School of Distance Education Calculus & Analytic Geometry 1 UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc Mathematics (2011 Admission Onwards) IV Semester Core Course CALCULUS AND ANALYTIC GEOMETRY QUESTION BANK 1. The natural logarithm of a positive number 1 is a.1 b. 2 c. o d. -1 2. The natural logarithm function is not defined for a. x ≤ 0 b. x > 0 c. x ≤ 1 d. x > 1 3. In = ? a. b. In a – In x c. d. In a + In x 4. Choose the correct one a. In x is an increasing function of x b. In x is a decreasing function of x c. In x is a constant function of x c. All the above. 5. The solution of the integral ∫ is a. not defined b. In 5 c. In 1 d. –In 5
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School of Distance Education
Calculus & Analytic Geometry 1
UNIVERSITY OF CALICUT
SCHOOL OF DISTANCE EDUCATION
B Sc Mathematics
(2011 Admission Onwards)
IV Semester
Core Course
CALCULUS AND ANALYTIC GEOMETRY
QUESTION BANK
1. The natural logarithm of a positive number 1 isa.1 b. 2 c. o d. -12. The natural logarithm function is not defined fora. x ≤ 0 b. x > 0 c. x ≤ 1 d. x > 13. In = ?a. b. In a – In x c. d. In a + In x4. Choose the correct onea. In x is an increasing function of x b. In x is a decreasing function of xc. In x is a constant function of x c. All the above.5. The solution of the integral ∫ isa. not defined b. In 5 c. In 1 d. –In 5
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6. The derivative of y with respect to the given independent variable y = = isa. ( ) ( ) b. ( ) ( )c. ( )( ) ( ) d. ( )( ) ( )7. The value of ‘e’ can be computed using the formulaa. e = → ∞ ! + !+ !+. . . . + ! b. e = → ∞ ∑= 0 !c. e = → ∞ 2 + + + . . . . + ! d. All the above
8. The value of In √ isa. b. √ c. d. √9. The value of isa. ln b. 2 c. d.10. The Solution of √ = for the value of y isa. y = 4 [ ] b. y = -2 [ ] c. y = 4 [ ] d. y = 4 [ ]11. The derivative of y = cos 5 with respect to θ isa. [3cos 5θ - 2θcos 5θ – 5θ sin 5θ]b. [3cos 5θ + 2θcos 5θ + 5θ sin 5θ]c. [3cos 5θ - 2θcos 5θ – 5θ sin θ]d. [cos 5θ - 2θcos 5θ – 5θ sin θ]12. The solution of the integral ∫(2 + 3 ) isa. 2 + 3 b. 2 −c. − d. 2 +13. The value of √ isa. 3√ b. 5√ c. √5 d. √3√14. The derivative of 3sin x with respect of x is
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Calculus & Analytic Geometry 3
a. sin x 3sin x – 1 b. 3Sin x cos xc. 3sin x (In 3) cos x d. sin x (In 3) cos x15. The second derivative of ax with respect to x isa. ax In a b. c. ( ) d. (In a)2 ax16. The derivative of xx with respect to x isa. x x-1 b. xx(1+In x) c. xx In x d. In x (1+xx)17. The value of ∫ 2 cos isa. 2sinx + c b. 2sinx In 2 + cc. + d. +18. The value of isa. x b. ax c. a d. a/x19. The value of ∫ 2 dx isa. ( ) + b. ( ) +c. + d. 2 +20. We invest an amount ‘A0’ of money at a fixed annual interest rate ‘r’ and if interest isadded to our account ‘k’ time a year, then the amount of money we will get at the endof ‘t’ year isa. At = A0 ert b. At = A0 (K +r)rtc. At = A0 (k/r)rk d. At = A0 (1 + r/k)kt21. Suppose you deposit 62,100/- in a bank account that pays 6% interest compoundedcontinuously. How much money will you get after 8 years.a. 62100e(0.06)(8) b. 62100e(8/6)c. 62100 × 8 × 0.06 d. 62100 × 8/0.0622. The no. of radio active Polonium-200 atoms remaining after t days in a sample thatstarts with y0 atoms is given by the Polonium decay equation y = × ,then thePolonium-200 half-life isa. In 2×5×10-3 b. ×c. × d. ×23. The value of → 0 isa. 0 b. -1 c.1 d.2
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24. The value of → 0 − 3 isa. ½ b. 1/3 c. ¼ d. 1/625. The value of → ∞ isa. 0 b. 1 c. ½ d. not exist26. If ‘a’ is any positive real number, then → ∞a.∞ b.0 c. 1 d. ½27. The value of → 1 − isa. ½ b. 1/3 c. ¼ d.1/528. The value of → (sin ) isa. 0 b. ½ c. 1 d. 3/229. The value of → 0 (1 + ) isa. e/4 b. e/2 c. √ d. e30. The hyperbolic tangent tanhx is equal toa. b.
c. ( − )( + ) d. −31. In hyperbolic function cosh(x-y) equal toa. coshx coshy - sinhx sinhy b. coshx sinhx – coshy sinhyc. coshx coshy + sinhx sinhy d. coshx sinhx + coshy sinhy32. If sinhx = -3/4 then tanhx isa. 3/5 b. -3/5 c. 5/4 d. -5/433. The derivative of tanh√1 + isa. √ b. x sech2 √1 +c. √ sech2 √1 + 2 d. √ sech2x
34. The value of ∫ 4 sin hx dx isa. 3 – 2 In 2 b. 4 + 2 In 2c. 3 + 2 In 2 d. 4 - 2 In 2
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35. The derivative of cosh-1(x2) with respect to x isa. √ b. √c. √ d. √36. The value of ∫ 23+ 4 2 isa. sinh-1 √ b. sinh √c. sinh √ d. sinh-1 √37. The nth term of a sequence is , then 4th term isa. 14/15 b. 15/16 c. 7/8 d. 31/32
38. The first few terms of a sequence {un} is 5, 7, 9, 11 ……. By assuming natural pattern,the formula for the nth term isa. n+2 b. (2n + 2) c. 2n +1 d. 2n39. Given u1 = 1, u2 = 2 and un + 2 = the u5 isa. 3 b. 5 c. -5 d. 440. The formula for the nth term of a sequence 1, -1/4, 1/9, -1/16, 1/25, …….. isa. ( ) b. ( )c. ( )( ) d. ( )41. The sequence {un} when un = isa. Converges to 2 as n →∞ b. Converges to 3 as n →∞c. Converges to 1 as n →∞ d. Diverges
42. The sequence {un} when un = isa. Converges to 2 as n →∞ b. Converges to 3 as n →∞c. Converges to 1 as n →∞ d. Diverges43. The sequence {un} when un = isa. Converges to 2e as n →∞ b. Converges to e2 as n →∞c. Converges to e/2 as n →∞ d. Diverges
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44. The sequence ½, 2/3, ¾, ……, isa. Converges to 1 b. Converges to 0c. Converges to 2 d. Diverges45. The series . + . + . +⋯+ ( )+ …. isa. Converges to 0 b. Converges to 1c. Converges to 1/2 d. Diverges46. The geometric series a+ ar+ ar2 + …… + arn-1 + …….a. converges if ∣r∣>∣ b. Converges if r = 1c. diverges of ∣r∣<∣ d. diverges of ∣r∣≥∣47. The series + + +⋯+ + …. isa. Converges to 0 b. Converges to 1c. Converges to 2 d. Diverges48. The series + + +⋯+ + …. isa. Converges if p<1 b. Converges if p = 1c. Converges if p ≤1 d. Diverges if p> 149. In D’Alembert’s Ratio-Test, if ∞∑= 1u is a series with positive terms, and if
n → ∞ = l , thena. ∞∑= 1u is convergent when l < 1 b. ∞∑= 1u is convergent when l ≤ 1c. ∞∑= 1u is divergent when l < 1 d. ∞∑= 1u is divergent when l ≥ 1
50. The series . . + . . + . . +⋯ isa. converges b. diverges c. oscillatory d. none of these51. The series . ! + .! + . ! +⋯ isa. Converges b. diverges c. oscillatory d. none of these52. In Cauchy’s nth root test, if ∑ un is a series with non-negative terms such thatn → ∞ = l then ∑una. converges if l > 1 b. converges if l = 1c. converge if l < 1 d. diverges
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53. The series ∑ isa. diverges b. converges c. oscillatory d. none of these54. The series 1 - + − + −⋯+ ( ) + …. isa. converges absolutely b. divergentc. oscillatory d. convergent55. The series − + − +.… (p > 0) isa. oscillatory b. convergent c. divergent d. none of these56. The sum of the power series 1 + x + x2 + x3 + ……+ xn + …….. ∣x∣<1 isa. b. ( ) c. ( ) d. ( )57. The sum of the power series 1 - ( − 2) + ( − 2) + ………+ ( − 2) + … isa. 1/x b. 2/x c. 3/x d.4/x58. The second derivative of the power series 1 + x + x2 + x3 + …..+ xn isa. 1 + 2x + 3x2 + 4x3 + …… b. 1 + 3x + 4x2 +5x3 + ……c. 2 + 6x + 12x2 + 20x3 + ……. d. 1 + 2x + 12x2+ 22x3 + ……59. The sum of the power series x - + − +⋯ isa. sin-1x b. cos-1x c. cot-1x d. tan-1x60. The radius of convergence of the series ∞∑= 0 ! isa. 0 b. 1 c. 2 d. 361.The interval of the convergence of the series ∞∑= 1(−1) (n + 1) ( ) isa. -3 ≤ x ≤ 1 b. -3 < x ≤1c. -3 < x < 1 d. -3 ≤ x < 162. The series sin x + h .cos x – !sin x – ! cos x + ……isa. sin (x – h) b. sin (x + h) c. cos (x-h) d. cos (x +h)63. log[1 + 1/x] is equal toa. − + − + −⋯ b. − + − +⋯
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c. 1− 1 + 12 2− 13 3+ 14 4−⋯ d. 1− 12+ 12 3− 13 4+ 14 5−⋯64. The series x+ + +⋯ is equal toa. sinx b. cosx c. tanx d. cotx65. esin x is equal toa. 1 + + + +⋯ b. 1− 22 − 36 − 412+⋯c. 1− + 22 − 48 +⋯ d. 1+ + 22 − 48 +⋯66. The series + . + .. . + . .. . . + ⋯ is equal toa. sin-1x b. cos-1x c. tan-1x d. sec-1x67. The approximate value of e is equal toa. 2.708282……. b. 2.718282c. 2.728282 d. 2.69828268. For any real number θ, eiθ is equal toa. sinθ + i cosθ b. sinθ – i cosθc. cosθ + i sinθ d. cosθ – i sinθ69. The standard form of the circle of radius ‘a’ centered at the point (h, k)isa. (x – h)2 + (y + k)2 = a2 b. (x – h)2 + (y – k)2 = a2c. (x + h)2 + (y – k)2 = a2 c. (x + h)2 + (y + h)2 = a270. A set that consist of all the points in a plane equidistant from a given fixed point and agiven fixed line in the plane is a parabola. The fixed point is.a. focus b. directrix c. centre d. radious71. The directrix of the parabola x2 = -6y isa. y +1/2 = 0 b. y – 6 = 0 c. y + 3/2 = 0 d. y – 3/2 = 072. The centre –to-focus distance of − = 1 isa. √1 b. √7 c. √25 d. √1273. The eccentricity of the ellipse + = 1 (a>b) is
a. e = √ b. e = √c. e = √ d. e = √
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74. The vertices of an ellipse of eccentricity 0.8 whose foci lie at the point (0, ±4) isa. (0, ±5) b. (0, 5) c. (0, -5) d. (5, 0)75. The eccentricity of the hyperbola 9x2 -16y2 = 144 isa. 4 b. 5/4 c. 4/5 d. 576. The Cartesian equation for the hyperbola centered at the origin that has a focus at(3, 0) and the line x = 1 as the corresponding directrix isa. + = 1 b. − = 1c. + = 1 d. − = 177. The x and y axes are rotated through an angle of π/4 radians about the origin. Thenthe equation for the hyperbola 2xy = 9 in the new coordinates isa. − = 1 b. + = 1c. − = 1 d. + = 178. The quadratic curve 4x2 – 8xy + 4y2 + 5x – 3 = 0 representsa. hyperbola b. parabola c. ellipse d. circle79. The quadratic curve 2x2 + xy + y2 – 1 = 0 representsa. hyperbola b. parabola c. circle d. ellipse80.The quadratic curve 3xy – y2 – 5y + 1 = 0 representsa. hyperbola b. parabola c. circle d. ellipse81. The tangent to the right-hand hyperbola branch x = sec t, y – tan t -π/2<t<π/2 at thepoint (√2 ,1) where t = π/4 isa. y = √2 x – 1 b. y = √2 x + 1c. y = √2 – 1 d. y = √2 + 1
82. If x = 2t – t3 and y = t – t2 then d2y/dx2 isa. ( ) b. ( )c. ( ) d. ( )83. The length of the arc of the curve x = a sin 2t (1 + cos 2t), y = a cos 2t (1 – cos 2t)measured from the origin to any point isa. ¾ a cos 3t b. ¾ a sin 4t c. 4/3 a sin 3t d. 4/3 a cos 3t
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84. The centroid of the first quadrant arc of the asteroid x = cos3t, y = sin3 t, 0 ≤ t ≤ 2π isa. (2/5, 3/5) b. (3/5, 2/5) c. (3/5, 3/5) d. (2/5, 2/5)85. If a smooth curve x = f(t), y = g(t), a ≤ t ≤ b ; is traversed exactly once as t increases froma to b, then the area of the surface generated by revolving the curve about the x-axis(y ≥ 0)isa. s = ∫ 2 + dt b. s = ∫ 2 + dt
c. s = ∫ + dt d. s = ∫ 2 − dt
86. The Cartesian equation equivalent to the polar equation r cos (θ – π/4) = √2 isa. x – y = 2 b. x + y = 2 c. x – y = √2 d. x + y = √287. The polar equivalent of the curve whose Cartesian equation is x2 – y2 = 1 isa. r2 cos2θ = 1 b. r2 sin2θ = 1 c. r cos2θ = 1 d. r sin2θ = 188.The angle between the lines whose equations are d = r cos (θ - ∝) and d1 = r cos (θ - ∝1) isa.θ=(∝1 ± ∝) b.θ=(∝±∝1) c. θ = -(∝1 ± ∝) d. θ = (∝1 - r)89. The equation of the circles passing through origin and having radius 3 and centre (3, 0) isa. r = 3 cosθ b. r = 4 sinθ c. r = 5 sinθ d. r = 6 cosθ90. The equation r = represents an ellipse ifa. 0 < e < 1 b. e = 1 c. e = 0 d. e > 191. The equation of the directrix of the parabola r =a. x = 2/5 b. x = 5/2 c. x = 1/5 d. x = 5/392. The polar equation of an ellipse with eccentricity e and semi major axis ‘a’ isa. r = ( ) b. r = ( )
c. r = ( ) d. r = ( )93. The area of the curve r = a + b coseθ, a > b isa. r = b. a2 +c. + d.
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94. The area of a loop of the curve r = a sin3θ isa. π/12 b. πa/12 c. πa2/4 d. πa2/1295. The area of the region that lies inside the circle r = 1 and outside the Cardioidr = 1 – cos θ isa. 2 + b. 2 - c. d.and if the point p(r, θ) traces thecurver=f(θ)exactlyonceasθrunsfrom∝toβ,thenthelengthofthecurveisa. L = ∫ + b. L = ∫ −
c. L = ∫ 2 + d. L = ∫ 2 −97. The length of the perimeter of the cardioid r = a(1 – cosθ) isa. 6a b. 7a c. 8a d. 9a98. In an equiangular spiral r = aeθcot∝, we havea.θcot∝=log ∝ −1 b.θcot∝=log ∝ −1
c.θcot∝=log ∝ +1 d.θcot∝=log ∝ +199. The perimeter of the cardioids r = 4(1 – cos θ) isa. 24 b.28 c.30 d. 32100.The area of the surface generated by revolving the right – hand loop of the lemniscatesr2 = cos 2θ about the y – axis isa. 2π√2 b. 4√ c.4π d.2π√3
