partial differential coefficient of f (x,y) with respect, to x means the ordinary differential coeffcient of, f(x,y) with respect to x keeping y constant., , sin–1 (–x) = – sin–1x., cos–1(–x) = – cos–1 x., tan–1 (–x) = – tan–1 x or  – tan–1 x., 1– x , (xvii) /4 –tan–1 x = tan–1 , , 1 x , Some suitable substitutions, Function, Substitution, (xiv), (xv), (xvi), , 5.2 Differentiation of logarithmic functions, In differentiation of an expression or an equation is, done after taking log on both sides, then it is called, logarithmic differentiation. This method is useful, for the function having following forms–, (i) When base and power both are the functions of x, i.e. the function is of the form [f(x)] g(x)., y = [f(x)]g(x), log y = g(x) log [f(x)], d, 1 dy, ., =, g(x).log [f(x)], dx, y dx, , (v), , = sin x 1 – y  y 1 – x , , , –1, –1, cos x cos y, = cos–1 xy  1 – x 2 1 – y 2 , , , , (vi), , xy, tan–1x ± tan–1 y = tan–1 , , 1  xy , , (vii), (viii), (ix), , 2, , , , 2 sin x = sin  2x 1 – x , , , –1, –1, 2, 2 cos x = cos (2x – 1),  2x , 2 tan–1x = tan–1 , 2 , 1– x , –1, , –1 , , , , PHOTON, , x = a sin  or a cos , , (ii), , x2  a2, , x = a tan  or a cot , x = a sec  or a cosec , , (vi), , a–x, ax, , x = a cos 2, , (v), , a2 – x2, a2  x2, , x2 = a2 cos 2 , , (vi), , ax – x 2, , x= a sin2 , , (vii), , x, ax, , x = a tan2 , , (viii), , x, a–x, , x = a sin2 , , (ix) ( x – a )(x – b), , x = a sec2  – b tan2 , , ( x – a )(b – x ), , x = a cos2  + b sin2 , , 5.4 Differentiation of infinite series, If y is given in the form of infinite series of x and, we have to find out dy/dx then we remove one or, more terms, it does not affect the series, (i) If y =, , , 2, , y =, , f (x)  f (x)  f (x)  ..... then, f (x)  y, ,  y2 = f(x) + y, , dy, dy, = f (x) +, dx, dx, dy, f ( x ), , =, dx 2 y – 1, , 2y, , , , 2 , ,  2x , –1  1 – x , = sin–1 , =, cos, , 2,  1 x2 , 1 x , , , , (x), (xi), , a2 – x2, , (x), , 2, , , , (i), , (iii) x 2 – a 2, , dy, d, , = [f(x)]g(x).  [g( x ) log f ( x )], dx,  dx, , 5.3 Differentiation by trigonometrical substitutions, Some times before differentiation, we reduce the given, function in a simple form using suitable trigonometrical, or algebric transformations. This method saves a lot of, energy and time. For this following formulae and, substitutions should be remembered., Formulae, (i), sin–1x + cos–1 x = /2, (ii), tan–1 x + cot–1x = /2, (iii), sec–1 x + cosec–1 x = /2, (iv), sin–1 x ± sin–1y, –1 , ,  3x – x 3 , , 3 tan–1 x = tan–1 , 2 ,  1 – 3x , tan–1 x + tan–1 y + tan–1 z,  x  y  z – xyz , , = tan–1 ,  1 – xy – yz – zx , , (xii), , 3 sin–1 x = sin–1 (3x – 4x3), 3 cos–1 x = cos–1 (4x3 –3x), DIFFERENTIATION, , (ii) If y = f ( x )f ( x ), , , f ( x )...., , then y = f(x)y., , log y = y log [f(x)], 1 dy y.f ( x ),  dy , =, + log f(x) .  , y dx, f (x),  dx , 99

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, , , dy, y2f (x), =, , dx, f (x)[1 – y log f (x)], , (iii) If y = f ( x ), , then, , PHOTON, , , , 1, f (x) , , 1, f (x), , 1, f ( x )...., , dy, yf( x ), =, dx, 2y – f (x), , DIFFERENTIATION, , 100

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SOLVED EXAMPLES, Ex.1, , If y = (1+x1/4) (1+x1/2) (1–x1/4), then dy/dx, equals(A) –1, , Sol., , (B) 1, , (C) x, , (D), , x, , ( x 2  a 2 ) 2 .2  2x.2( x 2  a 2 ).2x, , =–, , Ans.[A], , If x = a (+ sin ), y = a (1– cos),, , dx, then, d, , =, Ex.5, , dy/dx equals (A) tan , (C) tan, Sol., , dy, 2x, d2y,  2, , dx ( x  a 2 ) 2, dx 2, , (x 2  a 2 ) 4, , y = (1+x1/2) (1–x1/2) = 1– x,  dy/dx = –1, , Ex.2, , Sol., , (B) cot , , 1, , 2, , (D) cot, , 1, , 2, , Sol., , dx, dx, = a (1+ cos),, = a sin , d, d, , 2(3x 2  a 2 ), , If y =, (A), (B), (C), (D), , 2 sec x (sec x – tan x)2, – 2 sec x (sec x – tan x)2, 2 sec x (sec x + tan x)2, – 2 sec x( sec x + tan x)2, , y=, , sec x  tan x sec x  tan x, ., sec x  tan x sec x  tan x, , , , dy, = 2(secx – tanx) (sec x tan x–sec2 x), dx, , = –2 sec x (sec x– tan x)2, , Ans.[C],  ex , If y = log  x  , then dy/dx equals  e 1 , , (A), , e 1, , e 1, x, , (A), , (D) None of these, , (C), Sol., , = x – log (ex + 1), , Ex.4, , 1, , 3x  a, 2, , (A), , (C), , PHOTON, , Ans.[A], , , then, , d y, dx 2, , (x 2  a 2 )3, 2(3x 2  a 2 ), (x 2  a 2 )3, , (B), , (D), , 1 x2, , 1, (1  x ) 2, , (D) None of these, , Let us first express y in terms of x because all, alternatives are in terms of x. So,  x2` (1+ y) = y2 (1 + x),  (x – y) (x + y + xy) = 0, , equals -, ,  x + y + xy = 0, 3x  a, 2, , 2, , 1, , (B) –, , dy, equals dx, ,  x2 –y2 + x2y – y2 x = 0, , 2, , x2  a2, , 1, (1  x ) 2, , Ans.[B], , x 1  y  y 1  x, , dy, ex, 1, =1– x,  x, dx, e 1 e 1, , If y =, , If x 1  y  y 1  x = 0, then, , (e  1) 2, x, , y = log ex – log (ex + 1), , , , Ex.6, , 1, , (B), , ex 1, , (C), Sol., , 1, x, , sec x  tan x, dy, , then, equals sec x  tan x, dx, , = (sec x – tan x)2/ 1, , dy dy / d, 1, a sin , , , =, = tan , dx dx / d, a (1  cos ), 2, , Ex.3, , Ans.[C], , (x 2  a 2 )3, , ( x  y), , 2, , (x 2  a 2 ) 4, , y=–, , 2(3x 2  a 2 ), , , , (x 2  a 2 ) 4, , DIFFERENTIATION, , x, 1 x, , dy, (1  x )1  x.1, 1, , , Ans.[B], 2, dx, (1  x ), (1  x ) 2, , 100

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Ex.7, , If y = sin–1, , sin x , then, , Ex.10 If x2 ey + 2xyex + 13 = 0, then dy/dx equals -, , dy, equals dx, , (A) –, 2 sin x, , (A), , Sol., , 1  sin x, , 1, 1  cos ec x, (D), 2, , 1, 1  cos ec x, (C), 2, , (C) –, Sol., , 1, 1, dy, =, . cos x, ., dx, 1  sin x 2 sin x, , 1  sin x, , =, , , , 2 sin x, , x ( xe, , sin x, , (B), , 1  sin x, , 2xe y  x  2 y( x  1), yx, , (B), ,  2), , 2xe x  y  2 y( x  1), x ( xe x  y  2), , 2xe y  x  2 y( x  1), x ( xe y  x  2), , (D) None of these, , Let f(x,y) = x2ey + 2xyex + 13, , , dy, f f, =–, /, dx, x y, , 1, 1  cos ec x Ans.[C], 2, , =–, , 2xe y  2 yex  2xye x, x 2 e y  2xe x, , Dividing Numr and Denr by ex, Ex.8, , (A) 1/x, (C) –, , Sol., , dy, 2xe y  x  2 y( x  1), =–, dx, x ( xe y  x  2), , If y = logx10, then the value of dy/dx equals(B) 10/x, , (log x 10 ) 2, x log e 10, , y = logx 10 =, , (D), , 1, ( x log e 10 ), , Ex.11 If xy yx = 1, then, , log e 10, log e x, , dy, , = loge 10, dx, , (A), , , 1, 1, , , . , , 2, ,  (log e x) x , , , (C), , (log e 10 ), 1, ., x log e 10 (log e x ) 2, , =–, , Ex.9, , Sol., , If cos (xy) = x ,then, , y( y  x log y), x ( x  y log x ), , Ans.[C], Sol., , y  cos ec( xy ), x, , (B), , (C), , y  cos(xy ), x, , (D) –, , Taking log on both sides, we have, y log x + x log y = 0, , dy, is equal to dx, , (A), , Now using partial derivatives, we have, , y  sin(xy ), x, , y / x  log y, y( y  x log y), dy, , =–, Ans [D], dx, log x  x / y, x ( x  y log x ), , y  cos ec( xy ), x, ,  cos (xy) – x = 0, , , x ( x  y log y), y( y  x log x ), , y( y  x log y), x ( x  y log x ), , (D) –, , (log e 10 ) 2, x log e 10, , =–, , dy, equals dx, , x ( y  x log y), y( x  y log x ), , (B) –, , 2, , Ans.[A], , y  cos ec( xy ), dy,  y sin(xy )  1, =–, =–, dx, x,  x sin(xy ), , Ans.[D], , 2, 1  y  x , Ex.12 If x = e tan  2  , then dy/dx equals x , , (A) x [1+ tan (log x)] + sec2 (logx), (B) 2x [1 + tan (logx)] + x sec2 (logx), (C) 2x [1+ tan (log x)] + x sec (log x), (D) None of these, , PHOTON, , DIFFERENTIATION, , 101

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Sol., , 2, 1  y  x , x = e tan  2 ,  x , , Sol., , Taking logarithm of both the sides, we get, , x2 = cos , ,  y  x2 , log x = tan –1  2 ,  x , , = tan–1 , ,  cos  / 2  sin  / 2 , ,  cos  / 2  sin  / 2 ,  1  tan  / 2 , ,  1  tan  / 2 , ,  y = x2 + x2 tan (log x), , = tan–1 , , dy/dx = 2x+2x tan(log x) +x2 sec2(log x)., , 1, x, , = tan–1 [tan (/4 + /2)] = /4+ /2, , = 2x [1+ tan (log x)] + x sec2 (log x).Ans.[B], , Ex.13 If y = tan–1, , 3x  x 3, 1  3x, , (A) 3x, (C), , Sol., ,  1  cos   1  cos  ,  , where, y = tan–1 ,  1  cos   1  cos  , , , , , then dy/dx equals-, , 1, , +, cos–1 x2, 2, 4, , , , 1, dy, =–, dx, 2, , 3, 1 x, , (D) 3 tan–1 x, , 2, , 3x  x 3, 1  3x 2, , 1 x2, , (A), , dy,  2x , equals  , then, 2, dx, 1 x , , 2x, 1 x, , (B), , 2, , 2x, , 1 x2, , 1 x, , 2, ,  2cos, , 1 x2, , Ans.[B], , (C) –, , dy, , then, equals, 2, 2, dx, 1 x  1 x, , PHOTON, , 1, , x, 1 x, , 4, , 1 x2, ,  cot, , , , , , . cos,  2a cos, . sin, 2, 2, 2, 2, , , = a= 2 cot–1 a, 2, ,  sin–1 x – sin–1 y = 2 cot–1 a, 1 x2  1 x2, , 2 1 x, , 1 y2, , cos  + cos  = a (sin  – sin ), , (A) –, , 1 x2, , (D) –, , 1 y2, , 2, , 2, dy, , =, dx 1  x 2, , tan–1, , 1 y2, , Substituting x = sin  and y = sin  in the given, , y = 2 tan–1 x, , Ex.15 If y =, , Ans.[C], , equation, we get, , 2, , (D) –, , 1 x2, , Sol., , (B), , 1 y2, , Ans.[C], , Ex.14 If y = sin–1 , , Sol., , 1 x4, , of dy/dx is = 3 tan-1 x, , 3, dy, =, dx, 1 x2, , (C) –, , 1 x4, , x, , . 2x = –, , 1  x 2  1  y 2 = a (x – y), then the value, , Ex.16 If, , (C) –, , (A), , 1, , (B) tan 3x, , y = tan–1, , , 2, , =, , 2, , (B) –, (D) –, , 1, 1 x, , 4, , Differentiating with respect to x, we get, 1, 1 x, , , , 2, , , , 1, , dy, =0, 1  y dx, , 1 y2, dy, , dx, 1 x2, , 2, , Ans.[B], , x, 2 1 x4, DIFFERENTIATION, , 102

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Ex.23 If (a +bx)ey/x = x, then the value of x3,  dy, ,  x,  dx, , , 2, ,  dy, ,  y,  dx, , , (A)  y, (C) x, Sol., , d2y, dx, , 2, , is -, ,  2x  1 ,  and f ' (x) = sin2 x, then dy/dx,  x2 1, , Ex.25 If y = f , equals -, , 2, , (B)  x, , dy, –y, dx, , (A), , (1  x ), , (B), , (1  x 2 ) 2,  2x  1 , ,  x2 1, , dy, y, 1, dx, , 2, x, x, , dy, – y = x2, dx, ,  2x  1 , ,  x 2 1, , x3, , Ex.24, , d2y, dx 2, ,  a  bx  bx , ax, ,  ,  x (a  bx)  (a  bx), , Sol., , dy,  2x  1  d  2x  1 , = f ' 2 , , , dx,  x  1  dx  x 2  1 , , d2y, dx, , 2, , 2, , 2, ,  ax ,  dy, ,  y  Ans.[B],  = x, a, , bx, dx, , , , , , =, , 3/ 4, d   x  x  2  , log e ,   equals dx    x  2  , , , , (A), , (C), ,  2 x  1  ( x 2  1)2  (2x  1)2x, Ans.[A], ., ( x 2  1) 2,  x2 1, , = sin2 , , dy dy (a  bx)a  ax(b), , , dx dx, (a  bx) 2, , , , x 2 1, x2  4, x 2 1, x 4, 2, , 2, , (D) sin , , Again differentiating with respect to x, we get, x, , Ex.26 If f(x) = |x–2| and g(x) = f[f(x)], then for, x > 20, g'(x) is equal to -, , Sol., , (A) 1, , (B) –1, , (C) 0, , (D) None of these, ,  g(x) = f [f(x)], = f {|x–2|}, , (B) 1, (D) ex, , = || x –2| –2|, x 2 1, , But x > 20 |x–2| = x – 2, , x 4, ,  g(x) = |x –2–2| = x – 4, , 2, ,  g'(x) = 1, Sol., , Ans.[A], , Derivative, =, , =, , d , 3, , log e x  {log( x  2)  log(x  2)}, dx , 4, , , d , 3, , x  {log( x  2)  log( x  2)}, , dx , 4, , , =1+, , =1+, PHOTON, , 2, , sin , , (C) sin2 , , Now differentiating with respect to x, we get, , x, ,  2x  1 , ,  x 2 1, , 2(1  x  x 2 ), , log (a+bx) + y/x = log x, , x, , sin2 , , 2 2, , (D) None of these, , Taking logarithm of both the sides, , b, , a  bx, ,  2x  1 , ,  x2 1, , 2(1  x  x 2 ), , 1 , 3  1, , , , 4 x2 x2, , 3 4, x 2 1, , 4 x2  4 x2  4, , Ex.27 f(x) is a function such that f " (x) = – f(x) and, f '(x) = g(x) and h(x) is a function such that, h(x) = [f(x)]2 + [g(x)]2 and h(5) = 11, then the, value of h (10) is (A) 0, , (B) 1, , (C) 10, , (D) None of these, , Ans. [A], DIFFERENTIATION, , 104

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Sol., , h' (x) = 2f(x) f ' (x) + 2g(x) g'(x), , Sol., , When 1 < x  3,, , = 2f(x) g(x) + 2g (x) f " (x), , f(x) = (x–1) – (x–3) = 2, , = 2f (x) g(x) – 2f (x) g(x), ,  f ' (2 – 0) = 0, f ' (2 + 0) = 0, , [ f " (x) = –f(x)], , =0, ,  f ' (2) = 0, , Ans.[B], ,  h(x) = c,  h(10) =h (5) = 11, , Ans.[D], , Ex.30 If f(x) = logx (n x), then at x = e, f '(x) equals-, , Ex.28 If x = (sec – cos ) and y = secn  – cosn ,, 2, , Sol., ,  dy ,  equals  dx , , then , , 2, , n 2 ( y 2  4), , (C), Sol., , (B), , n ( x  4), 2, , (C) e, , (D) 1/e, ,  n x = loge x, so, , y2  4, n ( x 2  4), , log(log x ), log x, ,  1 , 1,   log(log x ), log x, x, log, x, x, , ,  f ' (x) =, 2, (log x ), , (D) None of these, , x2  4, ,  f ' (e) =, , dx, = sec  tan  + sin , d, , Here, , (B) 1, , f(x) = logx (logex) =, , y2  4, , (A), , (A) 0, , 1/ e  0, (1), , 2, , , , 1, e, , Ans.[D], , = tan (sec + cos ), = tan , , (sec   cos ) 2  4, , = tan , , x2  4, , (sin 2x cos 2x cos 3x+log2 2x+3) w.r.t. x at x = , is -, , dy, = n secn  tan + n cosn–1  sin , d, , and, , = n tan , , , , Ex.31 The first derivative of the function, , (secn, , +, , (secn   cosn ) 2  4, , = n tan , , y2  4, , n tan  y 2  4, dy, =, dx, tan  x 2  4, , (C) –2 + 2loge 2, , (D) –2+ loge 2, , Sol., , Let y = sin 2x cos 2x cos 3x + log2 2x+3, =, , 1, sin 4x cos 3x + (x+3) log22, 2, , =, , 1, [sin 7x + sin x] + x+ 3, 4, , , , n ( y  4),  dy ,  , x2  4,  dx , , , , 2, , (B) –1, , cosn ), , = n tan , , 2, , (A) 2, , 2, , Ans.[C], , 1,  dy , =, [7 cos 7 + cos ] + 1, ,  dx  x   4, ,  , , Ex.29 The value of the derivative of |x–1| + |x–3| at, =, , x = 2 is -, , PHOTON, , 1, dy, = [7 cos 7x + cos x] + 1, dx, 4, , (A) –2, , (B) 0, , (C) 2, , (D) Not defined, DIFFERENTIATION, , 1, [– 8] + 1 = – 1, 4, , Ans.[B], , 105

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LEVEL-1, Question, based on, , Q.1, , Q.2, , If f(x) = loga (logax), then f '(x) is equal to (A), , log a, x, , (B), , log a, x log x, , (C), , log a e, x log e x, , (D), , x, log a, , If f(x) =, , 2x  4, x 2 1, , equals(A) x2 – 8x –2, (C) 4x + 2, , Q.3, , If y =, (A), , and f ' (x) =, , 1, sec2 x/2, 2, , (D) (/180) sec x tan x, Q.8, , p, ( x 2  1) 2, , , then p, , (B) –2x2 + 8x + 2, (D) –2x2 + 8x –2, , If a cos2 (x + y) = b, then, , (C), Q.11, , If y =, (A), , Q.13, , , 1  cos x , , , If y = tan 1,  where  < x < 2 then, 1  cos x , , , , dy, =, dx, , (A) 1/2, (C) sec x, PHOTON, , (C), Q.14, , (B) –1/2, (D) cosec x, DIFFERENTIATION, , x, cot x, , (D), , 2 x, , dy, equalsdx, , (B) –1, , (D) –1/2, , (C) 1/2, , dx, x, , then, equals( x  5), dy, 5, , (1  y) 2, 1, (1  y) 2, , If y =, (A), , Q.6, , cot x, , If x = a (cos t + t sin t), y = a (sin t– t cos t),, , (C), , (D) cosec , , (B), , cot x, 2x, , (A) 1, , d  1  1  cos  ,  equalstan , d ,  sin  , , (C) sec , , tan x, , then at t = /4,, , (B) cosec x, (D) – cosec x, , (B) 1, , (D) –1, , (C) 1, , 2 x, , Q.12, , (A) 1/2, , (B) –2, , dy, equals dx, , d/dx [log sin x ] is equal to(A), ,  1  cos x ,  , then the value of,  1  cos x , , (A) sec x, (C) –sec x, , Q.5, , Q.10, , If y = log , dy, isdx, , (B) –1, (D) None of these, , (A) 2, , 1, cosec2 x/2, 2, , (D) cosec2 x/2, , , 2, , If f(x) = |cosx – sinx|, then f '   is equal to–, (A) 1, (C) 0, , Q.9, , (B), , d/dx (sec xº) equals (A) sec x tan x, (B) sec xº tan xº, (C) (/180) sec xº tan xº, , 1  cos x, dy, , then, equals1  cos x, dx, , (C) sec2 x/2, , Q.4, , Q.7, , Differential coefficient, , (B), , 5, (1  y) 2, , (D) None of these, , dy, 1 x, , then, equalsdx, 1 x, y, , 1 x, , 2, , (B), , 2, , (D), , y, 1 x, , y, x 1, 2, , y, y 1, 2, , If y = log (cosec x – cot x), then, (A) cosec x + cot x, (C) sec x + tan x, , dy, equalsdx, , (B) cot x, (D) cosec x, 106

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Q.15, , 1  sin x, dy, , then, equals 1  sin x, dx, , If y = log, , (B) – sec x, (D) sec x tan x, , (A) sec x, (C) cosec x, , Q.16, , Q.21, , If y = e3x sin 4x, then the value of, , , dy, isdx, , Q.22, , , , 4, 3, , (B) e3x cos  4x  tan 1 , , , Q.23, , , , 4, 3, , (D) 5e3x cos  4x  tan 1 , , , Q.17, , If y =, , sin, , 1, , x, , 1 x2, , Q.24, , (B) 1+ xy, (D) xy – 2, , If y = sin x + cos 2x, then, , d, dx, , (A), (B), (C), ,  1  x  x,  tan ,  1  x 3/ 2, , , 1, 2 x (1  x ), , 1, 2 x (1  x ), , d2y, dx 2, , –, +, , Q.25, , ,  equals , , , 1 , 2 , , Q.26, , 1 x2, , 1, 1 x, , (D) 2, , dy, equals dx, , 1 , 2 , , dy, equalsdx, , 1, 4x, , 2, , 2, 4  x2, , 2, , (B), , 4  x2, , (D) None of these, , If x = a sin3 t, y = a cos3 t then, (A) tan t, (C) – tan t, , 1, , (C) –1, , If y = sin–1  x  – cos–1  x  , then the value, , (C) –, , equals-, , dy, equalsdx, , (B) cot t, (D) – cot t, , The differential coefficient of sec tan–1 x w.r.t., x is (A) x/ (1+ x2), , (B) x 1 x 2, , (C) 1/ 1 x 2, , (D) x/ 1 x 2, , 2, , 1, 1, –, 1 x 1 x 2, , Q.27, , 1 x ,  equals d/dx log ,  1 x , , , , dy, equalsdx, , (A), , (B) cos x log10e, (D) cot x, , (C), , If y = log10 (sin x), then, (A) sin x log10 e, (C) cot x log10e, , PHOTON, , (B) 1, , If y = sinn x cos nx, then, , (A), , (D) None of these, Q.20, , 1, dy, , then the value of, dx, x, , of, , (A) sin x + 4 sin 2x (B) – (sin x + 4 cos 2x), (C) – sin x + 4 cos 2x (D) None of these, , Q.19, , If y = tan–1 x + tan–1, , , then (1 – x2) dy/dx equals-, , (A) x + y, (C) 1– xy, , Q.18, , (B) 5e5 log x, (D) None of these, , (A) n sinn–1 x. cos (n +1) x, (B) n sinn–1 x. sin (n +1) x, (C) n sinn–1 x. cos (n –1) x, (D) n sinn–1 x. cos nx, , , 1 4 ,  4x  tan, , 3, , , (C) 5e3x sin, , dy, isdx, , (A) 5 log x, (C) 5x4, , is(A) 0, , 4, 3, , (A) e3x sin  4x  tan 1 , , , If y = e5 log x , then the value of, , DIFFERENTIATION, , x, 1 x, 1, x (1  x), , (B), , 1, 1 x, , (D) None of these, 107

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Q.28, , d/dx [ex sin 3 x] equals-, , Q.34, , If y = sin–1 x , then, , (A) ex sin ( 3 x + /3), , 2, , (A), , Q.29, , (C), , 1 x, e sin ( 3 x + /3), 2, , (D), , 1 x, e sin ( 3 x – /3), 2, , a  bx , then, , If y = sec, (A), , (B), , b, b a  bx, b, 2 a  bx, , Q.35, , (B), , a  bx tan a  bx, , (C), , sec, , Q.31, , Q.36, , The derivative of x |x| is(B) –2x, , (C) 2 |x|, , (D) Does not exist, , (C) –, , Q.37, , (B) 0, 1, 1 x, , (D) –, , If t = aex / (x – b), then, , x  x2, , 1 dt, ., equals to t dx, , (A) b(x–b)2, , (B) – b(x–b)–2, , (C) b2(x–b), , (D) None of these, , PHOTON, , , ,  sin x ,  equals 1  cos x , , d/dx tan 1 , , , 1  tan x, , , sec2   x , 1  tan x, 4, , 1  tan x, , , sec   x , 1  tan x, 4, , , 1, 2, , 1, 1 t2, , (A) 1/2, , (B) –1, , (C) 1/4, , (D) –1/2, , 1, , dy, =, dx, , 1 t, 1 t2, , (D) 1, , 1 t2, , 1  cos x, dy, , then value of, is1  cos x, dx, , (B) 1/2, (D) –1, , x, m 1 x2, m2x, 1 x, , (B), , (D), , 2, , If x = t2 +, (A) 2x, (C) x2, , DIFFERENTIATION, , 1 t2, , , then, , The differential coefficient of tan–1 mx with, respect to m is(A), , Q.39, , t, , and sin y =, (B), , If y = tan–1, , (C), Q.33, , 1  tan x, , , sec2   x , 1  tan x, 4, , , 1, 2, , (A) – 1/2, (C) 1, , 1, , Q.38, Q.32, , 1 x, , 1  tan x, dy, , then, is equal to1  tan x, dx, , If cos x =, , (C), , d/dx [cos–1 x + sin–1 1 x ] equals(A) 1, , 1, , (D), , (A) –1, , (A) 2x, , x 1 x, , (D) None of these, , a  bx, , a  bx tan, , (D) None of these, , Q.30, , (A), , dy, equalsdx, , 1, 2 x 1 x, , If y =, , sec a  bx tan a  bx, , a  bx sec, , (C) 2b, , (C), , 2, , (B), , x 1 x, , (B) 2ex sin ( 3 x + /3), , dy, is equal todx, , 1, t2, , , y = t4 +, , 1, t4, , x, 1 m2x2, , m2x, 1  mx 2, , , then, , dy, equalsdx, , (B) x, (D) None of these, 108

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Q.40, , Q.41, , If g(x) = x tan–1 x, then the value of g(1), equals(A), , 1, 2, , (B), , , 4, , (C), , 1 , –, 2 4, , (D), , 1 , +, 2 4, , Q.48, , Q.49, , (B) 1/2e, (D) 1, , Q.43, , If f(x) = x2 –5x – 5 and f "(2) + f '(2) + f(2) = 0,, then  equals–, (A) 0, (B) 2, (C) 10/3 (D) 15/4, , of c is(A) 2, , (B) 4, , (C) 6, , 1, , (A) –, , a2  x2, 1, , (C) –, , Q.45, , Q.46, , 2 a2  x2, , a2  x2, , (B), 2, , 1 x, , (D), , 2, , Q.52, , 2 a2  x2, , If y = tan–1 x/2 – cot–1 x/2, then, , (C), , Q.53, , If y =, , 2, , (D), , 1 x2, xc, 1 x2, , 1, 4x, , 1 x2, , If x2/3 + y2/3 = 1, then, , dy, equalsdx, , (A) (y/x)1/3, (C) (x/y)1/3, , (B) –(y/x)1/3, (D) –(x/y)1/3, , d2y, dx, , (A) 0, (C) –1, , 2, , Q.55, , equals(B) 1, (D) 2, DIFFERENTIATION, , dy, =0, dx, , dy,  a sin x  b cos x , equals , then, a, cos, x, , b, sin, x, dx, , , , If y = tan–1 , , (B) –1, (D) None of these, , If f(x) = log x(x > 1), then, (A) x log x, (C) x/ log x, , 2, , 3, , , then the value of xy where, , 4, , Q.54, , 1 x2, , (B) 3/4, (D) None of these, , (A) 1, (C) 0, , dy, is equal todx, , 4  x2, , 1, , 1, , d, [f (log x)] equalsdx, , (B) (x log x)–1, (D) None of these, , If x = log (1+ t2), y = 2t –2tan–1t, then at, t=1, , PHOTON, , (D) –, , is-, , , ,, , 2 ,  1 x , , , , (B), , is(A) 1/2, (C) 5/4, , a, , a, , dx 2, , (B) 3x + 6x log x, (D) None of these, , , , (D) 8, ,  x 2  x 1 ,  , then dy at x = 0 equalsIf y = log  2, , dx,  x  x 1 , (A) 1, (B) –1, (C) 2, (D) –2, , (A) 0, , Q.47, , (B) –, , d2y, , , , (C), ,  1 a  x , tan,  equals a  x , , , d, dx, , (D) – a/xy, , (C) 0, , If y = sin–1  2x 1  x 2  + sec–1 , , (A) 0, , Q.51, Q.44, , equals-, , dx 2, , then dy/dx equals -, , 2x 2  c, and f '(1) = 0, then the value, x2, , If f(x) =, , d2y, , If y = x3 log x, then the value of, (A) 5x + 6x log x, (C) 6x + 5x log x, , Q.50, Q.42, , (B) –a, , (A) a, , If f(x) = log x 2 (log x), then f '(x) at x = e is (A) 1/e, (C) 0, , If x = at2 and y = 2at, then xy, , The derivative of even function is(A) odd function, (B) even function, (C) constant function (D) None of these, , 109

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Q.56, , Q.57, , If y = |log x|, then(A) y '(1+ 0) = 1, (C) y '(A) = 1, , Q.59, , (B) y '(1– 0) = –2, (D) y '(0) = , , If u = sin–1 (x–y), x = 3t, y = 4t3, then du/dt, equals1, 1, (A), (B), 2, 1 t, 3 1 t 2, 3, , (C), , Q.58, , Q.64, , of, , If f(x) = |x–2| and g(x) = f (f(x)), then for, x > 20, g'(x) is equal to (A) 1, (B) –1, (C) 0, (D) None of these, , Q.65, , (A) 1, , (B) 0, , (C) –1, , (D) –2, , dx, , dx 2, , Q.61, , (A) – 4t(t2 – 1)–2, (C) (t2 + 1)(t2 – 1)–1, Q.62, , x2), , (A) my2, (C) m2y2, , PHOTON, , dx 2, , x, , then which one is true –, , Q.67, , If x = a (cos  + sin ), y = a (sin  – cos ), d2y, dx 2, 1, a, , at  =  equals to –, (B), , 1, a, , (C) a, , (D) –a, , Q.68, , If y = sin (m sin–1x), then (1–x2) y2–xy1+ m2y, equals to –, (A) 0, (B) y, (C) –y, (D) None of these, , Q.69, , If y = (1–x2)–1/2 sin–1 x, then (1–x2) y1 – xy, equal to –, (A) 0, (B) 1, (C) –1, (D) 3, , Q.70, , d/dx (log10x) equals –, (A) 1/x, (B) (1/x) log10e, (C) log loge x, (D) None of these, , Q.71, , The derivative of loga x + log xa is equal to–, , equals –, , (B) – 4t3(t2 – 1)–3, (D) – 4t2(t2 – 1)–2, , If y1/m = x + 1 x 2 , then, (1 +, , Q.63, , d2y, , 1, , If y = (sin–1x)2, then which one is true –, (A) (1 – x2)y2 – xy1 = –2, (B) (1 – x2)y2 + xy1 = 2, (C) (1 – x2)y2 – xy1 – 2 = 0, (D) None of these, , (A), , If x = t +1/t , y = t –1/t , then, , (B) – 3/8, (D) None of these, , Q.66, , equals –, , (A) m2y, (B) – m2y, (C) – am2 sin x + bm2 cos x, (D) None of these, , 1, is–, 2, , (A) (1 + x )y2 – (2x – 1) y1 = 0, (B) (1 + x2) y2 + (2x + 1) y1 = 0, (C) (1 + x2) y2 + (2x – 1) y1 + y = 0, (D) (1 + x2) y2 + (2x – 1) y1 = 0, , If y = a sin mx + b cos mx, then the value of, d2y, , at t =, , If y = e tan, , then, Q.60, , 2, , 2, , If y = tan–1 (cot x) + cot–1 (tan x), then, dy, is equal todx, , d2y, , (A) 3/8, (C) 8/3, , (D) None of these, , 1 t 2, , If y = log(1– t2) and x = sin–1(t), then the value, , d2y, , dy, +x, equals –, dx, dx 2, , (B) m2y, (D) None of these, , If y = a cos (log x) + b sin (log x) then, (x2y2 + xy1) equals to –, (A) y, (B) ay, (C) by, (D) –y, , DIFFERENTIATION, , 1, (a+ logea), x, , (A), , 1, (a + loga e), x, , (B), , (C), , a, + logae, x, , (D) None of these, , 110

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Q.84, , If ex sin y – ey cos x = 1, then, (A), , (C), , Question, based on, , Q.85, , e x sin y  e y sin x, e cos x  e cos y, y, , x, , e x sin y  e y sin x, e y cos x  e x cos y, , dy, equals–, dx, , Q.90, , e x sin y  e y sin x, , (B), , If (cos x)y = (sin y)x, then, (A), , log sin y  y tan x, log cos x  x cot y, , (B), , log sin y  y tan x, log cos x  x cot y, , (C), , log sin y  y tan x, log cos x  x cot y, , e cos x  e cos y, y, , x, , (D) None of these, , Logarithmic function, , dy, equalsdx, , (D) None of these, , x, , The derivative of x a is -, , Q.91, , The derivative of (tan x)x is equal to(A) x (tan x)x–1, (B) (tan x)x [sec x + tan x], (C) (tan x)x [x sec x cosec x + log tan x], (D) (tan x)x [sec2x + x tan x], , Q.92, , d/dx (xlog x) is equal to(A) 2x log x–1.log x, (B) xlog x–1, (C) 2/3 (log x), (D) xlog x–1. log x, , Q.93, , If y  e logcos, , x, , (A) x a [ax x–1 + ax log a log x], x, , (B) x a [xax + ax log x], x, , (C) x a [ax + xax log x], (D) None of these, , Q.86, , Q.87, , Q.88, , If y = log (xx), then, , dy, equals dx, , (A) log (ex), , (B) log (e/x), , (C) log (x/e), , (D) 1, , (A) x1/x log (ex), , (B) x1/x log (e/x), , (C) x1/x log (x/e), , (D) None of these, , 1, , (A), , cos–1, , x), , x, , (C) 1/ 1 x, , , then, , dy, equalsdx, , (B), 2, , 1/sin–1, , x, , + e logsin, , 1, , x, , , 0 < x < 1, then which, , of the following statement is true –, (A) y1 = 0, (B) y2 = 5, (C) y1 does not exist (D) None of these, , The derivative of x1/x equals-, , If y = elog(sin, , 1, , Q.94, , If y = eax+b, then (y2)0 is equal to –, (A) aeb, (B) eb, (C) a2ea, (D) a2eb, , Q.95, , Differential coefficient of xlog x is –, , x, , (D) x/ 1 x, , (A) 2xlog x. log x, 2, , (B) 2xlog(x/e) log x, (C) 2xlog(ex). log x, , Q.89, , If xy = ey, then, x, (A), y ( y  1), , (C), , PHOTON, , x ( y  1), y, , dy, equalsdx, , (D) None of these, , y, (B), x ( y  1), , (D) –, , x, y( y  1), , DIFFERENTIATION, , Q.96, , If y = xx, then the value of, (A) xx, , dy, is–, dx, , (B) xx log (ex), x, e, , (C) xx log  , , (D) xx–1 log (ex), , 112

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Q.97, , , , 1, x, , x, , If y = 1   , then, , , x, , 2, , dy,  1  x  1, , equals , then, dx, x, , , , , , dy, equals–, dx, , , , 1 , , , , 1, , , ,  , , , , , , Q.102 If y = tan–1 , , 1 , , (A) 1   log 1   , , x, x 1 x, , (A), , , , x, ,  1 ,  1 , (B) 1   log 1  ,  x ,  x , x, , , , 1 ,  , , (C), , , , Q.103, , x, , x ,  1   1, (D) 1   log 1   , ,  x    x  1 x , , Question, based on, , Q.98, , Q.99, , 1, 1 x, , (C) –, , (D) x, , (B), , 2, , (C), , 1 x2, , a2  x2, , (B) a a 2  x 2, , 1, , (D) x a 2  x 2, , a x, , Q.101 If y = tan–1, (A), (C), , 1, 1 x2, 2, 1 x2, , 2, , 2, , , then, ,  1  1  x 2,  tan , 2, ,  1 x, 2x, , 1 x2, , (D) –, , 1, 2 (1  x 2 ), , (B) –, , 4, , (D) –, , 1, 1 x, , ,  equals, , , 2, , 1 x, , 2x, 1 x4, , x, 1 x4, ,  1 x  1 , cos,  equals2 , , 1, 2 1 x, , 2, , 1, 1 x, , 2, , dy, equalsdx, , (B), (D) –, , d  1  1  x 2, sin , 2, dx , 1 x, , (A) –, (C), , Q.106, , 2x, 1 x, , 2 (1  x ), 2, , 2, , (B) –, (D) –, , 1, 2 1 x2, , 1, 1 x2, , 1, , (D) None of these, , 1 x2, , 2, , 1, , (C), , Q.105, , 1, , d, dx, , (A), , , dy, 2, 2 , If y = log  x  (a  x )  , then the value of, isdx, a, , , , (A), , PHOTON, , (C) –1, , Q.104, , d, 1 x , cot–1 ,  is equal todx,  1 x , , (A), , Q.100, , (C), , dy, If y = cot–1  x 2 1  + sec–1 x, then, equals, , dx, , (B) 0, , d, dx, , (A), , Trigonometrical substitution, , (A) 1, , 1 x, , (B), , 2, , x , , (C) 1   log ( x  1) , 1 x , x, , , 1, , d, dx, , (A), , 2, 1 x2, 2, , (C), , 1, 1 x, , 2, , 1, 1 x, , ,  equals, , , (B) –, (D), , 2, , 2, 1 x2, 2, , 1 x2, ,  1  3x x 3 ,  , is equal tosin  , 2 , ,  2, , 3, 4x, , 2, , 1, 4  x2, , (B), , (D), , 3, 4  x2, , 1, 4  x2, , 1 x2, , DIFFERENTIATION, , 113

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 1 x  1 x , dy, Q.107 If y = tan–1 , equals , then, dx,  1  x  1  x , , (A) –, (C) –, , 1, , 1, , (D), , 1 x2, ,  x, Q.108 If y = tan–1 , , 2,  1 x, , (A) –, , 1 x2, , 1 x, , 2 1 x2, , 2, , 1, 1 x, , 2, , 1  4x, , 2, , (C) 0, , 16, , (A), , 1 x2, 1 x2, x, ,  1  x 2 1 ,  , then, , , x, , , , 1 x 2 ] , then, , x, , 2, 1 x, , 1, , (B), , 2, , 1, , (C), , + tan–1 , , , x  a2  x2, Q.110 d/dx tan 1, , x  a2  x2, , , 1, a x, , 1 x2, , –, , 1, 2 x 1 x,  2x, 1 x2, , Q.115 If y = sin–1 , , 1, 2(1  x 2 ), , (A), , Question, based on, , 1, a  x2, 2, , (D) –, , a2  x2, , (C), , ,  equals , , , (B) –, , 2, , a, , 1 x2, , (D) None of these, , 1 x2, ,  + sec–1 , 2, , 1 x, , 4, , (B) –, , 2, , (D) –, , 1 x, 2, 1 x, , 4, 1 x2, 2, 1 x2, , Differentiation of a function w.r.t., another function, , Q.116 If y = x – x2, then the derivative of y2 w.r.t. x2 is–, (B) 2x2 – 3x + 1, , a2  x2, , (C) 2x2 + 3x + 1, , (D) None of these, , 1 x2 ,  is sin–1 , 2 , 1 x , , (B) 1/2, , 2, , (A) 2x2 + 3x – 1, ,  2x ,  with respect to, 1 x2 , , Q.117 Derivative of cos–1 x with respect to, , (A), , (B) –1 / x, , x, , (C) – x, (C) –1, , Q.112 If y =, (A), , 1, 1 x2, , (C) 1 x, PHOTON, , (3x, , 2, , –4x3),, , (D) 1/ x, , (D) 1, , dy, then, equals –, dx, , (B), , 1 x, , is-, , 1, , sin–1, , ,  , then, , , , a, , Q.111 The derivative of sin–1 , , (A) 2, , 1  4x 2, , (D) None of these, , 1  4x 2, , (D) None of these, , 2, , 1, , (B), , 2, , dy, equals–, dx, , x, , (B) –, , 2(1  x ), , (C), , 4, , dy, equals–, dx, , 1, , (A), , 2, , Q.114 If y = sin–1 [x 1 x –, , dy, equals dx, , (A), , 4x, , 1 x2, , (D), , Q.109 If y = sin–1, , (C), , 1, , (B), , 1, , (C), , (A), , ,  , then dy is equal to , dx, , , 1, , dy, , equals –,  , then, dx,  1  4x , , 1, , (B), , 2 1 x2, , , , Q.113 If y = tan–1 , , Q.118 The differential coefficient of a sin, sin–1, , x is -, , (A) a sin, , 3, 1 x2, , (D) 3 1 x, , 2, , DIFFERENTIATION, , (C), , x, , 1, , a sin, , x, 1, , log e a, x, , (1  x ), 2, , 1, , (B) a sin, , 1, , (D) a sin, , x, , x, , (1  x 2 ), 114, , w.r.t.

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Q.119 Derivative of sin2 x w.r.t. cos2 x is (A) tan2 x, (B) tan x, (C) – tan x, (D) None of these, , Q.120 Derivative of, , tan 1 x, , Q.125 The derivative of, , w.r.t. tan–1 x is equal, , 1  tan 1 x, , to-, , ,  is 2 x 1 , , 1 x, , 1, , w.r.t. sec–1 , , 2, , 2, , (A) 1, , (B) –1, , (C) 1/2, , (D) –1/2, , Q.126 The differential coefficient of, 1, , (A), , (1  tan 1 x ) 2, , 1 x2, , (C), , (1  tan 1 x ) 2, , (B), , 1, , , , ,  w. r. t cos–1, , 2 ,  1 x , , (D) log (1+ tan–1 x), , sin 1 x, 1  sin 1 x, , with, , 1, (1  sin, , 1, , x), , 2, , (B), , 1, (1  sin 1 x ) 1  x 2, , 1 x2, , 1 x2, , , (A) 1/2, , (B) – 1/2, , (C) 1, , (D) –1, , ,  is , , , Q.127 The value of derivative of,  2x 1  x 2 ,  w.r.t sec–1 , tan–1 , 2, , respect to sin–1 x is(A), , x, , sin–1 , , (1  x 2 ) (1  tan 1 x ) 2, , Q.121 Differential coefficient of, , (C), , , , x, , tan–1, ,  1  2x, , , 1, , , ,  2 x 1 , , , , , 1, , 2, , at x = 1/2 equals-, , 1 x2, , (D) None of these, , (A) 1, , (B) –1, , (C) 0, , (D) None of these, , Q.128 The differential coefficient of sec–1[1/(2x2–1)], Q.122 The differential coefficient of, , with respect to, ,  1 , 2, sec–1  2,  w.r.t 1 x is  2x 1 , (A) 1/x2, (B) 2/x3 (C) x/2, (D) 2/x, , Q.123 If y =, , sin–1, , Question, based on, ,  2x , ,  , z = tan–1 x, then, 1 x2 , , (A) 2, , (A) 0, , (B) 1/2, , (C) 1/3, , (D) None of these, , Infinite Series, , Q.129 If y = x, , dy, equalsdz, , (B) 1, , (C), , 1, 1 x2, , (D), , 2, , 3x  1 at x = – 1/3 is -, , x, , x, , x ......, , , then the value of, , dy, isdx, , (A), , xy 2, 2  y log x, , (B), , x2, y (2  y log x ), , (C), , y2, x (2  y log x ), , (D), , y2, x (2  y log x ), , 1 x2, , Q.124 The differential coefficient of the function, tan–1, (A), (C), , PHOTON, , 2x, 1 x2, , w.r.t. x2 is-, , 1, 1 x, , 2, , 1, x (1  x 2 ), , (B), (D), , Q.130 If, 1, , y, , = log x  log x  log x  ...... ,, , 1 x2, , dy, equalsdx, , x, , (A) x/(2y+1), , (B) 1/x(2y–1), , (C) (2y–1)/x, , (D) x(2y–1), , 1 x2, , DIFFERENTIATION, , 115, , then

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Q.131 If y = (tan x ) tan x, ,  dy ,  equals dx , , , then tan x , , (A), , y 2 sec2 x, 1  y log tan x, , (B), , (C), , y sec2 x, 1  y log tan x, , (D) None of these, , Q.132 If y =, , x_________ , then, Q.133 If y = __________, a  __________, x______, b  __________, ______, x, a  __________, _____, x_, b  .......... .........., , y 2 sec2 x, 1  y log tan x, , x  x  x  ...... , then, , (A) 1, (C) 1/y–2, , PHOTON, , ........, , dy, equals–, dx, , (B) 1/2y, (D) 1/2y–1, , DIFFERENTIATION, , dy, equals dx, , (A), , b, a ( b  2 y), , (B), , (C), , a, b ( b  2 y), , (D) None of these., , b, b  2y, , 116

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LEVEL- 2, Q.1, , If g is inverse of f and f '(x) -, , 1, 1 xn, , , then g'(x), , Q.7, , x = 1, f ' (x) is equal to -, , equals -, , Q.2, , (A) 1 + xn, , (B) 1 + (f (x))n, , (C) 1 + (g(x))n, , (D) None of these, , If g is the inverse of f and f '(x) = sin x, then, , Q.8, , Let y = cot–1, , (D) None of these, , (A) 1/2, , (B) –1/2, , (C) Does not exist, , (D) None of these, , If y = logu |cos 4x| + |sin x|, u = sec 2x, then, Q.9, , 6 3, 3, , log e 2 2, , 6 3, 3, , log e 2 2, , If x3 cos (xy) + y3 sin (xy) + 1 = 0, then, dy, equals dx, , 6 3, 3, , log e 2 2, , (A), , (D) None of these, , (B), , (B) , , x 3 y tan(xy )  (3x 2  y 4 ), xy 3  (3y 2  x 4 ) tan xy, x 3 y tan(xy )  (3x 2  y 4 ), xy 3  (3y 2  x 4 ) tan xy, x 3 y  (3x 2  y 4 ) tan ( xy ), , If f(x) = cos x, g(x) = log x and y = (gof) (x),, , (C), , then the value of dy/dx at x = 0 is -, , (D) None of these, , (A) 0, , (B) 1, , (C) –1, , (D) None of these, Q.10, , If y = sin–1 x satisfy the equation, (A) –x, , (B) x, , (C) 0, , (D) None of these, , If y = tan x  tan x  tan x  ........, ,  3 cos x  4 sin x ,  , /2 < x <  then, 5, , , , If y = cos–1 , , (A) 1, (C), , , then, , Q.11, , (A) tan x sec x, , (B), , (C) sec x cosec x, , (D) cot x cosec x, , x, , DIFFERENTIATION, , (B) 0, 1, 1 x, , 2, , (D) –, , 1, 1 x2, , If sin y = x cos (a + y), then dy/dx equals(A), , (2y–1) dy/dx equals sec2, , xy 3 tan ( xy )  (3y 2  x 4 ), , dy/dx equals-, , (1–x2)y = f(x) y, then f(x) equals -, , PHOTON, , (D) 3/4, , (C) cosec (g(x)), , (C) , , Q.6, , (C) –1/2, , (B) sin (g(x)), , (A), , Q.5, , (B) 1/2, , (A) – cosec (g(x)), , dy, at x = – /6 is equal to dx, , Q.4, , (A) 0, , , dy,  1  sin x  1  sin x ,  , , , < x < . Then, dx, ,  1  sin x  1  sin x ,  2, , g'(x) =, , Q.3, , , 1  x , If f(x) = cos–1  sin, + xx, then at, , , 2, , , , (C), , cos 2 (a  y), cos a, cos (a  y), cos 2 a, , (B), , cos a, cos 2 (a  y), , (D) None of these, , 117

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Q.12, , If y =, , x1/ 2 (5  2x ) 2 / 3, (4  3x ) 3 / 4 (7  4x ) 4 / 5, 1, , 4, , ,, , dy, =, dx, , 9, , Q.17, 16, , , , (A) y  , , , ,  2x 3(5  2x ) 4(4  3x ) 5(7  4x ) , , (A), , 1, , 4, 9, 16, (B) y  , , , ,  2x 3(5  2x ) 4(4  3x ) 5(7  4x ) , 1, , 4, , 9, , 16, , , , (C) y  , , , ,  2x 3(5  2x ) 4(4  3x ) 5(7  4x ) , , If y =, , Q.18, , 2 (x  a ), , a, x  a2, 2, , (D) None of these,  2x  1 ,  and f (x) = sin x2, then,  x2 1, , If y = f , , (A), , 1, 1, 2 , (A) y  , , , x, 2, (, 2, x, , 1, ), (, 2, x,  1) , , 1, x, , (B), , 2, , dy/dx equals-, , x 2x  1, , then dy/dx equals 2x  1, , (B) y  , , a, 2, , (C) 1/2, , (D) None of these, Q.13, , 1/ 2, , 2,  2,  , ,  , is equal to 1  x  a  x , d/dx tan, ,  , 2, 2,  x  a  x  , , , (B), , 2(1  x  x 2 ), (1  x 2 ) 2, ,  2x  1 , ,  x 2 1, , 2, , sin , ,  2x  1 , ,  x 2 1, , 2(1  x  x 2 ), , 2, , sin , , (1  x 2 ) 2, 2, ,  1 x ,  2x  1 ,  sin  2, , 1 x2 ,  x 1, , 2, , (C) 2 , , 1, 2 , , , 2x  1 2x  1, , (D) None of these, , 1, 2 , 1, , (C)  , ,  x 2x  1 2x  1, , Q.19, ,  2x  3 ,  , then,  3  2x , , If f(x) = sin (log x) and y = f , , (D) None of these, dy/dx equals  1  (log x) ,  , then f ' (e) equalsIf f(x) =cos–1 , 2 ,  1  (log x) , , (A), , (A) 1/e, (C) does not exist, , (B) –, , 2, , Q.14, , Q.15, , (C), , ,  x 2 n  1 , d/dx cos 1  2 n  equals,  x  1 , , (A), (C), , Q.16, , (B) 1, (D) 2/e, , 2n x n 1, 1 x, , n x n 1, 1  x 2n, , (C), , (B) –, , x  y2, , 2n x n 1, 1  x 2n, , 2 y  2xy  1, 3, , x  y2, 2 y 3  2xy  1, , Q.20, , (D) None of these, , (B), , , , , , 12, 9  4x, , 2, ,  2 x  3 , ,  3  2 x , , cos log , , , , , 12, 9  4x 2, ,  2 x  3 , ,  3  2 x , , cos log , ,  2 x  3 , ,  3  2 x , , log cos , , , x  y2, , Q.21, , If f(x) = (x +1) tan–1 (e–2x), then f (0) equals, +5, 6, , (B), , (C), , , –1, 4, , (D) None of these, , If, , 1 x 6 + 1 y 6 = a3 (x3 – y3) , then, , dy/dx equals-, , 2 y  2xy  1, 3, , (A), (D) None of these, , DIFFERENTIATION, , , +1, 2, , (A), , (C), PHOTON, , 9  4x, , 2, , (D) None of these, , dy, x  y  x  ... , then, =, dx, , If y =, (A), , 2n, , , , 12, , x2, , 1 x6, , y2, , 1  y6, , x2, , 1  y6, , y2, , 1 x6, , (B), , (D), , y2, , 1  y6, , x2, , 1 x6, , y2, , 1 x6, , x2, , 1  y6, 118

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Q.22, , , 1,  2, 1, d/dx sin cot, 1 x, , 1 x, , , , (A) 0, , Q.23, , If y =, , (B) 1/2, a cos, 1 a, , 1, , , ,  , is equal to , , , , (C) – 1/2, , x, , cos 1 x, , and z = a cos, , 1, , x, , Q.29, y=, , (C), , 1, , 1, 1  x b a  x b c, , (B) –, , x, , 1, 1  x c a  x c  b, , , then, If f(x) = [x +, , x 2  a 2 ] n and f (x) =, , 1, 1  a cos, , 1, , x2  a2, , (A), , x, , (C) 1, 1, , +, , ,, , nf ( x ), ,, k, , then k equals-, , 1, (1  a cos, , +, , (A) ax–1 + bx–1+ cx–1 (B) 1, (C) 0, (D) a + b + c, Q.30, , 1, 1  a cos, , 1, 1  x a  b  x a c, , then dy/dx equals-, , (D) –1, , dy/dz is equal to(A), , If, , (B) x2 + a2, (D) None of these, , (D) None of these, , x 2, , Q.31, , ), , Let f(x) be a polynomial of degree three and, f(0) = 4, f (0) = 3, f (0) = 4, f (0) = 6 then, , x x, If f(x) = cot–1 , 2, , x, , Q.24, , x, , to(B) –1, , (A) 1, , Q.25, , , , (A) cosec x –1, , (B) cosec x, , (C) cosec x+ 1, , (D) x, , (B), , ey, (C) ey (y–2)(D) ey (y–1), 2 y, , If x = t2 + t +1 & y = sin, , , , t +cos t, then at t =1,, 2, 2, , (D) –2, , Q.33, , If y = a sin–1x + (sin–1 x)2, where a is a constant,, then the value of y2 is –, , If log (y–1) = log x + y, then dy/dx is equal toey, y 1, , (C) –140, , If y = (1 + x) (1 + x2) (1 + x4) … (1 + x2n), then, dy/dx at x = 0 is (A) 1, (B) –1, (C) 0, (D) None of these, , x, 2, , , , (B) 10, , Q.32, , If y = log  tan  + sin–1 (cos x), then dy/dx is, , (A), , Q.27, , (A) 2, , (D) –log 2, , (C) log 2, , equal to-, , Q.26, , f (–1) equal to, , ,  , then f ' (1) is equal, , , , Q.34, , (A), , xy 1  2, , (C), , y1  x, , 1 x, , 2, , 1 x2, , (B), , xy 1  2, 1 x2, , (D) None of these, , If y = ex (a cos x + b sin x), then, y2 – 2y1 + 2y is equal to (A) y, (B) –y, (C) 0, (D) None of these, , dy/dx equals(A) –, , Q.28, , (B), , , 2, , (C) –, , , 4, , (D), , , 3, , If f(x) = ex g(x), g(0) = 2, g' (0) = 1, then f (0), is(A) 1, , PHOTON, , , 6, , (B) 0, , (C) 2, , (D) 3, , DIFFERENTIATION, , 119

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LEVEL- 3, Q.1, , If x = (t), y = (t), then, (A), (C), , Q.2, , ' "   ' ", , (B), , ( ' ) 2, , ", ", , (D), , (C), , 1, , x0, x, , (A), , ", ", , (C), , 1 x2, , , x  0, ±1, then derivative of, , 2x, , (B), , (1  x ), , 2 2, , 1, , 1, (2  x 2 ) 3, 1, , (D), , (1  x ), , 2 2, , (2  x 2 ) 2, , d3y, , If x = a cos , y = b sin , then, , dx 3, , is, , equal to-, , (D) none of these, , If ƒ(x) =, is(A) 1, , (A) , , 3b, , (C) , , 3b, , x, , x, , (C) 2, , a3, , cos ec 4  cot 4  (B), , 3b, a3, , cos ec 4  cot , , / 1 x2, , (B) xe, , x 2 1, x2  4, , (C), , x 1, x2  4, , (D) ex, , x 1, , (B) cos(  3), (D) none of these, , If ƒ(x) = logx {n (x)}, then ƒ (x) at x = e is, equal to (A) e, (B) – e (C) e2, (D) e–1, , Q.13, , If y = sin x + ex, then, (A) (–sin x +ex)–1, , (C), , sin x  e x, (cos x  e ), , x 3, , d2x, dy 2, , (B), , (D), , is equals to sin x  e x, (cos x  e x ) 2, sin x  e x, (cos x  e x ) 3, , x2  4, , DIFFERENTIATION, , is-, , Q.12, , (B) 1, 2, , d 3, , Let ƒ(x) = sin x, g (x) = x2 and h(x) = logex., if F(x) = (hog of) (x), then F"(x) is equal to(A) 2 cosec3x, (B) 2 cot x2 – 4 cosec2 x2, (C) 2x cot x2, (D) – 2cosec2 x, , dy, is equal to dx, , 2, , d3y, , Q.11, , (D) none of these, , 3/ 4, d  , x  2  , log e x ,   is equal to dx    x  2  , ,  , , sin 3 (  ), cos , , (C) 0, , sin1 x, , 1 x2, , cos ec 4  cot  (D) none of these, , If y  sin 2   cos2 (  ), , (A), , dy, is du, , sin1 x, , a3, , 2 sin  sin  cos ( + ), then, , (D) 0, , and u = log x, then, , If y = logcosx sin x, then, , (A), , Q.10, , g(0) = 2, g'(0) = 1, then ƒ '(0), , (B) 3, 1, , xe, , (D) none of these, , ex g(x),, , If y = esin, 1, , (B) 1, , x2, , (A) (cot x log cos x + tan x log sin x) / (log cos x)2, (B) (tan x log cos x + cot x log sinx) / (log cos x)2, (C) (cot x log cos x + tan x log sin x) / (log cos x)2, (D) none of these, , PHOTON, , Q.9, , 3, , (C), , Q.7, , x2, , 2, , (A) esin, , Q.6, , ( ' ) 3, , If ƒ( x )  3e x , then ƒ' ( x )  2x ƒ( x )  1 ƒ( 0)  ƒ' (0), , (C) 7 / 3 e, , Q.5, , Let ƒ(x) =, , ƒ(x) with respect to x is equal to-, , ' "   ' ", , 1, (B), , x0, |x|, , is equal to(A) 0, , Q.4, , dx, , is equal to -, , 2, , Q.8, , The derivative of log |x| is 1, (A) , x  0, x, , Q.3, , d2y, , 120

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Q.14, , Let ƒ(x) be a polynomial in x. Then the second, order derivation of ƒ(e ), is x, , Q.19, , (A) ƒ(ex). ex + ƒ(ex), , 1/ 2, , , 4, , (C) ƒ(ex) e2x, (D), , Q.15, , 1/ 2, , e2x+, , , 4, , (B)  , , ƒ(ex).ex, , 2, , x, (A), y, , (C), , (B), , x 2  y2, , Q.20, , y, (D), x, , x 2  y2, ,  2 4 2 2, , , In ,  2, ,  , , , 1/ 2, ,  2  2 2, , , In ,  2, , 4, , , , , , 4, , x2, ,  2 4 2 2, , , In ,  2, ,  , , ,  2  2 2, , , In ,  2, 4,  , , , (D)  , , y, ,  ,  4, , , then ƒ'    is equals to-, , 1/ 2, , , (C)  , 4, , x y ,   log a , then dy is equal to, If sin 1  2, 2 , dx, x y , 2, , sin x, , (A)  , , (B) ƒ(ex). e2x + ƒ(ex).e2x, , ƒ"(ex)., , If ƒ(x)  x, , , x   x 2  , for 1 < x < 2 and [x], 3, , , If ƒ( x )  sin , , denotes the greatest integer less than or equal to, x, then ƒ' (  / 3 ) is equal to, , 2, , Q.16, , If y = x + ex, then, , Q.17, , dy 2, , e, , (B) –2  / 3, , x, , 1  e , , x 2, , (D), , ex, , x 3, , 1, , (D) none of these, Q.21, , 1  e , , If ƒ(x) = xn, then the value of, , x 2, , f (1) , , f (1) f (1) f (1), f ( n ) (1), , ,  ..... , 1!, 2!, 3!, n!, , If f(x) =|x2 –5x + 6|, then ƒ (x) is equals to-, , (A) n, , (B) 2n, , (A) 2x – 5 for 2 < x < 3, , (C) 2n–1, , (D), , (D) 5 – 2x for 2  x  3, , If f(x) = 1 sin 2x , then ƒ'(x) is equals to (A) – (cos x + sin x), for x  (/4, /2), (B) (cos x + sin x), for, , n (n  1), 2, ,  Assertion & Reason type Questions, , (C) 2x – 5 for 2  x  3, , x  (0, /4), , (C) –(cos x + sin x), for x  (0, /4), (D) cos x – sin x, for x  (/4, /2), , PHOTON, , (C) – , , 1  e , , (B) 5 – 2x for 2 < x < 3, , Q.18, , (A) (  / 3 ), , is equal to -, , (B) , , (A) ex, , (C) , , d x, , DIFFERENTIATION, , All questions are Assertion & Reason type questions., Each of these questions contains two statements:, Statement-I (Assertion) and Statement-II (Reason)., Answer these questions from the following four option., (A) Statement- I and Statement- II are true,, and Statement - II is the correct explanation of, Statement– I., (B) Statement - I and Statement - II are true, but Statement - II is not the correct explanation, of Statement – I., (C) Statement - I is true but Statement - II is false., (D) Statement - I is false but Statement - II is true., 121

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Q.22, , Statement I : f(x) = cos2x + cos2(/3 + x), ,  Column Matching Questions, , – cosx cos(/3 +x ) then f(x) = 0, Match the entry in Column 1 with the entry, in Column 2., , Statement II : Derivative of constant function, is zero., , Q.23, , Statement-I : For x < 0,, , 1, d, (ln |x|) = –, dx, x, , Statement – II :For x < 0, | x | = – x, , Q.27 Column Matching Column-I, (A) y = tan–1 (x  1  x 2 ), (B) y =, , Q.24, ,  2x , Statement-I : Derivative of sin–1 , with,  , 1 x , ,  1 – x2 ,  is 1 for 0 < x < 1., respect to cos–1 , 2, 1 x ,  1 – x2 ,  2x , , Statement – II :sin–1 , = cos–1 ,  , 2, 1 x , 1 x , , Q.25, , 2, , Statement – II : (xx)x = x x = e x, Q.26, , 2, , ln x, , Q.28, , Column Matching Column-I, , (A) x =, , du, dv, , 1, 2, , Statement – II :If u = f (x), v = g (x), then the, du du / dx, derivative of f with respect to g is, =, dv dv / dx, , PHOTON, , (P) y'(0) = 2, (Q) y'(0) = 0, (R) y'(0) = 1, (S) y'(0) = 1/2, , Column-II, , , (P) y'  t , , , ,  =–4, 2, , 2 cos t, 1, 1  cos t, 2, , (Q) y'(t = ) = 0, , tan–1, , =, x  / 4, , y=, , sin t, ,, 1, 1  cos t, 2, , (B) x = log (1 + t2),, , ., , Statement-I : If u = f (tan x), v = g (sec x) and, f (1) = 2, g ( 2 ) = 4, then, , 1 x2, , (C) f(x) = min(x2, x3), (D) If log (x + y) = 2xy, , for – 1  x  1., x, x, d, ( x x ) = x x .x (1 + 2 ln x), Statement-I :, dx, , 1  x tan 1 x, , Column-II, , DIFFERENTIATION, , y=t–, t, (C) x = t2 + 2,, y = t3/3 – t, (D) x = 4 tan2 t/2,, y = a sin t + b cos t, , (R) y'(t = 1) = 1/2, (S) y'(t = 1/2) = –3/4, , 122

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LEVEL- 4, (Question asked in previous AIEEE and IIT-JEE), dy, SECTION –A, Q.7, If xm. yn = (x + y)m + n, then, is, Q.1, , Q.2, , If y = logy x, then, , dx, , dy, =, dx, , [AIEEE 2002], , [AIEEE 2006], , (A), , 1, x  log y, , (B), , 1, log x (1  y), , (A), , xy, xy, , (B) xy, , (C), , 1, x (1  log y), , (D), , 1, y  log x, , (C), , x, y, , (D), , If x = 3 cos  – 2 cos3  and y = 3 sin  – 2 sin3 ,, dy, =, dx, , then, , Q.8, , (B) cos , , (C) tan , , (D) cot , , (A) ny2, (C) n2y2, , Let g(x) = [f(2f(x) + 2)]2,. Then g (0) =, , (B) n2y, (D) None of these, , is(A) 1, , Q.6, , (A) 4, (C) 0, , (B) 2, , n, , (C) 2, , n– 1, , Let f(x) be a polynomial function of second, degree. If f(A) = f(–1) and a, b, c are in A.P., then f ' (a), f '(b) and f '(c) are in[AIEEE 2003], (A) Arithmetic - Geometric Progression, (B) A.P., (C) G.P., (D) H.P., If x = e ye, , y.....to , , , x > 0, then, , (A), , x, 1 x, , (B), , 1, x, , (C), , dy, is, dx, , 1 x, x, , SECTION-B, Q.1, , [AIEEE 2003], (D) 0, , (D), , The derivative of function cot–1[(cos 2x)1/2] at, x=, , Q.2, , Q.3, , , is, 6, , [IIT-1992], , (A) (2/3)1/2, (C) 31/2, , (B) (1/3)1/2, (D) 61/2, , If y = sec tan–1 x then, , dy, =, dx, , [IIT-1993], , (A) x/(1+ x2), , (B) x, , (1  x 2 ), , (C) 1/ (1  x 2 ), , (D) x/ (1  x 2 ), , If f(x) = tan–1, , 1  sin x, 0  x  /2, then, 1  sin x, , f ' (/6) is, , [AIEEE 2004], , PHOTON, , [AIEEE 2010], (B) –4, (D) –2, , If f(x) = xn, then the value of, f ' " (1), f ' ' (1), f ' (1), (1) n f n (1), f(1) –, +, –, +.....+, 1!, 3!, 2!, n!, , Q.5, , Let f : (–1, 1)  R be a differentiable, function with f (0) = – 1 and f(0) = – 1., , If y = (x + 1 x 2 )n then (1+ x2) y2+ xy1 =, [AIEEE-2002], , Q.4, , [AIEEE 2009], (B) 1, (D) – log 2, , (A) –1, (C) log 2, Q.9, , Q.3, , Let y be an implicit function of x defined by, x2x – 2xx cot y – 1 = 0. Then y'(1) equals -, , [AIEEE 2002], , (A) sin , , y, x, , 1 x, x, , DIFFERENTIATION, , (A) –, (C), , 1, 4, , 1, 4, , [IIT- 1993], (B) –, (D), , 1, 2, , 1, 2, 123

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x3, , Q.4, , 1, p2, , Let f(x) = 6, p, , constant. Then, , Q.11, , sin x cos x, , d3, dx 3, , 0, p3, , 2, , [f (x)] at x = 0 is, , Q.5, , Q.6, , Q.7, , Let F(x) = f(x) g(x) h(x) for all real x, where, f(x), g(x) and h(x) are differentiable functions at, some point x0. F (x0) = 21 F(x0), f (x0) = 4f(x0),, g(x0) = – 7g(x0) and h(x0) = Kh(x0), then K =, [IIT-1997], (A) 12, (B) 24, (C) 6, (D) 18, If x2 + y2 = 1, then, (A) yy'' – 2(y')2 + 1 = 0, (B) yy'' + (y')2 + 1 = 0, (C) yy'' – (y')2 – 1 = 0, (D) yy'' + 2(y')2 + 1 = 0, ,  tan 1 x if, ,  2 | x | 1 if, , (A) R – {0}, (C) R – {– 1}, , Q.9, , Q.10, , PHOTON, , Q.12, , dy 2, , equals1, ,  d 2 y   dy  2, (C)  2   ,  dx   dx , , Q.13, , [IIT Scr. 2000], , [IIT -2007, AIEEE-2011],  d2y , (B) –  2 ,  dx , , 1, ,  dy ,  ,  dx , , 3, ,  d 2 y   dy  3, (D) –  2   ,  dx   dx , , Let f and g be real valued functions defined on, interval (–1, 1) such that g(x) is continuous,, g(0) 0, g(0) = 0, g(0)  0 & f(x) = g(x) sin x., STATEMENT-1, lim (g(x) cot x –g(0) cosec x) = f (0), x 0, , and, STATEMENT-2, f (0) = g(0), [IIT- 2008], (A) Statement–1 is True, Statement–2 is True;, Statement–2 is a correct explanation for, Statement–1., (B) Statement–1 is True, Statement–2 is True;, Statement–2 is NOT a correct explanation, for Statement–1, (C) Statement–1 is True, Statement–2 is False, (D) Statement–1 is False, Statement–2 is True, , | x | 1, | x | 1, , d2x, ,  d2y , (A)  2 ,  dx , , The domain of the derivative of the function, f(x) =  1, , Q.8, , 3, , (B) p + p, (D) Independent of p, , 2, ,   x ,   x , F(x) =  f    +  g    and given that,   2 ,   2 , F(5) = 5, then F (10) is, [IIT-2006], (A) 15, (B) 0, (C) 5, (D) 10, , where p is a, , [IIT-1997], (A) p, (C) p + p2, , If f  (x) = –f(x) and g(x) = f (x) and, , is–, , [IIT Scr. 2002], (B) R – {1}, (D) R – {– 1, 1}, , Let y be a function of x, such that, log (x + y) – 2xy = 0, then y (0) is[IIT Scr.2004], (A) 0, (B) 1, (C) 1/2, (D) 3/2, If x cos y + y cos x = , then y (0) =, [IIT Scr.2005], (A) , (B) – (C) 0, (D) 1, S is a set of polynomial of degree less then or, equal to 2, [IIT Scr.2005], f (0) = 0, f (1) = 1, f (x) > 0 ; x  [0, 1] then set S =, (A) , (B) ax + (1 – a) x2 ; a  R, (C) ax + (1 – a) x2 ; 0 < a < , (D) ax + (1 – a) x2 ; 0 < a < , DIFFERENTIATION, , Q.14, , ,  sin   ,   , where, Let f () = sin  tan –1, ,  cos 2  , , d, , , – <  < . Then the value of, f(f()), d (tan ), 4, 4, , [IIT-2011], , is, , (A) 0, (C) 5, , (B) 1, (D) 3, , 124

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ANSWER KEY, LEVEL-1, Q.No., , 1, , 2, , 3, , 4, , 5, , 6, , 7, , 8, , 9, , 10, , 11, , 12, , 13, , 14, , 15, , 16, , 17, , 18, , 19, , 20, , Ans., C, D, A, D, A, B, C, A, D, D, A, A, B, D, B, C, B, B, A, C, Q.No. 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40, Ans., C, A, A, B, D, D, C, B, B, C, D, B, A, C, A, D, B, B, A, D, Q.No. 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60, Ans., B, B, C, C, D, B, B, B, A, D, A, B, A, B, A, A, C, A, D, B, Q.No. 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80, Ans., B, B, D, D, D, C, B, A, B, B, A, B, B, B, B, B, A, C, A, D, Q.No. 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100, Ans., D, B, A, A, A, A, D, C, B, B, C, A, A, D, B, B, A, B, C, C, Q.No. 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120, Ans., C, C, B, B, B, A, A, C, B, A, C, B, A, C, A, B, D, A, D, A, Q.No. 121 122 123 124 125 126 127 128 129 130 131 132 133, Ans., A, D, A, C, D, A, B, A, C, B, A, D, A, , LEVEL-2, Q.No., , 1, , 2, , 3, , 4, , 5, , 6, , 7, , 8, , 9, , 10, , 11, , 12, , 13, , 14, , 15, , 16, , 17, , 18, , 19, , 20, , 21, , Ans. C, Q.No. 22, Ans. B, , C, 23, C, , C, 24, B, , A, 25, A, , B, 26, B, , B, 27, A, , D, 28, D, , B, 29, C, , A, 30, A, , A, 31, A, , A, 32, A, , A, 33, A, , B, 34, C, , A, , B, , D, , A, , A, , A, , C, , C, , LEVEL-3, Q.No., , 1, , 2, , 3, , 4, , 5, , 6, , 7, , 8, , 9, , 10, , 11, , 12, , 13, , 14, , 15, , 16, , Ans., Q.No., Ans., , B, 17, B, , C, 18, C, , B, 19, A, , B, 20, B, , C, 21, B, , A, 22, A, , D, 23, D, , A, 24, C, , C, 25, D, , C, 26, A, , D, , D, , C, , D, , D, , B, , Q.27 A  S ; B  Q ; C  Q ; D  R, , Q.28 A  P ; B  R ; C  S ; D  Q, , LEVEL- 4, SECTION-A, Q.No., Ans., , 1, C, , 2, D, , 3, B, , 4, D, , 5, B, , 6, C, , 7, D, , 8, A, , 9, B, , ,, , SECTION-B, Q.No., Ans., , PHOTON, , 1, A, , 2, D, , 3, D, , 4, D, , 5, B, , 6, B, , 7, D, , DIFFERENTIATION, , 8, B, , 9, A, , 10, D, , 11, C, , 12, D, , 13, B, , 14, B, , 125