WK Reg No : Question Paper Code : 78 BE/BTech DEGREE EXAMINATION, JANUARY 4 First Semester Marine Engineering MA 65 MATHEMATICS FOR MARINE ENGINEERING I (Regulation ) Time : Three hours Maimum : marks Answer ALL questions PART A ( = marks) Find the equation of the sphere having the points (,,4) and (,, ) ends of diameter Define right circular cylinder log Differentiate ( ) y= sin 4 State Lagrange s mean value theorem dy 5 If + y+ 6y + y =, find d 6 Find the critical points of 7 Evaluate tan d f (, y) = y + 8 State the theorem of perpendicular ais as 9 Change the order of integration in (, y) f dyd Sketch the region of integration for the double integral (, y) f dyd
WK PART B (5 6 = 8 marks) (a) (i) Find the equation of sphere having its centre on the plane 4 5y z = and passing through the circle + y + z y+ 4z+ 8=, + y + z + 4+ 5y 6z+ = (8) Find the equation of the right circular cylinder whose ais is y z = = and radius (8) (b) (i) Show that the plane y+ z+ = touches the sphere (a) (i) Find d + y + z 4y+ z = and find the point of contact (8) Find the equation to the right circular cone whose verte is at (,,5), ais makes equal angles with the coordinate aes and semi-vertical angle is (8) If dy if ( )( ) = ( )( 4) a sin (b) (i) Epand tan y (8) y= e prove that ( ) y + ( n+ ) yn+ ( n + a ) y= (8) Evaluate n in powers of as far as the terms containing ( + ) 5 (8) e lim (8) (a) (i) A rectangular bo open at the top is to have volume of cubic feet Find the dimensions of the bo requiring least material for its construction (8) y If u= show that u u = (8) y y y (b) (i) Verify the Euler s theorem for the function u= sin + tan y (8) Eamine for maimum and minimum values of sin + siny+ sin+ y (8) ( ) 78
WK sin e 4 (a) (i) Evaluate sin d (8) Determine the area between the cubic y= and the parabola y= 4 (8) (b) (i) Find the volume of a sphere of radius a (8) Find the work done on a spring when you compress it from its natural length of meter to a length of 75 meter if the spring constant is k = 6 N/m (8) 5 (a) (i) Change the order of integration in I = ydyd (8) Find the area between the parabolas y = 4a and = 4ay by using double integration (8) (b) (i) Evaluate yzddydz over the positive octant of the sphere + y + z = a by transforming into spherical polar coordinates (8) Compute the mass of a sphere of radius b if the density varies inversely as the square of the distance from the centre (8) 78
Reg No Question Paper Code : 57497 Time : Three Hours BE/BTech DEGREE EXAMINATION, MAY/JUNE 6 First Semester Marine Engineering MA 65 MATHEMATICS FOR MARINE ENGINEERING I (Regulations ) Answer ALL questions PART A ( = Marks) Find the centre and radius of the sphere + y + z 5 7 5 7 y 7 z = Maimum : Marks Find the equation of the cone whose verte is at the origin and guiding curve is 4 + y 9 + z =, + y + z = State Taylor s theorem n 4 Evaluate lim e 5 If u = sin + y u find the value of + y + y u y 6 The base diameter and altitude of a right circular cone are measured as 4 cm and 6 cm respectively The possible error in each measurement is cm Find approimately the maimum possible error in the value computed for the volume 7 Calculate by double integration, the volume generated by the revolution of the cardiod r = a( cos θ) about it ais 8-6 57497
8 Evaluate d 9 Sketch the region of integration of the integral y d dy State Steiner s theorem PART B (5 6 = 8 Marks) (a) (i) Show that the plane y + z + = touches the sphere + y + z 4y + z = and find also the point of contact (8) Find the equation of the right circular cone whose verte is the origin, whose ais is the line = y = z and which has semi-vertical angle of (8) OR (b) (i) Find the equation of the right circular cylinder describe on the circle through the points (a,, ), (, a, ), (,, a) as guiding curve (8) Find the equation of the sphere which touches the plane y z = 7 at the point (,, ) and passes through the point (,, ) (8) (a) (i) Find the n th derivative of + a (6) Trace the curve a y = (a ) ( a) () OR (b) (i) Find the Maclaurin s theorem with Lagrange s form of remainder for f() = cos (7) Find dy d when () y = n ( + ) ; () y = a log ; () y = πe sin (9) 57497
(a) (i) If a + hy + by =, show that d y d = h ab (h + by) (8) Discuss the maima and minima of the function f(, y) = 4 + y 4 + 4y y (8) OR (b) (i) A rectangular bo, open at the top, is to have a volume of cc Find the dimensions of the bo, that requires the least material for its construction () If z = f (, y) where = r cos θ and y = r sin θ,prove that z + z y = z r + r z θ (6) 4 (a) (i) Evaluate e d (6) Find the volume of the solid obtained by revolving the cissoid y (a ) = about it asymptote () OR (b) (i) Find the limit, when n, of the series n n + n n + + n n + + + n n +(n ) (8) Find the area of the top of the curve ay = (a ) (8) ( ) ( y ) 5 (a) (i) Evaluate yz d dy dz (8) Find the volume of the portion of the sphere + y + z = a lying inside the cylinder + y = ay (8) OR (b) (i) Using double integration, find the centre of gravity of a lamina in the shape of a quadrant of curve a + y b =, the density being ρ = ky, where k is constant () Compute the mass of a sphere of radius a if the density varies inversely as the square of the distance from the centre (6) 57497
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WS 7 Reg No : Question Paper Code : 76 BE/BTech DEGREE EXAMINATION, APRIL/MAY 7 Time : Three hours First Semester Marine Engineering MA 65 MATHEMATICS FOR MARINE ENGINEERING I (Regulations ) Answer ALL questions PART A ( = marks) Maimum : marks Find the equation of the sphere which has (,, 4) and (,, ) as the etremities of a diameter Define right circular cone Differentiate tan ( log( tan ) ) with respect to 4 State Taylor s theorem y y 5 If u=, epress du in terms of d and dy 6 Find the minimum point of f (, y) = + y + 6+ 7 Find the area of the cardiod r = a( cosθ) 8 State parallel ais theorem y e 9 Change the order of integration in dyd y Write the formula for total mass of the lamina
WS 7 PART B (5 6 = 8 marks) (a) (i) Find the equation of the sphere for which the circle + y + z + 7y z+ =, + y+ 4z = 8 is a great circle (8) Find the equation of the right circular cone whose verte is the y z origin and ais is the line = = and which has semi-vertical angle (8) (b) (i) Find the equations of the spheres passing through the circle + y + z 6 z+ 5=, y = and touching the plane y + 4z+ 5= (8) Find the equation of the right circular cylinder of radius and y z whose ais is the line = = (8) (a) (i) If y = sin + sin + sin +, prove that If y sin, n+ n+ n dy cos = (8) d y = prove that ( ) y ( n+ ) y n y = (8) (b) (i) Epand loge in powers of ( ) by using Taylor s series (8) Trace the curve y ( a ) = (8) (a) (i) If y u= tan y tan, find y u and y u (8) y A rectangular bo open at the top is to have volume of cubic feet Find the dimensions of the bo requiring least material for its construction (8) 8 8 8 (b) (i) If u = sin [( + y+ z) /( + y + z )], find the value of u u u + y + z (8) y z In a triangle ABC, find the maimum value of cos A cos B cosc (8) 76
WS 7 4 (a) (i) Evaluate ( + ) d by substitution method (8) Find the area common to the parabola y = a and + y = 4a (8) (b) (i) Find the first moment of area of a circular area about an ais touching its edge in terms of its diameter d (8) A cup is made by rotating the area between y= and y = + with around the ais Find the volume of the material needed to make the cup Units are cm (8) y 5 (a) (i) Change the order of integration in y yddy (8) Using double integration, find the centre of gravity of a lamina in the positive quadrant of the curve y + a b =, the density being ρ = ky where k is a constant (8) (b) (i) Using double integration, find the area bounded by the parabolas y = 4 and y = (8) Evaluate y / / z d dydz taken throughout the volume of the sphere + y + z =, by transforming into spherical polar coordinates (8) 76
wk4 Reg No : Question Paper Code : 7786 BE/BTech DEGREE EXAMINATION, APRIL/MAY 5 First Semester Marine Engineering MA 65 MATHEMATICS FOR MARINE ENGINEERING I Time : Three hours (Regulation ) Answer ALL questions PART A ( = marks) Find the equation of the sphere whose center is (,,4) Define cone and what is verte of a cone? Evaluate log Lt cot 4 For a cardioid ( ) =a cosθ Maimum : marks and radius is 5 r, prove that φ = θ /, where φ is the angle between the radius vector and the tangent at any point of the curve 5 For a given function does limit eist at all its points? If it eists, is limit unique at any point? 6 Show that the function f ( y) = y,, has a saddle point 7 What is the value of the definite integral d? a a 8 State the Guldinus theorem for surface areas 4a a 9 Change the order of integration of / 4a dyd [do not evaluate] Sketch the region of integration for = I yddy
wk4 PART B (5 6 = 8 marks) (a) (i) Find the equation of the sphere which passes through the points (, 4,), (, 5,), (,,) and whose center lies in the plane + y+ z= (8) Show that the plane y+ z+ = touches the sphere + y + z 4y+ z = and find its point of contact (8) (b) (i) Find the center, radius and area of the circle in which the sphere + y + z + y 4z 9= is cut by the plane + y+ z+ 7= (8) (a) (i) If y sin ( ) Find the equation of the right circular cylinder of radius and y z whose ais is the line = = (8) = ( ), show that ( ) y ( n+ ) y n y = n+ n+ n (8) Using Maclaurin s series, epand tan upto the term containing 5 (8) / m / m (b) (i) If y + y =, prove that ( ) yn+ + ( n+ ) yn+ + ( n m ) yn = Trace the curve y 8a /( + 4a ) (8) (a) (i) Show that V( y, z) cos cos4y sinh4z =, given in its standard form (8), = satisfies the Laplace s V V V equation, + + = (8) y z The altitude of a right circular cone is 5 cm and is increasing at m/sec The radius of the base is cm and is decreasing at cm/sec How fast is the volume changing? (8) =, prove that 6 V 4V + V = (8) (b) (i) If V f( y, y 4z, 4z ) + y z Find the dimensions of a rectangular bo of maimum capacity whose surface area is given as 54 sq units Consider a closed bo (8) 7786
wk4 4 (a) (i) Find the area of the segment cut off from the parabola = 8y by the line y+ 8= (8) Find the volume formed by the revolution of loop of the curve y a+ = a about the ais (8) ( ) ( ) (b) (i) A force of N compress a spring from its natural length of 8 cm to a length of 6 cm How much work is done in compressing it from 6 cm to 4 cm? (8) 5 (a) (i) Evaluate For the first quadrant area bounded by the curve y=, find the moment of inertia with respect to y ais and mass of the area (8) ( + y e ) ddy by changing to polar coordinates (8) Find the area of the position of cylinder +z = 4 lying inside the cylinder +y = 4 (8) (b) Change the order of integration in I = y ddy and hence evaluate the same (6) 7786
WK Reg No : Question Paper Code : 864 BE/BTech DEGREE EXAMINATION, NOVEMBER/DECEMBER 6 Time : Three hours First Semester Marine Engineering MA 65 MATHEMATICS FOR MARINE ENGINEERING I (Regulations ) Answer ALL questions PART A ( = marks) Maimum : marks Show that the plane y+ z = touches the sphere + y + z + + y 7= Define right circular cylinder dy Find if d log y= (sin) 4 Find the n th derivative of 4 4 u u 5 If u = + y + y, find + y y dy 6 Using partial differentiation, find if d 7 Evaluate tan d + y = y 8 Find the area bounded by the -ais and the curve y= log, the lines = and = 9 Evaluate dy d + y Find the area of the circle + y = a
WK PART B (5 6 = 8 marks) (a) (i) Find the equation of the sphere passing through the points (,, 4), (, 5, ), (,, ) and having its centre on the plane + y+ z = (8) Find the equation of the right circular cylinder of radius whose ais passes through (,, ) and has direction cosines proportional to (,, 6) (8) (b) (i) Find the equation of the sphere passes through the circle + y + z + + y+ z = and + y z+ = and cuts orthogonally the sphere + y + z + z = (8) Find the equation of the cone whose verte (,, ) and passing through the circle + y + z = 4, + y z = (8) (a) (i) Differentiating n times the equation y + y + y= by Leibnitz s theorem and obtain the resulting equation (8) Trace the curve ) (8) y ( a = (b) (i) Using Maclaurin s theorem, epand tan in a series of ascending 5 powers of as far as term containing (8) e + sin e Evaluate lim 5 y (a) (i) If u= sin, prove that 4 (8) u u = (8) y y Eamine for maimum and minimum values of the function y = ( ) ( ) (8) y u u (b) (i) If u= sin, show that + y = tanu (8) + y y The radius of a sphere is found by measurement to be 85 inches with a possible error of inch Find the consequent error in the surface area and the volume as calculated from this measurement (8) 864
WK tan e 4 (a) (i) Evaluate sin d and + d (8) Find the area of an ellipse of semi aes a and b and deduce the area of a circle of radius a (8) / (b) (i) The area of the curve + y = a lying in the first quadrant revolves about the -ais Find the volume of the solid generated (8) 5 (a) (i) Evaluate / Find the moment of inertia of the area bounded by the curve r = a cos θ about its ais (8) y / d dy by changing the order of integration (8) Find the moment of inertia about z-ais of a homogeneous tetrahedron bounded by the planes =, y =, z = + y and z = (8) y (b) (i) Evaluate ddydz y z by changing to spherical polar coordinates (8) Find, by double integration, the centre of gravity of the area of the cardiod r =a( + cosθ ) (8) 864