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Q.1 If y = tan n e2 x + tan n ex (A) Q.2 d 2y nx then = dx nx 1 (B) (C) (D) u ( x) u' ( x) u(x) = If = p and v ' ( x ) v( x ) v( x ) Let u(x) and v(x) are differentiable functions such that p q has the value equal to p q (A) (B) ' = q, then Q.3 f ' (x ) f ' ' (x) Suppose (C) (D) – f ( x) = where f (x) is continuously differentiable function with f '(x) f ' (x) and satisfies f (0) = and f ' (0) = then f (x) is (B) 2ex – (C) e2x (A) x2 + 2x + Q.4 (A) & f (x) = tan x2 then If x = t3 tan x2 sin t 3t2 3t2 (C) 3x tan 5x (5x 6)2 (D) none d 2y + t + & y = sin t then = dx (A) 3t2 3t2 t cos t 1 sin t t cos t 1 sin t (B) (D) Let g is the inverse function of f & f (x) = (A) 10 Q.7 dy = dx (B) tan x tan x 3t2 Q.6 tan x3 (C) f Q.5 3x 5x If y = f (D) 4ex/2 – (B) 3t2 x2 If g(2) = a then g (2) is equal to a 10 (C) a2 a2 a 10 cos t 3t2 x10 t cos t (D) a 10 a2 cot (e x ) dx is equal to : ex (A) ln (e2x + 1) cot (e x ) +x+c ex cot (e x ) 2x (B) ln (e + 1) + +x+c ex (C) ln (e2x + 1) cot (e x ) ex cot (e x ) 2x (D) ln (e + 1) + ex x+c x+c ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.8 If y = (A) Q.9 38 27 is : (C) 27 38 (D) none dx equals : x (1 x ) (A) ln x + ln (1 + x7) + c (B) ln x ln (1 x7) + c (C) ln x ln (1 + x7) + c (D) ln x + ln (1 x7) + c If f (x) = a2 x2 a a2 x2 x where a > and x < a, then f ' (0) has the value equal to a x (B) a a (C) a Suppose that f (0) = and f ' (0) = 2, and let g (x) = f to (A) (B) (C) xdx Q.13 x2 (A) ln (C) Q.14 38 27 (B) x7 (A) Q.12 d 2y at x = dx cos x (cos x )1 (cos 3x )1 is not defined at x = If f (x) is continuous The function f (x) = x2 at x = then f (0) equals (A) (B) (C) (D) – Q.10 Q.11 then If (1 x ) (D) a x f f ( x ) The value of g ' (0) is equal (D) is equal to : x2 + c (B) 1 x2 + c x2 + c (D) none of these x a = b cot–1(b ln y), b > then, value of yy'' + yy' ln y equals (A) y' (B) y' (C) (D) ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.15 (A) P Q.16 (x) + P (x) (B) P (x) P Q.18 (x) If f (x) = x (A) Let f(x) = g (x) cos x1 if x 0 if x equals (D) a constant (D) none where g(x) is an even function differentiable at x = 0, passing (C) is equal to (D) does not exist cos x sin x x dx = ln f ( x ) + g(x) + C where C is the constant of integration and f (x) e x sin x x is positive, then f (x) + g (x) has the value equal to (B) ex + sin x (C) ex – sin x (D) ex + sin x + x (A) ex + sin x + 2x If Q.20 3x2 Let f (x) = x 2x 5x (A) is equal to If y = (A) 1 x n m x for x for x p m emnp + (B) 1 x m n emn/p : then f (B) is equal to 27 x p n (C) is equal to 27 + 1 x (C) m p x n p then enp/m (D) does not exist np dy at e m is equal to: dx (D) none f f x2 x 2 x (C) 10 If f is differentiable in (0, 6) & f (4) = then Limit (A) Q.23 (x) & g (x) = f [ f (x)] then for x > 20, g (x) = (B) (C) Q.19 Q.22 (C) P (x) P d 2y dx 3x w.r.t x If F(10) = 60 then the value of F(13), is x (B) 132 (C) 248 (D) 264 through the origin Then f (0) (A) is equal to (B) is equal to Q.21 y3 Let F(x) be the primitive of (A) 66 Q.17 d dx If y2 = P(x), is a polynomial of degree 3, then Integral of (A) ln cos (C) (B) 5/4 (D) 20 2cotx(cotx cos ecx ) w.r.t x is : x +c x ln cos + c 2 (B) ln sin (D) ln sin x x +c ln(cosec x cot x) + c ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.24 cos x sin x cos x Let f(x) = cos 2x sin 2x cos 2x then f cos 3x sin 3x cos 3x (A) Q.25 (B) – 12 (C) f (x h) f (x) where f (x) means [f(x)]2 If f(x) = x lnx then h D * f (x ) x e has the value (A) e (B) 2e x (A) (C) x ln x x ln2 x x2 x2 x + c (B) x ln2 x x2 x + x If (x) = x sin x then Limit x /2 (A) Q.28 (D) 8e dx equals : x2 (C) 4e x2 ln x Q.27 (D) 12 People living at Mars, instead of the usual definition of derivative D f(x), define a new kind of derivative, D*f(x) by the formula D*f(x) = Limit h Q.26 = +c (x) (D) x x ln x x +c x2 x2 +x+c = (B) (C) (D) none Let f (x) = x + sin x Suppose g denotes the inverse function of f The value of g' has the value equal to (A) Q.29 2 (B) (C) (D) 2 A differentiable function satisfies 3f 2(x) f '(x) = 2x Given f (2) = then the value of f (3) is (A) Q.30 24 If y = x + ex then (A) ex (B) (C) 6 (D) d 2x is : dy (B) ex e x (C) ex e x (D) 1 ex ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.31 Primitive of f (x) = x ·2ln ( x (A) (C) Q.32 ln ( x w.r.t x is 2( x 1) ( x 1) 2ln ( x (B) ln +C ( x 1) ln +C 2(ln 1) Let y = ln (1 + cos x)2 f (x) g (x) f (x ) 1) +C d2y then the value of + y / equals e dx (B) cos x 2 ( x 1) ln +C 2(ln 1) (D) (C) (1 cos x ) Let g (x) be an antiderivative for f (x) Then ln (A) Q.34 1) 1) (A) Q.33 g( x ) f (x) g (x) (B) g (x) (D) (1 cos x ) is an antiderivative for f (x ) (C) If f is twice differentiable such that f (x ) f (x), f (x) h (x) f (x ) h (0 ) then the equation y = h(x) represents : (A) a curve of degree (C) a straight line with slope 2 g(x) g(x ) , h (1) (D) none f (x ) and (B) a curve passing through the origin (D) a straight line with y intercept equal to Q.35 If f(x) is a twice differentiable function, then between two consecutive roots of the equation f (x) = 0, there exists : (A) atleast one root of f(x) = (B) atmost one root of f(x) = (C) exactly one root of f(x) = (D) atmost one root of f (x) = Q.36 A function y = f (x) satisfies f "(x) = – f x – sin( x) ; f '(2) = + is (A) ln (B) (C) – ln cos Q.37 Let a, b, c are non zero constant number then Lim r (A) and f (1)=0 The value of a2 b2 c2 2bc (B) c2 a b2 2bc (C) b2 (D) – ln a b c cos cos r r r equals b c sin sin r r c2 a 2bc (D) independent of a, b and c ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.38 Q.39 cos3 x cos5 x dx sin x sin x (A) sin x (C) sin x tan (sin x) + c (sin x) tan (sin x) + c If f (x) = + x 2 x + x 2 x , then the value of 10 f ' (102 ) (B) is (C) is (D) does not exist (A) is – Q.40 Which one of the following is TRUE dx x (A) x (C) Q.41 (B) x tan x C (D) dx x x ln | x | cos x dx cos x Cx x C cos 1 cos x sin x 13 (B) sin cos x sin x 13 (C) 10 w.r.t x at x is (D) Let f (x) be a polynomial function of second degree If f (1) = f (–1) and a, b, c are in A.P., then f '(a), f '(b) and f '(c) are in (A) G.P (B) H.P (C) A.G.P (D) A.P (2x 1) dx ( x 4x 1)3 / 2 (A) (C) Q.44 C The derivative of the function, (A) Q.43 x ln | x | cos x dx cos x f x Q.42 sin x + c (sin x) + tan (sin x) + c (B) sin x (D) sin x (x (x x3 4x 1)1 / C (B) x2 x 1)1 / C (D) y If x2 + y2 = R2 (R > 0) then k = (A) – R2 (B) – R y (x (x x 4x 1)1/ C 4x 1)1 / C where k in terms of R alone is equal to (C) R (D) – R2 ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.45 sin(101x ) ·sin 99 x dx equals sin(100 x )(sin x )100 (A) +C 100 (C) cos(100 x )(sin x )100 (B) +C 100 cos(100 x )(cos x )100 +C 100 (D) sin(100 x )(sin x )101 +C 101 Q.46 If f & g are differentiable functions such that g (a) = & g(a) = b and if fog is an identity function then f (b) has the value equal to : (A) 2/3 (B) (C) (D) 1/2 Q.47 Given f(x) = x3 + x2 sin 1.5 a x sin a sin 2a (A) f(x) is not defined at x = sin (C) f (x) is not defined at x = sin Q.48 Q.49 Given: f(x) = 4x3 Q.51 If y = (A + Bx) emx + (m 1) (A) ex (B) emx If In = (sin x ) n dx n N Q.52 xp d 2y then dx product ab is equal to (A) 25 (B) 2m (C) e n 2a a2 xp xp q C then dy + m2y is equal to : dx mx (D) e(1 m) x (B) sin2x · cos2x + C sin 2x [cos22x + – cos2x] + C Suppose f (x) = eax + ebx, where a q C (D) (B) f (1/2) < (D) f (1/2) > Then I4 – I6 is equal to (A) sin x · (cos x)5 + C (C) xq 6x2 cos 2a + 3x sin 2a sin 6a + ex 8a + 17) then : (B) f (sin 8) > (D) f (sin 8) < P X p 2q q x q dx is The evaluation of X p q 2x p q xq xp C C (B) p q (A) – p q (C) x x (A) f(x) is not defined at x = 1/2 (C) f (x) is not defined at x = 1/2 Q.50 arc sin (a2 (D) sin 2x [cos22x + + cos2x ] + C b, and that f '' (x) – f ' (x) – 15 f (x) = for all x Then the (C) – 15 (D) – ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.53 Let h (x) be differentiable for all x and let f (x) = (kx + ex) h(x) where k is some constant If h (0) = 5, h ' (0) = – and f ' (0) = 18 then the value of k is equal to (A) (B) (C) (D) 2.2 Q.54 e tan x (1 x ) (A) e tan x tan (C) e 1x sec 1 x2 tan x sec cos 1 x2 x2 dx e tan 1x tan (D) e 1x (B) C x2 C (x > 0) tan x 2 cos ec C x2 C Q.55 Let f(x) = xn , n being a non-negative integer The number of values of n for which f (p + q) = f (p) + f (q) is valid for all p, q > is : (A) (B) (C) (D) none of these Q.56 Let ef(x) = ln x If g(x) is the inverse function of f(x) then g (x) equals to : (A) ex Q.57 ( x 1) dx x2 ( x 3x 1) tan x (A) ln x Q.58 x (C) e = ln | f (x) | + C (B) tan–1 x x (x ex ) (D) e(x + ln x) then f (x) is (C) cot–1 x x (D) ln tan x x A non zero polynomial with real coefficients has the property that f (x) = f ' (x) · f ''(x) The leading coefficient of f (x) is (A) Q.59 (B) ex + x Let f (x) = (B) (C) 12 (D) 18 sin x cos x ( sin x 1) + then sin x cos x e x f ( x ) f ' ( x ) dx where c is the constant of integeration) (A) ex tanx + c Q.60 (B) excotx + c (C) ex cosec2x + c (D) exsec2x + c The function f(x) = ex + x, being differentiable and one to one, has a differentiable inverse d –1 f–1(x) The value of (f ) at the point f(l n2) is dx (A) n2 (B) (C) (D) none ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.61 The ends A and B of a rod of l ength are sliding along the curve y = 2x2 Let xA and xB be the dx B x-coordinate of the ends At the moment when A is at (0, 0) and B is at (1, 2) the derivative has the dx A value(s) equal to (A) 1/3 (B) 1/5 (C) 1/8 (D) 1/9 Q.62 If y = (A) Q.63 (a x) a x (b x ) x b a x x x b (a b) (B) (a x) (x b) 2x then dy wherever it is defined is equal to : dx (a b) (a x) (x b) (a b ) (C) (D) (a x) (x b) 2x (a b) (a x) (x b) cotn x d x , then I0 + I1 + (I2 + I3 + + I8) + I9 + I10 equals to : If In = (where u = cot x) u2 (A) u + (C) Q.64 u u9 u2 2! u9 9! (D) u u2 2u u9 9u 10 dy is dx (D) non existent For the curve represented implicitly as 3x – 2y = 1, the value of Lim x d x dy If dy dx (B) equal to (A) + (C) equal to log23 d 2y = K then the value of K is equal to dx (B) –1 (C) e Q.66 (B) (A) equal to Q.65 u x2 if x if x (D) Let y = f(x) = Then which of the following can best represent the graph of y = f(x)? (A) Q.67 (B) Let f (x) = sin3x + sin3 x (C) + sin3 x sin 3x cos 3x C (B) – 4 where C is an arbitrary constant (A) – C (C) (D) then the primitive of f (x) w.r.t x is sin 3x C (D) cos 3x C ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 10 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.68 (A) Q.69 Q.70 n x (B) The integral cot x e (A) tan x e sin x +C cos x (C) e sin x sin x m n n m n n m m x (C) – w.r.t x is x mn (D) cos x dx equals (B) 2e +C (D) (B) 24 a (ax + b)2 sin x +C cot x e sin x +C cos x If y = at2 + 2bt + c and t = ax2 + 2bx + c, then (A) 24 a2 (at + b) d 3y equals dx (C) 24 a (at + b)2 (D) 24 a2 (ax + b) x (1 ln x ) dx equals ln x x Q.71 x (A) ln ln x Limit x ln ln x x ln x x ln ln x x (C) Q.72 x Differential coefficient of m m n (A) x x a b a arc tan tan x a C ln x x C b arc tan x b (B) ln x x (B) ln ln x x tan ln x x (D) ln ln x x tan 1 ln x x C ln x x C has the value equal to (C) (a b ) 6a b (D) a b2 3a b2 (2 x 3) dx = C– where f (x) is of the form of ax2 + bx + c then x ( x 1)( x 2)(x 3) f (x) (a + b + c) equals (A) (B) (C) (D) none Q.73 If Q.74 Suppose A = dy dy of x2 + y2 = at ( , ), B = of sin y + sin x = sin x · sin y at ( , ) and dx dx dy of 2exy + ex ey – ex – ey = exy + at (1, 1), then (A + B + C) has the value equal to dx (A) – (B) e (C) – (D) C= ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 11 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) t dy dy x Q.75 A function is represented parametrically by the equations x = ; y = then dx dx t t 2t has the value equal to (A) (B) (C) –1 (D) –2 Q.76 Suppose A = x2 If 12(A + B) = (A) dx and B = 6x 25 x2 dx 6x 27 x x + ! · ln + C, then the value of ( + !) is x (B) (C) (D) · tan–1 Q.77 Suppose the function f (x) – f (2x) has the derivative at x = and derivative at x = The derivative of the function f (x) – f (4x) at x = 1, has the value equal to (A) 19 (B) (C) 17 (D) 14 Q.78 If x + y = 3e2 then D(xy) vanishes when x equals to (A) e (B) e2 (C) ee Q.79 dx Let x 2008 where p, q, r (A) 6024 Q.80 Q.82 N and need not be distinct, then the value of (p + q + r) equals (B) 6022 (C) 6021 (D) 6020 " (B) " (C) " d2y The value of at the point where t = is dx (A) (B) (C) – If F (t) = ( x y) dt then the value of F (A) Q.83 +C A curve is represented parametrically by the equations x = et cos t and y = et sin t where t is a parameter Then The relation between the parameter 't' and the angle " between the tangent to the given curve and the x-axis is given by, 't' equals (A) Q.81 xq = p ln xr x (D) 2e2 (B) – (D) " (D) – F (0) is (C) e /2 (D) Consider the following statements Statement-1: f (x) = x ex and g (x) = ex(x + 1) are both aperiodic function because ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 12 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Statement-2: Derivative of a differentiable aperiodic function is an aperiodic function (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1 (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1 (C) Statement-1 is true, statement-2 is false (D) Statement-1 is false, statement-2 is true x Q.84 Statement-1: The function F (x) = ( x 1)( x 1) dx is discontinuous at x = because Statement-2: If F (x) = f ( x ) dx and f (x) is discontinuous at x = a then F (x) is also discontinuous at x = a (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1 (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1 (C) Statement-1 is true, statement-2 is false (D) Statement-1 is false, statement-2 is true Q.85 If y x y x = c (where c 2x (A) c Q.86 0), then dy has the value equal to dx x (B) y2 x y If y = tan x tan 2x tan 3x then (C) y y2 x x2 c2 (D) 2y dy has the value equal to dx (A) sec2 3x tan x tan 2x + sec2 x tan 2x tan 3x + sec2 2x tan 3x tan x (B) 2y (cosec 2x + cosec 4x + cosec 6x) (C) sec2 3x sec2 2x sec2 x (D) sec2 x + sec2 2x + sec2 3x Q.87 n (tan x ) dx equal: sin x cos x (B) ln (sin x sec x) + c (D) (A) ln2 (cot x) + c (C) Q.88 If 2x + 2y = 2x + y then (A) 2y 2x ln (sec x) + c 2 ln (cos x cosec x) + c dy has the value equal to dx (B) 1 2x (C) 2y 2x (D) y 2x 2y ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 13 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.89 For the function y = f (x) = (x2 + bx + c)ex, which of the following holds? (A) if f (x) > for all real x # $ f ' (x) > (B) if f (x) > for all real x # f ' (x) > (C) if f ' (x) > for all real x # f (x) > (D) if f ' (x) > for all real x # $ f (x) > Q.90 If eu sin 2x dx can be found in terms of known functions of x then u can be: (A) x (B) sin x (C) cos x (D) cos 2x Q.91 Let f (x) = x x x 1 x then (A) f (10) = (C) domain of f (x) is x % Q.92 Let f (x) = 3x2 sin x x cos (B) f (3/2) = (D) none , if x x ; f(0) = and f(1/ ) = then : (A) f(x) is continuous at x = (C) f (x) is continuous at x = Q.93 If y = x ( nx ) (A) nx (C) Q.94 Q.95 n( nx ) dy is equal to : dx nx n nx (B) f(x) is non derivable at x = (D) f (x) is non derivable at x = , then nx y ((ln x)2 + ln (ln x)) x nx (B) (D) (ln x)ln (ln x) (2 ln (ln x) + 1) y ny (2 ln (ln x) + 1) x nx Which of the following functions are not derivable at x = 0? (A) f (x) = sin–12x (C) h (x) = sin–1 x2 x2 x2 (B) g (x) = sin–1 2x 1 4x (D) k (x) = sin–1(cos x) sin x sin x cos x cos x dx and K = dx If C is an arbitrary constant of Suppose J = sin x cos x sin x cos x integration then which of the following is/are correct? (A) J = (x – sin x + cos x) + C (C) J = x – K + C (B) J = K – (sin x + cos x) + C (D) K = (x – sin x + cos x) + C ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 14 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.1 C Q.2 A Q.3 C Q.4 B Q.5 A Q.6 B Q.7 C Q.8 A Q.9 B Q.10 C Q.11 D Q.12 C Q.13 B Q.14 B Q.15 C Q.16 B Q.17 A Q.18 B Q.19 B Q.20 B Q.21 D Q.22 D Q.23 B Q.24 C Q.25 C Q.26 A Q.27 A Q.28 C Q.29 B Q.30 B Q.31 C Q.32 A Q.33 B Q.34 C Q.35 B Q.36 D Q.37 C Q.38 C Q.39 C Q.40 B Q.41 C Q.42 D Q.43 B Q.44 B Q.45 A Q.46 D Q.47 D Q.48 C Q.49 D Q.50 A Q.51 C Q.52 C Q.53 C Q.54 C Q.55 C Q.56 C Q.57 B Q.58 D Q.59 A Q.60 B Q.61 D Q.62 B Q.63 B Q.64 C Q.65 D Q.66 C Q.67 D Q.68 B Q.69 B Q.70 D Q.71 B Q.72 D Q.73 B Q.74 C Q.75 C Q.76 B Q.77 A Q.78 B Q.79 C Q.80 C Q.81 B Q.82 C Q.83 C Q.84 C Q.85 A, B, C Q.86 A, B, C Q.87 A, C, D Q.88 A, B, C, D Q.89 A, C Q.90 A, B, C, D Q.91 A, B Q.92 A, C, D Q.93 B, D Q.94 B, C, D Q.95 B, C ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 15 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005)