Tiêu đề MP DPP Sheet 01 (1).pdf ✅

1 North Delhi : 56-58, First Floor, Mall Road, G.T.B. Nagar (Near Metro Gate No. 3), Delhi-09, Ph: 011-41420035 South Delhi : 28-A/11, Jia Sarai, Near-IIT Metro Station, New Delhi-16, Ph : 011-26851008, 26861009 CSIR-NET/JRF JUNE 2024 SECTION: MATHEMATICAL PHYSICS Daily Practice Problem (DPP) Sheet 01: DIFFERENTIAL EQUATIONS (Seperable ODE, Reducible to Seperable ODE) Q.1. Solve the following differential equations: (i) 2 2 x y dx y x dy 1 1 0     given that y(0) 3  [Ans. 2 2 1 1 3     x y ] (ii) ( 1)cos sin 0 y y e xdx e xdy    given that 0 3 y         [Ans. sin 1 3   y x e   ] (iii) ln 2 dy x x y dx  given that    2 y 2 ln 2  [Ans.  2 y x  ln ] (iv) 1 1 x dx e dt         given that x 0 1   [Ans. 1 1  x t e e e    ] (v)     2 2 2 given that 0 1 6 x x e dy y dx y    [Ans. 2 3 2 4 2 3 x y e x    ] Q.2. Solve the following differential equations: (i) 2 (4 1) dy x y dx    given that y 0 1   [Ans. 1 4 1 tan 2 2 4 x y x             ] (ii)     4 3 sec given that 1 6 dy x x y xy y dx      [Ans.   2 1 sin 2 xy x  ] (iii) 1 y x dy e e dx         given that y 1 1   [Ans.   1 2 2 2 x y x e e e    ] Q.3. Solve the following differential equations: (i) sin sin 0 y y y x dx x dy x x          given that y 4   [Ans. 1 cos ln 4 ln 2 y x x          ] (ii) (1 ) 1 0 x y x y x e dx e dy y           given that y 0 1   [Ans. x y/ x 1 e y y   ] (iii) 2 2 dy y x y dx x    given that y 1 0   [Ans. 2 2 2 y x y x    ]
2 North Delhi : 56-58, First Floor, Mall Road, G.T.B. Nagar (Near Metro Gate No. 3), Delhi-09, Ph: 011-41420035 South Delhi : 28-A/11, Jia Sarai, Near-IIT Metro Station, New Delhi-16, Ph : 011-26851008, 26861009 (iv) 2 2 2 dy x xy x y dx    given that y 1 0   [Ans. ln 1 x x x y    ] (v)     3 3 2 x y dx xy dy y     3 0 given that 1 0 [Ans. 3 3 x y x   2 ] Q.4. The particular solution of differential equation   2 sin 3 dy x x dx  given y 0 0   , is (a)   2 cos 3 6 x y   (b)   2 cos 3 1 6 x y   (c)   2 cos 3 1 6 x y    (d)   2 1 cos 3 6 x y   Q.5. Biotransformation of an organic compound having concentration (x) can be modelled using the ordinary differ- ential equation 2 0 dx kx dt   , where k is the reaction rate constant. If x = a at t = 0, the solution will be (a) kt x ae  (b) 1 1 kt x a   (c) 1  kt x a e   (d) x a kt   Q.6. Consider the following differential equation:   3 2 such that 0 1 1 dy xy y dx x     The range of x for which the solution will be real, will be (a) 3 3 2 2    x (b) 5 5 2 2    x (c)    2 2 x (d) 0 2  x Q.7. The solution of the differential equation  2 2 y x dx xydy    2 0 will be represented by (a) Family of circles having center at the origin. (b) Family of circles having center on x-axis and touching the origin (c) Family of circles having center on y-axis and touching the origin (d) Family of circles having center on y-axis but not touching the origin Q.8. The geometrical interpretation of the graph of the differential equation:   dy x dx y is a family of (a) Hyperbola (b) Ellipse (c) Parabola (d) Circle Q.9. A drop of liquid evaporates at a rate proportional to it’s area of surface. If the radius of the drop is initially 4 mm and 5 minutes later, the radius is reduced to 2 mm, then the radius of the drop after 10 minutes will be (a) 0 mm (b) 0.5 mm (c) 0.8 mm (d) 1 mm Q.10. Which of the following graph best represents the function y f x  ( ) , where f x( ) is the solution of differential equation dy dx xy / 2   with y(0) 2  is (a) 0 x y 2 (b) 0 x y 1 (c) –2 x y (d) –1 x y
3 North Delhi : 56-58, First Floor, Mall Road, G.T.B. Nagar (Near Metro Gate No. 3), Delhi-09, Ph: 011-41420035 South Delhi : 28-A/11, Jia Sarai, Near-IIT Metro Station, New Delhi-16, Ph : 011-26851008, 26861009 Q.11. The geometrical interpretation of he solution of the differential equation ' 6 25 0 yy x   is a family of (a) Hyperbola (b) Ellipse (c) Parabola (d) Circle Q.12. The rate at which a hot body cools is proportional to the difference in temparature between it and surroundings (Newton’s law of Cooling). A body is heated to 0 110 C and placed in air at 0 10 . It’s temparature becomes 0 60 after 1 hour. The additional time after which it cools down to 0 30 , is (a) ln5 ln 2 hrs (b) ln5 1 ln 2 hrs        (c) ln 2 1 ln5 hrs        (d) ln 2 1 ln5 hrs        Q.13. A radio active substance disintegrates at a rate which is proportional to the amount of the substance present. 50% of the substance disintegrates in 1000 years. The percentage of the substance that will disintregrate in 50 years, will be (Yon can use 1 x e x    for small x) (a) 1.5 % (b) 2.5 % (c) 3.5 % (d) 4.5 % Q.14. The rate at which the ice melts is proportional to the amount of ice at the instant. If the half of the initial quantity of the ice, melts in 30 minutes, then the fractional amount of the ice left after 2 hours will be (a) 1/8 (b) 1/16 (c) 1/24 (d) 1/32 QUESTIONS ON NEW PATTERN OF GATE EXAMINATION Numerical Answer Type (NAT) Questions Q.15. Suppose that, a tumour in a rat is approximately spherical and its rate of growth is proportional to its diam eter. If the tumour has diameter 5 mm. When detected and 8 mm, after three months later, what will the diameter be after another three months ______________ mm (Your answer should be upto ONE DECIMAL PLACES) Q.16. Consider the linear differential equation dy xy dx  . If y = 2 at x = 0, then the value of y at x = 2 is __________________ (in the units of 2 e ) (Your answer should be an INTEGER) Q.17. The growth rate of bacteria population is proportional of it’s population. Initially the population is 10000, while after 10 days it’s population is 25000. The population after 20 days will be ________________ (Your answer should be an INTEGER) Q.18. A particle moves on a straight line so that it’s acceleration is equal to four times it’s velocity. At time t = 0, it’s displacement from the origin is 1 km and it’s velocity 1.5 km/sec. The time at which the displacement of the particle will 10 km, will be _____________ sec. (Your answer should be upto ONE DECIMAL PLACES)
4 North Delhi : 56-58, First Floor, Mall Road, G.T.B. Nagar (Near Metro Gate No. 3), Delhi-09, Ph: 011-41420035 South Delhi : 28-A/11, Jia Sarai, Near-IIT Metro Station, New Delhi-16, Ph : 011-26851008, 26861009 Multiple Select Questions (MSQ) Q.19. Consider the following differential equation: 2 ;     dy x f x x R dx    , satisfying y (0) = 0 ; where   0 for 0 1 for 0 x f x x          . The solution of the initial value problem is (a) continuous (b) not differentiable at x = 0 (c) differentiable at x = 0 (d) discontinuous Q.20. The solution of dy 2 y dx  with initial conditions y 0 1   , will bounded in the interval (a) x 1 (b) x 1 (c) x 1 (d) x  0 Answer Key 4. (d) 5. (b) 6. (b) 7. (b) 8. (d) 9. (a) 10. (a) 11. (a) 12. (b) 13. (c) 14. (b) 15. (10.1) 16. (2) 17. (62500) 18. (0.8) 19. (a, b) 20. (a, b)