# FINALTERM EXAMINATION 2011 Calculus &. Analytical Geometry-I

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1 FINALTERM EXAMINATION 011 Clculus &. Anlyticl Geometry-I Question No: 1 { Mrks: 1 ) - Plese choose one If f is twice differentible function t sttionry point x 0 x 0 nd f ''(x 0 ) > 0 then f hs reltive... At Minim Mxim None of these Question No: { Mrks: 1 ) - Plese choose one If f is twice differentible function t sttionry point x 0 x 0 nd f ''(x 0 ) < 0 then f hs reltive... At Minim Mxim None of these Question No: 3 { Mrks: 1 ) lim sin x = x 0 x - Plese choose one 4

2 1 Question No: 4 { Mrks: 1 ) - Plese choose one lim ln x x x = 1 o e None of these Question No: 5 { Mrks: 1 ) - Plese choose one d (tn x) = dx sec x sec x co sec x co sec x Question No: 6 { Mrks: 1 ) - Plese choose one If xy = 4 o 1 x 4 x 4 x then dy dx =

3 Question No: 7 { Mrks: 1 ) - Plese choose one Consider function h(x) nd constnt c then

4 d ((c) {h(x)}) = dx o d (h(x)) dx d dx (h(cx)) c d (h(x)) dx Question No: 8 { Mrks: 1 ) - Plese choose one Suppose tht f nd g re differentible functions of x then d [ f ][g] = dx [ f ][g] [ f ][g ] g [ f ][g ] [ f ][ g ] + [ f ][ g ] [ f ][g ] [ f ][g ] Question No: 9 { Mrks: 1 ) - Plese choose one The power rule, d [x n ] = nx n 1 dx holds if n is An integer A rtionl number An irrtionl number All of the bove

5 Question No: 10 { Mrks: 1 ) - Plese choose one Let function f be defined on n intervl, nd let x 1 nd x denotes two distinct points in tht intervl. If f (x 1 ) = f (x ) for ll points x 1 nd x then

6 which of the following sttement is correct? f is decresing function f is n incresing function f is constnt function Question No: 11 { Mrks: 1 ) - Plese choose one f (x) < 0 If on n open intervl (,b) then which of the following sttement is correct? f is concve up on (, b). f is concve down on (, b) f is liner on (, b). Question No: 1 { Mrks: 1 ) Wht does 'n' represent in Riemnn Sum No. of Circles No. of Rectngles No. of Loops No. of Squres Question No: 13 { Mrks: 1 ) f If is continuous function such tht f (, + ) then hs on mximum vlue but no minimum minimum vlue but no mximum both mximum nd minimum vlue - Plese choose one n f (x * ) x k k L. k =1 - Plese choose one lim x? f (x) = + nd lim x + f (x) = + Question No: 14 { Mrks: 1 ) The expression t f x t - Plese choose one

7 x Both t nd x dx, represents function of :

8 Question No: 15 { Mrks: 1 ) f cf (x)dx = - Plese choose one if c is constnt o c f f (cx)dx cf f (x)dx Question No: 16 { Mrks: 1 ) - Plese choose one Sigm nottion is represented by which of the following Greek letter? χ η Σ ψ Question No: 17 { Mrks: 1 ) - Plese choose one In the following figure, the re enclosed is bounded below by : y = x + 6 y = x

9 x =

10 x = 0 Question No: 18 { Mrks: 1 ) - Plese choose one In the following figure, the re bounded on the sides by the lines re : x = 0 x = x = 0 nd x = x = 6 Question No: 19 { Mrks: 1 ) - Plese choose one Wht is the re of the region in the following figure? A = l( x + 6) ( x )l dx f L 0 J

11 L A = f l ( x + 6) ( x )l dx x J

12 0 A = f L x A = f l ( x + 6) ( x )l dx 0 L J Question No: 0 { Mrks: 1 ) - Plese choose one Which of the following is pproximte re under the curve over the intervl [, 4], evluted by using the formul y = f (x) = 3x + 1 * * Are = f (x 1 ) x + f (x ) x If the intervl [, 4] is divided into two sub-intervls of equl * * length nd x1 nd x re left endpoint of ech subintervl. 17 o 3 Question No: 1 { Mrks: 1 ) - Plese choose one Which of the following is pproximte re under the curve [0, ] over the intervl, evluted by using the formul y = f (x) = x + 3 * * Are = f (x 1 ) x + f (x ) x [0, ] If the intervl * is divided into two sub-intervls of equl * length nd x 1 nd x re right endpoint of ech sub-intervl. 8 1o 1

13 stion No: { Mrks: 1 ) - Plese choose one If x > 0 then d [ln x] = dx Que

14 1 x 1 x ln 1 x Question No: 3 { Mrks: 1 ) - Plese choose one Suppose f nd g re integrble functions on [,b] nd c is constnt, then b f c [ f (x) + g(x)] dx = b f f (cx)dx + f g(cx)dx b b b f f (x) dx + f b g(x)] dx c f f (x)dx + c f g(x)dx b o Question No: 4 { Mrks: 1 ) - Plese choose one If the function f is continuous on [,b] nd if f (x) 0 for ll x in [,b], then which of the following gives re under the curve y = f (x) over the intervl [,b]? n lim L.[x k ][ f (x k )] where n is x k =1 π [rdius] number of subdivisions of b f f (x) dx

15 (Width) (Height) [, b]

16 Question No: 5 { Mrks: 1 ) - Plese choose one y = 3x nd y = x Let region R in the first qudrnt enclosed between is revolved bout the x-xis.which of the following eqution gives the volume of solid by cylindricl shells? 3 V = f π x(3x x )dx 0 3 V = f x(3x x )dx 0 3 V = f π (3x x )dx 0 3 V = f π (3x x )dx 1 Question No: 6 { Mrks: 1 ) - Plese choose one Let f is smooth function on [, b]. Wht will be the rc length L of the curve y = f(x) from x = to x = b? L = f b b L = f

17 1 + [ f '(x)]dy 1 + [ f '(x)] dx

18 L = f 0 b L = f 1 + [ f '(x)]dy 1 + [ f '(x)]dx Question No: 7 { Mrks: 1 ) - Plese choose one If f is continuous on (, b] but does not hve limit from the right then the b b f (x)dx = lim f (x)dx f f l l integrl defined by is clled : Improper Proper Line Question No: 8 { Mrks: 1 ) - Plese choose one n+1 > 1 n For sequence is known s: { n } if the rtio of successive terms then the sequence Incresing Decresing Nondecresing Nonincresing Question No: 9 { Mrks: 1 ) - Plese choose one n+1 < 1 n For sequence is known s: { n }

19 if the rtio of successive terms the n the seq uen ce Incresing Decresing Nondecresing Nonincresing

20 Question No: 30 { Mrks: 1 ) - Plese choose one 3x + 4x + 1 f x 3 + x + x 3 dx Consider the indefinite integrl t = x 3 + x + x 3 Let Is the following substitution correct? 3x + 4x f x 3 + x + x 3 dx = f t dt Yes No Question No: 31 { Mrks: 1 ) - Plese choose one L.u k The series be series with positive terms nd suppose tht ρ = 1 if, then which of the following is true? ρ = lim u k +1 k u k Converges Diverges My converges or diverges Gives no informtion Question No: 3 { Mrks: 1 ) - Plese choose one L.u k The series be series with positive terms nd suppose tht 1 ρ = lim k u k = lim(u k ) k k k ρ = 1 if, then which of the following is true?

21 Converges Diverges My converges or diverges Gives no informtion Question No: 33 { Mrks: 1 ) - Plese choose one

22 If the series the following is true for L. u k = u 1 + u + u u k +... k =1 L.u k = u 1 + u + u +...u k +... k =1 converges, then which of? Converges Diverges Gives no informtion Question No: 34 { Mrks: 1 ) - Plese choose one Let L. u k ρ = + if be series with nonzero terms nd suppose tht, then which of the following is true? lim u k +1 ρ = k u k Then the series The series L.u k L.uk diverges My converges or diverges Gives no informtion converges bsolutely nd therefore converges Question No: 35 { Mrks: 1 ) - Plese choose one 1 f (x 1) dx = 1 - o

23 4 Question No: 36 { Mrks: 1 ) - Plese choose one f How mny criticl points exist for function if

24 f '(x) = (x 3)(x ) Zero One Two Four Question No: 37 { Mrks: 1 ) - Plese choose one log b c = log b + log b c log b log b c log b log b c (log b )(log b c) Question No: 38 { Mrks: 1 ) - Plese choose one log b r = log b r r log b log b log b r log b + log b r

25 Question No: 39 { Mrks: 1 ) - Plese choose one 3 y = x ; 0 x 3 Let then which of the following is the length of the curve?

26 l d L = f I x dx 0 IL dx 3 ) J 3 l L = f l d 3 l 1+ x dx I I L dx 3 )J 3 l l d L = f 1+ I x dx 0 IL dx 3 ) J l L = d f 1+ I 0 IL dx 3 3 l x ) J dx Question No: 40 { Mrks: 1 ) - Plese choose one Which of the following re first two terms for the Tylor series of = o? 1+ (1)( x 0) 1+ ( 1)( x + 0) 1+ ( 1)( x 0) ( 1)( x 0) f (x) = e x t x Question No: 41 { Mrks: ) 3 f (1 x)dx Evlute the integrl

27 3 f (1 x)dx = x x 1 3 = 1 x x 3 = 1 ((3 ) (3 ) ) = 1 ( 1) = 1 Question No: 4 { Mrks: ) Evlute the improper integrl + dx f x Question No: 43 { Mrks: ) f (x) = x 4x 9 A function hs criticl point in n intervl [o, 5]. Find the mximum vlue of the function nd point hving this vlue. Question No: 44 { Mrks: 3 ) 5 6 sin x f dx sin x Evlute: 5 6 sin x f dx sin x Question No: 45 { Mrks: 3 ) Find the re of the region bounded by the curve y = x on the sides by the lines y = 1 nd y = 4, x > 0, nd bounded y = x, x > 0 So we hve

28 4 A = f 1 x dx = x3 L = 1 (4 1) 3 3 = 1 (3) 3 3 = 9 Question No: 46 { Mrks: 3 ) Determine whether the following sequence converges or diverges. If it converges, find the limit. 5n 1 lim n 0n + 7n Question No: 47 { Mrks: 5 ) Use the Alternting series Test to determine whether the given series converges ( 1) n 1. n! 1 n Question No: 48 { Mrks: 5 ) Evlute the integrl 0 1+ cos t dt f π Solution

29 0 1+ cos t f dt π u = t du = dt dt du = dt so 1 0 = 4 f 1+ cos udu π = 1 u + sin u 0 π 4 = 1 t + sin t 0 π 4 = 1 ( π + sin π ) 4 = 1 (π + sin π ) 4 = 1 (π + 0) 4 = π 4 Question No: 49 { Mrks: 5 ) Evlute the sums 5 L. k (3k + 5) k =1 = 1(3 + 5) + (6 + 5) + 3(9 + 5) + 4(1 + 5) + 5(15 + 5) = (45) + 4(60) + 5(75) = = 780 Question No: 50 { Mrks: 10 ) Find the volume of the solid tht results when the region enclosed by the given

30 curves is revolved bout the x - xis.

31 y = 1+ x 3, x = 1, x =, y = 0 b from V= f π [ f (x)] dx V= f π [1+ x3 ] dx 1 V= f π [1+ x5 + x 3 ]dx 1 V=π f (1+ x5 + x 3 )dx 1 V=π (x + 1 x x 4 ) 6 V=π (( 1) + 1 ( 1) ( 1) 4 ) 6 V=π {(( 1) + 1 ( 1) ( 1) 4 )} 6 V=π ((1) )) 6 V= π ( ) 6 V= π (10) = π

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