North Maharashtra University 2007 B.Sc Mathematics FY 2 - Question Paper
FYBSC (Mathematics) Paper 2
 Paper 2_files/12868-22736-FYBSC (Mathematics) Paper 2-1.png)
Paper II (CALCULUS )
Prof. R. B. Patel
Dr. B. R. Ahirrao Prof. S. M. Patil
Prof. A. S. Patil
Prof. G. S. Patil
Prof. A. D. Borse
 Paper 2_files/12868-22736-FYBSC (Mathematics) Paper 2-2.png)
Art, Science & Comm. College, Shahada
Jaihind College, Dhule Art, Science & Comm.
College, Muktainagar Art, Science & Comm.
College, Navapur Art, Science & Comm.
College, Shahada Jijamata College,
Nandurbar
Limit, Continuity, Differentiability and Mean Value Theorem Q.1 Objective Questions Marks - 02
x - 4 x - 5 1. lim--
x5 x + 2x - 35is equal to
b)l
d) none of these
a) 1
c) T
cos x
lim- is equal to
x1 x -1
d) none of these
a) 0
b) 1
c) -1
. x - tan x Evaluate lim-3
x x
d) 1
c) 0
a) T
The value of the limlog(sin2 x) is
x0 log(sin x)
d) -1
a) 2
b) 0
c)1
n
lim is equal to
e
a) 1
b) -1
c) 2
d) 0
.. log(sin ax) , 7 _ .
lim- ,(a, b > 0) is equal to
x0 log(sin bx)
c) 0
d) none of these
a) -1
b) 1
lim x log x is equal to
x0
d) -1
a) 0 b) 1
1 1
c) 2
lim
x0
is equal to
x sin x
d) none of these
a) 0
b)1
c) -1
1__1_
x ex -1
lim
xQ
a) 1
limxx is equal to
10.
x0
a) 1
b) -1
c) 2
d) none of these
... / \tan2 x
lim (tan x) is
11.
a) e
d) - e
e
, for x 0 and , for x = 0
at the point x = 0
, for x 0 and , for x = 0
is
, . f (x) = x sin1
12. The function x
f (0) = 0
a) Continuous and derivable
b) Not continuous but derivable
c) Continuous but not derivable
d) Neither continuous nor derivable
, f (x) = x2 sin1
13. The function x
f (0) = 0
a) Continuous and derivable
b) Not continuous but derivable
c) Continuous but not derivable
d) Neither continuous nor derivable
14. For which value of c e (a,b), the Rolls theorem is verified for the function
defined on [a, b]
x + ab x(a + b)
f (x) = log
a) Arithmetic mean of a & b b) Geometric mean of a & b
c) Harmonic mean of a & b d) None of these .
b)
c)
d)
4
2
3
a) 0
For which value of c e| 0, |, the Rolles theorem is applicable for the
16.
function f (x) = sin x + cos x in
\ 71 c) 7
d) f
a) 0
b) 4
For which value of c e (1,5), the Rolles theorem is verified for the function f (x) = x2 - 6x + 5 in[1,5]
17.
a) 1 b) 2 c) 3 d) 4
for which value of ce (-2, 3) . the L.M.V.T. is verified for the function
18.
f (x) = x2 - 3x + 2 in[-2,3]
1
-1
a) 1
b)
c)
2
2
d) 0
19. L.M.V.T is verified for the function f (x) = 2x2 -7x +10 in[2,5]
a) 5 b) 1
a) - b) -
c) 0
2 2
71
For which value of c e | 0, | C.M .V.T. is applicable for the function
20.
f(x) = sin x , g(x) = cos x in [0, n/2]
a) 0
d) 4
b) -Jab
a)
c) a + b
If the C.M.V.T. is applicable for the function
f(x) = ex , g(x) = e-x , in [a, b] find the value of c e (a, b)
a + b
d) none of these
f(x) =1/x2 , g(x) = 1/x , in [a, b] find the value of C.
a + b
2ab a + b
b) yfab
d) none of these
a)
c)
2
log x - log 5
x - 5
23. If f (x) =
, x 5 is continuous at x = 5 then find f(5)
a) 5
b) -5
1 - sin x (n - 2x)2
24. If f (x) =
, x is continuous at x = then f(n/2) is 2 2
1
2
b)
c) 1
d) -1
a)
3
1 - cos x
If f (x) =
, x 0 is continuous at x = 0 then value of f(0) is
25.
sin x
a) 0 b) 1
c) -1
d) none of these
x ~a
a - a
I f f (x) =
, x a is continuous at x = a , then find f(a)
26.
a - x
a) aa log a b) -aa log a
c) log a d) none of these
Evaluate lim sin x log x
x0
27.
a) 0
b) 1
c) -1
d)
2
Evaluate lim tan x log x
x0
28.
a) 0
b) 1 1 1
c)-1 is equal to
d) none of these
lim
x>1
29.
log x x-1
c) 2
d)-2
30. lim I I is equal to
x01 x
31. Evaluate limxa(x a)x
-1
a)
2
b)
32. Evaluate lim.2 (---"-)
x * \x-2 log (x-l)}
-1
a)
b)
c) 1
d) -1
2
2
33. Evaluate lim(1 + x)
c) -2
d) 1
a) -1
b) 2 sin 4( x - 3)
34. If f (x) = x2 - 2
f(3)
, x 3 is continuous at point x = 3 find
d) none of these
c) 3 c) 2
a) !
ex - esmx
35. Evaluate lim-
x x - sin x
a) 1
b) -1
c) 2
d) -2
. tan x - x
1. Evaluate lim-
x x - sin x
ex -1 - x
2. Evaluate lim-
x0 log(1 + x) - x
x0 log(tan x)
4. Evaluate lim I -1 - cot2 x
x0i x
5. Evaluate lim I 2
xa[ a
6. Evaluate lim(cot x)x , x > 0
x0
1/
/log x
7. Evaluate lim(cot x)
x0
1/
n -1
--tan x
2
8. Evaluate lim
9. Examine for continuity, the function
x 2
f (x) =--a, for 0 < x < a
a
= 0 , forx = 0
a3
= a - , forx > a
x2
10. Using e -5 definition , prove that
2 1
f (x) = x cos , if x * 0
is continuous at x = 0
x
= 0 , if x = 0
11. Examine the continuity of the function
e/x -1
f (x) = -7, if x * 0 x +1 = 0 , if x = 0
at the point x = 0.
x2 - 9
f (x) =
for 0 < x < 3 for x = 3 for x > 3
x - 3
= 6
18
,2 :
at the point x = 3.
13. Examine the continuity of the function
2
f(x) = - 4, for 0 < x < 4
for x = 4
= 2
64
= 4 -, for x > 4
x2
at the point x = 4..
14. If the function
r, , sin4x
f (x) =--+ a, for x > 0
5 x
= x + 4 - b = 1 ,
for x < 0 for x = 0
is continuous at x = 0 , then find the values of a & b.
15. If f(x) is continuous on [-n, n]
-n
2
f (x) = -2sin x,
= a sin x + f ,
= cos x,
Find a & f.
for -n < x <
-n n
for < x < 2 2
n
for < x < n 2
Define differentiability of a function at a point and show thatf (x) = |x| is
16.
continuous, but not derivable at the point x = 0.
Discuss the applicability of Rolles Theorem for the function f (x) = (x - a)m (x - b)n defind in[a,b] where m, n are positive integers.
f (x) = ex (sin x - cos x) in
n 5n
4 4
19. Verify Langranges Mean Value theorem for the function f (x) = (x -1)( x - 2)( x - 3) defined in the interval [0,4].
20. Find 0 that appears in the conclusion of Langranges Mean Value theorem
3 1 for the function f (x) = x , a = 1, h = 3.
b - a , , -1 b - a
21. Show that -- < tan b - tan a <--, if 0 < a < b.
1 + b 1 + a
And hence deduce that + < tan-11 | < + -1
4 25 3) 4 6
22. For 0 < a < b , Prove that 1 - a < logb b -1 and hence show that
b a a
1,61 -< log <6 5 5
, b - a . , 7 . -1 b - a
23. If < a < b <1 , then prove that , < sin b - sin a <
n 1 -1 1 n 1
Hence show that---1= < sin <---;=
6 2V3 4 6 V15
24. Show that-- < tan 1 x < x, x > 0
1 + x2
x2 x2
25. For x > 0 , prove that x--< log(1 + x) < x--
2 2(1 + x)
26. Separate the interval in which f (x) = x3 + 8 x2 + 5x - 2 is increasing or decreasing.
x
27. Show that -< log(1 + x) < x, Vx > 0
1+x
f (x) = sinx, g (x) = cosx in 0 < x <
2
|
sina-sin B Show that -= cot 6, 31. cos B - cos a |
where 0 <a <6 < B < 2 |
If f (x) = -1 and g(x) = in Cauchys Mean Value Theorem, Show that
32.
33.
x x
C is the harmonic mean between a & b.
Discuss applicability of Cauchys Mean Value Theorem for the function f (x) = sinx and g (x) = cosx in [a,b] .
34. Verify Cauchys Mean value theorem f (x) = Vx, g (x) = -1= in [a, b]
yjx
35. Find c e (0,9) such that
f (9) - f (0) = f '(C) g (9) - g (0) g '(c)
where f (x) = x3 and g (x) = 2 - x
36. Discuss the applicability o Rolles Theorem for the function
x2 +12
f (x) = log
x
in (3,4) .
37. Verify Langranges Mean Value theorem for the function
f (x) = x( x -1)( x - 2) in
'5
38. Discuss the applicability o Rolles Theorem for the function
-n n 2 , 2
f (x) = ex cos x in
39. Verify Langranges Mean Value theorem for the function f (x) = 2x2 -10x + 29 in[2,7] .
1. If a function f is continuous on a closed and bound interval [ a, b] ,then show that f is bounded on [a, b].
2. Show that every continuous function on closed and bounded interval attains its bounds.
3. Let f: [a,b] R be a continuous on [a ,b] and if f (a) < k < f (b), then show that there exists a point c e (a, b) such that f (x) = k.
4. If f (x) is continuous in [a, b] and f (a) * f (b) , then show that f assume every value between f (a) and f (b).
5. If a function is differentiable at a point then show that it is continuous at that point. Is converse true? Justify your answer.
6. State and Prove Rolles theorem OR If a function f(x) defined on [a,b] is
i)continuous on [a,b] ii) Differentiable in (a, b) iii) f (a ) = f( b)
then show that there exists at least one real number c e (a, b) such that f(c)=0.
7. State and Prove Langranges Mean Value Theorem. OR If a function f(x) defined on [a,b] is i) continuous on [a,b]
ii) differentiable in (a, b) then show that there exixt at least one real number c e (a, b) such that
f ,c)=mzm
b - a
8. State and Prove Cauchys Mean Value Theorem. OR
If f(x) and g(x) are two function defined on [a,b] such that
i) f(x) and g(x) are continuous on [ a, b]
ii) f(x) and g(x) are differentiable in (a,b)
iii) g'(x) * 0, V x e (a, b)
then show that there exist at least one real number c e (a, b) such that
f '(c) = f (b) - f (a) g '(c) g (b) - g (a)
9. State Rolles Theorem and write its geometrical interpretation.
10. State Langranges Mean Value Theorem and write its geometrical interpretation.
11. If f(x) is continuous in [a,b] with M and m as its bounds then show that f(x) assumes every value between M and m.
12. Using Langranges Mean Value Theorem show that
cos aO - cos bO
< b - a, if O 0
O
13. If f(x) be a function uch that f'(x) = 0, Vx e (a, b) then show that f(x) is a constant in this interval.
14. If f(x) is continuous in the interval [a,b] and f'(x) > 0, Vx e (a, b) then show that f(x) is monotonic increasing function of x in the interval [a,b].
15. If a function f(x) is such that i) it is continuous in [a, a+h]
ii) it is derivable in (a, a+h)
iii) f(a) = f(a+h)
then show that there exist at least one real number O such that f'(a + Oh) = 0, where 0<O <1.
16. If the function f(x) is such that i) it is continuous in [a, a+h]
ii) it is derivable in (a, a+h) then show that there exists at least one real number O such that
f (a + h) = f (a) + hf '(a + Oh), where0 <O< 1
17. If f(x) is continuous in the interval [a,b] and f'(x) < 0, Vx e (a, b) then show that F(x) is monotonic decreasing function of x in the interval [a, b].
Successive Diff. And Taylors Theorem, Asymptotes, Curvature and Tracing of Curves
1. State Leibnitz theorem for the nth derivative of product of two functions.
2. Write nth derivative ofeax.
3. Write ntth derivative of sin(ax + b).
4. Write ntth derivative of cos(ax + b).
5. State Taylors theorem with Langranges form of reminder after nth term.
6. State Maclaurins infinite series for the expansion of f(x) as power series in [0,x].
7. Define Asymptote of the curve.
8. Define intrinsic equation of a curve.
9. Define point of inflexion.
10. Define multiple point of the curve.
11. Define Double point of the curve.
12. Define Conjugate point of the curve.
13. Define Curvature point of the curve at the point.
( 4- marks each)
Q-2 Examples
1 If y = 3 o 2- fnd yn
x + 2 x - x - 2
2. If y = ea cos2 x sin x, find yn.
3. If y = x2sin(3x + 7), find y8.
y = (sin-1 x)2 Provethat
4. If
(1 - x2)yn+2 - (2n + 1)yn+1 - n"yn = 0
If y = cos(m sin 1 x) Prove that
5.
(1 - x2)yn+2 - (2n + 1)xyn+1 + (m2 - n2)yn = 0
If y = tan(log y) Prove that
6.
(1+x 2) yn+1 +(2 nx -1) yn + n( n -1) yn -1 =0
1/ -1/
If y +y =2x Prove that (x2 -1)yn+2 + (2n +1)xyB+1 + (n2 - m2)yn = 0
7.
If cos 1 (yb) = log () Prove that x yn+2 + (2n + 1)xyn+1 + 2n" yn = 0
8.
x2
9. Find yn if y =-
n (x + 2)(2 x + 3)
10. Find yn if y = cos4 x
If y = a cos(log x) + b sin(log x) Prove that
11.
x yn+2 +(2n+1) xyn+1 +(n +1) yn = 0
If y = tan 1 x Prove that
(1 + x 2) yn+2 + 2( n + 1) xyn+1 + n( n + 1) yn = 0
13. Find yn if y = ex log x
14. Find yn if y = cos x cos 2x cos3x
If y = sin2 x cos2 x Prove that
15.
 Paper 2_files/12868-22736-FYBSC (Mathematics) Paper 2-3.png)
If y=(x2-1)n Prove that
16. 2
(x - 1)yn+2 + 2xyn+1 - n(n + 1)yn = 0
If y = emcos"x Prove that (x 2 -1) yn+2- (2n+1) xyn+i- (n 2+m2) yn =0
18 If y = ( x + v x2 - a1) Prove that
(x2 - a2)yn+2 + (2n +1)xyn+i + (n2 - m2)yn = 0
19 If y = sin(m sin-1 x) Prove that
(1 - x2)yn+2 - (2n + 1)xyn+1 - (n2 - m2)yn = 0
If y = cos(log x) Prove that x" yn+2+(2n+1) xy+1 +(n 2 +1) yn =0
21. Use Taylors theorem to express the polynomial 2 x3 + 7 x2 + x - 6 in powers
of ( x-2 ) .
 Paper 2_files/12868-22736-FYBSC (Mathematics) Paper 2-4.png)
22. Expand sinx in ascending powers of (x -;
23. Assuming the validity of expansion , prove that
ex cos x = 1 + x-------+
3 6 30
x2 2 x4 16 x5
sec x = -
2! 4! 6!
25. Expand log(sinx) in ascending powers of ( x- 3).
26. Expand tanx in ascending powers of (x - )
27. Prove that tan-1 x = x -1 x3 +1 x5........ and hence find the value of n
3 5
28. Prove that sin 1 x = x + 12. + 12 32 + 12 32 52 =----
3! 5! 7!
2 (tan x - sin x)-x
29. Use Taylor s theorem ,Evaluate lim-5---
x0 x
30. Expand ex in ascending powers of ( x- 1).
31. Expand 2 + x2 - 3 x5 + 7x6 in power of ( x-1 ).
32. Obtain by Maclurins theorem the first five term in the expansion of log(1 + sin x).
33. Obtain by Maclurins theorem the expansion of log(1 + sin2 x) upto x4.
34. Assuming the validity of expansion , prove that
esin x = 1 + x +1 x2-1 x4 +-----
2 8
x
35. Find the asymptotes of the curve y =
x2 - 4
x2
36. Find the asymptotes of the curve y = x - 2
y/x2 + 9
/x" +
37. Find the asymptotes of the curve y = Wx2 -.
38. Find the asymptotes of the curve x = t, y = t + 2 tan-11
-x3
x2
39. Find the asymptotes of the curve y = - 2 4
x3
40. Find the asymptotes of the curve y = - 2
x + x - 2
41. Find the differential arc and also the cosine and sine of the angle made by the tangent with positive direction of X-axis . for the curve y4 = 4ax.
42. Find the differential of the arc of the curve r = a cos2 (92) Also find the sine ratio of the angle between the radius vector and the tangent line.
43. Find the point on the parabola y2 = 8x at which curvature is 0.128.
44. Find the curvature of r2 = 2a2 cos 29, at 9 = n.
45. Find the curvature and radius of curvature at a point t on the curve, x = a(cos t +1 sin t), y = a(sin t -1 cos t).
46. Find the curvature of the curve, y = x - x2 at P(1,0).
47. Find the curvature of the curve, y = x5 - 4x6 -18x2 at origin.
48. Find the curvature of the curve, y3 = x at P(1,1).
2
49. Examine for concavity and point of inflection of Guassian Curve y = e~x
50. Trace the curve y = (x -1)2 (x + 2)
51. Trace the curve y = x(1 - x)3
52. Find the asymptotes parallel to co-ordinate axes for the curve
y 2(x2 - a2) = x
53. Find the radius of curve of y = c tan .
54. Show that the curvature of the point (32, 32) on the folium
3 3 o -8V2
x + y = 3axy is -
3a
Integration of Irrational Algebraic and Transcendental Functions, Applications of Integration
Q-1
Marks - 02
The proper substitution for
dx
the integral of the
type
1
is -
(px + q)y[ax + b
dx
3. Evaluate
4. Evaluate
5. Evaluate
6. Evaluate
(1 - 3 x)yj x + 2
dx
(2 - x)V 1 - x
dx
W 3x + 2
dx
(1 - 2 x)yf2.
7. Evaluate
dx
(2x - 3)Vx
(4x + 1)V x - 2 cos x.dx
(2 sin x - 1)V2 - sin;
dx
(2ex + 3)Vex - 4
exdx
8. Evaluate
9. Evaluate
10. Evaluate
/ 2
11. Reduction formula for j sinn xdx
is----
2
13. Evaluate j sin6 xdx
2
14. Evaluate j sin7 xdx
. 2
15. Reduction formula fpr j cosn xdx
2
16. Evaluate j cos8 xdx
2
17. Evaluate j cos9 xdx
/ 4
18. Evaluate j sin4 2xdx
n
x
19. Evaluate j sin5 dx
o 2
20. Evaluate j r
a x5
a2 - x2
CO
dx
21. Evaluate j
o (a2 + x2)4
22. Evaluate j
dx
o(1 + x2)52
23. Evaluate j sin3 x. cos4 xdx
24. Evaluate j sin6 x. cos5 xdx
25. Evaluate j sin4 x. cos6 xdx
26. Evaluate j sin5 x. cos7 xdx
27. Evaluate j sin5 x. cos4 xdx
0
28. Evaluate j sin8 x.cos5 xdx
0
29. Evaluate j sin4 x. cos8 xdx
0
30. Evaluate j sin5 x. cos9 xdx
0
31. The proper substitution for the integral of the type
dx
(px + qx + r )V ax + bx + c
32. The length of the arc of the curve y = f(x) between the points x = a , x = b is given by S = ................ with usual notation.
33. The length s of the arc of the curve x = f (t) ,y = /(t) between the points where t = a , t = b is given by S = ------------ with usual notations.
34. The equation of the Catenary is ..........
35. The equation of the Astriod is ..........
36. The volume of the solid generated by revolving about X-axis , the area bounded by the curve y = f(x) , the X- axis and the ordinate x = a , x = b is given by V = -------------with usual notation .
37. The volume of the solid generated by Revolving about X-axis ,the area bounded by the curve x = g (y) , the Y-axis and the abscissas y = c , y = d is given by V= ---------------with usual notation .
38. The volume of the solid generated by revolving about X-axis , the area bounded by the parametric curve X = (t) ,Y = / (t ) and the ordinate t = a , t = b is given by V =.............with usual notation .
39. The volume of the solid generated by revolving about Y-axis , the area bounded by the parametric curve Y= (t) ,Y = / (t ) and the abscissas t = a , t = b is given by V =.............with usual notation.
40. The volume of the solid generated by revolving about X-axis , the area bounded by the curve Y1 = (x ) ,Y2 = / (x ) and the ordinates x = a , x = b is given by V =.............with usual notation.
41. The Volume of the sphere of radius a is ..............
2 2 x V
42. The volume of the ellipsoid formed by revolving the ellipse + = 1
b2
about Y-axis is
43. The area of the curved surface of the solid generated by revolving about X-axis , the area bounded by the continuous curve y = f( x) , the X-axis and the ordinates x = a , x = b is S=.............
44. The area of the curved surface of the solid generated by by revolving about Y-axis , the area bounded by the continuous curve g = f( y) , the Y-axis and the abscissae y = c , y = d is S=.............
45. The area of the curved surface of the solid generated by by revolving about X-axis , the area bounded by the curve x = (t), y = / ( t) , the X-axis and
the ordinates t = a , t = b is S =............. where =-----
dt
46. The surface area of the sphere of radius a is .........
47. Write down the parametric equation of the cycloid.
1 xB
48. brrdx=J----
o V1 X 0
n
TO 1 /2
49. J-2 dx = J----
id+x2)B J
50. Define i) A rectification
ii) A cap of the sphere.
Integral of the form j
dx
(px2 + qx + r )V ax + b
1. Evaluate j -dx;= (x +1 )vx
2. Evaluate
6. Evaluate
Integral of the form j
|
dx | |||||||||||||||||
|
3. Evaluate 4. Evaluate 5. Evaluate |
| ||||||||||||||||
|
(x2 - 4x + 5)Vx - 2 dx | |||||||||||||||||
(px + q)V ax2 + bx + c dx
(x > 1)
|
8. Evaluate 9. Evaluate 10. Evaluate 11. Evaluate 12. Evaluate 13. Evaluate |
|
14. Evaluate j j=
xs 1
dx
- 2 x - x2 dx
15. Evaluate j
Integral of the form j dx
(px2 + qx + r )V ax2 + bx + c
16. Evaluate j-dx,
J (1 + x2)yj\
x2
17. Evaluate j d
(x2 + 4)V x2 +1 dx
18. Evaluate j
(x2 - 1)Vx2 +1 dx
19. Evaluate j-.
(x2 + 2)V x2 +1
Reduction formula type examples-
Xj3 d , f dx
20. Evaluate j
(1 + x 2)-\/1 - x2
21. Evaluate j x2(1 - x)2 dx
0
a
22. Evaluate jx4Va2 -x2dx
23. Evaluate I f=
1 x4
1
26. Evaluate j x1 - x2dx
27. Evaluate j x7J-- dx
o '1 - x
4
28. Evaluate j W 4 x - x2 dx
OT 3
x
29. Evaluate j- dx
o(1 + x2)72
OT 2
x
30. Evaluate j- dx
J . 2\/A
(1 + x2/2
o
OT 7
x
31. Evaluate j- dx
J /'x . 2\ A
o
OT 5
x
32. Evaluate j-dx
J /'x . 2 \ />
(1 + x2/2
o
OT 4
x
33. Evaluate j-Tirdx
o (1 + x2)4
34. Evaluate j-t-tdx
36. Evaluate 11 ' dx
o 11 + x2
4
37. Show that j x24 x - x2 dx = 1on
5n
38. show that j x2\J x - x2 dx =
I 128
39. show that j W 6 x - x2dx =
40. show that j x3*Jl x - x2 dx = J 8
27n
Let In = J : dx ,n> 1 show that
41.
J sin*
2sin(n-l)x
In-1 Where n is a positive integer.
In =
n1
sin 5 x sin3x
= 2
sin x
, rsin6x 7
Show that I-dx
sin x
_, , } sin 6x 7
42.
Show that j-dx = 0
Hence
sin x
|
sin7x Show that j-dx = 2 J sin x |
sin 6 x sin 4 x sin 2 x x --1---1---1-- 6 4 2 2 |
43.
Hence Show that jsin7 x dx sin x
f sin
Let I22 = j :
J C1
Show that I22 = 2
sin 22 x
sin x
' = n
dx,
44.
sin21x sin19 x
21
45. Show that jsin5x dx = sin 2x(3 - 2 sin2 x) + x J sin x
1. Evaluate j sinm x. cos xdx, where m, n are positive integers.
2. Evaluate j sin xdx, where n is positive integers.
Vi
3. Evaluate j cos xdx, where n is positive integers.
/2
4. Evaluate j (sin x)m.(cos x) dx, where m and n are positive integers.
to i
5. Evaluate j--dx, where n is a positive integers.
J /I 2\ + /2
0 (1 + x ) /2
Application of Integration.
Rectification -
6. Show that the length of an arc of the parabola y2 = 4ax cutoff by the y = 2x is "J2 + log(1 + >/2)
7. Show that the length of an arc of the parabola x2 = y form the vertex to any
extremity of the latus rectum is +1 log(1 + -n/2).
2V2 4
8. Show that the length of the arc of the curve y = x2 cutoff by the line x - y = 0 is 1 2>/5 + log(2 + V5) .
C ( x/ -x/\
9. Find the length of an arc of the catenary y = (e/c + e /C) measured from the vertex (0 , c) to any point (x, y ).
10. Find the length of an arc of the curve y = sin 1 ex between the points where
n , n
y = and y = .
6 2
11. Using theory of integration , obtains the circumference of the circle
x2 + y2 = 25.
12. Find the length of an arc of the cycloid x = a(0- sin 6), y = e6\ cos - 2sin | between the cups 6 = 0 and 0 = 2n.
13. Find the length of an arc of the curve x = e sin + 2cos |,
y = e6 cos 66 - 2sin | between the cups 6 = 0 and 6 = n.
14. Find the length of an arc of the curve x = a(2cos6- cos26), y = a(2 sin 6 - six26), measured from the points, where 6 = 0 and 6 = n is 89.
15. Find the length of an arc of the curve x = a(cos6 + 6sin6), y = a (sin 6-6 cos 6), from the points, where 6 = 0 and 6 = 2n is 2n2 a .
Volumes of Solids of Revolution
16. Using theory of integration , show that the volume of sphere of radius a is
4 3
na cubic units.
3
x2 v2
17. Show that the volume of solid genered by revolving the ellipse 2 + ? = 1 ,
a b
4 2
about X-axis is nab cubic units .
3
18. Find the Volume of the solid formed by revolving the arch of the cycloid x = a(6 - sin 6), y = a(1 - cos 6) about its base.
19. The area enclosed by the hyperbola xy = 12 and the line x + y =7 is revolved
n
about X-axis , Show that the volume of the solid generated is cubic units
20. Compute the volume of the solid generated by revolving about Y-axis , the region enclosed by the parabolas y = x2 and 8x = y2.
Areas of surface s of revolution-
21. The are of the parabola y2 = x between the origin and the point (1,1) is revolved about X-axis , Find the area of the surface of revolution of the solid formed .
22. Find the surface area of the solid generated by the revolution about the X-axis
13
of the loop of the curve x = t2, y = t -
3
.2
23. The arc of the parabola y = 4 x between its vertex and an extremity of its latus rectum revolves about its axis. Find the surface area traced out.
24. If the segment of a straight line y = 2x between x = 0 to x = 1 is revolved about Y-axis .show that surface area of the solid so formed is 4>/5n square units .
25. Find the area of the surface generated when the segment of the straight line y = x between x = 0 to x = 1 is revolved about Y-axis.
Differential Equation of First Order & First Degree
Q-1 04 or 06 Marks
1. Explain the method of solving homogeneous diff. Equation of the type Mdx + Ndy = 0, where M = M(x, y), N = N(x, y)
2. Explain the method of solving non-homogeneous diff. Equation
dy = a1x+b1y+c1 , where a1, b1, c1, a2, b2, c2 are real numbers. dx a2x+b2y+c2 2 2 2
3. Explain the method of solving exact diff. Equation Mdx + Ndy = 0, where M = M(x, y), N = N(x, y)
l
4. If the diff. Eq. Mdx + Ndy = 0 is homogeneous then - = 0 is an
M J Mx+Ny
integrating factor, where Mx + Ny 0 and M = M(x, y), N = N(x, y)
5. If the diff. Eq. Mdx + Ndy = 0 is of type f1(x, y)ydx + f2(x, y)xdy = 0 then
l
-= 0 is an integrating factor, where Mx - Ny 0.
Mx-Ny & & J
dM dN
6. IF dy n dX is a function of x alone then e$ fWdx is an integrating factor of equation Mdx + Ndy = 0 where M = M(x, y), N = N(x, y)
dN M
7. IF is a function of y alone then e$ f(y)dy is an integrating factor of
equation Mdx + Ndy = 0 where M = M(x, y), N = N(x, y)
dy
8. Solve the linear diff. Equation + Py = Q , where P & Q are functions of x
dx
only.
dx
9. Solve the linear diff. Equation + Px = Q , where P & Q are functions of y only.
10. Explain the method of solving the diff. Equation F(x, y, p ) = 0, which is
solvable for p, where p = .
dx
11. Explain the method of solving the diff. Equation F(x, y, p ) = 0, which is
solvable for y, where p = .
dx
12. Explain the method of solving the diff. Equation F(x, y, p ) = 0, which is
solvable for x, where p = .
dx
Solve the following differentials equations
1. sec2x tany dx + sec2y tanx dy = 0
2. y sec2x dx + (y+7) tanx dy = 0
dy x(2logx+l)
3. =-:--+ycosy
dx siny
5. (x2-yx2)dy + (y2+xy2)dx = 0 Solve the homogeneous diff. Eq.
6. (x3+y3)dx - 3xy2dy = 0
7. x2dy + (y2-xy)dx = 0
8. (x2+xy-y2) dy + (2xy -3y2)dx = 0
9. xdy - ydx = jx2 + y2dx
10. x2 = y(x+y)/2
11. (x2-y2) dx + 2xy dy = 0 dy (x2 - xy +y2)
12.
dx xy
dx 2xy
! ,2,2 dy
14. (x +y) =xy
15. ( x + y cotx/y ) dy - y dx = 0 Solve the Non-homogeneous diff. Eq.
16 dy = (2x- 5y+3)
" dx (2x+4y-6)
17. (2x - y + 1) dx + (2y - x - 1)dy = 0
dy (6x - 4y+3)
18.
dx (3x-2y+l)
19 dy = (x+ y+1)
. dx (x+y1)
20 dy = (x +2y+l) " dx (2x+4y-6)
21 = (x+ 2y +3)
" dx (2x+3y+4)
22 dy = (y- x +1)
" dx (y+x+5)
23 dy = (2x - y+l)
" dx (x+2y-3)
24 = (4x-6y +3)
" dx (6x-9y-l)
Solve the exact diff. Eq.
25. (2x2 + 3y)dx + (3x + y - 1) dy = 0
dy y cosx+siny+y
26. + -- = 0
dx sinx+xcosy+x
27. (x2 + y2 - a2) x dx + (x2 - y2 - b2) y dy = 0
X X
28. (1 + ey) dx + [ey (1 - x/y] dy = 0
29. (secx tanx tany - ex) dx + secx sec2x dy = 0
30. (x2 - 4xy - 2y2) dx + (y2 - 4xy + 2x2) dy = 0
31. (ey + 1) cosx dx + ey sinx dy = 0
32. (sinx cosy + e2x) dx + (cosx siny + tany) dy = 0
33. [x jx2 + y2 - y] dx + [y*2 + y2 - x] dy = 0
34. [cosx tany + cos(x + y)] dx + [sinx sec2y + cos(x + y)] dy = 0
Solve the Non-exact diff. Eq.
35. (x2y - 2xy2) dx - (x3 - 3x2y) dy = 0
36. (x2 - 5xy + 7y2) dx + (5x2 - 7xy) dy = 0
37. (x2y2 + 4xy + 2) x dx - (x2y2 + 5xy + 2 ) ydy = 0
38. (3xy2 - y3) dx - (2x2y - xy2) dy = 0
39. (1 + xy) ydx + (1 - xy) xdy = 0
40. (xy sinxy + cosxy) ydx + (xy sinxy - cosxy) xdy = 0
41. y(xy + 1) dx + x(1 + xy + x2y2) dy = 0
42. (xy + 2x2y2) ydx + (xy - x2y2) xdy = 0
43. (1/x+y) dx + (1/y-x) dy = 0
44. (x4y4 + x2y2 + xy) ydx + (x4y4 - x2y2 + xy) xdy = 0
45. (x2 + y2) dx - 2xy dy = 0
46. (x2y2 + 2xy + 1) ydx + (x2y2 - xy + 1) xdy = 0
47. (1 + xy) ydx + (1 - xy) xdy = 0
48. (xy3 + y) dx + 2(x2y2 + x + y4) dy = 0
49. (y4 + 2y) dx + (xy3 + 2y4 - 4x) dy = 0
50. (x - y2) dx + 2xy dy = 0
51. (3x2y4 + 2xy) dx + (2x2y3 - x2) dy = 0
52. (x2y + y3) dx + (2/3 x3 + 4xy2) dy = 0
53. (x4ex - 2mxy2) dx + 2mx2y dy = 0
54. (x2 + y2 + x) dx + xy dy = 0
55. (x2 + y2 + 2x) dx + 2y dy = 0
56. (x - y2) dx + 2xy dy = 0
57. (x3 + xy4) dx + 2y3 dy = 0
58. (2y2 + 3xy - 2y + 6x) dx + x(x + 2y - 1) dy = 0
59. 2y (x + y + 2) dx + (y2 - x2 - 4x - 1) dy = 0
60. (7x4y + y + 2) dx + (x4 + xy) x dy = 0
Solve the Linear diff. Eq.
61. - 2y = e2x dx
2 5
62 .--+ x y = x
dx
dy
63. sinx--+ 3y = cotx
dx
64. + 2xy + xy4 = 0 dx
65. 3y2 + 2xy3 = 4xe*2
dx
66. (x2y3 - xy) dy = dx
dy 3 -*2
67. xy - = y e
dx
68. = x(x2 - 2y) dx
69. = (2x + 3y - 7) 2 dx
dy
70. cosx + 2y sinx = sinx cosx
dx J
71. p2 - 5p + 6 = 0
72. p - 1/p = x/y - y/x
73. p(p + y) = x(x+ y)
74. p(p - y) = x(x+ y)
75. p2 - 7p + 12 = 0
76. 2y = ax/p + px
77. 4y = x2 + p2
78. 3x - y + logp = 0
79. y = 2px + x2p4
80. y - 2px = f(xp2)
81. y = 2px + p2y
82. p3 - 2xyp + 4y2 = 0
83. y = 3px + 6y2p2
84. y = 2px + y2p3
85. xyp2 + (x2 + xy + y2)p + x(x + y) = 0
86. 3x - y + log p = 0
87. y = (1 + p)x + p2
88. y2 logy = xyp + p2
3
89. xp = m + np
Write the definition of following
1. Homogeneous differential equation
2. Non- homogeneous differential equation
3. Exact differential equation
4. Linear differential equation
5. Bernaolls differential equation
6. Clarauts differential equation
Find the integrating factor of the following differential equation
7. (1 + y2) dx + (x - etan~ly) dy = 0
dv 4x 1
8. --1--y =-
dx x2+ 1 (x2+ l)3
dy 1
9. - -tany = (1 + x) ex secy
QX
dy
10. (x cosx) + (x sinx + cosx) y = 1
n dy 3 3
11. = xy - xy dx
12. (xy3 + y) dx + 2(x2y2 + x + y4)dy = 0
13. (x2 + y2 + 2x) dx + 2ydy = 0
14. (y4 + 2y) dx + (xy3 + 2y4 - 4x) dy = 0
Multiplying by appropriate integrating factor, make following diff. Eq. Exact.
15. (x2y2 + 2) ydx + (2 - 2x2y2) dy = 0
16. (x2y2 + xy + 1) ydx + (x2y2 - xy + 1) dy = 0
17. (3xy2 - y3) dx - (2x2y - xy2) dy = 0
18. (x2 + y2) dx - 2xy dy = 0
19. (7x4y + 2xy2 - x3) dx + (x4 + xy) xdy = 0
20. (x2 + y2 + x) dx + xy dy = 0
Write the appropriate answer of the following, where P & Q are functions of x only.
|
21. The diff. Eq. + Py dy = Q is dx A) Linear D. E. B) C) Exact D. E. D) 22. The diff. Eq. (x2 + y2) = xy is -- dx A) Linear D. E. B) C) Bernaolls D. E. D) 22. The diff. Eq. (1 + xy) ydx + (1 - xy) xdy = A) Not exact D. E. B) C) Linear D. E. D) dy 2 + - .7 dx x+1 y3 24. The diff. Eq. 3 + y = is --- A) Not exact D. E. B) C) Linear D. E. D) 25. The diff. Eq. y = px + J4 + p2 is A) Non-homogeneous D. E. B) C) Bernaolls D. E. D) |
Bernaolls D. E. Not exact D. E. Homogeneous D. E. Non- homogeneous D. E. 0 Clairauts D. E. Non- homogeneous D. E. Clairauts D. E. Homogeneous D. E. Clairauts D. E. Homogeneous D. E. |
Differential Equations
Q-1. Questions 2 - Marks
1. Let f( D)y = X be the L.D.E. If x = 0 with constant coefficient. Then
i) f ( D)y = 0 is called
ii) f (D ) = 0 is called --
2. If m1,m2,-----mn are n distinct real roots of A.E. f(D) = 0 then G.S. of
the equation f(D)y = 0 is --
3. If m1 = m2 two root of f(D) = 0, then C. F. of f(D)y = 0 is
4. If m1 = a + ip and m2 = a-ip are the complex roots of the f(D) = 0 , then G.S. of f(D)y = 0 is --
5. If f (D) = (D-m1)(D-m2)-----(D-mn) , then the G.S. of the L.D.E.
f(D)y = 0 is --- .
6. If -j- + 4 + 4 y = e2 x ,then what is its complementary function ?
dx dx
7. If f (D2) is polynomial in D2 with constant coefficients and
F(-a)2 0 then i) cos (ax + b) = ?
f\P)
ii) ysin (ax + b) = ?
8. If D =d and f(D) is a polynomial in D with constant coefficients then
dx
i)Dex V = ?
where V is function of x .
ii)1 xV = ? f (D)
i) cos(ax) = ? (D2 + a 2)r
9. V '
ii) 2- sin(ax) = ?
(D2 + a2)
10. Let (D2 + 4) y = cos 2 x , find P.I.
11. If D = d and f(D) is a polynomial in D with constant coefficients then
dx
-L- eax = ?,/(a) * 0
i )-1-ea = ?
(D - a)r
ii)If f (D) = (D - a)rf(D) and tf>(a) 0, then ea = ?
1. Linear differential equation with constant co-efficients of order n.
2. Associated D.E. and Auxiliary equation.
3. Inverse Operator
4. Homogeneous Linear Differential equation of the order n.
Q-3. Multiple choices
1. If - 2 + 4y = e2 x is a linear differential equation ,then C.F. is
dx dx
a )(Cj + c2 x)ex
b)(c1 x + c2 x 2)ex
c)(c1 + c2)ex
d)none of these
2. If (D3 + 3D2 + 3D +1)y = e~x is a linear differential equation then C.F. is
a)(c1 x + c2 x + c3 x 2)e~ x
b)(c1 + c2 x + c3 x 2)e_ x
c)(c1 + c2 + c3 x)e_ x
d)none of these
3. If (D2 + 2D + 3) y = x - 2 x2 is a linear differential equation then C.F. is
a)e~x (c1 cosflx + c2 sin V2x)
b)e~x + (c1 cos V2x + c2 sin V2x)
c)e~x (c1 cosflx + ic2 sin yjlx)
d)none of these
4. If (D2 + 4) y = cos2x is a linear differential equation then C.F. is......
a)c1 cos 2 x + c2 sin 2 x
b)c1 cos 2x + ic2 sin 2x
c)c1 sin 2x + ic2 cos 2x
d)none of these
5. If (D2 + 2) = cos 2 x is a linear differential equation then P.I. is......
x sin\Jlx
a)
b)
c)
d)
sin-\/2x
x sin-\/2x
x cos 2 x
dx dx
is......
a) y = c1e2 z + c2e- z
b) y = c1e4 z + c2e~4 z
c) y = c1e2 z + c2e2 z
d)none of these
7. If (D2 + 4)2 y = cos2 x is a linear differential equation then C.F. is.......
a)(c1 + c2)cos2 x + (c3 x + c4 x)sin2 x
b)(c: + c2 x) cos 2 x + (c3 + c4 x) sin 2 x
c)(c x + c2 x2) cos 2 x + (c3 x + c4 x2) sin 2 x
d)none of these
d2 y
8. If 2 + 4 y = 0 is a linear differential equation then G.S. is.......
dx
a)A cos 2x + B sin 4x
b)A cos 2x + B sin 2x
c) A sin 2x + B cos 4x
d)none of these
9. If (D2 -6D +13)y = 0 is a linear differential equation then G.S. is.......
a)e3x (A cos 2x + B sin 2x)
b)e3x (A cos 4x + B sin 4x)
c)e3x (cos 2 x + B sin 2 x)
d)none of these
10. If x2 - 3x + 4y = 0 is a homogeneous L.D.E. , then G.S. is......
i) (c1 + c2 log x) x2
3 3x
ii) x e
iii) x e3z
iv) z2 ez
Q-4. Numerical Examples 04 Marks
1) Solve -5+6y = 0
dx2 dx
dy
dx3 dx
2) Solve4- -13- +12 y = 0
' t-3 j
3) Solve + 2 = 0
dx dx dx
4) Solve 2 + 5 dy - 12y = 0
dx2 dx
5) Solve + 4y = 0 dx4 '
6) Solve d2y + 4 + 4 y = e2
dx2 dx
7) Solve x2 -4- + x-4y = 0
dx2 dx
8) Solve + y = 0
dx2
9) Solve (D3 - 6D2 + 9D)y = 0
10) Solve (D4 + 8D2 +16) y = 0
11) Solve (D - 1)2(D2 +1) y = 0
12) Solve (D2 + 4) y = cos 2 x
13) Solve d2y - 2 +y = e=
dx2 dx
14) Solve - - 6y = ex cosh 2x
dx2 dx
15) Solve - 3 + 2 y = e5
dx2 dx
16) Solve 4 (- + 4 + y = 4e
dx2 dx
17) Solve - 9 y = e2 x + x2 dx
18) Solve - 5- + 6 y = x
J2z - 5 dy
dx2 dx
19) Solve di + dJy - iy - y = cosh x
dx dx dx
20) Solve (dxy - y = (1 + ex )2
+4 +4y
-2x 2
= e x + x
21) Solve
dx dx
22) Solve + 8 y = x4 + 2 x +1
dx3 '
23) Solve dy - 2 + 5y = x2
dx2 dx
d3y - 3 d2y + 3 dy_ dx3 dx2 dx
24) Solve f - 32. + 3- y = 2x3 - 3x2 +1
dh+6dy
ix3 dx'
j4,. >2.
25) Solve + 64 +12- - 8 y = e ~2x + x2 dx dx dx
26) Solve + 8 yy +16 = cos2 x
dx4 dx2
27) Solve - a4y = cos ax
dx
d4 y
28) Solve 4 + y = sin x sin 2 x
dx
d3 y
29) Solve - + y = cos 2 x
dx
j2y ?dy
-2 + 3
dx2 dx
30) Solve 2 + 3JL + 2 y = sin ex
2
, d2 y dy
31) Solve - 2--+ y = x sin x
dx2 dx
32) Solve ( D2 - 5D + 6 )y = e3x
33) Solve (D2 + 13D + 36)y = e~4x + sinh x
|
34) |
Solve |
(D3 + 3D2 + 3D +1) y = e~x | |
|
35) |
Solve |
(D3 |
- 5D2 + 8D - 4) y = e2 x |
|
36) |
Solve |
(D2 |
- 2D +1)y = ex |
|
37) |
Solve |
(D2 |
- 4D + 4) y = sinh 2x |
|
38) |
Solve |
(D3 |
- 4 D) y = 2 cosh 2 x |
|
39) |
Solve |
(D3 |
- 5D2 + 8D - 4) y = e2 x + 3ex |
|
40) |
Solve |
(D3 + 3D2 + 2D) y = x2 | |
|
41) |
Solve |
(D2 + 2 D + 3) y = x - 2 x2 | |
|
42) |
Solve |
(D2 |
- D - 2) y = 1 - 2x - 9e_ x |
|
43) |
Solve |
(D3 + 3D2 + 2D) y = x2 + 4 x + 8 | |
|
44) |
Solve |
(D2 |
- 4D + 4) y = 8( x2 + e2 x) |
|
45) |
Solve |
(D2 |
- 3D + 2)y = 2x2 - 9x2 + 6x |
|
46) |
Solve |
(D2 |
- 4D + 3) y = 2 cos x + 4sin x |
|
47) |
Solve |
(D3 |
+ D2 - D -1)y = sin x |
|
48) |
Solve |
(D3 + D) y = sin3x | |
|
49) |
Solve |
(D2 + 4) y = cos 2 x | |
|
50) |
Solve |
(D4 |
-1) y = cos x cos3x |
|
51) |
Solve |
(D2 + 4) y = sin 3x + ex + x2 | |
|
52) |
Solve |
(D3 + D) y = cos x | |
|
53) |
Solve |
(D2 |
-1) y = 10sin2 x |
|
54) |
Solve |
(D2 +1) y = 12cos2 x | |
|
55) |
Solve |
(D3 |
- D2 - 6D)y = cos x + x2 |
|
56) |
Solve |
(D3 |
- D2 - D +1)y = cosh x + sin x |
|
57) |
Solve |
(D2 |
- 2D + 2)y = x2e3x |
|
58) |
Solve |
(D2 |
- 4D + 3)y = ex cos 2x | |
|
59) |
Solve |
(D2 |
- 6D +13)y = e3x sin 2x | |
|
60) |
Solve |
(D2 |
- 2D + 4)y = ex cos2 x | |
|
61) |
Solve |
(D2 |
- 2 d+1) y=e | |
|
62) |
Solve |
(D2 |
-1)y = x2 cos x | |
|
63) |
Solve |
(D2 |
-1) y = x2 sin x | |
|
64) |
Solve |
(D4 |
-1) y = cos x cosh x | |
|
65) |
Solve |
(D4 |
-1)y = ex cos x | |
|
66) |
Solve |
(D3 +1)y = e2x sin x + e2 sin |
f x ] | |
|
2 V J | ||||
|
67) |
Solve |
(D3 |
- 7 D - 6) y = e2 x (1 + x2) | |
|
68) |
Solve |
(D4 |
- 2D3 - 3D2 + 4D + 4)y |
= exx2 |
|
69) |
Solve |
(D2 |
-1) y = x sinh x | |
|
70) |
Solve |
(D2 |
1 y = x | |
|
71) |
Solve |
(D2 +1)y = x cos2 x | ||
|
72) |
Solve |
(D2 + 4) y = x sin x | ||
|
73) |
Solve |
(D2 |
-1)y = x2 cos x | |
|
74) |
Solve |
(D2 +1) y = x cos 2 x | ||
|
75) |
Solve |
(D2 + 2D + 2) y = x cos x | ||
|
76) |
Solve |
(D2 + 3D + 2) y = x sin 2 x | ||
|
77) |
Solve |
(D2 + D) y = (1 + e )-1 | ||
|
78) |
Solve |
(D2 + 5D + 6) = e 2x sec2 x(1 + 2 tan x) | ||
|
79) |
Solve |
(D2 |
- 2D +1) = xex sin x | |
|
80) |
Solve |
(D |
- 9D +18) y = eex | |
|
81) |
Solve |
D2 + 3 D + 2 )y = sin ex |
|
82) |
Solve |
D2 + 3 D + 2 )y = ee* |
|
83) |
Solve |
D2 + 4 )y = tan 2x |
|
84) |
Solve |
D2 + 3 D + 2 )y = sin e -x |
|
85) |
Solve |
D2 - 2 D + 2 )y = x ex cosx |
|
86) |
Solve |
D2 - 1 )y = ( 1 + e-x ) -2 |
|
87) |
Solve |
x2D2 + x D - 4 )y = 0 |
|
88) |
Solve |
x2D2 - 3x D + 4 )y = 2x2 |
|
89) |
Solve |
D2 - 1/x D + 1/x2 )y = (2/x2) Logx |
|
90) |
Solve |
x2D2 - x D - 3 )y = x2 Logx |
|
91) |
Solve |
x2D2 - 3x D + 5 )y = x2 sin (Logx) |
|
92) |
Solve |
( 2x+ 1 )2 D2 - 2 (2x + 1 ) D - 12 ]y = 6x |
|
93) |
Solve |
(1 + x)2D2 + ( 1 + x ) D + 1 ]y = 4 cos [ Log (2 + x ) ] |
|
94) |
Solve |
x2D2 + 4x D + 2 )y = ex |
|
95) |
Solve |
( 2x- 1 )3 D3 + ( 2x- 1 ) D - 2 ]y = 0 |
|
96) |
Solve |
( 3x+ 2 )2 D2 + 3 (3x + 2 ) D - 36 ]y = 3x2 + 4x + 1 |
|
97) |
Solve |
( 1+x )2 D2 + (1+x ) D + 1 ]y = 2 sin[ log(1+x) ] |
|
98) |
Solve |
( x + 3 )2 D2 - 4 ( x + 3 ) D + 6 ]y = log ( x + 3 ) |
|
99) |
Solve |
( x + 2 )2 D2 - ( x + 2 ) D + 1 ]y = 3x + 4 |
1) If D = d and f(D) is a polynomial in D with constant coefficients ,then
dx
Prove that eax = eax, iff(a) 0 f{p) fid) i i j
2) Prove that
i eax = x_e_ (D - a)r r!
r ax
1 xreax
Hence- eax = , if f(D) = (D-a)rf(D) & f(a) * 0 f (D) r !f(a)
3) If f (D2) is polynomial in D2 with constant coefficients and f (-a2) 0
then prove that cos(ax + b) = cos(ax + b) f (D ) f (-a )
4) If f (D2) is polynomial in D2 with constant coefficients and f (-a2) 0
1 sin(ax + b)
then - sin(ax + b) =-;
f (D ) f (-a )
1 (-1)rxr f r I
5) Prove that ;-;cos ax =-cos I ax +--I, r e N
1 (-1)rxr . f rn i
6) Prove that ;-;sin ax =-sin I ax +--I, r e N
(D2 + a )r r !(2a)r 2 1
7) If D = d and f(D) is a polynomial in D with constant coefficients ,then
dx
Prove that 1 eacV = ea-1-V, where V is a function of x.
f (D) f (D + a)
8) If D = d and f(D) is a polynomial in D with constant coefficients ,then
dx
9) Define a homogeneous linear differential equations & explains the methods solving it.
tan 1 x
28. Show that -T <-< 1, Vx > 0
1 + x2 x
29. With the help of Langranges formula Prove that
a-B n a-B n -2 < tana- tan p<-2, where0 <p<a<
55. Find the point on the parabola y = 8x at which radius of curvature is 7 /16.
56. Examine the nature of the origin of x3 + y3 - 3axy = 0.
58. Trace the curve xy2 = a 2(a - x).
57. Trace the curve x3 + y3 = 3axy.
- V, where Vis a function of x.
f (D)
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Attachment: |
| Earning: Approval pending. |
