Math 1

Moment Of Truth

Math 1

Fashion For Next Generation

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UNIT-I

MATRICES

PART-A

1. Find the constants a and b such that the matrix

has 3 and -2 as its

eigen values.

− 2

2. The product of two eigen values of the matrix A = − 2

− 1

is 16,

− 1

Find

the third eigen value.

3. Find the sum and product of the eigen values of the matrix

− 2

A =

using properties.

− 2

− 1

− 3

4. If the sum of two eigen values and trace of a 3X3 matrix A are equal, find A .

4 1 3

5. Given that A = , find the eigen values of A .

3 2

				2	5	− 1

6. Find the eigen values of A^-1 if the matrix A is A = 0					3	2 .

				0	0	4
	2	2	1

7. Two eigen values of the matrix	A = 1	3	1	are equal to 1 each. Find

	1 2 2
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the eigen values of A^-1.

8.If 1 & 2 are the eigenvalues of a 2X2 matrix A, what are the eigenvalues of A² and A^-1.

9. Let λ be an eigen value of a non-singular matrix A with eigen vector x.

Show	that	1	is an eigen value of A^-1 with eigen vector x.

		λ

10.	If λ₁ , λ₂ , λ₃ ,………, λ_n				are the eigen values of an nXn matrix A, then show
	3	3		3	3
	that λ₁ , λ₂		, λ₃	,………, λ_n are the eigen values of A³.
						3	1	4

11.	Find the sum of the squares of the eigenvalues of					A = 0	2	6	.

						0	0	5

12. State Cayley-Hamilton theorem.

13. Give two uses of Cayley-Hamilton theorem.

					1	4
14.	using	Cayley-Hamilton theorem		find the inverse of		.

					2	3
				3		− 1
15.	Verify Cayley-Hamilton theorem for the matrix A =					.

				− 1		5
		1	0
16.	If A =		write A³ interms of A and I, using Cayley-Hamilton theorem.

		0	5

17. Define orthogonal matrix.

18. write down the quadratic form corresponding to the symmetric matrix

	2	1	− 2

	1	2	− 2	.

− 2		− 2	3

19. Classify the quadratic forms x₁² + x₃² and x₁² − x₂ ² .

20. Find the index and signature of the quadratic form x₁² + 2x₂ ² − 3x₃² .

PART-B
				11		− 4	− 7

1.a. Find the eigen values and eigen vectors of the matrix A = 7						− 2	− 5	and

				10		− 4	− 6
hence find the eigen values of A²,5A and A^-1 using properties.								(8)
		8	− 6	2

b. Find the eigen values and eigen vectors of	− 6		7	− 4	.			(8)
		2	− 4	3
		2	− 4	3

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															4		1		1

2.a. Find the eigen values and the eigen vectors of the matrix 1																	4		1 .			(8)

															1		1		4
																2			1	− 1

b. Find the eigen values and eigen vectors of the matrix A = 1																			1	− 2 .(8)

																− 1		− 2		1
																2		2	0

3.a. Find the eigen values and the eigen vectors of the matrix																2		1	1 .			(8)

															− 7			2	− 3
																1			1	1

b. Using Cayley-Hamilton theorem, find A^-1 given the matrix A = 1																			2	− 3 .(8)
																			− 1
																2				3
			7												3
4.a. Verify Cayley-Hamilton theorem for the matrix A =															and hence find A^-1

			2												6
and A³.																						(8)
			7												2	− 2

b. Using Cayley-Hamilton theorem, find A^-1 if A = − 6															− 1	2 .						(8)

			6												2	− 1
1	0		0

5.a. If A = 1	0		1 , find A^-1 and A⁴ using Cayley-Hamilton theorem.																			(8)

0	1		0
															1			2		− 2

b. Verify Cayley-Hamilton theorem and hence find A^-1 if A = − 1																		3		0 . (8)
																		− 2
															0					1
				2		2	0

6.a. Diagonalise				2		1	1 .							(8)

		− 7				2	− 3
									1		0		0

b.Diagonalize the matrix A = 0											3	− 1 using an orthogonal transformation. (8)
											− 1
									0				3
					6			− 2			2

7.a. Diagonalize A = − 2									3	− 1		by an orthogonal transformation.									(8)
								− 1
					2						3
					10				− 2		− 5

b. Reduce the matrix − 2									2		3		to diagonal form.								(8)

					− 5				3		5

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8.a. Reduce the quadratic form x ² + 2 y ² + z ² − 2 xy + 2 yz into canonical form.																			(8)
b. Reduce the quadratic form 10x₁² + 2x₂ ² + 5x₃² + 6 x₂ x₃ − 10x₃ x₁ − 4 x₁ x₂ .																			(8)
9. Reduce 6 x ² + 3 y ² + 3z ² − 4 xy − 2 yz + 4 xz into a canonical form by an
orthogonal reduction and find the rank, index, signature and the nature of																			the
quadratic form.																			(16)
10. Reduce the quadratic form x₁² + 5x₂ ² + x₃															²+ 2x₁ x₂ + 2x₂ x₃ + 6x₃ x₁			to canonical
form through an orthogonal transformation.																			(16)
11. Reduce the quadratic form												to canonical 3x ² + 5 y ² + 3z ² − 2 xy − 2 yz + 2 zx
form through an orthogonal transformation.																			(16)
12. Reduce the quadratic form 8x₁² + 7 x₂														²+ 3x₃² − 12x₁ x₂ − 8x₂ x₃ + 4x₃ x₁ to
	canonical form through an orthogonal transformation and hence show that it
	is positive semi-definite.																		(16)
													UNIT - II
							THREE DIMENSIONAL GEOMETRY
													PART – A
1. Find the angle between the lines ^x =														y	= ^z and	x − 4	₌ y − 1 ₌ z + 6
1. Find the angle between the lines ^x =														− 2	= ^z and		₌ y − 1 ₌ z + 6
												2		− 2	1	2	1	2
2. Find the angle between the lines whose direction cosines are
	1		1		1			1		1		1
		,		,		& −			,		,		.
		,		,		& −			,		,		.
	3		3		3			3		3		3

3. Find the direction cosines of the line joining the points (2,3,-6) and (3,-4,5).

4. Find the equation of the line joining the points (1,2,3) and (-3,4,5).

5. Find the acute angle between the lines			x	= ^y =	z	and
5. Find the acute angle between the lines				= ^y =	− 1	and
			1	2	− 1
x ₌	y	= ^z .
x ₌		= ^z .
2	1	1

6. Prove by direction ratios, the points (1,2,3), (4,0,4), (-2,4,2) are collinear.

7. Find the direction cosines of a line perpendicular to the two lines whose direction ratios are (1,2,3) and (-2,1,4).

8. Find the projection of the segment joining A(1,2,3) and B(6,7,9) on the line whose direction ratios are (1,2,-3).

9. Find the values of K, if the lines ^x ⁻ ² = ^y ⁻ ¹ = ^z ⁻ ³ and

		3	2	K
x − 3	= ^y ⁻ ² = ^z ⁻ ⁴ are coplanar.
	= ^y ⁻ ² = ^z ⁻ ⁴ are coplanar.
K	3	5

10. Find the centre and radius of the sphere 2( x ² + y ² + z ² ) + 6 x − 6 y + 8z + 9 = 0 .

11. Find the equation of the sphere concentric with

x ²+ y ² + z ² − 2 x − 2 y − 2 z − 1 = 0 and passing through the point (-2,1,-5).

12.	Find the equation of the sphere whose centre is same as that of	the
	sphere x ² + y ² + z ² − 2 x − 4 y − 6 z + 7 = 0 and which passes through the point
	(1,-1,1).
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13. Write down the equation of the sphere whose diameter is the line joining the points (1,1,1) and (-1,-1,-1).

14. Find the centre and the radius of the sphere

7( x ² + y ² + z ² ) + 28x − 42 y + 56 z + 3 = 0 .

15. Check whether the two spheres x ² + y ² + z ² + 6 y + 2 z + 8 = 0

and x ² + y ² + z ² + 6 x + 8 y + 4 z + 20 = 0 intersect each other orthogonally.

16. Find the equation of the sphere having the circle

x ²+ y ² + z ² = 9 and x − 2 y + 2 z = 5 as a great circle.

17. Find the equation of the tangent plane at the point (1,-1,2) to the sphere x ² + y ² + z ² − 2 x + 4 y + 6 z − 12 = 0 .

18. Write down the general equation of the cone whose vertex is at the origin.

19. Find the equation of the cone with vertex at the origin and which passes through the curve x ² + y ² = 4 , z = 2 .

20. Write down the equation of the right-circular cylinder whose axis is the z-axis and radius “a” units.

PART-B

1. a) Find the length and the equation of shortest distance between

	The lines ^x ⁻ ³ = ^y ⁻ ⁸ = ^z ⁻ ³ and				x + 3 ₌ y + 7 ₌ z − 6 _.(8)
	3	− 1	1		− 3	2		4
	b)Show that the lines		x − 4 ₌	y − 5	₌ z − 6	and	x − 2 ₌ y − 3 ₌ z − 4
	b)Show that the lines		x − 4 ₌		₌ z − 6	and	x − 2 ₌ y − 3 ₌ z − 4
			2	3	4		3	4	5
	are coplanar.							(8)
2.	a) Show that the lines		x − 5 ₌ y − 7 ₌ z + 3			and	x − 8 ₌ y − 4 ₌ z − 5
			4	4	− 5		7	1	3
	are coplanar and find their point of contact.							(8)
	b)Prove that the lines		x + 1 ₌ y − 3 ₌ _z ₊ ₂			and x = ^y ⁻ ⁷ = ^z ⁺ ⁷
			− 3	2				− 3	2
	intersect. Find the co-ordinates of the point of intersection and equation of the
	plane containing them.							(8)

3. a)Find the angle between the straight lines whose direction cosines are given

by the relation 3l+m+5n=0 and
6mn-2nl+5lm=0.	(8)

b)Find the equation of the sphere described on the line joining

the points (2,-1,4) and (-2,2,-2) as diameter. Find also the area of the circle in which the sphere is cut by the plane 2x+2y-z=3. (8)

4. a) Find the equation of the sphere passing through the points (1,1,-2) and (-1,1,2) and having its centre on the line

x+y-z-1=0=2x-y+z-2.	(8)
b) Find the equation of the sphere through the circle x ² + y ² + z ²		= 9
2 x + 3 y + 4 z = 5 and the point (1,2,3).	(8)
5. a)Show that the plane 2x-2y+z=9 touches the sphere
x ² + y ² + z ² + 2 x + 2 y − 7 = 0 and find the point of contact.	(8)
b)Find the equation of the tangent plane to the sphere
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MA1101-MATHEMATICS-I 3( x ² + y ² + z ² ) − 2 x − 3 y − 4 z − 22 = 0 at the point (1,2,3). Also find

the equation of the normal to the sphere at (1,2,3). (8)

6. a) Find the equation of the tangent plane to the sphere

x ²+ y ² + z ² − 2 x − 4 y − 6 z − 2 = 0 which passes through the line

9x-3y+25=0=3x+4z+9. (8)

b)Find the equation of the sphere which touches the plane 3x+2y-

z+2=0 at	the point (1,-2,1)and also cuts orthogonally the sphere
x ² + y ² + z ² − 4 x + 6 y + 4 = 0 .		(8)

7. a)Prove that a point at which the sum of the squares of whose distances from the planes x+y+z=0 , x-z=0, x-2y+z=0 is 9, lies on the sphere

	x ² + y ² + z ² = 9 .			(8)
	b)Find the area of the circle in which the sphere
	x ² + y ² + z ² + 12 x − 2 y − 6 z + 30 = 0 is cut by the plane x-y+2z+5=0. (8)
8. a)Find the centre and radius of the circle given by
	x ² + y ² + z ² + 2 x − 2 y − 4 z − 19 = 0 and x+2y+2z+7=0.			(8)
	b)Find the equation of the cone with vertex at the origin and the guiding
	curve is the circle x ² + y ² + z ² + 4 x + 2 y − 6 z + 5 = 0 ,2x+y+2z+5=0. (8)
9.	a)The plane ^x +	y	+ ^z = 1 meets the axes in A,B,C. Find the equation of the
9.	a)The plane ^x +
	a	b	c
	cone whose vertex is the origin and the guiding curve is the circle ABC. (8)
	b)Show that the equation to the right-circular cone whose vertex is O, axis
	OZ and semivertical is α is x ² + y ² = z ² tan ² α .			(8)

10. a)Find the equation of the cylinder whose generator are parallel to the line

b)Find the equation of the right-circular cylinder of radius 2 whose axis passes through the point (1,2,3) and has direction cosines proportional to 2,-3,6. (8)

UNIT III

DIFFERENTIAL CALCULUS

PART - A

1. Find the radius of curvature of the curve y = log sin x at x = π/2 .

2. Find the radius of curvature of the curve y = e^x at the point where it crosses the y-axis.

3. Find the radius of curvature of the curve xy = c² at (c,c).

4. Find the curvature at (3,-4) to the curve x² + y² = 25.

5. What is the curvature of x²+y²-4x-6y+10 = 0 at any point on it

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6. Define the curvature of a plane curve and what is the curvature of the straight line.

7. Find the radius of curvature at any point on the curve r = e^θ

8. Find the radius of curvature at y = 2a on the curve y² = 4ax.

9. Find the radius of curvature of the curve y = a cosh(x/a) at the point where it crosses the y axis.

10. Find the radius of curvature of the curve y = clog(sec(x/c)).

11. Find the radius of curvature at x = π/2 on the curve y = 4sinx – sin2x.

12. Write any two properties of evolute.

13. Define evolute.

14. Find the envelope of the lines x/t + yt = 2c, t being the parameter.

15. Show that the family of straight lines 2y – 4x + λ = 0 has no envelope where λ is the parameter.

16. Find the envelope of	^xcosθ +	^ysinθ = 1, where θ is the parameter.
	a	b

17. Find the envelope of the family of straight lines x cosθ + y sinθ = 6, where θ is the parameter.

18. Find the envelope of the family of straight lines x cosα + y sinα = a secα , where α is the parameter.

19. Find the envelope of the family 1- x²+(y-k) ² = 0 where k is the parameter.

20. Find the envelope of x² + y²-ax cosθ - by sinθ = 0 where θ is a parameter.

PART – B
1. (a) Find the radius of curvature of the curve x = 3acos θ – acos3θ,
y = 3asin θ – asin3θ at θ.	(8)

(b) Find the radius of curvature of the parabola x = at² , y = 2at at t. (8)

2. (a) For the curve x = a(cos θ + θ sinθ), y = a(sinθ – θ cosθ), prove that

the radius of curvature is aθ.	(8)
(b) Find the radius of curvature at any point (a cos3θ, a sin3θ) on the
curve x^2/3 +y^2/3 = a^2/3.	(8)
3. (a) Prove that the radius of curvature at any point of the cycloid
x = a(θ+sinθ), y=a(1- cosθ) is 4a cos θ/2.	(8)
(b) Find the center of curvature of the curve √x+√y = √a at (a/4,a/4).	(8)

4.(a)Find the circle of curvature of the parabola y² = 12x at the point(3,6). (8)

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	(b) Show that the circle of curvature of √x + √y = √a at (a/4,a/4) is
	(x-3a/4)² + (y-3a/4)² = a² /2.			(8)
5.	(a) Find the evolute of the hyperbola ^x ²	− ^y ² = 1 .		(8)
	a ²	b ²
	(b) Show that the equation of the evolute of the parabola x² = 4ay is
	4(y-2a)³ = 27ax².			(8)
6.	(a) Find the equation of the evolute of the parabola y² = 4ax.			(8)
	(b) Find the equation of the evolute of the ellipse ^x ²		+ ^y ² = 1 .	(8)
		a ²	b ²
7.	(a)Obtain the equation of the evolute of the curve x = a(cos θ + θ sinθ),
	y = a(sinθ – θ cosθ).			(8)
	(b)Find the evolute of the four cupsed hypocycloid x^2/3 + y^2/3 = a^2/3.			(8)
8.	(a) Show that the evolute of the cycloid x=a(θ -sin θ),y=a(1-cos θ) is
	another cycloid.			(8)
	(b) Find the evolute of the rectangular hyperbola xy = c².			(8)

9. (a) Prove that the envelope of ^x + ^y = 1 . where the parameters a

a b

and b are connected by a + b=c is √x +√y = √c. (8)

(b) Find the equation of the envelope of ^x + ^y = 1 , where the

a b

parameters a and b are connected by the relation a²+b² = c² and c is

a constant.		(8)
10. (a) Find the envelope of y cos θ-x sin θ = a cos2 θ where θ is a
parameter.		(8)
(b) Find the envelope of	ax	− *^by* = a ² − b ² , θ is a parameter. (8)
(b) Find the envelope of		− *^by* = a ² − b ² , θ is a parameter. (8)
	cosθ	sin θ
		UNIT- IV
FUNTIONS OF SEVERAL VARIABLES
		PART-A

1. If x = u(1-v),y = uv find ^∂^(u, ^v⁾ .

∂( x, y)

2. Find the minimum point of f(x,y)=x²+y²+6x+12.

3. Find the stationary point of f(x,y)=xy + ⁹ + ³ .

x y

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4.	Find Taylor’s series expansion of e^x siny near the point (-1,				π	) up to the first
4.					4	) up to the first
					4
	degree terms.
5. If u=f(x-y, y-z, z-x) find		∂u	₊ ∂u ₊ ∂u _.
5. If u=f(x-y, y-z, z-x) find			₊ ∂u ₊ ∂u _.
		∂x	∂y	∂z

6. Find ^du , if u=x³y²+x²y³, where x=at²,y=2at,using partial derivatives. dt

7. Find the stationary points of the function f(x,y)=x³+y³-12xy.

8. If u = acoshx cosy, v = asinhx siny, then show that

∂(u, v)

a²(cosh2x – cos2y),

∂( x, y)

9. If

u=x+y

and

y=uv

find the jacobian ^∂⁽ ^x^, ^y⁾ .

∂(u, v)

10.Find.

, if u = , where x=e^t , y = logt.

∂(u, v)

11.

Find

, if u=2xy ,v = x²-y² , x= r cosθ , y=r sinθ .

∂(r,θ )

12.

If z = sin^-1(x-y), x = 3t,y=4t² find

13.

IF u= f(x+ay)+g(x-ay) prove that ^∂ ² ^u = a ²

∂ ² u _.

∂y ²

∂x ²

14. Find the Taylor’s series expansion of x^y near the point (1,1) up to the second degree terms.

15. If u=e^xyz² find du.

16.If x = rcosθ			,y = rsinθ		_find∂(r,θ ) _.
						∂( x, y)
17. If u =	x	+ ^y + ^z		find	x ^∂^u + y ^∂^u + z ^∂^u .
17. If u =		+ ^y + ^z		find	x ^∂^u + y ^∂^u + z ^∂^u .
	y	z	x		∂x	∂y	∂z

18. If x = u(1+v) and y=v(1+u) , find ^∂⁽ ^x^, ^y⁾ .

∂(u, v)

19.	If u =	x _,	prove that	x	∂u	+ y ^∂^u	= 0.

		y			∂x	∂y
20.	Find the stationary points of x²-xy+x²-2x+y.

		PART – B
1.(a) Find the maxima and minima for xy²z³ subject to			x+y+z=6. (8)
(b)Are the functions u=	x + y	and v=tan^-1(x)+tan^-1(y)	functionally
(b)Are the functions u=	1 − xy	and v=tan^-1(x)+tan^-1(y)	functionally
	1 − xy
If so, find the relation between them.(8)

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2.(a) Expand f (x,y) = 4x²+xy+6y²+x-20y+21 in Taylor’s series about (-1,1).

(8)

∂(u, v)

x=rcosθ ,

y=rsinθ .Evaluate _∂_(r,_θ ₎ .

(b) If u = 4x +6xy , v = 2y +xy

(8)

3.(a) FIind the minimum value of xy²z² subject to x +y +z=24 using lagrange

multiplier.(8)

(b) Expand e^xlog(1+y) in powers of x and y upto the terms of third degree.

(8)

4.(a)Given the transformation u=e^x cosy , v = e^x siny

and that φ is a function of u

and v also

of x and y , prove that

^∂ ^φ ₊ ^∂ ^φ ₌ _(u 2 ₊ _v 2 ₎₍ ^∂ ^φ ₊ ^∂ ^φ ₎

(8)

∂x ²

∂y ²

∂u ²

∂v ²

(b)Investigate the maxima and minima ,if any,of the function y²+4xy+3x²+x³ (8)

5 (a) If u = sin^-1(

x ² + y ²

) prove x

∂u

+ y

∂u

=tanu.

(8)

x + y

∂x

∂y

(b). Using lagrange’s

multiplier

method , determine the maximum capacity of

a rectangular tank , open at the top , if the surface area is 108 sq.m.

(8)

6(a). If u= ^yz , v = ^zx , w = ^xy , find

∂( x, y, z) _.

(8)

∂(u, v, w)

(b) Expand x²y+siny+e^x in powers of (x-1) and (y-π ) through quadratic

forms.(8)

7(a). Find the maximum and minimum values of 2(x²-y²)-x⁴+y⁴. (8)

(b) Examine the function f(x, y)=x³y²(12-x-y) for extreme values.

(8)

8 (a). Find the jacobian of y₁, y2, y3 with respect to x1,x2, x3 if

y1= ^x² ^x³ , y 2 = ^x^3x1 , y3 = ^x^1x2 .

(8)

(b).If u=cos^-1(

x + y

) , prove that x

∂u

+ y

∂u

= -

cotu.

(8)

x +

∂x

∂y

9(a). Expand x²y+3y-2 in powers of (x-1) and (y+2) using Taylor’s expansion.(8)

(b). Examine f(x,y)=x³y³-12x-3y+20 for its extreme values.	(8)
10(a). Using Taylor’s expansion, express f(x,y) = e^ax cos(by)	in powers of x and y
upto second degree terms.	(8)
(b)Obtain terms upto the third degree in the taylor series expansion of e^xsiny
around the point [1,π/2]	(8)

UNIT-V

ORDINARY DIFFERENTIAL EQUATIONS

PART-A

Solve (D² + 4)y = 0.

Find the P.I of (D²-2D+5)y = e^x sin2x.

Solve ^dx − y = 0; ^dy + x = 0

dt dt

Write Euler’s homogeneous linear differential equation .How will you convert it to a linear differential equation with constant coefficients?

5. Solve (D²+2D+1)y=e^-x.

KINGS COLLEGE OF ENGINEERING-PUNALKULAM 10

MA1101-MATHEMATICS-I

6. Find the P.I of (D³+8)y = cosh2x.

7. Solve (x²D²-xD+1)y=0.

8^. Solve	d ² y	= y
8^. Solve	dx ²	= y
	dx ²
9. If x = e^z , express			d ² y	in terms of the derivatives of y w.r.to z.
9. If x = e^z , express			dx ²	in terms of the derivatives of y w.r.to z.
			dx ²

10. Solve (D²+D+1)y=0

11. Find the P.I. Of the (D² +1)y = x³.

12. Find the P.I. of (D² +4)y = cos2x.

13. Solve (x²D²+4xD+2)y=0.

14. Find the P.I. Of the (D³ -1)y = e^2x.

15. Find the P.I. Of the (D -1)²y = e^x sinx.

16. Find the P.I. Of the ^{d y} = xe^x. dx ²

17. Solve (D³ -3D²+3D-1)y = x²e^x.

18. Solve (D³ -6D²+11D-6)y = 0

	Find the C.F. of x² ^d	2	^y-x ^dy +y = 0.
19.	Find the C.F. of x² ^d		^y-x ^dy +y = 0.
	dx ²			dx

20. Solve ((3x+2)²D² -(3x+2)D +1)y = 0.

PART - B

1. a)solve (D²+2D-1)y=x²+e^2x (8).

b) Solve (x²D²-2xD-4)y=32(logx)². (8)

2. a) Solve (D+4)x+3y = t

(D+5)y+2x = e^2t

(8)

b)Solve y^’’ + y = secx

by method of variation of parameters

(8).

a)Solve

by method of variation parameters y’’-

y‘+

y = x ² + 1

(8)

_x2

b)Solve ^d ² ^y +4y = 4tan2x by using method of variation of parameters.

dx ²

a)Solve (D²+6D+8)y = e^-2x +cos²x

(8)

b)Solve x² ^d

^y+4x ^dy +2y = sin(logx)

(8).

dx ²

a) Solve

+ y = e ²^t ,

+ 4 x = t. given that x(0) = 2 and y(0) =

(8)

b)Solve ⁽x²D²+4XD+2)y=xlogx

(8).

a) Solve (D²+5D+4)y=e^-xsin2x+x²+1.

(8)

KINGS COLLEGE OF ENGINEERING-PUNALKULAM 11

				MA1101-MATHEMATICS-I
	b)Solve	d ² y	+y=xcosx, by using method of variation of parameters		(8).
	b)Solve	dx ²	+y=xcosx, by using method of variation of parameters		(8).
		dx ²
7	a).Solve	*^dx+ 2 y = − sin* t, *^dy* − 2 x = cos t. .		(8)
		dt	dt
	b)Solve	(2x+3)²y’’-(2x+3)y’-12y=6x. (8)

8. a)Solve (D²-2D+2)Y=e^xx²+5+e^-2x. (8) b) Solve (D²+4D+3)y=e^-xsinx+xe^3x. (8)

9.	a) Solve by method of variation of parameters ^d ² ^y +y=xsinx				(8).
				dx ²
	2
	b) Solve (x+2)² *^{d y}* -(x+2) *^dy* +y=x+2.			(8)
	dx ²	dx
	2
10. a) Solve x² *^{d y}* -3x *^dy* +4y=x²+cos(logx)				(8).
	dx ²	dx
	b)Solve (D²-2D+1)Y=xe^xsinx		(8).

KINGS COLLEGE OF ENGINEERING-PUNALKULAM 12

and whose guiding curve is the ellipse x ² + 2 y ²

= 1 , z=3. (8)