School of Distance Education UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION. B Sc Mathematics. (2011 Admission Onwards) IV Semester.

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1 School of Dtance Education UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc Mathematics 0 Admsion Onwards IV Semester Core Course CALCULUS AND ANALYTIC GEOMETRY QUESTION BANK The natural logarithm of a positive number o The natural logarithm function not defined for x 0 x > 0 x 3 In =? In a In x 4 Choose the correct one In x an increasing function of x In x a constant function of x 5 The solution of the integral not defined In 5 - x > In a In x In x a decreasing function of x All the above In In 5

2 School of Dtance Education 6 The derivative of y with respect to the given independent variable y = 7 The value of e can be computed using the formula e = e = 9 The value of 0 The Solution of ] y = 4 [ 8 The value of In ln = =0 All the above for the value of y ] y = - [ y = 4 [ ] y = 4 [ ] [3cos 5θ - θcos 5θ 5θ sin 5θ] [3cos 5θ θcos 5θ 5θ sin 5θ] [3cos 5θ - θcos 5θ 5θ sin θ] [cos 5θ - θcos 5θ 5θ sin θ] The solution of the integral 3 3 The value of 3 cos 5 with respect to θ The derivative of y = e = = The derivative of 3sin x with respect of x 5 3

3 School of Dtance Education sin x 3sin x 3sin x In 3 cos x 3Sin x cos x sin x In 3 cos x 5 The second derivative of ax with respect to x ax In a 6 The derivative of xx with respect to x x x- xxin x 7 The value of sinx c cos 8 The value of x 9 The value of xx In x dx In x xx sinx In c ax In a ax a a/x 0 We invest an amount A0 of money at a fixed annual interest rate r and if interest added to our account k time a year, then the amount of money we will get at the end of t year At = A0 ert At = A0 K rrt At = A0 k/rrk At = A0 r/kkt Suppose you deposit 6,00/- in a bank account that pays 6% interest compounded continuously How much money will you get after 8 years 600e e8/ /006 The no of radio active Polonium-00 atoms remaining after t days in a sample that starts with y0 atoms given by the Polonium decay equation y =,then the Polonium-00 half-life In The value of 0 0-3

4 4 The value of ½ 5 The value of 0 0 School of Dtance Education 3 /3 ¼ /6 ½ not ext 6 If a any positive real number, then 7 The value of ½ 0 8 The value of 9 The value of 0 0 e/4 /3 sin ½ e/ 30 The hyperbolic tangent tanhx equal to 3 In hyperbolic function coshx-y equal to coshx coshy - sinhx sinhy coshx coshy sinhx sinhy 3 If sinhx = -3/4 then tanhx 3/5-3/5 33 The derivative of tanh sech 34 The value of 3 In 3 In 4 sin hx dx ½ ¼ /5 3/ e coshx sinhx coshy sinhy coshx sinhx coshy sinhy 5/4-5/4 x sech sech x 4 In 4 - In 4

5 School of Dtance Education 35 The derivative of cosh-x with respect to x 36 The value of sinh- sinh The nth term of a sequence 4/5 5/6 sinh sinh-, then 4th term 7/8 3/3 38 The first few terms of a sequence {un} 5, 7, 9, By assuming natural pattern, the formula for the nth term n n n n 39 Given u =, u = and un = 3 the u The formula for the nth term of a sequence, -/4, /9, -/6, /5, 4 The sequence {un} when un = Converges to as n Converges to as n 4 The sequence {un} when un = Converges to as n Converges to as n 43 The sequence {un} when un = Converges to e as n Converges to e/ as n Converges to 3 as n Diverges Converges to 3 as n Diverges Converges to e as n Diverges 5

6 School of Dtance Education 44 The sequence ½, /3, ¾,, Converges to Converges to 45 The series Converges to 0 Converges to / Converges to 0 Diverges Converges to Diverges 46 The geometric series a ar ar arn- converges if r > Converges if r = r < diverges of diverges of r 47 The series 48 The series Converges to 0 Converges to Converges if p< Converges if p Converges to Diverges Converges if p = Diverges if p> 49 In D Alembert s Ratio-Test, if u a series with positive terms, and if = =l, then n u convergent when l < = u divergent when l < = 50 The series converges 5 The series 5 Converges diverges diverges u convergent when l = u divergent when l = oscillatory none of these oscillatory none of these In Cauchy s nth root test, if un a series with non-negative terms such that = l then un n converges if l > converge if l < converges if l = diverges 6

7 School of Dtance Education 53 The series diverges converges 54 The series - converges absolutely oscillatory 55 The series oscillatory oscillatory none of these divergent convergent p > 0 convergent divergent none of these /x /x 56 The sum of the power series x x x3 xn x < 57 The sum of the power series - 3/x 58 The second derivative of the power series x x x3 xn x 3x 4x3 6x x 0x3 59 The sum of the power series x sin-x cos-x 0-3 x -3 < x < 6 The series sin x h cos x sin x h 63 log[ /x] equal to sin x sin x h cot-x 6The interval of the convergence of the series 4/x tan-x n = 3-3 < x -3 x < cos x 3x 4x 5x3 x x x3 =0 60 The radius of convergence of the series cos x-h cos x h 7

8 64 The series x sinx 65 esin x equal to 3 3 School of Dtance Education 4 4 equal to cosx 4 sin-x 3 tanx cotx 8 66 The series cos-x 67 The approximate value of e equal to equal to tan-x sec-x For any real number θ, eiθ equal to sinθ i cosθ cosθ i sinθ sinθ i cosθ cosθ i sinθ 69 The standard form of the circle of radius a centered at the point h, k x h y k = a x h y k = a x h y k = a x h y h = a 70 A set that const of all the points in a plane equidtant from a given fixed point and a given fixed line in the plane a parabol The fixed point focus directrix centre radious 7 The directrix of the parabola x = -6y y / = 0 y 6 = 0 7 The centre to-focus dtance of 7 73 The eccentricity of the ellipse e = e = y 3/ = 0 y 3/ = 0 5 = = a>b e = e = 8

9 School of Dtance Education 74 The vertices of an ellipse of eccentricity 08 whose foci lie at the point 0, ±4 0, ±5 0, 5 0, -5 5, 0 75 The eccentricity of the hyperbola 9x -6y = /4 4/ The Cartesian equation for the hyperbola centered at the origin that has a focus at 3, 0 and the line x = as the corresponding directrix = = = = 77 The x and y axes are rotated through an angle of π/4 radians about the origin Then the equation for the hyperbola xy = 9 in the new coordinates = = = = 78 The quadratic curve 4x 8xy 4y 5x 3 = 0 represents hyperbola parabola ellipse 79 The quadratic curve x xy y = 0 represents hyperbola parabola circle circle ellipse 80The quadratic curve 3xy y 5y = 0 represents hyperbola parabola circle ellipse 8 The tangent to the right-hand hyperbola branch x = sec t, y tan t -π/<t<π/ at the point, where t = π/4 y = x y = x y = y = 8 If x = t t3 and y = t t then dy/dx 83 The length of the arc of the curve x = a sin t cos t, y = a cos t cos t measured from the origin to any point ¾ a cos 3t ¾ a sin 4t 4/3 a sin 3t 4/3 a cos 3t 9

10 School of Dtance Education 84 The centroid of the first quadrant arc of the asteroid x = cos3t, y = sin3 t, 0 t π /5, 3/5 3/5, /5 3/5, 3/5 /5, /5 85 If a smooth curve x = ft, y = gt, a t b ; traversed exactly once as t increases from a to b, then the area of the surface generated by revolving the curve about the x-ax y 0 s = dt s = dt s = s = dt dt 86 The Cartesian equation equivalent to the polar equation r cos θ π/4 = x y = x y = x y = x y = 87 The polar equivalent of the curve whose Cartesian equation x y = r cosθ = r sinθ = r cosθ = r sinθ = 88The angle between the lines whose equations are d = r cos θ - and d = r cos θ - θ = ± θ = ± θ = - ± θ = - r 89 The equation of the circles passing through origin and having radius 3 and centre 3, 0 r = 3 cosθ r = 4 sinθ r = 5 sinθ r = 6 cosθ 90 The equation r = represents an ellipse if 0 < e < e = e = 0 e > x = /5 x = 5/ x = /5 x = 5/3 9 The equation of the directrix of the parabola r = 9 The polar equation of an ellipse with eccentricity e and semi major ax a r = r = 93 The area of the curve r = a b coseθ, a > b r = r = r = a 0

11 School of Dtance Education 94 The area of a loop of the curve r = a sin3θ π/ πa/ πa/4 πa/ 95 The area of the region that lies inside the circle r = and outside the Cardioid r = cos θ - and if the point pr, θ traces the curve r = f θ exactly once as θ runs from to β, then the length of the curve L = L = L = L = 97 The length of the perimeter of the cardioid r = a cosθ 6a 7a 8a 98 In an equiangular spiral r = aeθ cot, we have θ cot = log θ cot = log θ cot = log θ cot = log 99 The perimeter of the cardioids r = 4 cos θ a 3 00The area of the surface generated by revolving the right hand loop of the lemncates r = cos θ about the y ax π 4 4π π3

12 School of Dtance Education ANSWER KEYS c 4 d 47 d 3 c 6 b 49 a a 4 a 5 d 6 b 7 d 8 a 9 b 0 c a b 3a 4 c 5 d 6 b 7 c 8 a 9 b 0 d a b 3 c 5 a 7 a 8 c 9 d 30 b 3 a 3 b 33 c 34 a 35 b 36 d 37 b 38 c 39 d 40 b 4 a 4 d 43 b 44 a 45 b 46 d 48 c 50 a 5 a 5 c 53 b 54 d 55 b 56 a 57 b 58 c 59 d 60 a 6 c 6 b 63 a 64 c 65 d 66 a 67 b 68 c 69 b

13 70 a 7 d 7 b 73 c 74 a 75 b 76 d 77 a 78 b 79 d 80 a School of Dtance Education 8 a 9 a 83 c 94 d 8 b 93 c 84 d 95 b 85 a 96 a 86 b 97 c 87 a 98 c 88 c 99 d 89 d 00a 90 a 9 b Reserved 3