Question 1

Describe the surface represented in cylindrical coordinates (r, $\theta$, z) by the equation r = 2sin($\theta$), 0 $\le$ $\theta$ $\le$ $\pi$.

Accepted Answer

A)  A cylinder of radius 1 and axis along the z-axis 
B)  A cylinder of radius 1 and axis parallel to the z-axis through the point (0 , 1 , 0) 
C)  A cylinder of radius 2 and axis along the z-axis 
D)  A cylinder of radius 1 and axis parallel to the z-axis through the point (1 , 0 , 0) 
E)  A cylinder of radius 2 and axis parallel to the z-axis through the point (1 , 0 , 0) 
A)  A cylinder of radius 1 and axis along the z-axis 
B)  A cylinder of radius 1 and axis parallel to the z-axis through the point (0 , 1 , 0) 
C)  A cylinder of radius 2 and axis along the z-axis 
D)  A cylinder of radius 1 and axis parallel to the z-axis through the point (1 , 0 , 0) 
E)  A cylinder of radius 2 and axis parallel to the z-axis through the point (1 , 0 , 0)

Question 2

Find the equation of the cone z =   expressed in spherical coordinates [ R,  , $\theta$].

Accepted Answer

A)  R =   
B)    =   
C)  $\theta$ =   
D)  R =   
E)    =   
A)  R =   
B)    =   
C)  $\theta$ =   
D)  R =   
E)    =

Question 3

Find the eigenvalues of the matrix A =   .

Accepted Answer

A)  0, -3, 4 
B)  0, 3, 4 
C)  0, -3, -4 
D)  0, 3, -4 
E)  -3, 3, 4 
A)  0, -3, 4 
B)  0, 3, 4 
C)  0, -3, -4 
D)  0, 3, -4 
E)  -3, 3, 4

Question 4

Describe the intersection of the graphs of y = x and y = 5 in 3-space.

Accepted Answer

A)  line through the point (5, 5, 0) and perpendicular to the xy-plane 
B)  line through the points (5, 5, 5) and (0, 0, 0) 
C)  line through the point (0, 5, 0) and perpendicular to the xy-plane 
D)  line through the points (5, 5, 0) and (0, 0, 0) 
E)  line through the points (0, 0, 0) and (5, 5, 0) 
A)  line through the point (5, 5, 0) and perpendicular to the xy-plane 
B)  line through the points (5, 5, 5) and (0, 0, 0) 
C)  line through the point (0, 5, 0) and perpendicular to the xy-plane 
D)  line through the points (5, 5, 0) and (0, 0, 0) 
E)  line through the points (0, 0, 0) and (5, 5, 0)

Question 5

Determine the equation of the plane that contains the points P = (1, -2, 0), Q = (3, 1, 4), and R = (0, -1, 2).

Accepted Answer

A)  2x - 8y + 5z = 18 
B)  2x + 8y + 5z = -14 
C)  2x + 8y - 5z = -14 
D)  2x - 8y - 5z = 18 
E)  2x + 8y + 5z = 18 
A)  2x - 8y + 5z = 18 
B)  2x + 8y + 5z = -14 
C)  2x + 8y - 5z = -14 
D)  2x - 8y - 5z = 18 
E)  2x + 8y + 5z = 18

Question 6

When does the equation x2 + y2 + z2 + Ax + By + Cz + D = 0 represent a sphere?

Accepted Answer

A) if and only if A2 + B2 + C2 + D > 0 B) if and only if A2 + B2 - C2 > D C) if and only if A2 - B2 + C2 > 0 D) if and only if A2 + B2 + C2 > 4D E) if and only if A = 0, B = 0, C = 0, and D < 0 A) if and only if A2 + B2 + C2 + D > 0 B) if and only if A2 + B2 - C2 > D C) if and only if A2 - B2 + C2 > 0 D) if and only if A2 + B2 + C2 > 4D E) if and only if A = 0, B = 0, C = 0, and D < 0

Question 7

Describe the intersection of the graphs of y = x and y2 + z2 = 9.

Accepted Answer

A)  the ellipse in which the vertical plane containing the z-axis and the point (1, 1, 0) intersects the horizontal circular cylinder of radius 3 with central axis along the x-axis 
B)  the circle in which the vertical plane containing the z-axis and the point (1, 1, 0) intersects the horizontal circular cylinder of radius 3 with central axis along the x-axis 
C)  the hyperbola in which the vertical plane containing the z-axis and the point (1, 1, 0) intersects the horizontal circular cylinder of radius 3 with central axis along the x-axis 
D)  the parabola in which the vertical plane containing the z-axis and the point (1, 1, 0) intersects the horizontal circular cylinder of radius 3 with central axis along the x-axis 
E)  the ellipse in which the vertical plane containing the z-axis and the point (1, 1, 0) intersects the horizontal circular cylinder of radius 3 with central axis along the y-axis 
A)  the ellipse in which the vertical plane containing the z-axis and the point (1, 1, 0) intersects the horizontal circular cylinder of radius 3 with central axis along the x-axis 
B)  the circle in which the vertical plane containing the z-axis and the point (1, 1, 0) intersects the horizontal circular cylinder of radius 3 with central axis along the x-axis 
C)  the hyperbola in which the vertical plane containing the z-axis and the point (1, 1, 0) intersects the horizontal circular cylinder of radius 3 with central axis along the x-axis 
D)  the parabola in which the vertical plane containing the z-axis and the point (1, 1, 0) intersects the horizontal circular cylinder of radius 3 with central axis along the x-axis 
E)  the ellipse in which the vertical plane containing the z-axis and the point (1, 1, 0) intersects the horizontal circular cylinder of radius 3 with central axis along the y-axis

Question 8

Find the scalar and vector projections of 2i -5j + k in the direction of 3i - 4k.

Accepted Answer

A)  scalar 2, vector   (3i - 4k) 
B)  scalar   , vector   (3i - 4k) 
C)  scalar   , vector   (3i - 4k) 
D)  scalar   , vector   (3i - 4k) 
E)  scalar   , vector   (3i - 4k) 
A)  scalar 2, vector   (3i - 4k) 
B)  scalar   , vector   (3i - 4k) 
C)  scalar   , vector   (3i - 4k) 
D)  scalar   , vector   (3i - 4k) 
E)  scalar   , vector   (3i - 4k)

Question 9

For what values of the constants k and c does the line    =  =   lie in the plane x - y + 2z = c?

Accepted Answer

A)  k = -10, c = 1 
B)  k = 10, c = 0 
C)  k = 10, c = -1 
D)  k = -8, c = 2 
E)  k = -2, c = 0 
A)  k = -10, c = 1 
B)  k = 10, c = 0 
C)  k = 10, c = -1 
D)  k = -8, c = 2 
E)  k = -2, c = 0

Question 10

Describe the graph of 25x2 - y2 - z2 = 25.

Accepted Answer

A)  a hyperboloid of one sheet 
B)  a hyperboloid of two sheets 
C)  a hyperbolic paraboloid 
D)  an elliptic paraboloid 
E)  an ellipsoid (or a sphere)
F) a cylinder (circular, elliptic, parabolic, or hyperbolic)
G) a cone (circular, elliptic, parabolic, or hyperbolic)
H) none of the above 
A)  a hyperboloid of one sheet 
B)  a hyperboloid of two sheets 
C)  a hyperbolic paraboloid 
D)  an elliptic paraboloid 
E)  an ellipsoid (or a sphere)
F) a cylinder (circular, elliptic, parabolic, or hyperbolic)
G) a cone (circular, elliptic, parabolic, or hyperbolic)
H) none of the above

Question 11

Describe the surface with equation x2 + y2 + z2 - 6x + 2y - 12z = 75.

Accepted Answer

A)  sphere with centre at (3, -1, 6) and radius   
B)  sphere with centre at (-3, 1, -6) and radius   
C)  sphere with centre at (3, 1, 6) and radius 11 
D)  sphere with centre at (3, -1, 6) and radius 11 
E)  sphere with centre at (3, 1, 6) and radius   
A)  sphere with centre at (3, -1, 6) and radius   
B)  sphere with centre at (-3, 1, -6) and radius   
C)  sphere with centre at (3, 1, 6) and radius 11 
D)  sphere with centre at (3, -1, 6) and radius 11 
E)  sphere with centre at (3, 1, 6) and radius

Question 12

Find parametric equations of the straight line containing the point (2, 1, -4) and is parallel to the line of intersection of the planes 3x +2y - z = 0 and x + y + z = -9.

Accepted Answer

x = 2 + 3t

Question 13

Describe the set of points in 3-space satisfying z2 = x2 + y2 and z = x + y.

Accepted Answer

A)  two straight lines 
B)  one straight line 
C)  an ellipse 
D)  a parabola 
E)  a hyperbola
F) none of the above 
A)  two straight lines 
B)  one straight line 
C)  an ellipse 
D)  a parabola 
E)  a hyperbola
F) none of the above

Question 14

Find the equation of a plane that contains the line   =   =   and is parallel to the plane 2x - 3y + 2z = 0.

Accepted Answer

A)  2x - 3y + 2z = 15 
B)  2x - 3y + 2z = -15 
C)  2x - 3y + 2z = 12 
D)  2x - 3y + 2z = -12 
E)  2x + 3y + 2z = 15 
A)  2x - 3y + 2z = 15 
B)  2x - 3y + 2z = -15 
C)  2x - 3y + 2z = 12 
D)  2x - 3y + 2z = -12 
E)  2x + 3y + 2z = 15

Question 15

Find the cosine of the angle $\theta$ between the planes 3x + 3y - 6z = 12 and 7x - 5y + 2z = -8.

Accepted Answer

A)  ±   
B)  ±   
C)  ±   
D)  ±   
E)  ±   
A)  ±   
B)  ±   
C)  ±   
D)  ±   
E)  ±

Question 16

Find the equation for the plane that passes through the point (1, 2, 3) and is normal to the vector joining (1, 3, 2) and (2, 3, 1).

Accepted Answer

A)  x - z + 2 = 0 
B)  x + z +2 = 0 
C)  2x - z +1 = 0 
D)  x - 2z -1 = 0 
E)  x + z -2= 0 
A)  x - z + 2 = 0 
B)  x + z +2 = 0 
C)  2x - z +1 = 0 
D)  x - 2z -1 = 0 
E)  x + z -2= 0

Question 17

If u = 3i + j + 4k, v = -i + 2j, and w = - 2i - 3j + 5k, evaluate u × (3v - w).

Accepted Answer

A)  41i + 11j + 28k 
B)  41i - 11j + 28k 
C)  -41i - 11j + 28k 
D)  -41i + 11j + 28k 
E)  41i - 11j - 28k 
A)  41i + 11j + 28k 
B)  41i - 11j + 28k 
C)  -41i - 11j + 28k 
D)  -41i + 11j + 28k 
E)  41i - 11j - 28k

Question 18

Let u and v be non-zero vectors in 2 or 3-space. Find a simplified expression for   .

Accepted Answer

A)    
B)    + 2   +   
C)    + 2     +   
D)    + 2   +   
E)    + 2u.v +   
A)    
B)    + 2   +   
C)    + 2     +   
D)    + 2   +   
E)    + 2u.v +

Question 19

Which of the following is an equation of a hyperboloid of two sheets?

Accepted Answer

A)  x2 + y2 - z2 = 0 
B)  x2 - y2 - z2 = -1 
C)  z2 = 1 - x2 - y2 
D)  x2 + y2 + 1 = z2 
E)  z = - x2 - y2 
A)  x2 + y2 - z2 = 0 
B)  x2 - y2 - z2 = -1 
C)  z2 = 1 - x2 - y2 
D)  x2 + y2 + 1 = z2 
E)  z = - x2 - y2

Question 20

Describe the set of points in 3-space satisfying x2 + 2y2 + 3z2 = 4 and z = x + y.

Accepted Answer

A)  two straight lines 
B)  one straight line 
C)  an ellipse 
D)  a parabola 
E)  a hyperbola
F) none of the above 
A)  two straight lines 
B)  one straight line 
C)  an ellipse 
D)  a parabola 
E)  a hyperbola
F) none of the above

Describe the surface represented in cylindrical coordinates (r, $\theta$ , z) by the equation r = 2sin( $\theta$ ), 0 $\le$ $\theta$ $\le$ $\pi$ .

Find the equation of the cone z = $Find the equation of the cone z = expressed in spherical coordinates [ R, , \theta ].$ expressed in spherical coordinates [ R, $Find the equation of the cone z = expressed in spherical coordinates [ R, , \theta ].$ , $\theta$ ].

Find the eigenvalues of the matrix A = .

Describe the intersection of the graphs of y = x and y = 5 in 3-space.

Determine the equation of the plane that contains the points P = (1, -2, 0), Q = (3, 1, 4), and R = (0, -1, 2).

When does the equation x² + y² + z² + Ax + By + Cz + D = 0 represent a sphere?

Describe the intersection of the graphs of y = x and y² + z² = 9.

Find the scalar and vector projections of 2i -5j + k in the direction of 3i - 4k.

For what values of the constants k and c does the line = = lie in the plane x - y + 2z = c?

Describe the graph of 25x² - y² - z² = 25.

Describe the surface with equation x² + y² + z² - 6x + 2y - 12z = 75.

Find parametric equations of the straight line containing the point (2, 1, -4) and is parallel to the line of intersection of the planes 3x +2y - z = 0 and x + y + z = -9.

Describe the set of points in 3-space satisfying z² = x² + y² and z = x + y.

Find the equation of a plane that contains the line = = and is parallel to the plane 2x - 3y + 2z = 0.

Find the cosine of the angle $\theta$ between the planes 3x + 3y - 6z = 12 and 7x - 5y + 2z = -8.

Find the equation for the plane that passes through the point (1, 2, 3) and is normal to the vector joining (1, 3, 2) and (2, 3, 1).

If u = 3i + j + 4k, v = -i + 2j, and w = - 2i - 3j + 5k, evaluate u × (3v - w).

Let u and v be non-zero vectors in 2 or 3-space. Find a simplified expression for .

Which of the following is an equation of a hyperboloid of two sheets?

Describe the set of points in 3-space satisfying x² + 2y² + 3z² = 4 and z = x + y.

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Exam 11: Vectors and Coordinate Geometry in 3-Space

Describe the surface represented in cylindrical coordinates (r, θ\thetaθ , z) by the equation r = 2sin( θ\thetaθ ), 0 ≤\le≤ θ\thetaθ ≤\le≤ π\piπ .

Find the equation of the cone z = expressed in spherical coordinates [ R, , θ\thetaθ ].

Find the eigenvalues of the matrix A = .

Describe the intersection of the graphs of y = x and y = 5 in 3-space.

Determine the equation of the plane that contains the points P = (1, -2, 0), Q = (3, 1, 4), and R = (0, -1, 2).

When does the equation x2 + y2 + z2 + Ax + By + Cz + D = 0 represent a sphere?

Describe the intersection of the graphs of y = x and y2 + z2 = 9.

Find the scalar and vector projections of 2i -5j + k in the direction of 3i - 4k.

For what values of the constants k and c does the line = = lie in the plane x - y + 2z = c?

Describe the graph of 25x2 - y2 - z2 = 25.

Describe the surface with equation x2 + y2 + z2 - 6x + 2y - 12z = 75.

Find parametric equations of the straight line containing the point (2, 1, -4) and is parallel to the line of intersection of the planes 3x +2y - z = 0 and x + y + z = -9.

Describe the set of points in 3-space satisfying z2 = x2 + y2 and z = x + y.

Find the equation of a plane that contains the line = = and is parallel to the plane 2x - 3y + 2z = 0.

Find the cosine of the angle θ\thetaθ between the planes 3x + 3y - 6z = 12 and 7x - 5y + 2z = -8.

Find the equation for the plane that passes through the point (1, 2, 3) and is normal to the vector joining (1, 3, 2) and (2, 3, 1).

If u = 3i + j + 4k, v = -i + 2j, and w = - 2i - 3j + 5k, evaluate u × (3v - w).

Let u and v be non-zero vectors in 2 or 3-space. Find a simplified expression for .

Which of the following is an equation of a hyperboloid of two sheets?

Describe the set of points in 3-space satisfying x2 + 2y2 + 3z2 = 4 and z = x + y.

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Describe the surface represented in cylindrical coordinates (r, $\theta$ , z) by the equation r = 2sin( $\theta$ ), 0 $\le$ $\theta$ $\le$ $\pi$ .

Find the equation of the cone z = $Find the equation of the cone z = expressed in spherical coordinates [ R, , \theta ].$ expressed in spherical coordinates [ R, $Find the equation of the cone z = expressed in spherical coordinates [ R, , \theta ].$ , $\theta$ ].

When does the equation x² + y² + z² + Ax + By + Cz + D = 0 represent a sphere?

Describe the intersection of the graphs of y = x and y² + z² = 9.

Describe the graph of 25x² - y² - z² = 25.

Describe the surface with equation x² + y² + z² - 6x + 2y - 12z = 75.

Describe the set of points in 3-space satisfying z² = x² + y² and z = x + y.

Find the cosine of the angle $\theta$ between the planes 3x + 3y - 6z = 12 and 7x - 5y + 2z = -8.

Describe the set of points in 3-space satisfying x² + 2y² + 3z² = 4 and z = x + y.