Maths

BIRLA INSTITUTE OF TECHNOLOGY INTERNATIONAL CENTER, RAK - UAE FORMAT FOR QUESTION BANK Department: Mathematics Class: Diploma -III Question Bank :ALL Semester: III Subject: Mathematics -III Subject Code :DMA1003 Question Bank Module: 1 Differential Equations: Degree and order of D.E ., Formation . Variable separable method. Homogeneous differential equations. D.E reducible to homogeneous forms. Linear D.E. ( Bernoulli’s forms not required ). Higher order differential equation with constant co -effici ents complementary functions and rules for finding particular integrals. Module : 2 Statics: Parallelogram law of forces, Resolution for forces, Triangle of forces, Lami’s theorem. Parallel forces, Moments, Couples. Module : 3 Dynamics : Rectilin ear motion projectiles, Projectiles, Motion on inclined planes, Work, Energy and Power , Collision of elastic bodies ( direct impact ). Module : 4 Statistics: Frequency distribution , Histogram, Polygon, Measures of central tendencies ( Mean , Med ian and Mode ) , S.D and variance . Module: 1 Differential Equations: Degree and order of D.E ., Formation . Variable separable method. Homogeneous differential equations. D.E reducible to homogeneous forms. Linear D.E. ( Bernoulli’s forms not required ). Higher order differential equation with constant co -effici ents complementary functions and rules for finding particular integrals. Degree and order of D.E : 1.Find the degree and order of the following differential equations. (i) ( dy/dx) 2 - 4x = d 2 y/ dx 2 (ii) Y = x dy/ dx + a 2 (iii) 3/2 = 5 d 2y / d x 2 (iv) x 2 ( d 2 y / dx 2 )3 + y ( dy/dx ) 4 + y 4 = 0 2. Form a differential equation: (i) y = A cos x + B sin x. (ii) y = 2 ( x 2 - 1 ) + C e -x2 (iii) y = A e 3x + B e5x where A and B are arbitrary constants. (iv) Y=a cos (x+3) ) ( 1 dx dy         2 ) ( 1 dx dy BIRLA INSTITUTE OF TECHNOLOGY INTERNATIONAL CENTER, RAK - UAE (v) Y= (vi) Y= (vii) Find the differential equation of the family curves where a and b are arbitrary constants. (viii) Show that the function is a solution of the differential equation Variable separable method. Solve the following equations using variable separable method. 3. (e x + e -x ) dy = (e x - e-x) dx 4. 5. 6. (1 + cos x) dy = (1 – cos x) dx 7. ( x + 1) dy /dx = x ( y 2 + 1 ) 9. sec 2 x tan y dx + sec 2 y tanx dy = 0 10. dy/ dx = e x+y 11 . 12 . 13 . 14 . 15. dy / dx = cot 2 ( x + y ) Homogeneous differential equation 16 . 17. 18. (xy + x)dy – (xy +y)dx =0 ) sin cos ( x B x A e x  x x be ae  3 ) sin( ) cos( x b x a y     c Be Ae y x x    2 0 2 3 2 2 3 3    dx dy dx y d dx y d 0 sin cos ) 1 (    xdy e xdx e y y 0 sec ) 1 ( tan 3 2    ydy e ydx e x x ) 1 ( ) 1 ( 2 2 x y dx dy    y y x e x e dx dy     2 2 4 , 0 cot    wherex giveny y dx dy x  dx dy y x   ) cos( 0 ) ( 2    dx y x y dy x     0 2 2     dy y yx dx x xy BIRLA INSTITUTE OF TECHNOLOGY INTERNATIONAL CENTER, RAK - UAE 19. 20. (x2 - y2 ) dx + 2xy dy = 0 given that y = 1 when x = 1 21. 22. dy/ dx = y /x + sin (y/x). 23. ( 1 + e x/y ) dx + e x/y ( 1 - x/y ) dy = 0 24. 25 . Reducible to Homogeneous Equation 26. 27. ( x + 2y ) ( dx – dy ) = dx + dy 28. 29. dy = x + 2y – 3 30. ------- -------------- dx 2x + y – 3 31 . Linear differential equation of 1 st order 32. 33. cos 3 x dy/dx + y cos x = sin x 34. dy/dx + y tan x = 2x + x 2 tan x 35. dy/ dx + y sec x = tan x . 36. 37. 0 ) ( 2 2    dy x dx xy y x y x y dx dy   tan 2 2 3 3 x y xy dx dy   xydx dy y x   ) ( 2 2       b y x a y x dx dy b y x a y x          ) ( dx y x dy y x ) 3 ( ) 3 (      0 ) 1 2 4 ( ) 1 2 (       dy y x dx y x 10 2 6 20 9 2      y x y x dx dy 2 ) 1 ( ) 1 (     x e y dx dy x x x y dx dy x x log 2 log   y x dx dy x   BIRLA INSTITUTE OF TECHNOLOGY INTERNATIONAL CENTER, RAK - UAE 38. 39. 41. 40. If , and y=0 for x= , show that maximum value of y is 1/8 41 . Higher order differential equation with constant co -efficients 42 . Solve: d 2 y / dx 2 - 8 dy/dx + 15 y = 0 43. Solve: 44 . d2y / dx 2 - 6 dy /dx + 9y = 6 e 3x + 7 e -2x - log 2. 45. 46 . 47 . 48 . 50 . 51 . 52. 53. x y dx dy 4   2 1 2 2 ) 1 ( 2 ) 1 ( x x xy dx dy x     x x y dx dy sin tan 2   3  x x y dx dy cos cot   x e y dx dy dx y d 3 2 2 5 9 6    x e y dx dy dx y d dx y d     2 4 3 2 2 3 3 x y dx y d 4 cos 6 2 2   x y dx y d 2 sin 4 2 2   x y dx dy dx y d 2 cos 2 2    2 2 2 3 3 4 4 2 x dx y d dx y d dx y d    x x y D D cos 7 3 ) 2 3 ( 3 2     x e e x y dx dy dx y d x x 2 sin 4 4 2 2 2 2       x x y dx dy dx y d cos 2 3 2 2    BIRLA INSTITUTE OF TECHNOLOGY INTERNATIONAL CENTER, RAK - UAE Module : 2 Statics: Parallelogram law of forces, Resolution for forces, Triangle law of forces, Lami’s theorem. Parallel forces, Moments, Couples. (51) Show that the resultant of two perpendicular forces P+Q and P -Q is equal in magnitude to the resultant of two perpendicu lar forces and . (52) The resultant of two forces P and Q is of magnitude R. Show that if P is doubled, Q remaining unaltered, then the new resultant S will be of magnitude (53)Two equal forces act on a particle. Find the angle between them when the square of their resultant is equal to three times their product. (54)With two forces acting at a point, the maximum effect is obtained when their resultant is of 4 N .

If they act at right angles, their resultant is of 3 N. Find the forces. (55) Find the magnitude and direction of the resultant of two forces of magnitudes 10 N and 15 N acting at a point and inclined to each other at an angle of 60 ◦ . (56) Find the angle between two equ al forces so that their resultant is also equal to either of them in magnitude. (57) When two forces of equal magnitudes are inclined at an angle 2 , the magnitude of their resultant is twice as great as when they are inclined at an angle 2 . Prove that cos = 2 cos .

(58) Two forces p and Q acting at a point have resultant R . If the magnitude of P be doubled, the magnitude of R is doubled and if the magnitude of Q is doubled and P is reversed in direction, even then the magnitude of R is doubled. Show that P : Q: R = : : (59) The direction of the resultant R of two concurrent forces P and Q act at right angle to P , Show that the angle between the direction of the forces is cos -1 ( -P/ Q ) . Also show that Q 2 - P2 = R 2 (60) The resolved part of a force of magnitude 32 N in a direction is 16 N. Find its inclination with the force . Also find the other resolved part (61) A block of weight 80 N is placed on a wi ghting machine and a force F of magnitude 30 N is applied to the block at 30 as shown in the adjoing fig. What would the weighing machine read. (62) D,E,F are the mid points of the sides of a triangle ABC and G is its centroid. Show that three forces acting at a point and represented by GD, GE, GF are in equilibrium. (63) State and Prove Lamis theorem and its conversse. (64) Three coplanar forces acting on a particle are in equilibrium . the angle between the first and the second is 90 ◦ and that b etween the second and the third is 120 ◦ . Find the ratio of the magnitude of the forces. P 2 Q 2 2 2 2 2 2 Q R P       6 2 5 BIRLA INSTITUTE OF TECHNOLOGY INTERNATIONAL CENTER, RAK - UAE (65) A weight of 10 N hangs by a string from a fixed point the string drawn out of the vertical by applying a force of 5N to the weight. In what direction must this force be applied, in order that in equilibrium, the deflection of the string from the vertical may be 30 ◦ ? Also find the tension in the string ? (66) Three forces P,Q,R act at a point O inside a triangle ABC , these forces act along OA , OB, OC respectively and are in equilibrium. Show the following , (i) if O is the centroid of triangle ABC then P/OA = Q / OB = R/ OC (ii) if O is the orthocenter of triangle ABC , then P / a = Q / b = R/ c (iii) if O is the incentre of the triangle ABC then P/ cos A/2 = Q/cosB/2 = R/cosC/2 (iv) if O is the circumcentre of the triangle ABC, then P/ sin 2A = Q/ sin 2B = R/ sin 2C or equivalently. P/a 2 ( b 2 + c 2 – a2 ) = Q/ b 2 (c2 + a 2 – b2 ) = R / c 2(a2 + b 2 – c2 ) (67) A force P acting along a smooth inclined plane support a weight w 1 amd when applied horizontally balances a weight w 2 on the same plane , show that P 2 = w 12 - w 22. Parallel forces : (68) Two men are carrying a straight uniform bar 10 m long and weighing 140 Kg . One man supports it at a distance of 1 m from one end. And the other man at a distance of 2 m from the other end . What weight does each man bear ? (69) A man carries a bundle at the end of a stick which is placed over his shoulder, if the distance between his hand and his shoulder be changed, how does the pressure on his shoulder vary ? (70) Two parallel forces of magnitude 12 N and 8 N act at two points 45 cm apart on a rigid body.

Find the magnitude, direction and the position of the point of application of their res ultant, when the forces are (i) like (ii) unlike. (71).Like parallel forces P and Q act at a points A and B of a rigid body. Their resultant meets AB in C when the forces are interchanged, the resultant meets AB in D. if AC = CD show that P:Q = 2:1 (72) Two unlike parallel forces P and Q act at x units apart on a rigid body, if the direction of P be reversed . Show that the resultant is displaced through the distance 2PQ / ( P 2 – Q2) x units. (73) Two like parallel forces P and Q act on a rigid body at A and B respectively. (a). if Q be changed to P 2/ Q , show that the line of action of the resultant is the same as it would be if the force were simply interchanged. (b) If P and Q be interchanged in position, show that the point of applicatio n of the resultant will be displaced along AB through a distance d , where d =( P -Q / P+Q ) AB Moments and Couples: (75) Find the moment about the point O of a force of magnitude 14 N acting at the point A along the line AB where OA = 3m and ∟OAB = 2 /3 . (76) A force of magnitude 18N acts in the xy plane and its direction makes angle 150 ◦ with the positive direction of the x - axis if OA = 5 m. Find the moment of the force about the origin. (77) ABCD is a square of si de 1.5 m , forces of magnitude 3,4,7 and 12 N act along AB, BC, AD and DB respectively , Find the algebraic sum of moments of the forces about the centre of the square.  BIRLA INSTITUTE OF TECHNOLOGY INTERNATIONAL CENTER, RAK - UAE (78) The side of a square lamina ABCD us 2 a metres, along AB, CB, CD ,AD and BD a cts forces of magnitude 1N, 2N, 3N, 4N and 5N respectively. Find the algebraic sum of their moments (i) about the centre of the square (ii) about A . (79) The side of a regular hexagon ABCDEF is 2√3 m . along the sides AB, CB,DC,EF and FA act forces of magnitude 1, 2,3,4,5,6N respectively. Find the algebraic sum of the moments of the forces (i) about A (ii) about the centre of the hexagon. (80) Forces forming a couple are each 2.4 N and its arm is 6 cm. Find (i) the arm of an equivalent couple each of whose force is 3N (ii) the force of an equivalent couple whose arm is 9 cm. (81) Two unlike parallel forces each of magnitude 20√ 3 units acting on a rigid body form an anti - clockwise couple If these fo rces lie in the xy plane and act at the points A ( -1, 0 ) ,B ( 3, 0 ) and are inclined at 60 ◦ to the x - axis , find the moment of the couple. (82) Two forces each of magnitude 30 units form a couple, If one of them acts at the ( 3,0) inclined at 60 ◦ to the positive direction of x - axis . Find whether the line of action of the other force cuts x - axis . given that the moment of the couple is 60√ 3 units. Module : 3 Dynamics : Rectilinear motion projectiles, Projectiles, Motion on inclined planes , Work, Energy and Power , Collision of elastic bodies ( direct impact ). Module : 4 Statistics: Frequency distribution , Histogram, Polygon, Measures of central tendencies ( Mean , Median and Mode ) , S.D and variance . (79). A particle moving with uniform acceleration passes over 720 cm, in the 11 th second and 960 cm in the 15 th second of its motion. Find the initial velocity of the particle. (80). A ball rolling down a slope with a uniform acceleration passes three posts driven in the ground at equal distances. The velocities when passing the three successive posts are x, y, z. Show that x 2 , y2, z2 are in A.P. (81). A body start from rest and at the end of 10 seconds, it is moving at the rate of 22 m/sec, 5 seconds later, i ts velocity is 105.6 km/hr . Is the acceleration constant ? (82). A particle moving with uniform acceleration described in the last second of its motion, 9/25 th of the whole distance. I f it started from rest, how long was it in motion and thro ugh what distance did it move if it described 10 m in the 3 rd second of its motion ? (83). If a, b, c be the distances described by a particle during the p th , q th, r th seconds respectively, show that (q -r) a + ( r -p) b + (p -q) c = 0. The part icle is assumed to be moving with uniform acceleration. (84). A bullet fired into a target loses half its velocity after penetrating 3 cm. How much further will it penetrate ? (85). A train runs between two stations which are 4 km apart and takes 4 mi nutes for the journey. If its motion is first of uniform acceleration X and then that of uniform retardation Y , prove that 1/ x + 1/y = 2. (86)A particle starting from rest moves in a straight line, first with uniform acceleration a and then with uniform retardation b. It comes to rest in time t , measured from the beginning , after having described a distance s, show that t 2 = 2s ( 1/ a + 1/ b ) BIRLA INSTITUTE OF TECHNOLOGY INTERNATIONAL CENTER, RAK - UAE (87) A body is projected vertically upward with a velocity of 24.5 m/sec. (i) How high will it go? (ii) How long will it take to return to the point of projection ? (iii) When will its velocity be 4.9 m/sec ? (iv) At what time will it be 29.4 m above the point of projection ? (v) At what time will it be 29.4 m below the point o f projection? (88)Two balls are projected vertically upward at the same instant along parallel lines from the ground . One ascends 34.3 m higher than the other and returns to the ground 2 seconds later. Find the velocities of projection of the balls and the heights attained by them. (89) A particle is dropped from the top of a tower 60 m high and another particle is projected at the same time vertically upwards from the foot of the tower so as to meet the first particle at a height of 40 m. Find the velocity of projection of the second particle. ( Assume g = 10 m/s 2 ) (90)It is safe to jump from the height of 2m on the surface of the earth. What is the corresponding safe height on the surface of a planet, where the gravitational acceleratio n is 1.96 m/sec 2 ? The surfaces of earth and the planet are equally hard. (91) A,B,C and D are points in a vertical line such that AB = BC = CD . If a body falls from rest from A, Prove that the times taken in describing the distances AB, BC,and CD are in the ration 1 : √2 – 1 : √3 - √2 . (92) A body falls freely from the top of a tower and during the last second of its motion , it is observed to fall 16/25 of the whole distance , Find the height of the tower. (93)A particle is projected vertically upward with a velocity of u m/sec and after t seconds another particle is projected upward from the same point and with the same velocity. Prove that particles will meet at a height 4u 2 – g 2t 2 / 8g metres after ( t/2 + u/g ) seconds. (94) Derive the equation of the trajectory , Latus rectum, vertex , Focus, Directrix and axis of the trajectory of a projectile. (95)A particle is projected with a velocity of 24 m/sec at an angle of elevation of 60 ◦ Find (i) the equati on of its path, (ii) the length of the latus rectum of the path, (iii) its time of flight , (iv) height of its directrix (v) maximum height attained by it and ( vi) its horizontal range. (96)A boy can throw a cricket ball 100 metres . How lo ng is the ball in the air and what height does it attain ? (97) Two balls are projected from the same point in directions inclined at 60 and 30 to the horizontal, it they attain same height, what is the ration of their velocities of projection ? what is this ratio, if they have the same horizontal range ? (98) A stone is dropped from a balloon moving horizontally with a velocity of 30 m/sec and reaches the ground in 4 seconds. Find the height of the balloon and the velocity of the ston e on striking the ground . ( take g = 10 m/sec 2 ) (99) A fountain jet projects stream in all directions with a velocity of 5 m/ sec from a point 2m above the centre of the circular basin. What must be the diameter of the basin of catch all the water ? (100) A particle is projected so as to pass through two points whose horizontal distances from the point of projection are 12 m and 24 m and which are at vertical heights of 4m and 5m above the horizontal BIRLA INSTITUTE OF TECHNOLOGY INTERNATIONAL CENTER, RAK - UAE plane through the point of projecton. Find the velocity and the direction of projection . ( take g = 10 m/sec 2 ) (101) From the foot of an inclined plane, whose rise is 7 in 25, a shot is projected with a velocity of 196 m per second at an angle of 30 with the horizontal, (i) up the pla ne, (2) down the plane . Find the range in each case. (102) A particle is projected at an angle α with the horizontal from the foot of a plane , whose inclination to the horizon is β, show that it will strike the plane at right angles, if cot β = 2 ta n (α - β ) (103). A ball of mass 8 kg and moving with velocity 4 metres per second , overtakes a ball , of mass 10 kg moving with velocity 2 metres per second in the same direction , if e be ½ , find the velocities of the balls after impact. If the second ball be moving in a direction opposite to the first , find the velocities. (104) a train , of mass 100 metric tons is ascending uniformly an incline of 1 in 250 , and the resistence due to friction ,etc is equal to 8 kg per metric tons, if the engine be of 1.176 10 5 watts and be working at full power , find the rate at which the train is going. (105) A particle moving in a straight line is acted by a force which works at a constant rate and changes its velocity from u to v in passing over a distance x , prove that the time taken is 3 ( u + v ) x / 2 (u 2 + uv + v 2 ) Mean , Media n and mode (106)Find the A.M for the following distribution C.I 0 -10 10 -20 20 -30 30 -40 40 -50 F 7 8 20 10 5 (107)Calculate the median for the following data C.I 5 -10 10 -15 15 -20 20 -25 25 -30 30 -35 35 -40 40 -45 F 5 6 15 10 5 4 2 2 Verify grap hically. (108)Find the mode for the following data Weight 93 -97 98 -102 103 -107 108 -112 113 -117 118 -122 123 -127 No.of .people 5 5 12 17 14 6 3 (109)De termine the mode for the following data X 20 21 22 23 24 25 26 27 28 29 F 6 9 4 2 10 8 7 5 1 3 and verify graphically (110) the interquartile range and coefficient of variation for gi ven data C.I 30 -50 50 -70 70 -90 90 -110 110 -130 130 -150 150 -170 F 100 140 300 230 125 51 1000 (111 )coefficient of variation of two series are 75% and 90%. Their standard Deviations 15 and 18 respectivly. Find their means. (112)Wages in Rs: 0 -10 10 -20 20 -30 30 -40 40 -50 No.of workers : 12 23 35 20 10 Draw the Histogram, frequency polygon and cumulative frequency curve. (113) From the following frequency distribution compute the std of 100 students BIRLA INSTITUTE OF TECHNOLOGY INTERNATIONAL CENTER, RAK - UAE Mass in Kg : 60 -62 63 -65 66 -68 69 -71 72 -74 No. of students : 5 18 42 27 8 (114 )A candidate obtained the following percentage in the Examination: English – 60, Hindi – 75, Math ematics – 63, Physics -59, Chemistry -55. Find the weighted A.M if weights 1,2,1,3,3. respectively allotted to the Subjects. (115).The taste below gives the number (F) of candidates obtaining marks(x) or higher in a certain examination: X: 10 20 30 40 50 60 70 80 90 100 F: 140 133 118 100 75 45 25 9 2 0 Calculate the mean and median marks obtained by the candidates. (116) Compute th e standard deviation from the following distribution of marks obtained by 90 students: Marks: 20 -29 30 -39 40 -49 50 -59 60 -69 70 -79 80 -89 90 -99 No.of students: 5 12 15 20 18 10 6 4 Books /Course Materials Referred: 1. Modern Approach to intermediate vector analysis – Das Gupta & Prasad 2. Engineering Mathematics -Bali, saxena, Iyengar -Laxmi publication. 3. Statics Via Vector -A.Ghatuary, Dynamics Via Calculus -A.Ghatuary.