3D & Vect

Analytical Geometry of Three Dimensions and Vector Calculus Semester-III

Code: 2K6M3:4/U03MA3:M4 Level : K Unit : 1.1 Type : MCQ

1. The number of compartments into which the whole space is divided by the coordinate planes is (a) 4 (b) 6 (c) 8 (d) 12

2. The compartments into which the whole space is divided by the coordinate planes are called(a) quadrants (b) octants (c) squares (d) sectors

3. If a point lies in the xy plane, its z-coordinate is (a) x (b) y (c) z (d) zero

4. The equation of the xoy plane is (a) x=0 (b) y=0 (c) z=0 (d) x+y+z=0

5. The equation of the yoz plane is (a) x=0 (b) y=0 (c) z=0 (d) x+y+z=0

6. The equation of the zox plane is (a) x=0 (b) y=0 (c) z=0 (d) x+y+z=0

7. x=0 is the equation of (a) xoy plane (b) yoz plane (c) zox plane (d) x+y+z=1.

8. The equation y=0 represents (a) xoy plane (b) yoz plane (c) zox plane (d) x+y+z=1

9. If three points A,B and C are collinear, then (a) (b) AB2+BC2=CA2

(c) (d) AB+BC=CA

10. The y coordinate of the point dividing the line joining the points (x1,y1,z1) and (x2,y2,z2) internally in the ratio m:n is

(a) (b) (c) (d)

11. The x coordinate of the point dividing the line joining the points (x1,y1,z1) and (x2,y2,z2) externally in the ratio m:n is

(a) (b) (c) (d)

Code: 2K6M3:4/U03MA3:M4 Level : UUnit: 1.1 Type : MCQ

1. The midpoint of the line joining the points (3,0,-4) and (1,-2,-3) is(a) (4,-2,-7) (b) (2,2,1) (c) (2,1,-1) (d) (2,-1,-7/2)

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2. The centroid of the triangle with vertices (0,1,3), (-1,-2,5) and (4,7,1) is (a) (1,2,3) (b) (-1,2,-3) (c) (1,3,2) (d) (-1,0,-3)

Code: 2K6M3:4/U03MA3:M4 Level : KUnit: 1.1 Type : VSA

1. Write the equation of zox plane.2. Write the formula to find the distance of the point (x,y,z) from the origin.3. Write the coordinates of the point which divides the line joining the points (x1,y1,z1)

and (x2,y2,z2) internally in the ratio l:m.4. Write the coordinates of the centroid of a triangle whose vertices are (x1,y1,z1),

(x2,y2,z2) and (x3,y3,z3).

Code: 2K6M3:4/U03MA3:M4 Level : UUnit: 1.1 Type : VSA

1. Find the distance between the points (2,3,5) and (-1,5,-1).2. Find the distance of the point (3,-4,7) from the origin.3. Find the midpoint of the line joining the points (4,-2,3) and (2,-3,1).4. Find the point which divides the line joining (1,2,3) and (-1,3,5) externally in the

ratio 2:3.

Code: 2K6M3:4/U03MA3:M4 Level : KUnit: 1.1 Type : PARA

1. Obtain the coordinates of the point which divides the line joining the points P(x1,y1,z1) and Q(x2,y2,z2) internally in the ratio m:n.

2. Obtain the coordinates of the point which divides the line joining the points P(x1,y1,z1) and Q(x2,y2,z2) externally in the ratio m:n.

3. Derive the formula to find the distance between the points P(x1,y1,z1) and Q(x2,y2,z2).

Code: 2K6M3:4/U03MA3:M4 Level : UUnit: 1.1 Type : PARA

1. Find the perimeter of the triangle whose vertices are the points (3,1,5), (-1,-1,9) and (0,-5,1).

2. Show that the points (10,7,0), (6,6,-1) and (6,9,-4) form an isosceles right-angled triangle.

3. Show that the four points (4,-1,2), (0,-2,3), (1,-5,-1) and (2,0,1) lie on a sphere whose centre is (2,-3,1). Find the radius of the sphere.

4. Find the ratios in which the straight line joining the points (1,-3,5) and (7,2,3) is divided by the coordinate planes.

5. Show that the points A(3,2,1), B(0,5,4) and C(2,3,2) are collinear and find the ratio in which C divides the line AB.

6. The line through A(-2,6,4) and B(1,3,7) meets the coordinate plane yoz at C. Find the ratio in which C divides AB and also the coordinates of C.

Code: 2K6M3:4/U03MA3:M4 Level : UUnit: 1.1 Type : ESSAY

1a. The points (2,1,-3), (5,4,3) and (1,4,7) are three of the corners of a parallelogram. Find the coordinates of the remaining corner which is opposite to the point (2,1,-3).

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1b. Find the perimeter of the triangle whose vertices are the points (2,1,0), (5,-2,7) and (-1,2,3).

2.a. Obtain the coordinates of the centroid of a triangle, given the coordinates of its vertices.

2.b. Show that Q(-6,5,4) is a point on the line joining the points P(-4,3,2) and R(-10,9,8). Find the ratio in which Q divides PR.

3.a. Show that the points (a,b,c), (b,c,a) and (c,a,b) form an equilateral triangle.3.b. Show that the points A(-4,5,2), B(-3,4,1) and C(-6,7,4) are collinear and find the ratio

in which B divides AC.4.a. Obtain the centroid of the triangle whose vertices are (-2,5,8), (-3,4,4) and (-6,7,4).4.b. Find the ratios in which the straight line joining the points (1,-3,5) and (7,2,3) is

divided by the coordinate planes.5.a. Obtain the formula to find the distance between two points (x1,y1,z1) and (x2,y2,z2).5.b. If two vertices and centroid of a triangle are respectively (4,2,1), (5,1,4) and (5,2,3)

find the coordinates of the third vertex.

Code: 2K6M3:4/U03MA3:M4 Level : KUnit: 1.2 Type : MCQ1. If be the angle between two lines AB and CD, the projection of AB on CD is

(a) AB Sin (b) AB Cos (c) CD Cos (d) CD Sin

2. If P is the point (x,y,z) and O is the origin with OP=r, the d.c.'s of OP are given by

(a) xr,yr,zr (b) (c) (d)

3. The d.c.'s of x axis are(a) (1,1,1) (b) (1,0,0) (c) (0,1,0) (d) (0,0,1)

4. The projection of the line joining (x1,y1,z1) and (x2,y2,z2) on x axis is(a) x2-x1 (b) x2+x1 (c) y2-y1 (d) z2-z1

5. If is the angle between the lines whose d.c.'s are (l1,m1,n1) and (l2,m2,n2) then Cos is given by(a) l1l2-m1m2-n1n2 (b) l1m1+m1n1+n1l1

(c) (d) l1l2+m1m2+n1n2

6. Two lines whose d.c.'s are l1,m1,n1 and l2,m2,n2 are parallel if (a) l1l2+m1m2+n1n2=0 (b) l1l2+m1m2+n1n2=1

(c) l1m1+m1n1+n1l1=0 (d)

7. Two lines whose d.c.'s are l1,m1,n1 and l2,m2,n2 are perpendicular if (a) l1l2+m1m2+n1n2=0 (b) l1l2+m1m2+n1n2=1

(c) (d)

Code: 2K6M3:4/U03MA3:M4 Level : KUnit: 1.2 Type : VSA1. Define the projection of a point on a line.2. Define the projection of a finite line on another line.3. Define direction cosines of a line.

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4. Define direction ratios of a line.5. Find the d.r.'s of the line joining the points (1,2,3) and (4,-1,-2).6. Write the condition for two lines whose d.c.'s are (l1,m1,n1) and (l2,m2,n2) to be

parallel.7. Write the condition for the two lines whose d.c.'s are (l1,m1,n1) and (l2,m2,n2) to be

perpendicular.8. Find the value of “a” such that the lines whose d.c."s are (1,a,-2) and (-1,3,1) are

perpendicular.


1. Obtain the projection of the line joining P(x1,y1,z1) and Q(x2,y2,z2) on another line with d.c.'s l,m,n.

2. Obtain the direction cosines of the line joining the points (x1,y1,z1) and (x2,y2,z2).3. Derive the formula for finding the angle between the lines whose direction cosines are

l1,m1,n1 and l2,m2,n2.


1. Find the acute angle between the lines whose d.c.'s are and .

2. Find the angle between the lines joining the points (3,1,-2), (4,0,-4) and (4,-3,3), (6,-2,2).

3. A,B,C,D are the points (-1,2,3), (-3,3,5), (3,1,2) and (7,3,5). Show that AB is at right angles to CD.

4. Find the angles of the triangle whose vertices are the points (-1,1,0), (3,2,1) and (1,3,2).

5. A,B,C,D are the points (4,3,5), (6,4,3), (2,-1,4) and (0,1,5). Find the projection of AB on CD.

6. Find the angle between two diagonals of a cube.7. A line makes angles ,,, with the four diagonals of a cube. Prove that

Cos2+Cos2+Cos2+Cos2= .

Code: 2K6M3:4/U03MA3:M4 Level : KUnit: 1.2 Type : ESSAY

1. Derive the formula to find the angle between the lines whose direction cosines are (l1,m1,n1) and (l2,m2,n2). Deduce the conditions for the lines to be parallel and perpendicular.

2. a) Find the direction cosines of the line joining the points (3,-5,4) and (1,-8,-2). b) Find the angle between two diagonals of a cube.


1. Prove that each pair of opposite edges of a tetrahedron with vertices at the points (0,0,0), (1,1,0), (0,1,-1) and (1,0,-1) are perpendicular.

2. Find the angle between the lines whose direction cosines are given by the equations 3l+m+5n=0, 5lm+6mn-2nl=0.

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3. A line is equally inclined with the x,y,z axes. Find the equal angles.4. Find the angles between the line joining (5,3,-2), (1,1,4) and the line joining (3,-3,3),

(2,0,1).5. Show that the opposite edges of the tetrahedron whose vertices are (1,2,3), (2,2,3),

(1,3,3), (1,2,4) are perpendicular.6. Find the acute angle between the lines whose d.r.'s are 2,1,-3 and 1,-3,2.7. Show that the acute angle between the lines whose direction cosines l,m,n are given

by l+m+n=0, l2+m2-n2=0 is 60.

Code: 2K6M3:4/U03MA3:M4 Level : KUnit: 1.3 Type : MCQ

1. The plane is represented by an equation in x, y and z of degree (a) 1 (b) 2 (c) 3 (d) zero

2. The equation of the plane in the intercept form is

(a) ax+by+cz=1 (b)

(c) (d)

Code: 2K6M3:4/U03MA3:M4 Level : UUnit: 1.3 Type : MCQ1. The angle between the planes 2x-4y+z=6 and x+y+2z=3 is

(a) 0 (b) 45 (c) 60 (d) 90

2. The distance of the plane 2x+y-2z=6 from the origin is (a) 2/3 (b) 2 (c) 6 (d) 3

3. The angle between the planes 2x+4y-6z=11 and 3x+6y+5z+4=0 is(a) 0 (b) 45 (c) 60 (d) 90

4. The x-intercept of the plane 4x-3y+2z-7=0 is

(a) (b) 4 (c) (d) –7

Code: 2K6M3:4/U03MA3:M4 Level : KUnit: 1.3 Type : VSA1. Write the equation of the plane in the intercept form.2. Write the equation of the plane in the normal form.3. Give the general form of equation to a plane.4. Write the condition for the planes a1x+b1y+c1z+d1=0 and a2x+b2y+c2z+d2=0 to be

parallel.

Code: 2K6M3:4/U03MA3:M4 Level : KUnit: 1.3 Type : PARA1. Prove that every equation of the first degree in x,y,z represents a plane.2. Derive the equation of a plane in the intercept form.3. Derive the equation of a plane in the normal form.Code: 2K6M3:4/U03MA3:M4 Level : UUnit: 1.3 Type : PARA

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1. Find the equation of the plane passing through the points (3,1,2), (3,4,4) and perpendicular to the plane 5x+y+4z=0.

2. Find the equation of the plane which passes through the point (4,1,1) and is perpendicular to each of the planes x-3y+5z+4=0 and 3x-y+7z-8=0.

3. Find the equation of the plane through the points (2,2,-1), (3,4,2) and (7,0,6).4. Find the equation to a plane through the point (-10,5,4) perpendicular to the line

joining the points (4,-1,2) and (-3,2,3).5. The foot of the perpendicular drawn from the origin to a plane is (12,-4,-3). Find the

equation to the plane.6. A variable plane which remains at a constant distance 3p from the origin cuts the

coordinate axes at A,B,C. Show that the locus of the centroid of the triangle ABC is x-2+y-2+z-2=p-2.

7. A variable plane is at a constant distance p from the origin and meets the axes in A,B,C. Show that the locus of the centroid of the tetrahedron OABC is

.


1.a) Obtain the equation of the plane passing through the points (x1,y1,z1), (x2,y2,z2) and (x3,y3,z3).

b) Find the equation of the plane which passes through the point (-1,3,2) and perpendicular to the planes x+2y+2z=5 and 3x+3y+2z=8.

2.a) Find the angle between the planes 2x+4y-6z=11 and 3x+6y+5z+4=0.2.b) Find the equation of the plane which bisects the line joining the points (-1,2,3) and

(3,-5,6) at right angles.3.a) Show that the points (0,-1,-1), (-4,4,4), (4,5,1) and (3,9,4) are coplanar and find the

equation of the plane on which they lie.3.b) A plane meets the coordinate axes in A,B,C such that the centroid of the triangle ABC

is (p,q,r). Show that the equation of the plane is .

4.a) Find the equation to the plane through the point P(-2,3,-4) at right angles to OP where O is the origin.

4.b) Find the equation of the plane through the points (9,3,6) and (2,2,1) and perpendicular to the plane 2x+6y+6z-9=0.

5.a) Find the angle between the planes 6x-3y-2z=7 and x+2y+2z+9=0.5.b) Show that the points (0,-1,0), (2,1,-1), (1,1,1) and (3,3,0) are coplanar and find the

equation to the plane containing them.


1. The ratio in which the plane ax+by+cz+d=0 divides the line joining the points (x1,y1,z1) and (x2,y2,z2) is :1 where equals

(a) (b)

(c) (d)

2. The length of the perpendicular from the origin to the plane ax+by+cz+d=0 is

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(a) (b)

(c) (d)


1. The distance of the point (2,1,0) from the plane 2x+y+2z-17=0 is(a) 2 (b) 4 (c) –17 (d) 8

2. The perpendicular distance of the plane x+y+z=3 from the origin is (a) 3 (b) –3 (c) (d)


1. Write the condition for the planes a1x+b1y+c1z+d1=0 and a2x+b2y+c2z+d2=0 to be at right angles.

2. Write down the length of the perpendicular from the origin to the plane ax+by+cz+d=0.

3. Give the form of the equation of a plane through the line of intersection of two given planes.

4. What is the length of the perpendicular from the point (x1,y1,z1) on the plane ax+by+cz+d=0?

5. Find the distance of the plane 2x+3y-5z-7=0 from the origin.6. Find the distance of the plane 2x-3y+4z+8=0 from the point (1,0,-1).


1. Find the equation of the plane through the line of intersection of the planes 2x+y+3z-4=0 and 4x-y+5z-7=0 and which is perpendicular to the plane x+3y-4z+6=0.

2. Find the equation of the plane passing through the line of intersection of the planes x-2y-z+3=0 and 3x+5y-2z-1=0 and perpendicular to the yoz plane.

3. Obtain the length of the perpendicular from the point (x1,y1,z1) on the plane ax+by+cz+d=0.

4. Find the locus of the point whose distance from the plane 3x-2y+6z-3=0 is thrice the distance from the plane 12x+4y-3z+4=0.

5. Find the equation of the plane through the point (2,3,-1) parallel to the plane 3x-4y+7z=0 and also find the distance between the two planes.

6. Find the bisector of the acute angle between the planes x+2y+2z-3=0 and 3x+4y+12z+1=0.

7. Find the bisector of the obtuse angle between the planes 3x+4y-5z+1=0 and 5x+12y-13z=0.

8. Find the equation of the plane through the line of intersection of the planes x-y+z+3=0 and x+y+2z+1=0 and parallel to the line whose d.r.'s are 4,-5,6.

9. Find the bisector of the angle between the planes 3x-6y+2z+5=0 and 4x-12y+3z-3=0 containing the origin.

10. Show that the plane 2x+7y-5z-21=0 bisects the obtuse angle between the planes x+2y+2z-3=0 and 3x+4y+12z+1=0.

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1.a) Find the equation of the plane passing through the line of intersection of the planes 2x-y+5z-3=0 and 4x+2y-z+7=0 and parallel to the z-axis.

b) Find the equation of the plane through the point (2,3,-1) parallel to the plane 3x-4y+7z=0 and find the distance between the two planes.

2.a) Find the locus of the point whose distance from the plane 3x-2y+6z-3=0 is thrice the distance from the plane 12x+4y-3z+4=0.

b) Find the bisector of the acute angle between the planes 3x+4y-5z+1=0 and 5x+12y-13z=0.

3.a) Find the equations of the planes passing through the line of intersection of the planes 5x-3y+4=0 and x+y-2z+10=0 and which are 1unit distance from the origin.

b) Find the bisector of the obtuse angle between the planes 3x+4y-5z+1=0 and 5x+l2y-13z=0.

4.a) Find the equation of the plane through the line of intersection of the planes x+2y+3z+2=0 and 2x+3y-z+3=0 and parallel to the line whose d.r.'s are 1,1,1.

b) The plane 2x-y+3z+5=0 is rotated through a right angle about its line of intersection with the plane 5x-4y-2z+1=0. Show that its equation in the new position is 27x-24y-26z=13.

5.a) Obtain the ratio in which the plane ax+by+cz+d=0 divides the line joining the points (x1,y1,z1) and (x2,y2,z2).

b) Find the bisector of the acute angle between the planes x+2y+2z-3=0 and 3x+4y+12z+1=0.


1. The equation of the x-axis is (a) y=0,z=0 (b) x=0,y=0 (c) x=0,z=0 (d) x'=0,z'=0

2. The equation of the planes taken together give the equation of the line of intersection of two(a) spheres (b) planes (c) circles (d) straight lines

3. The equation represent a pair of planes passing

through the (a) plane (b) line (c) circle (d) sphere

4. The d.c.’s l,m,n of the line of intersection of the planes ax+by+cz+d=0 and a1x+b1y+c1z+d1=0 are given by(a) la+mb+nc=0 and la1+mb1+nc1=0 (b) ll1+mm1+nn1=0 and aa1+bb1+cc1=0(c) la+mb+nc=0 and ll1+mm1+nn1=0 (d) la1+mb1+nc1=0 and aa1+bb1+cc1=0

5. Equations of a straight line passing through two given points (x1,y1,z1) and (x2,y2,z2) are

(a) (b)

(c) (d)

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1. What is the intersection of xz and xy planes?2. Write down the equation of the y-axis.


1. Write the equation of a straight line passing through the point (2,1,0) and whose direction ratios are 2,-1,-3.

2. Write the equation of the straight line through the points (0,0,0) and (5,-2,3).


1. Find the equation to the line which intersects the lines 2x+y-4=0=y+2z and x+3z-4 = 0 = 2x+5z-8 and passes through the point (2,-1,1).

2. Put in the symmetrical form the line 4x+4y-5z-12=0=8x+12y-13z-32.3. Put in the symmetrical form the line 3x-2y+z-1=0=5x+4y-6z-2.4. Prove that the lines 3x-4y+2z=0=-4x+y+3z, x+3y-5z+9=0=7x-5y-z+7 are parallel.5. Find the equations to the line through the point (2,3,1) parallel to the line

–x+2y+z =5, x+y+3z=6.6. Show that the lines 2x+y+3z-7=0=x-2y+z-5 and 4x+4y-8z=0=10x-8y+7z are at right

angles.

7. Find the length of the perpendicular from the point (5,4,-1) to the line .

8. Find the length of the perpendicular drawn from (3,4,5) to the line

.

9. Find the co-ordinates of the foot of the perpendicular from the origin to the line x+2y+3z+4=0=2x+3y+4z+5.

10. Find the equations to the line which intersects the lines 2x+y-4 = 0 = y+2z and

x+3z-4=0=2x+5z-8 and parallel to the line .


1.a. Put in the symmetrical form the lines 4x+4y-5z-12=0=8x+12y-13z-32.1.b. Find the equation to the line which intersects the lines 2x+y-4=0=y+2z and x+3z-

4=0=2x+5z-8 and passes through the point (2,-1,1).2.a. Show that the lines 2x+y+3z-7=0=x-2y+z-5 and 4x+4y-8z=0=10x-8y+7z are at right

angles.2b. Find the coordinates of the foot of the perpendicular from the origin to the line

x+2y+3z+4=0=2x+3y+4z+5.3.a) Find the equations to the line which intersects the lines 2x+y-4=0=y+2z and x+3z-

4=0=2x+5z-8 and parallel to the line .

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3b. Find the length of the perpendicular drawn from (3,4,5) to the line


1. The condition for the line to be parallel to the plane

ax+by+cz+d=0 are(a) al+bm+cn0 & ax1+by1+cz1+d=0(b) al+bm+cn=0 & ax1+by1+cz1+d0(c) al+bm+cn=0 & ax1+by1+cz1+d=0 (d) al+bm+cn0 & ax1+by1+cz1+d0

2. If the line lies in the plane ax+by+cz+d=0then

(a) al+bm+cn=0 & ax1+by1+cz1+d=0 (b) al+bm+cn0 & ax1+by1+cz1+d=0(c) al+bm+cn0 & ax1+by1+cz1+d0 (d) al+bm+cn=0 & ax1+by1+cz10

3. The equation of any plane containing the line is

(a) a(x-x1)+b(y-y1)+c(z-z1)0subject to the condition al+bm+cn=0(b) a(x+x1)+b(y+y1)+c(z+z1)=0(c) a(x+x1)+b(y+y1)+c(z+z1)=0(d) a(x-x1)+b(y-y1)+c(z-z1)=0 subject to the condition al+bm+cn=0

4. If the angle between the plane ax+by+cz+d=0 & the line is, Sin

is equal to

(a) (b)

(c) (d)


1. Write the condition for the line to be parallel to the plane

ax+by+cz+d=0.2. Write the condition if the line lies in the plane ax+by+cz+d=0.

3. Write the equation of any plane containing the line .

4. What is the angle between the plane ax+by+cz+d=0 and the line

5. Write the condition for the line to be parallel to the plane ax+by+cz+d=0.


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1. Find the equation of the plane which contains the line and is parallel

to the plane x-2y-4z+7=0.

2. Find the equation to the plane through the line parallel to the line

.

3. Find the equations of the line passing through the point (3,1,-6) and parallel to each of the planes x+y+2z-4=0 and 2x-3y+z+5=0.

4. Find the equation of the plane passing through the line 5x-2y+7=0=x-3y+z-4 and

parallel to the line .

5. Find the equation of the plane which contains the two parallel lines

and .

6. Find the angle between the lines x+y+z-6=0=3x-y+4z-13 and 2x-y+2z=0=x-2z. Find also the equation of the plane through the first line parallel to the second.

7. Find the equations of the orthogonal projection of the line on to the

plane 8x+2y+9z-1=0.

8. If is the line , find the equation of the plane through which is

parallel to the line of intersection of the planes 5x+2y+3z=4 & x-y+5z+6=0.

9. Show that the straight lines ; ; will lie in one plane

if =0.

10. Prove that the equation of the plane through the line u1=a1x+b1y+c1z+d1=0,

u2=a2x+b2y+c2z+d2=0 and parallel to the line is u1(a2l+b2m+c2n) =

u2(a1l+b1m+c1n).


1.a) Show that the straight lines ; ; will lie in one plane

if .

1b. Find the equation of the plane passing through the line 5x-2y+7-0=x-3y+z-4 and

parallel to the line .

2.a) Prove that the equation of the plane through the line u1=a1x+b1y+c1z+d1=0,

u2=a2x+b2y+c2z+d2=0 and parallel to the line is u1(a2l+b2m+c2n) =

u2(a1l+b1m+c1n).

2b. Find the equation of the plane which contains the two parallel lines

and .

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3.a) Find the angle between the lines x+y+z-6=0=3x-y+4z-13 and 2x-y+2z=0=x-2z. Find also the equation of the plane through the first line parallel to the second.

3b. If is the line find the equation of the plane through which is

parallel to the line of intersection of the planes 5x+2y+3z=4 and x-y+5z+6=0.


1. The condition for the two given straight lines and

to be coplanar is

(a) (b)

(c) (d)

2. If two lines are coplanar they must be(a) parallel (b) perpendicular (c) similar (d) intersecting

3. The co-ordinates of the points on the two coplanar lines = r ;

= r1 are

(a) –3r-1=-4r -3,8r-10=7r1-1,2r+1=r1+4.(b)–3r-1-4r -3,8r-107r1-1,2r+1r1+4.

(c)

(d)

4. The condition for the lines ax+by+cz+d = 0 = a1x+b1y+c1z+d1, a2x+b2y+c2z+d2=0 =a3x+b3y+c3z+d3 to be coplanar is

(a) (b)

(c) (d)

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5. The condition for the two given straight lines ;

to be coplanar is

(a) (b)

(c) (d)


1. Write the condition that two given straight lines and

to be coplanar.

2. Write the conditions for the line to be parallel to the plane

ax+by+cz+d=0.

3. Write the condition for the two given straight lines and

to be coplanar.

4. Write the condition for the two given straight lines and

to be coplanar.


1. Prove that the lines , ax+by+cz+d = 0 = a1x+b1y+c1z+d1 are

coplanar, if .

2. Show that the lines and are coplanar and find

the equation of the plane determined by them.

3. Show that the lines and are coplanar. Find

their common point and find the equation of the plane in which they lie.

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4. Prove that the lines ax + by + cz + d=0= a1x + b1y + c1z + d1, a2x + b2y + c2z + d2=0 =

a3x+b3y+c3z+d3 are coplanar, if .

5. Show that the lines x+y+z+3=0=2x+3y+4z-5 and –4x-y+5z-7=0=2x-5y-z-3 are coplanar.

6. Prove that the lines ; are coplanar. Find also

their point of intersection and the plane through them.

7. Find the values of k, if the lines and are

coplanar.

8. Show that the following lines are coplanar l1: 3x-y-z+2=0=x-2y+3z-6,l2: 3x-4y+3z-4=0=2x-2y+z-1.

9. Show that the following lines are coplanar and find the equation of the plane of

coplanarity, .

10. Show that the following lines are coplanar, find their point of intersection and equation of the plane of coplanarity,

l1: ; l2: x+2y+z+2=0, 4x+5y+3z+6=0


1.a) Show that the lines are coplanar and find

the equation of the plane determined by them.1b. Show that the lines x+y+z+3=0=2x+3y+4z-5 and –4x-y+5z-7=0=2x-5y-z-3 are

coplanar.

2.a) Show that the lines and are coplanar, find

their common point and find the equation of the plane in which they lie.

2b. Find the values of k, if the lines are

coplanar.


1. Two straight lines which do not lie in the same plane are called(a) parallel (b) intersecting (c) skew lines (d) similar

2. The length of the segment of the common perpendicular intercepted between the skew lines is called(a) shortest distance (b) coplanar (c) collinear (d) parallel

14

3. If u1=0=v1 and u2=0=v2 be two straight lines, then the general equations of a straight line intersecting them both are.(a) u1+1v1=0=u1+1v2 (b) u1-1v1=0=u1-1v2

(c) –u1+1v1=0=-u1+1v2 (d) where 1, 2 are constants.

4. The two lines are coplanar if the

shortest distance between them is (a) 1 (b) 2 (c) 0 (d) 3

5. The line of shortest distance is (a) perpendicular (b) parallel (c) similar (d) collinear to both the lines

Code: 2K6M3:4/U03MA3:M4 Level : KUnit: 2.4 Type : VSA1. Define skew lines.2. Define shortest distance between the skew lines.3. If u1=0=v1 and u2=0=v2 be two straight lines then write the general equation of a

straight line intersecting them.4. Write the equation of the shortest distance between two given lines

and .

5. What is the shortest distance between them if the two lines are coplanar?


1. Find the shortest distance between the lines ;

and find the equations of the line of shortest distance also.2. The equations of two straight lines are x=y+2a=6z-6a and x+a=2y=-12z. Show that

the shortest distance between the lines is 2a and find the equations of the line along which it lies.

3. Find the shortest distance between the z-axis and the line ax+by+cz+d=0=a1x+b1y+c1z+d1.

4. Find the shortest distance between the lines ; .

5. Find the length and the equations of the common perpendicular drawn to the lines

.

6. Find the length and the equations of the common perpendicular drawn to the

following lines: .

7. Find the feet of the common perpendicular drawn to the following lines and find the

length of the common perpendicular: .

8. Find the feet, length and equations of the common perpendicular of each of the

following pairs of lines: .

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9. Find the S.D. between the lines

l1: ; l2: 2x-2y+z-3=0=2x-y+2z-9.

10. Find the shortest distance between the lines3x-9y+5z=0=x+y-z, 6x+8y+3z-10=0=x+2y+z-3.


1.a) Find the shortest distance between the lines ;

and find the equations of the line of shortest distance also.

1b. Find the shortest distance between the z-axis and the line ax+by+cz+d = 0 = a1x+b1y+c1z+d1.

2.a) Find the shortest distance between the lines ; .

2b. Find the shortest distance between the lines 3x-9y+5z=0=x+y-z6x+8y+3z-10=0=x+2y+z-3


1. The number of conditions necessary to determine a unique sphere is a) 2 b) 3 c) 4 d) 5


1. The Centre of the sphere 2x2+2y2+2z2+4x-6y-8z+1=0 is (a) (-2,3,4) (b) (-1,3/2,2) (c) (-1,3,2) (d) (-1,-3,2)

2. ax2+by2+cz2+2gx+2fy+2hz+d=0 will represent a sphere if (a) a=b=c (b) a=bc (c) ab=c (d) abc

3. The equation 5x2+5y2+5z2+2x+7y-9z+kxy+7=0 will denote a sphere if k is (a) 3 (b) –3 (c) 0 (d) 1


1. Define a sphere.2. What is the general form of the equation of the sphere?3. When will a second degree equation in x,y,z represent a sphere?4. Given the sphere x2+y2+z2+2ux+2vy+2wz+d=0 what will be its centre and radius?5. For the sphere a(x2+y2+z2)+2ux+2vy+2wz+d=0 find its centre and radius.6. How many conditions are necessary to determine a unique sphere?7. Write the equation of a sphere with the extremities of a diameter at the points

(x1,y1,z1) and (x2,y2,z2).8. If a sphere passes through the origin, what will be its equation?9. What is the condition for two spheres to be orthogonal?

Code: 2K6M3:4/U03MA3:M4 Level : U

16

Unit: 3.1 Type : VSA

1. What is the equation of a sphere with centre at (1,1,1) and radius 2?2. Write the equation of a unit sphere with centre as the origin.3. Find the centre and radius of the sphere x2+y2+z2-6x+8y-10z+1=0.4. Obtain the centre and radius of the sphere 2(x2+y2+z2)+6x-6y+8z+9=0.5. Find the radius of the sphere 3x2+3y2+3z2+6y-4x-9z+1=0.6. If A(-3,1,0) and B(5,3,-2) are the extremities of the diameter of a sphere, find its

centre and radius.7. Find the equation of a sphere with centre at (0,1,1) and radius 5.


1. Derive the equation of a sphere of radius r and centre at (a,b,c).2. Determine the equation of a sphere with the extremities of a diameter at the points

(x1,y1,z1) and (x2,y2,z2).


1. Find the equation of the sphere whose centre is at (1,-1,2) and radius 3 units. Find also the points where the sphere cuts the x axis.

2. Obtain the equation of the sphere described on the line joining (-1,2,1) and (-2,3,1) as diameter and locate its centre.

3. Find the equation of the sphere circumscribing the tetrahedron with faces x=0, y=0,

z=0, .

4. Find the equation of the sphere with centre at (3,2,-1) and passing through (-1,1,2).5. Obtain the equation of the sphere with centre at (6,-1,2) and touching the plane 2x-

y+2z=0.6. Find the equation of the sphere passing through the origin and cutting off positive

intercepts 5,7,9 on the co-ordinate axes.7. Obtain the equation of the sphere passing through the origin and cutting off equal

intercepts on the axes.


1. Find the equation of the sphere through the points (4,-1,2), (0,-2,3), (1,-5,-1) and (2,0,1). Find also the centre and radius of the sphere.

2. Obtain the centre and radius of the sphere through the points (0,0,0), (-a,b,c), (a-b,c) and (a,b,-c).

3. Determine the sphere passing through the points (2,3,1), (5,-1,2), (4,3,-1) and (2,5,3)4. A plane passes through a fixed point (a,b,c). Show that the locus of the foot of the

perpendicular from the origin to the plane is x2+y2+z2-ax-by-cz=0.5. Find the sphere through (1,1,-1), (-5,4,2) (0,2,3) and having its centre on the plane

3x+4y+2z=6.


17

1. If x12+y1

2+z12+2ux1+2vy1+2wz1+d >0, then we can get (i) 2 (ii) 1 (iii) no tangents from

(x1,y1,z1) to the sphere x2+y2+z2+2ux+2vy+2wz+d=0.

2. The origin will lie on the sphere 5x2+5y2+5z2-8x+11y-2z+d=0 if (a) d = 0 (b) d = 1 (c) d = 2 (d) d 0

3. The length of the tangent from (0,0,0) to the sphere x2+y2+z2+2x+7y+5z+16=0 is (a) 16 (b) 4 (c) 1.6 (d) 0.4

4. The distance of the point (1,1,1) from the point of contact on x2+y2+z2+x+y+z+3=0 is (a) 9 (b) 6 (c) 3 (d) 1


1. Considering the sphere x2+y2+z2+2x+7y+8z+9=0, the point (1,-1,1) is (i) inside the sphere (ii) on the sphere(iii) outside the sphere (iv) the centre of the sphere

2. The origin lies outside the sphere x2+y2+z2+2ux+2vy+2wz+d=0 acording asa)d>0 b) d<0 c) d=0 d) d 0

3. The length of the tangent from (3, 0, 0) to the sphere x2+y2+z2=1 isa) 1 b) 2 c) 3 d) 4


1. What is the length of the tangent from (x1,y1,z1) to the sphere x2+y2+z2+2ux+2vy+2wz+d=0?

2. When will a point (x1,y1,z1) lie outside a sphere x2+y2+z2+2ux+2vy+2wz+d=0?3. What is the condition which makes a point (x1,y1,z1) to lie inside a sphere

x2+y2+z2+2ux+2vy+2wz+d=0?4. Specify the term in the sphere x2+y2+z2+2ux+2vy+2wz+d=0, which decides the

position of the origin with respect to the sphere.5. Write the equation of a tangent plane to a sphere x2+y2+z2+2ux+2vy+2wz+d = 0 at the

point (x1,y1,z1).6. When will the plane lx+my+nz=p touch the sphere x2+y2+z2+2ux+2vy+2wz+d = 0?


1. Find the length of the tangent from (4,5,6) to the sphere x2+y2+z2=2.2. Where will the point (1,1,1) lie with respect to the sphere x2+y2+z2-4x+2y-4=0?3. What is the equation of the tangent plane at (4,-2,2) to the sphere x2+y2+z2-4x+2y-

4=0?4. Identify the tangent plane at (0,0,-2) to the sphere x2+y2+z2-4x+2y-4=0.


18

1. Derive the equation of a tangent plane to a sphere x2+y2+z2+2ux+2vy+2wz+d = 0 at the point (x1,y1,z1).

2. Obtain the condition for a plane to touch the sphere x2+y2+z2+2ux+2vy+2wz+d = 0.3. Determine the length of the tangent from (x1,y1,z1) to the sphere

x2+y2+z2+2ux+2vy+2wz+d = 0.4. Find the value of k for which the plane x+y+z-k touches the sphere x2+y2+z2-2x-

2y-2z-6=0.5. Show that 2x-2y+z+12=0 touches the sphere x2+y2+z2-2x-4y+2z-3=0.


1. Find the tangent planes to the sphere x2+y2+z2-4x+2y-6z+5=0 which are parallel to 2x+2y-z=6.

2. Show that 2x-y+2z+6=0 touches the sphere x2+y2+z2-2x-4y-6z-2=0.3. Show that 4x-3y+2z=0 touches the sphere x2+y2+z2+8x-6y-4z=0.4. Find the tangent planes at (-3,1,0), (5,3,-2) to the sphere x2+y2+z2-2x-4y+2z-12=0 and

show that they are parallel. Hence find the diameter of the sphere.5. Find the equation of the tangent plane to the sphere x2+y2+z2-5x-7y-9z=0 at the point

where it cuts the x axis.


1. Find the points of contact of the tangent planes to the sphere x2+y2+z2-4x+2y-4=0, which are parallel to x2+y2+z2-2x-4y+2z-3=0.

2. Show that the sphere x2+y2+z2-2x-4y+2z-3=0 touches the plane 2x-2y+z+12=0 and find the point of contact.

3. Find the equation of the sphere which has its centre at (5,-2,3) and which touches the

line .

4. Show that touches the sphere x2+y2+z2-2x-4y-4=0 and find the

point of contact.5. Find the equations of the sphere which has its centre at the origin and which touches

the line 2(x+1) = 2-y = z+3.


1. The intersection of a sphere and a plane is (i) sphere (ii) circle (iii) line (iv) point

2. If S1=0 and S2=0 are 2 spheres, then the equation S1+S2=0 represents (i) circle (ii) plane (iii) sphere (iv) line


19

1. How is a circle identified in three dimensional space?2. What can you say about the intersection of two spheres?3. Write the equations representing a circle.4. What will be the equation of a sphere through a given circle.5. Given two spheres S1 = 0 and S2 = 0 find the plane in which the circle of intersection

lie. 6. If S1=0 and S2=0 are two spheres, what will be S1+S2=0, where is any constant?7. If S=0 is a sphere and u=0 is any plane, then what can you say about S+u=0 where

is a constant?


1. If S = x2+y2+z2+2x-4y+3z+5=0 and u = x+2y+3z+8=0, then write the equation of a sphere through the intersection of S=0 and u=0.

2. If S1=x2+y2+z2=0, S2=2x2+2y2+2z2+7x-8y+9z+11=0, find the equation of a sphere through the intersection of S1=0, S2=0.


1. Prove that the intersection of a sphere and a plane is a circle.2. Prove that the intersection of two spheres is a circle.


1. Find the equation of the sphere which has its centre on 5x+y-4z+3=0 and passing through the circle x2+y2+z2-3x+4y-2z+8=0, 4x-5y+3z-3=0.

2. Find the equation of the sphere through the circle x2+y2+z2+2x+3y+6=0, x-2y+4z=9 and through the centre of the sphere x2+y2+z2-2x+4y-6z+5=0.

3. Obtain the equation of the sphere having x2+y2+z2+10y-4z-8=0, x+y+z=3 as the great circle.

4. Find the sphere through the circle x2+y2+z2+2x+3y+6=0, x-2y+4z-9=0 and passing through the point (1,-2,3).

5. Obtain the equation of a sphere having its centre on 4x-5y-z=3 and passing through the circle with equations x2+y2+z2+4x+5y-6z+2=0, x2+y2+z2-2x-3y+4z+8=0.

6. Find the sphere through the circle x2+y2+z2=1, 2x+4y+5z=6 and touching the plane x=0.

7. Find the sphere which touches x-2y-2z-7=0 at (3,-1,-1) and passes through (1,1-3).8. Show that the circles x2+y2+z2-2x+3y+4z-5=0, 5y+6z+1=0 and x2+y2+z2-3x-4y+5z-

6=0 x+2y-7z=0 lie on the same sphere and find its equation.

Code: 2K6M3:4/U03MA3:M4 Level : UUnit: 3.3 Type : ESSAY1. Find the centre and radius of the circle x+2y+2z=15, x2+y2+z2-2y-4z-11=0.2. Show that the intersection of the sphere x2+y2+z2-2x-4y-6z-2=0 and the plane

x+2y+2z=20 is a circle with centre (2,4,5) and radius .3. Find the sphere which passes through (-2,-1,-4), (0,3,0) and cuts orthogonally the two

spheres x2+y2+z2+x-3z-2=0 and 2x2+2y2+2z2+x+3y+4=0.

20


1. If a scalar function (x,y,z) is defined at each point (x,y,z) of a region then is called(a) vector point function (b) scalar point function(c) gradient of a function (d) divergence of a function

2. Velocity vector of a particle at time t is

(a) (b) (c) (d)

3. Acceleration vector of a particle at time t is given by

(a) (b) (c) (d)

4. The magnitude of velocity is vector

(a) (b) (c) (d)

5. The magnitude of acceleration is

(a) (b) (c) (d)

6. is

(a) (b)

(c) (d)

7. is

(a) (b)

(c) (d)

8. If = 2 Sin 3t +2 Cos 3t +8t then velocity is

(a) 6 Cos 3t – 6 Sin 3t+8 (b) 6 Sin 3t +6 Cos 3t +8t

(c) 6 Cos 3t -6 Sin 3t +8 (d)

9. If = -e+t -e-t their velocity is

21

(a) e+t +et (b) (c) -et +et (d) -et -e-t


1. Define scalar point function.2. Define vector point function.3. Define velocity of a particle.4. Define acceleration of a particle.5. What is the magnitude of velocity?6. What is the magnitude of acceleration?7. Write the expression for unit tangent vector.8. Write the expression for unit tangent vector in terms of velocity.

9. What is ?

10. What is ?

11. What is ?

12. What is ?


1. If =e-t +log(1+t2) -tan t find (a) (b) (c) (d)

2. Find the values of scalar field defined by (x,y,z)=4yz2x+3xyz-z2+2 at the points (1,-1,-2); (0,-3,1); (-2,-3,1) and (1,-3,-5)

3. Given = sin t + cos t +t find

(a) (b) (c) (d)

4. Find the values of scalar field defined by (x,y,z)=2x2yz-8y2z3-z4+2x at (2,-2,0) and (-3,5,-2) and (1,-1,5).

5. If and find .

6. Show that where and are constant vectors is a solution

of differential equation .

7. Find the velocity and acceleration of a particle which moves along the curve x=2 sin 3t, y = 2 cos 3t, z = 8t at any time t. Find also the magnitudes of velocity and acceleration.

8. If and find

22

(i) (ii) (iii)

9. Find the unit tangent vector at any point on the curve x=1+t2, y=4t-3; z=2t2-6t. Determine the unit tangent vector at the point where t=2.

10. If where are constants prove that and

.

11. A particle mover along the curves x=2t2, y=t2-4t, z=3t-5 where t is the time. Find the

components of its velocity and acceleration at time t=1 in the direction of .

12. Prove that .

13. If has constant magnitude show that and are perpendicular.

14. If are differentiable functions of scalar u prove that

.

15. If are differentiable functions of scalar u, prove that

.


1.a. Prove that .

1.b. If are differentiable functions of scalar prove that

.

2.a. If has constant magnitude, show that and are perpendicular.

2b. A particle moves along the curve where ‘t’ is the time. Find

the components of velocity and acceleration at t=1 in the direction +2 +3


1. (a) If find

(i) (ii) (iii) (iv)

1b. Find the values of scalar field defined by (x,y,z)=14yz2x+3xyz-4z2+2 at the points (1,-1,2) and (0,3,-1).

2. (a) Given find

(i) (ii) (iii) (iv)

2b. Find the values of scalar field defined by (x,y,z)=3xyz-8y2z3-z4+2xy at (2,-2,0)

23

and (5,-3,-2).

3. (a) If find at t = 0.

3b. Show that where and are constant vectors is a solution

of differential equation .

4. (a) Find the velocity and acceleration of a particle which moves along the curve x=3sin2t, y=2cos 3t, z=8t2 at any time t. Find also the magnitudes of velocity and acceleration.

4b. If and find

(i) and (ii)

5. (a) Find the unit tangent vector at any point on the curve x=1+t2, y=4t-3, z=2t2-6t. Determine the unit tangent vector at the point where t=2.

5b. If where are constants prove that

.

6. (a) A particle moves along the curves x=2t2, y=t2-4t, z=3t-5 where t is the time. Find the

components of its velocity and acceleration at time t=1 in the direction of .

6b. If are differentiable functions of scalar u, prove that

.


1. Gradient of is

(a) (b)

(c) (d)

2. Grad () is (a) grad grad (b) grad - grad (c) grad grad (d) grad grad

3. Grad () is (a) grad + grad (b) grad + grad (c) grad + grad (d) grad + grad

4. Directional derivative of is maximum in the direction of (a) (b) (c) . (d)

5. Maximum value of directional derivative is (a) (b) (c) (d)

24

6. The normal to the surface at (x0,y0,z0) is (a) at (x0,y0,z0) (b) grad at (x0,y0,z0)(c) at (x0,y0,z0) (d) . at (x0,y0,z0)

7. The equation of the tangent plane to the surface at is

(a) (b)

(c) (d)

8. Cartesian form of equation of the tangent plane is given by

(a)

(b)

(c)

(d)

9. The cartesian form of equation of the normal is

(a)

(b)

(c)

(d)


1. Define vector differential operator.2. Define gradient of scalar point function .3. Define directional derivative.4. What is the maximum value of directional derivative?5. What is the direction in which grad is maximum?6. Write the equation of tangent plane at (x0,y0,z0).7. Write the cartesian form of equation of tangent plane at (x0,y0,z0).8. Write the vector equation of normal at a given point.9. Write the cartesian form of equation of normal.10. How grad and normal to the surface at (x0,y0,z0) are related?

25

11. What can you say about the directions of and grad at .


1. Prove that grad () = grad + grad .2. Find grad (rn) where r2=x2+y2+z2.3. Prove that 2(rn ) = n(n+3) rn-2 .

4. Show that grad .

5. Show that is a vector perpendicular to the surface (x,y,z)=c where c is a constant.


1. Find the equation of the tangent plane and normal to the surface xyz=4 at the point (1,2,2) on it.

2. If and =2x2yz3 find (i) grad (ii) grad (iii)

3. Find the unit vector normal to the surface x2+2y2+z2=7 at (1,-1,2).

4. Find the directional derivative of =x2yz+4xz2 at (1,-2,-1) in the direction .

5. Find the direction and magnitude of the maximum directional derivative of = x2yz3

at (2,1,-1).6. Find the angle between the surfaces x2+y2+z2=9 and z=x2+y2-3 at the point (2,-1,2).7. Find an equation of tangent plane to the surface xz2+x2z=z-1 at the point (1,-

3,2).8. Find equations for the tangent plane and the normal the surfacez=x2+y2 at the point

(2,-1,5).

9. Find the directional derivative of =3xyz-x2y2z3 at (1,2,-1) in the direction .


1.a. Find the equation of the tangent plane and normal to the surface at (x0,y0,z0).1.b. Prove that .2.a. Prove that grad ()= grad + grad .2.b. Find grad (rn) where r2=x2+y2+z2.

3.a. Prove that grad (1/r)= .

3.b. Show that is a vector perpendicular to the surface (x,y,z) = c where c is a constant.

Code: 2K6M3:4/U03MA3:M4 Level : UUnit: 4.2 Type : ESSAY1.a. Find the equation of the tangent plane and normal to the surface x2yz=4 at the point

(1,2,2) on it.

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1.b. If and = x2yz2 find

(i) grad (ii) (iii) 2.a. Find the unit vector normal to the surface x2+2y2+z2=7 at (1,-1,2).

2.b. Find the directional derivative of = x2yz-4yz2 at (1,-2,-1) in the direction .

3. (a) Find the direction and magnitude of the maximum directional derivative of = x2y+z2

at (2,1,-1).3b. Find the angle between the surfaces x2+y2+z2=9 and z=x2+y2-3 at the point

(-2,1,-2).4.a. Find an equation of tangent plane to the surface xy2+y2z=x-1 at the point (1,2,3).

4.b. Find the directional derivative of = x2y-x3y2z2 at (1,-2,3) in the direction .

Code: 2K6M3:4/U03MA3:M4 Level : KUnit: 4.3 Type : MCQ1. Div is

(a) 2 (b) 3 (c) 0 (d) 1

2. . is

(a) div +div (b) div A + div B

(c) div + div (d) .div + .div

3. div( ) is

(a) div + .grad (b) grad + . (div )

(c) .div +grad (d) .div + . grad

4. div is

(a) .curl - .curl (b) .curl - .curl

(c) . div + .div (d)

5. div (grad ) is (a) (b) (c) (d) 0

6. If is solenoidal then(a) curl = (b) curl =0 (c) div =0 (d) grad F=0


1. Define divergence of vector point function.2. Define solenoidal vector.3. When a vector is said to be solenoidal?

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4. What is the value of if is a constant vector?

5. What is the value of if is a solenoidal vector?

6. What is div ?

7. What is divergence of ( )?

8. What is the value of div (curl )?9. What is laplacian operator?10. Define curl of a vector point function.11. When a vector is said to be irrotational?.12. Define an irrotational vector.


1. Prove that div where and are differentiable vector functions.

2. Prove that div( ) = div .3. If prove that div =0 if is a constant vector.4. Show that div (r3 ) = 6r3.5. Show that div (grad rn) = n (n+1)rn-2

6. If and f(r) is differentiable prove that div (f(r) )= . Hence (or)

otherwise show that div =0.

7. Prove that div (rn )=(n+3)rn. Deduce div and div .


1. Show that is solenoidal.

2. If and = 3x2-yz find

(i) (ii) (iii) at (1,-1,1).

3. If .

4. Find whether is solenoidal.

5. Find whether xyz2 where is solenoidal.

6. Find the value of ‘a’ if the vector is solenoidal.

7. If evaluate div at (1,1,1).

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8. If = evaluate div at (1,2,0).

9. If =sinr and .

10. Find the divergence of .


1.a. Prove that div are differentiable vector functions.1.b. Show that div (r3 )=6r3.2.a. Prove that div ( ) = div + . grad .2.b. Show that div (grad rn) = n(n+1)rn-2.

3.a. If prove that div =0 if is a constant vector.

3.b. If and f(r) is differentiable prove that div (f(r) ) = idence (or)

otherwise show that div .


1.a. Show that is solenoidal.

1.b. If and = x2-3yz find (i) (ii) (iii) at

(0,1,-1).

2.a. If find at (1,-2,5).

2.b. Find whether is solenoidal.

3.a. Find whether xyz2 where is solenoidal.

3.b. If .

4.a. Find the value of ‘a’ if is solenoidal.

4.b. If Find div at (2,-1,5).

5.a. Find the divergence of

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5.b. If find div at (1,1,1).


1. curl is (a) 3 (b) 0 (c) 1 (d) 2

2. If and are irrotational vectors then div is(a) 0 (b) 1 (c) –1 (d) 2

3. If =x2 +y2 +z2 then curl at (1,1,1) is (a) 0 (b) 3 (c) –1 (d) 2

4. If is irrotational then (a) curl = (b) curl =0 (c) grad =0 (d) div =0

5. curl (grad ) is (a) 0 (b) (c) div (grad ) (d) 3


1. Define curl of a vector point function.2. When a vector is said to be irrotational?3. Define an irrotational vector.4. What is the value of curl (grad )?


1. Prove that curl where and are differentiable vector functions.

2. Prove that divergence of curl of a vector is zero and curl of gradient of scalar is zero.

3. Prove that curl ( ) = curl +(grad ) .4. Prove that curl .


1. Prove that is irrotational.

2. If = x2y3z4 find curl grad at (1,-1,1).

3. If find curl curl .

4. If find curl curl .

5. Show that is irrotational if a = 4.

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6. If find curl at (1,2,-1).

7. If find curl .

8. If where .

9. If show that and find if where is

some scalar point function.10. If = 3x2z-y2z3+3x-2y find .

Code: 2K6M3:4/U03MA3:M4 Level : KUnit: 4.4 Type : ESSAY1.a. Prove that curl where are differentiable

vector functions.1.b. Prove that divergence of curl of a vector is zero and curl of grudient of scalar is

zero.2.a. Prove that curl .2.b. Prove that curl ( ) = curl +(grad ) .


1.a. Prove that is irrotational.

1b. If = x2y3z4 find curl (grad ) at (1,-1,1).

2.a. If find curl curl .

2b. Show that is irrotational if a = 4.

3.a. If find curl curl .

3.b. If find curl at (1,2,-1).

4.a. If where find .

4.b. If show that = 0 and find if where is

some scalar point function.


1. is equivalent to

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(i) (ii) r2+c (iii) r+c (iv) 0

2. is equivalent to

(i) (ii) (iii) (iv) 0

3. is equivalent to

(i) (ii) (iii) (iv)

4. is equal to

(i) 0 (ii) (iii) (iv)

5. If acceleration of a particle is its velocity is given by

(i) (ii)

(iii) (iv)


1. If denotes acceleration of a particle at time ‘t’ define velocity at time ‘t’.2. Define the displacement of a particle at time ‘t’ in terms of its velocity.


1. What is the value of .

2. Write down the value of .



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1. If find

(i) (ii)

2. The acceleration of a particle at any time ‘t’ given by . Find the

velocity and displacement at any time assuming at t = 0.

3. If and find

(i) (ii)

4. The acceleration of a particle at time ‘t’ is given by . If the

velocity and displacement are which are zero at t = 0, find and at any time.

5. Evaluate if and .

6. Find from the equation where and are constant vectors, given that

both and vanish at t = 0.


1.a. If , evaluate dt.

1.b. If , where is a constant, prove that is constant.

2.a. If and find .

2.b. If and evaluate and


1. where along the time joining (0,0) and (1,1) is

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(i) 2 (ii) 1 (iii) 0 (iv) –1


1. If is irrotational then is

(i) 0 (ii) –1 (iii) 1 (iv) 2

2. If is a conservative field, then for some scalar (i) (ii) (iii) (iv)


1. Define line integral.2. Define circulation.3. When is a field said to be conservative?4. When is a vector said to be solenoidal?

5. State the necessary and sufficient condition for to be independent of the path.


1. Given the vector field , evaluate from (0,0,0) to (1,1,1) where

C is the curve x=t, y=t2, z=t3.

2. Evaluate where c is any path from (1,0,0) to (2,1,4).

3. Show that zexdx+2yzdy+(ex+y2)dz is exact and hence evaluate

.

4. Show that is conservative and find it’s scalar potential.

5. Show that the vector field is irrotational and obtain it’s

scalar potential.

6. Evaluate if and .

7. Evaluate where and c is the curve y2=4x in the xy plane from

(0,0) to (4,4).8. Find the total work done in moving a particle in a force field given by

along the curve x = t2 +1, y = 2t2, z = t3 from t = 1 to t = 2.9. Find the work done in moving a particle in the force field

from t=0 to t=1 along the curve x = 2t2, y = t, z = 4t3.

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1.a. Prove that is a conservative field. Find its scalar potential and evaluate the work done in moving a particle from (2, 0, 0)

to (0, 2,

1.b. Find along the curve x = cost, y = 2 sint, z = cost from t = 0 to t =

given .2.a. Find the circulation of around the rectangle whose vertices

are (0, 0), (1, 0), (i, /2) amd (0, /2).2.b. Show that (y2 cosx + z3) dx + (2ysinx-4)dy + (2xz2+2)dz is an exact differential and

hence solve the equation (y2cosx +z3) dx + (2ysinx-4)-dy +(3xz2+2) dz = 0.3.a. Compute the circuit integral over the triangle whose vertices are

(1, 0), (0,1), (-1, 0).3.b. Show that the vector function defined by = (ysinz – sinx) + (x sinz + 2yz) +

(xycosz + y2) is irrotational and find a function such that = .


1. If the region of integration in a surface integral is a cylinder x2+y2 = 16 in the I octant between z = 0 and z = 5 then the unit normal vector to the surface is

(i) (ii) (iii) (iv)


1. Define surface integral.2. Define flux.

3. Write an equivalent double integral to while projecting the surface on xy

plane.

4. Write an equivalent double integral to while projecting on yz plane.

5. Write an equivalent double integral to while projecting on xz plane.

6. Define volume integral.


1. Write down the limits for the volume integral if V is the region in the I octant bounded by the cylinder x2+y2 = 4 and the plane z= 3.

2. If V denotes the volume of the region bounded by the planes x = 0, y = 0, z=0 and 4x +2y +z = 8, write down the limits for the volume integral.

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1. If and S is the surface of the cuboid bounded by the

planes x = 0, x= 2, y = 0, y = 1, z=0, z=3 evaluate .

2. Show that where S is the surface of the sphere

x2+y2+z2 = 1 in the I octant.

3. Evaluate where and S is the part of the plane

2x +3y+6z = 12 located in the I octant.

4. Evaluate where and S is the surface of the plane

2x+y=6 in the I octant cut off by the plane z= 4.

5. Evaluate if and S is the surface of the cylinder x2+y2

= 4 included between z = 0 and z = 3.

6. If and S is the surface of the cylinder x2+y2=9 in the I octant between

z = 0 and z = 4.

7. Evaluate if and S is the surface of the plane

2x+y+2z=6 in the I octant.

8. Evaluate where and S is that portion of the plane

x+y+z=1 which lies in the I octant.

9. If , evaluate where V is the region

bounded by x=0,y=0,z=0 and 2x+2y+z=4.

10. Evaluate where V is the region bounded by the surfaces x = 0, y=0, y = 6,z =

x2, z = 4.


1.a. Evaluate where S is the part of the sphere x2+y2+z2=1 above

the xy plane.

1.b. If , evaluate where V is the region

bounded by x=0, y=0, z=0 and 2x+2y+z=4.

2.a. Evaluate where and S is the part of the plane

2x+3y+6z=12 in the I octant.

2.b. Evaluate where =4x2y and V is the closed region bounded by the planes

4x+2y+z=8, x = 0, y = 0, z = 0.

36


1. By divergence theorem is equivalent to

(i) V (ii) 2V (iii) 3V(iv) 0

2. is equivalent to

(i) (ii) (iii) (iv)

3. =

(i) (ii) r2 (iii) 0 (iv) 1

4. If is the unit outward drawn normal to a closed surface of area S =

(i) 2S (ii) S (iii) 3S (iv) 0

5. If S is a closed surface is equal to

(i) 1 (ii) 2 (iii) 0 (iv)


1. State gauss’ Divergence theorem.2. State Greens’ theorem.3. State Stokes’ theorem.


1. If S is a closed surface enclosing a volume V and , prove

that =(a+b+c)v. Deduce that if S is the surface of

a sphere of unit radius.

2. If and S is a rectangular parallelopiped bounded by x = 0, y

= 0, z = 0, x = 2, y = 1, z = 3, evaluate .

3. Evaluate over the surface bounded by z=0, z=c, x2+y2=a2.

4. Find by Greens’ theorem the value of along the closed curve c

formed by y2 = x, and y = x between (0,0) and (1,1).

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5. Find the area between y2=4x and x2=4y.6. Evaluate over the surface bounded by z=0, z=4,

x2+y2=a2.


1.a. By the divergence theorem show that .

1.b. Verify divergence theorem for taken over the cube bounded by the planes, x=0, x=1, y=0, y=1, z=0, z=1.

2.a. Verify Greens’ theorem for where C is the closed curve of the

region bounded by y=x, y=x2.

2.b. Evaluate around the square whose vertices are (1,0), (-1,0), (0,1), (0,-1) using Greens’ theorem.

3.a. Verify Stokes’ theorem when where S is the surface of the region bounded by the planes x=0, y=0, z=0 and x+y+z=1.

3.b. Evaluate over the upper half of the hemisphere of radius ‘a’ with the

centre at the origin if .

4.a. If S is any closed surface enclosing a volume V and , prove that

.

4.b. If and S is a rectangular parallelopiped bounded by x=0,

y=0, z=0, x=2, y=1, z=3 evaluate .

5.a. Verify divergence theorem for taken over the rectangular parallelopiped 0 x a, 0 y b, 0 z c.

5.b. If is the unit outward drawn normal to any closed surface of area S, show that

.

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