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ln Q.1 cos e x The value of the definite integral, 2 ·2x e x dx is (A) (B) + (sin 1) (C) – (sin 1) (D) (sin 1) – Q.2 sin x The value of the definite integral dx where [0, ] (A) (B) cos (C) cos (D) cos 12 Q.3 ( sin (3x 4x ) Value of the definite integral cos (4x 3x ) ) dx 12 (A) (B) x Q.4 Let f (x) = dt t4 (C) (D) and g be the inverse of f Then the value of g'(0) is (A) (B) 17 (D) none of these (C) 17 t Q.5 (1 a sin bx ) c x dx equals If a, b and c are real numbers then the value of Lim ln t t0 (A) abc (B) ab c (C) bc a (D) ca b dx Q.6 The value of the definite integral a (A) (B) (1 x )(1 x ) (C) n Q.7 n (1 sin t ) sin 2t dt then Lim n Let an = (A) 1/2 (a > 0) is (B) n an n (D) some function of a is equal to (C) 4/3 (D) 3/2 Q.8 The value of the definite integral (1 x ) sin x (1 x ) cos x dx , is (A) tan (B) 2tan (C) tan (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) n Q.9 Let Cn = n tan (nx ) n ·C n equals dx then Lim n sin (nx ) (A) (B) (C) – 1 Q.10 If x satisfies the equation t dt t cos (D) t sin t dt x – = (0 < t x2 – < ), then the value x is (A) ± (B) ± sin x Q.11 If f (x) = eg(x) and g(x) = (A) equals 2/17 sin (C) ± sin (D) ± sin t dt then f (2) t4 (B) equals (C) equals (D) cannot be determined x Q.12 f ' ( t ) (2 sin t – sin2t) dt then f (x) is A function f (x) satisfies f (x) = sin x + (A) x sin x (B) sin x sin x (C) cos x cos x (D) tan x sin x Q.13 Suppose the function gn(x) = x2n + + anx + bn (n ( px q ) g n ( x ) dx = N) satisfies the equation for all linear functions (px + q) then (A) an = bn = (C) an = 0; bn = – (B) bn = 0; an = – 2n (D) an = r 4n Q.14 n The value of Lim n (A) 35 r r r (B) 14 If F (x) = f ( t ) dt where f (t) = (A) 17 (B) x Q.16 15 17 3 ; bn = – 2n 2n is equal to (C) t2 x Q.15 n 2n 10 (D) u4 du then the value of F '' (2) equals u (C) 257 (D) 15 17 68 t Let f (x) = e dt and h (x) = f g( x ) , where g (x) is defined for all x, g'(x) exists for all x, and g (x) < for x > If h'(1) = e and g'(1) = 1, then the possible values which g(1) can take (A) (B) – (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) x Q.17 The value of x > satisfying the equation (A) e t ln t dt = , is (C) e2 (B) e (D) e – Q.18 f ( x ) dx = 17, then Let f be a one-to-one continuous function such that f (2) = and f (5) = Given f the value of the definite integral ( x ) dx equals (A) 10 Q.19 (B) 11 (C) 12 (D) 13 Let f (x) be a function satisfying f ' (x) = f (x) with f (0) = and g be the function satisfying f (x) + g (x) = x2 The value of the integral f ( x )g ( x ) dx is (A) e – e – 2 g( x ) Q.20 (B) e – e2 – t and f (0) = then f ' where g (x) = e – 2 (D) e – (1 sin t ) dt Also h(x) = e– | x | and f (x) = x sin if x x equals (B) h ' (0–) (A) l ' (0) (e – 3) cos x dt Let f (x) = (C) (C) h ' (0+) (D) Lim (C) (D) x cos x x sin x Q.21 Lim t | sin( x t ) sin x | dx equals |t| (A) (B) 1 Q.22 dx The value of (2 (A) Q.23 Lim x) x (B) n 6n (A) 3 sec 6n is (C) sec 2 · (B) 6n sec (n 1) (D) cannot be evaluated 6n (C) has the value equal to (D) Q.24 f (cos x )dx and I2 = For f (x) = x4 + | x |, let I1 = (A) (B) 1/2 f (sin x )dx then (C) 2 I1 has the value equal to I2 (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) x Q.25 If g (x) = cos t dt , then g (x + ) equals (A) g (x) + g ( ) e Q.26 x/2 (B) g (x) g ( ) (C) g (x) g ( ) (D) [ g (x)/g ( ) ] sin x dx is cos x (A) e /2 (C) 2e /2 e e /4 /3 (B) 2e /6 e /3 /4 2e e /6 /3 (D) 2e k Q.27 k I2 is I1 Then (A) k Q.28 If Lim a a x ax 1 x4 ·tan f x (1 x ) dx , where 2k – > k (B) 1/2 (A) Q.29 k x f x (1 x ) dx ; I2 = Let f be a positive function Let I1 = /4 2e (C) (D) 2 dx is equal to where k x k (B) N equals (C) 16 (D) 32 Suppose that the quadratic function f (x) = ax2 + bx + c is non-negative on the interval [–1, 1] Then the area under the graph of f over the interval [–1, 1] and the x-axis is given by the formula (B) A = f (A) A = f (–1) + f (1) (C) A = [ f ( 1) f (0) f (1)] (D) A = f [ f ( 1) f (0) f (1)] f (x) Q.30 t dt = x cos x , then f ' (9) If (A) is equal to – Q.31 Let I (a) = x a (B) is equal to – (C) is equal to (D) is non existent a sin x dx where 'a' is positive real The value of 'a' for which I (a) attains its minimum value is (A) (B) (C) 16 (D) 13 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) /2 Q.32 cos sin x dx and v = Let u = (A) 2u = v Q.33 cos (B) 2u = 3v /4 Let f (x) = (A) /2 sin x dx x x sin x dx , then the relation between u and v is (C) u = v (B) dt t 3t 13 11 x dx sin x (C) (D) u = 2v (B) 11 x dx sin x (C) 13 Domain of definition of the function f (x) = (B) R+ (A) R /2 (D) dt (D) {0} R R+ (B) a 13 (D) R – {0} The set of values of 'a' which satisfy the equation ( t log a ) dt = log2 (A) a x dx sin x is x2 t2 (C) R+ Q.36 /4 If g (x) is the inverse of f (x) then g'(0) has the value equal to x Q.35 tan x dx = x (A) Q.34 /2 a2 is (C) a < (D) a > (C) (D) 1x Q.37 Lim x x ln (1 t ) dt equals t e 1x (A) 1/3 (B) 2/3 y Q.38 Variable x and y are related by equation x = d2y The value of is equal to dx t2 dt y (A) y 2y (B) y (C) y2 (D) 4y Q.39 The value of the definite integral (A) /2 Q.40 dx is (1 e )(1 x ) x (B) /4 (C) /8 (D) /16 If f & g are continuous functions in [0, a] satisfying f (x) = f (a x) & g (x) + g (a x) = then a f ( x ).g( x )dx = a (A) f (x)dx 20 a a (B) f (x)dx (C) a f (x)dx (D) f (x)dx 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) x Q.41 1 f ( t ) dt = x + t ·f ( t ) dt , then the value of the integral If x (A) (B) /4 (C) /2 Q.42 f ( x ) dx is equal to (D) x e x The value of the definite integral e (1 x ·e )dx is equal to (A) ee (B) ee – (C) ee – e a Q.43 x ·a If the value of definite integral [loga x ] (D) e dx where a > 1, and [x] denotes the greatest integer, is e then the value of 'a' equals (A) e ee (B) e (C) ee Q.44 ee e dx x ln x ·ln (ln x ) ·ln ln (ln x ) equals (A) (B) e (C) e – 1 Q.45 (D) e – e Let f be a continuous functions satisfying f ' (ln x) = (D) + e for x x for x and f (0) = then f (x) can be defined as (A) f (x) = (C) f (x) = if x e x if x x ex if x 0 e x if x if x x if x if x e x if x (B) f (x) = (D) f (x) = 2008 Q.46 x | sin x | dx is equal to The value of (A) (B) 2008 n n 2008 (C) 1004 (D) 2008 Q.47 Lim Q.48 tan (x ) tan ( x ) (C) (D) x x2 The interval [0, 4] is divided into n equal sub-intervals by the points x 0, x1, x2, ., xn – 1, xn n k 1n (A) x tan–1(x) k2x2 , x > is equal to (B) tan–1(x) n where = x0 < x1 < x2 < x3 < xn = If x = xi – xi – for i = 1, 2, 3, n then Lim x x i x is i equal to 32 (D) 16 Industrial Area, End of Evergreen Motors ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha (A) (B) (C) (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) 19 Q.49 The absolute value of 10 (A) 10 (sin x ) dx is less than ( x ) 10 11 (B) 10 (C) 10 (D) 10 a Q.50 f ( x ) dx Let a > and let f (x) is monotonic increasing such that f (0) = and f (a) = b then f ( x ) dx Q.51 b equals (A) a + b (B) ab + b n Lim is equal to n ( n!)1 n (C) ab + a (D) ab 1 (B) e (A) e (D) ln x dx (C) 1 Q.52 n ·x n The value of the limit, Lim dx is equals n x (A) (B) 1/2 (C) (D) non existent 37 Q.53 {x}2 3(sin x ) dx where { x } denotes the fractional part function The value of the definite integral 19 (A) Q.54 (B) (D) can not be determined 3 x4 x4 2x cos dx dx then 'k' equals = k x4 x2 x If (A) f x Q.55 (C) (B) (C) (D) (B) is equal to one (C) is equal to ln x · dx x x (A) is equal to zero (D) can not be evaluated Q.56 The value of the definite integral tan x dx , is (A) (B) (C) 2 a2 Q.57 Positive value of 'a' so that the definite integral a (A) tan2 (B) tan2 dx x x (C) tan2 (D) 2 achieves the smallest value is 12 (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.58 n n (x r) The value of k 1x r (A) n k dx equals (B) n ! (C) (n + 1) ! (D) n · n ! x Q.59 (ln2t + lnt) dt where f ' (x) vanishes is The value of the function f (x) = + x + (A) e (B) Lim (1 x ) dx (D) + e is equal to (A) ln (B) x2n + 1· e Q.61 1 Q.60 (C) e x2 e (C) ln dx is equal to (n e (D) N) (A) (n – 1)! (B) n ! (C) n! (D) ( n 1)! Q.62 The true set of values of 'a' for which the inequality (3 2x x) dx is true is: a (B) ( ! , 1] (A) [0 , 1] Q.63 If the value of the integral e x dx is , then the value of (A) e4 Q.64 e ! e4 n x dx is : (C) (e4 e) ! (D) e4 – – [1, 5], where f (1) = and f (5) = 10 then the 10 f ( x ) dx g ( y) dy equals (A) 48 Q.65 e ! (B) e4 [0, ) e If g (x) is the inverse of f (x) and f (x) has domain x values of (D) ( ! , 1] (C) [0, ) (B) 64 (C) 71 (D) 52 Which one of the following functions is not continuous on (0, )? x (B) g(x) = t sin t dt (A) f(x)= cotx x (C) h (x) = sin Q.66 x x sin x , x (D) l (x) = x sin( x ), 2 x If f (x) = x sinx2 ; g (x) = x cosx2 for x [ 1, 2] 2 A = f ( x ) dx ; B = g( x ) dx then (A) A > ; B < (B) A < ; B > (C) A > ; B > (D) A < ; B < 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.67 dx is |x| The value of (A) (B) (A) (D) undefined x dx = x ln Q.68 (C) 3 2ln (B) ln 2 (C) 1 ln 54 (D) 27 ln 2 /2 Q.69 For < x < (A) (C) ln (ecos x) d (sin x) is equal to : , 1/ (B) 12 sin sin1 (D) sin sin1 x Q.70 5x x The true solution set of the inequality, (A) R (B) ( 1, 6) 20 dz > x sin x dx is : (C) ( 6, 1) (D) (2, 3) Q.71 ( | cos t | sin t | sin t | cos t ) dt has the value equal to The integral, (A) (C) 1/ (B) 1/2 (D) /2 Q.72 The value of the definite integral sin x sin 2x sin 3x dx is equal to : (A) (B) (C) Q.73 If the value of the definite integral (A) 2 Q.74 For Un = x (B) 2 cot x dx , is equal to ae– e sin x (C) 2 (D) /6 + be– /4 then (a + b) equals (D) xn (2 ture ? (A) Un = 2n Vn x)n d x ; V n= xn (1 x)n d x n Let S (x) = (B) Un = n Vn l n t d t (x > 0) and H (x) = x2 N , which of the following statement(s) is/are x3 Q.75 (C) Un = 22n Vn (D) Un = 2n V n S (x) Then H(x) is : x (A) continuous but not derivable in its domain (B) derivable and continuous in its domain (C) neither derivable nor continuous in its domain (D) derivable but not continuous in its domain 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.76 sin x Let f (x) = , then x f ( x ) dx (A) f (x) f x dx = (B) f ( x ) dx 0 f ( x ) dx (D) Q.77 f ( x ) dx (C) Statement-1 : If f(x) = ( x f ( t ) 1) dt , then f ( x ) dx = 12 0 because Statement-2 : f(x) = 3x + (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.78 dx sin x Statement-1: I = because Consider I = a f ( x ) dx Statement-2: , wherever f (x) is an odd function 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 x Q.79 t dt is an odd function and g (x) = f ' (x) is an even function Statement-1: The function f (x) = because Statement-2: For a differentiable function f (x) if f ' (x) is an even function then f (x) is an odd 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 Q.80 Given f (x) = sin3x and P(x) is a quadratic polynomial with leading coefficient unity P( x ) ·f ' ' ( x ) dx vanishes Statement-1: because f ( x ) dx vanishes Statement-2: (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 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) x Suppose Lim = l where p bx sin x If l exists and is non zero then (A) b > (B) < b < x Q.81 Q.82 Q.83 Q.84 t dt r 1p (a t ) N, p 2, a > 0, r > and b 0 (C) b < (D) b = If p = and l = then the value of 'a' is equal to (A) (B) (C) (D) 3/2 If p = and a = and l exists then the value of l is equal to (A) 3/2 (B) 2/3 (C) 1/3 (D) 7/9 Let the function f satisfies f (x) · f ' (– x) = f (– x) · f ' (x) for all x and f (0) = The value of f (x) · f (– x) for all x, is (A) (B) (C) 12 (D) 16 51 Q.85 dx f ( x ) has the value equal to 51 (A) 17 Q.86 (B) 34 Number of roots of f (x) = in [–2, 2] is (A) (B) (C) 102 (D) (C) (D) Suppose f (x) and g (x) are two continuous functions defined for x 1 Given f (x) = e x t ·f ( t ) dt and Q.87 Q.88 The value of f (1) equals (A) (B) (C) e–1 (D) e The value of g (0) – f (0) equals (A) Q.89 x t g (x) = e ·g ( t ) dt + x e2 The value of (A) (B) e g (0 ) equals g ( 2) (B) 2 (C) (C) e e2 (D) (D) e2 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) Consider the function defined on [0, 1] f (x) = R sin x x cos x if x and f (0) = x2 f ( x ) dx equals Q.90 (A) – sin (1) Q.91 Lim t t2 (B) sin (1) – (C) sin (1) (D) – sin (1) (C) 1/12 (D) 1/24 t f ( x ) dx equals (A) 1/3 (B) 1/6 Suppose a and b are positive real numbers such that ab = Let for any real parameter t, the distance from the origin to the line (aet)x + (be–t)y = be denoted by D(t) then Q.92 The value of the definite integral I = Q.93 is equal to e2 (B) a b2 e2 e2 (C) a b2 e2 b (D) a2 e2 [5] e2 The value of 'b' at which I is minimum, is (B) e e (C) (D) e [4] Minimum value of I is (B) e – e (C) e (D) e + e (D) cos x dx [3] Which of the following definite integral(s) vanishes /2 ln (cot x ) dx (A) The equation 10x4 e sin x dx (B) Q.96 a2 e2 (A) e – Q.95 D( t ) e2 (A) b (A) e Q.94 dt (C) dx 1/ 1/e x (ln x ) 3x2 = has (A) at least one root in ( 1, 0) (C) at least two roots in ( 1, 1) (B) at least one root in (0, 1) (D) no root in ( 1, 1) 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.97 Which of the following are true? a a x f (sin x) dx = (A) a a f (s in x ) dx a a n (x 1) x 2x (A) + ln2 tan (B) x2 (A) 2n In + = (C) I2 = x Q.101 If f(x) = ;n n /2 n cos( sin x ) dx , then cos( sin x ) dx and I3 = (C) I1 + I2 + I3 = (D) I2 = I3 N, then which of the following statements hold good ? + (2n 1) In (B) I2 = + ln4 + cot (B) I2 + I3 = dx cos( sin x ) dx ; I2 = Q.100 If In = /2 (A) I1 = + ln2 tan (D) /2 Suppose I1 = c dx is : (C) ln2 cot Q.99 b 2x 3x The value of f (x) dx f (x c) dx = f (x) dx (D) Q.98 a b c f cos x dx = n f cos x dx (C) f (x) dx = (B) (D) I3 = 16 48 nt dt where x > then the value(s) of x satisfying the equation, t f(x) + f(1/x) = is : (A) (B) e (C) e (D) e2 1 | t | cos( xt ) dt then which of the following hold true? Q.102 Let f (x) = (A) f (0) is not defined (B) Lim f ( x ) exists and equals x (C) Lim f ( x ) exists and is equal to x (D) f (x) is continuous at x = 0 Q.103 The function f is continuous and has the property f f ( x ) = – x for all x [0, 1] and J = f ( x ) dx then (A) f +f =1 4 (C) f ·f =1 3 (B) the value of J equal to (D) sin x dx has the same value as J (sin x cos x ) 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.104 Let f(x) is a real valued function defined by : 1 t ·f ( t )dt + x f ( t )dt x2 + x2 f(x) = 1 then which of the following hold(s) good ? 10 11 t ·f ( t )dt (A) 1 30 11 (D) f(1) – f(–1) = 20 11 t ·f ( t )dt > (C) (B) f(1) + f(–1) = f ( t )dt 1 x Q.105 Let f (x) and g (x) are differentiable function such that f (x) + g ( t ) dt = sin x (cos x – sin x), and f ' (x ) " + g( x ) " = then f (x) and g (x) respectively, can be (A) sin 2x, sin 2x (B) cos x , cos 2x (C) sin 2x, – sin 2x (D) – sin2x, cos 2x x t sin at bt c dt where a,b, c are non zero real numbers, then Lim f ( x ) is x x x (A) independent of a (B) independent of a and b and has the value equals to c (C) independent a, b and c (D) dependent only on c Q.106 Let f (x) = Q.107 L et L = Lim n n dx 2 where a a1 n x (A) Q.108 (A) (B) R then L can be (C) Column I Suppose, f (n) = log2(3) · log3(4) · log4(5) logn–1(n) 100 (D) Column II f ( k ) equals (P) 5010 Let f (x) = x ( x 1) ( x 2)( x 4) (Q) 5050 (R) 5100 (S) 5049 then the sum k (B 100 f ( x )dx is then (C In an A.P the series containing 99 terms, the sum of all the odd numbered terms is 2550 The sum of all the 99 terms of the A.P is 100 (1 rx ) (D) Lim x r x equals 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) T Q.109 Let Lim (sin x sin ax ) dx = L then T T0 (A) (B) (C) (D) Column I for a = 0, the value of L is for a = the value of L is for a = – the value of L is # a R – {–1, 0, 1} the value of L is Q.110 Column II (P) (Q) 1/2 (R) (S) Column I Column II e x cos x x (A) The function f (x) = is not defined at x = sin x The value of f (0) so that f is continuous at x = is (B) The value of the definite integral dx x x equals a + b ln (P) –1 (Q) (R) 1/2 (S) where a and b are integers then (a + b) equals n (C) Given e n sec $ tan $ d$ = then the value of tan (n) is equal to $ e n (D) Let an = tan (nx ) dx and bn = n Lim n an bn Q.111 n sin (nx ) dx then n has the value equal to Column–I g( x ) (A) If f (x) = Column–II cos x dt t3 then the value of f ' (B) (1 sin t )dt where g (x) = (P) (Q) (R) (S) –1 If f (x) is a non zero differentiable function such that x f ( t )dt = f ( x ) for all x, then f (2) equals b (C) ( x x ) dx is maximum then (a + b) is equal to If a (D) sin x a x x3 the value equal to If Lim b = then (3a + b) has the x2 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.112 Column-I (A) Lim x sin x Column-II ln (1 x ) (1 tan y)1 y dy equals (B) Lim (e x ex x )1 x equals (C) Let f (x) = Lim x n n k n 1n k2x2 then Lim f ( x ) equals x e (R) e2 (S) e–2 [0, ] 0 The quantity f ($) – g($) # $ in the interval given in column-I, is Column-I (Q) 1 , Q.113 Let f ($) = ( x sin $) dx and g ($) = ( x cos $) dx where $ (A) (P) Column-II (P) negative (B) , (Q) positive (C) , (R) non negative (D) 0, (S) non positive Q.114 ,2 4 Column-I Column-II 1 ( 2008) x 2008 e x (A) 2008 dx equals (P) e–1 (Q) e–1/4 (R) e1/2 (S) e (B) The value of the definite integral e x2 1e ln x dx is equal to dx + 1 (C) Lim n ·2 ·3 (n 1) n1 n n ·n n n2 equals 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.115 Column-I Column-II x sin (sin x ) cos (cos x ) dx (A) (P) (B) x dx sin x 2 (Q) 2 sin x (C) x cos x dx equals (R) (S) Q.116 Column-I (A) 2 Column-II Let f (x) = x sin x x cos x ·ln x sin x dx and f 2 = (P) rational (Q) irrational (R) integral (S) prime then the value of f ( ) is (B) Let g (x) = cos x (cos x 2) dx and g (0) = then the value of g (C) is If real numbers x and y satisfy (x + 5)2 + (y – 12)2 = (14)2 then the minimum value of (x y ) is ( x 1) dx (D) Let k (x) = x 3x and k (–1) = then the value of k (– 2) is 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.1 C Q.2 A Q.3 B Q.4 C Q.5 A Q.8 A Q.9 D Q.10 D Q.11 A Q.12 B Q.13 B Q.14 C Q.15 C Q.16 C Q.17 A Q.18 C Q.19 D Q.20 C Q.21 Q.22 B Q.23 A Q.24 C Q.25 A Q.26 D Q.27 D Q.28 C Q.29 D Q.30 A Q.31 A Q.32 A Q.33 C Q.34 B Q.35 D Q.36 B Q.37 A Q.38 B Q.39 B Q.40 B Q.41 C Q.42 A Q.43 A Q.44 A Q.45 D Q.46 D Q.47 C Q.48 B Q.49 C Q.50 D Q.51 A Q.52 B Q.53 B Q.54 A Q.55 A Q.56 B Q.57 A Q.58 D Q.59 D Q.60 B Q.61 C Q.62 D Q.63 B Q.64 A Q.65 D Q.66 A Q.67 C Q.68 A Q.69 A Q.70 D Q.71 A Q.72 D Q.73 A Q.74 C Q.75 B Q.76 A Q.77 C Q.78 D Q.79 C Q.80 A Q.81 D Q.82 A Q.83 B Q.84 B Q.85 A Q.86 A Q.87 A Q.88 A Q.89 B Q.90 A Q.91 B Q.92 C Q.93 D Q.94 B Q.95 ABC Q.96 ABC Q.97 ABCD ACD Q.99 ABC Q.100 AB Q.101 CD Q.102 CD Q.105 CD Q.106 AD Q.107 ABC Q.98 Q.6 Q.103 ABD A Q.7 A D Q.104 BD Q.108 (A) S; (B) R; (C) S; (D) Q Q.109 (A) Q; (B) S; (C) P; (D) R Q.110 (A) R; (B) P; (C) S; (D) R Q.111 (A) S; (B) R; (C) R; (D) Q Q.112 (A) S; (B) R; (C) P; (D) Q, R Q.113 (A) Q; (B) R; (C) S; (D) P Q.114 (A) S; (B) P; (C) Q Q.115 (A) Q; (B) S; (C) Q Q.116 (A) Q; (B) P; (C) P, R; (D) P, R, S 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)