1. Which of the following is a vector that lies on the line through (0, 0) and (2, 4)?

1
A. u⃗ =¿ (2, 1) C. u⃗ =¿( , 2)
2
B. u⃗ =¿ (−1 ,−2 ) D. u⃗ =¿(−2 ,−6 )
i j
2. Let and be the standard unit vectors in the direction of positive X-axis and positive Y-
axis, respectively, and ⃗ AB be a vector from the point B(2,2). If ⃗v ¿ 3⃗ AB +2 j , the unit
vector in the direction of ⃗v is equal to:
A. ( ) 3 4
5 5
, (
B.
−3 −4
5 5
,) C. ( −3 4
5 5
,) D.( 3 −4
5 5
, )
3. Which of the following is a vector equation of the line tangent to the circle
2 2
x + y + 2 x−7=0 at (1, 2)
A. ( x , y ) =¿ ( 0 , 3 ) + λ (−1 , 2 ) C. ( x , y ) =¿ ( 0 , 3 ) + λ ( 1 ,−1 )
B. ( x , y ) =¿ ( 1 , 2 )+ λ ( 2 ,−1 ) D. ( x , y ) =¿ ( 1 , 2 )+ λ (−1 ,2 )
4. Let l be the line whose equation is 2 x− y=10 . Which one of the following is the equation
of the image of l after a reflection in the line y=2 x−5 followed by a rotation through the
angle of 90° about the origin?
A. x +2 y=0 B. 2 x + y=0 C. x +2 y=5 D. x−2 y=5
5. If u⃗ ¿ (−3 , x ) and ⃗v ¿ ( x , y −2 ) are vectors, what is the value of y so that
1
u⃗ + ⃗v =3 ⃗u− ⃗v ?
2
2 −10 −22
A. B. C. −4 D.
3 3 3
6. A line given by a vector equation r ( x )=( 0 , 3 ) +t ( 1 , 1 ) is tangent to a circle at point( 0 , 3 ) . If
the radius of the circle is √ 2, which one of the following is the of the circle?
A. ( 1 , 4 ) B. ( 1 ,−4 ) C. (−1 , 2 ) D. ( 1 , 2 )
2 2
7. What is the image of the ellipse whose equation is 2 ( x+ 2 ) + ( y−1 ) =2 under a translation
that takes (2, 1) to (4, 0) followed by a rotation of 90° ?
A. x 2+ 2 y 2 =2 C. 2 ( x−4 )2 + y 2=2
B. 2 x 2+ y 2 =2 D. ( x−4 )2 +2 y 2=2
8. If ⃗ A is perpendicular to ⃗ B, what is the cosine of the angle between ⃗ A and ⃗ A−⃗ B?
|⃗A −⃗B| |⃗A| |⃗A −⃗B| |⃗B|
A. B. C. D. ⃗
|⃗A| |⃗A −⃗B| |⃗B| |⃗A −⃗B|
9. Which of the following is necessarily true?
A. If |⃗ A|=|⃗
B|, then⃗A=⃗B. C. If u⃗ is a unit vector in the direction of ⃗
A , then
A . ⃗u=|⃗
⃗ A|
B. |k ⃗A|=k|⃗ A|, for any real number. D. If ⃗ B, then ⃗
A is parallel to ⃗ A .⃗
B=0
10. If a point (2, 5) is reflected under a line to the point (−3 , 1), what is the line of reflection?
A. 2 x+3 y =7 C. 8 y +10 x=19
B. x +3 y=7 D. 2 x+3 y +5=0
11. If A¿(−2 ,3), B¿(3 ,1) and C is any other point on the plane, then which one of the
following is the coordinate form of ⃗ AC −⃗ BC ?
A. (−5 , 2 ) B. ( 5 ,−2 ) C. ( 1 , 4 ) D.(−1 ,−4 )
12. What is the equation of a line that passes through the point (−1 , 2), and parallel to the vector
(1,−1)?
A. 2 x− y=1 C. x−2 y=3
B. x + y−1=0 D. y−2 x +1=0
13. What is the image of the line given by ( x , y ) = ( −1 ,0 )+ t ( 3 , 6 ) , t ∈ R , under the translation
that takes (1, 0) to (0, 1) followed by the reflection about the line y=2 x ?
A. y=2 x +3 C. y=2 x +6
B. y=2 x−3 D. y=2 x−3
2 2
14. If a translation T takes the circle x + y −2 x+6 y+3=0 into the circle whose equation is
( x +2 )2+ ( y−4 )2=7 , then what is the image of the origin under T?
A. (−3 , 7) B. (1 , 2) C. (1.−3) D. (−2 , 4)
15. If l is the line that passes through (0, 2) and parallel to ⃗v =i+ 3 j which of the following is
true about l and the circle ( x−2 )2 + ( y−1 )2=5?
A. l is tangent to circle at (0, 2).
B. l is tangent to the circle at some point p, where p≠(0 ,2).
C. l Intersects the circle at exactly two points.
D. The distance between l and the center of the circle is greater than √ 5.
16. Suppose ⃗ A=3i−4 j and ⃗ B is a vector in the xy-plane such that the angle between ⃗ A and ⃗B
π
is . u⃗ is a unit vector in the direction of ⃗ B, then ⃗ A .( ⃗
A −2 ⃗u) is equal to:
3
A. 20 B. 5 C. 15 D. 30
17. If A=(1 ,−2), B=(−3 , 2) and V is a position vector such that 2 V + AB=0, then⃗
⃗ ⃗ ⃗ V is equal
to:
A. ( 2 , 0 ) B.(−1 , 0 ) C. (−2 , 2 ) D.( 2 ,−2 )
2
18. If ⃗ A=4 i−3 j and u⃗ is a unit vector such that |⃗ A +u| =27 , then the cosine of the angle
between ⃗ A and u⃗ is equal to…………
A. 0.1 B. 0.2 C. 0.3 D. 0.4
2 2
19. What is the image of the ellipse ( x−1 ) +4 y =1 under the translation that takes (1, 1) and
(0, 2) followed by the reflection through the x-axis?
A. x 2+ 4 ( y −1 )2=1 C. x 2+ 4 ( y +1 )2=1
B. 4 x 2+ ( y −1 )2=1 D. 4 x 2+ ( y +1 )2=1
20. What is the work done (in Joules) when a force of 50N is used to pull a crate 20m along a
level path if the force is at an angle of 60° ?
A. 360 B. 500 C. 760 D. 1500√ 2
 21. The image of a figure with vertices A(1 , 2), B(3, 6) C(−1 , 2) and D(−2 ,−2 ) after reflection
across the x-axis is:
A. A’( 1 ,−2 ) , B’(−3 ,−6 ), C’( 1 ,−2 ) and D’( 2 , 2 )
B. A’( 1 ,−2 ) , B’( 3 ,−6 ), C’(−1 ,−2 ) and D’(−2 , 2 )
C. A’(−1 , 2 ) , B’(−3 , 6 ) , C’( 1 ,−2 ) and D’( 2 ,−2 )
D. A’( 1 ,−2 ) , B’( 3 , 6 ) , C’(−1 , 2 ) and D’(−2 ,−2 )
22. If ⃗
V =⃗ AB+3 ⃗ BA where A and B are distinct points in the coordinate plane, then which one
of the following is equal to 3⃗ V?
A. 6 ⃗ AB B.−6 ⃗ AB C. 12 ⃗ AB D. −12 ⃗AB
23. If u⃗ is a unit vector in the direction of ⃗ A and |⃗A|=4, ⃗A .u⃗ is equal to………………………
1 1
A. B. 4 C. D. 2
4 2
24. If ⃗A and ⃗ B are parallel vectors with opposite direction and |⃗ B|=|2 ⃗
A|, then ⃗
B− ⃗A is equal
to:
A. ⃗ A B. −⃗A C. −3 ⃗ A D.3 ⃗
A

25. What is the translation vector u¿(h, k) so that the equation x 2+ 2 y 2 +6 x−8 y+15=0 is
transformed to an equation of the form x 2+ 2 y 2 +d =0 where d is constant?
A. u¿(−3 ,2) C. u¿(−2 ,3)
B. u¿(3 ,2) D. u¿(2,−3)
26. A line given by the vector the equation ( x , y ) =(−t , 6+2 t ) ,t ∈ R , is tangent to a circle at
point (1, 4). What is the radius of the circle if its center is on the y-axis?
1
A. √5 B. √ 5 C. 2 √ 5 D. √ 10
2
27. For two non-zero vectors a⃗ and b⃗ if |⃗a + ⃗b|=|a⃗| , then which of the following is true?
A. 2 ⃗a . ⃗b= ⃗b . ⃗b ⃗
C.2 ⃗a + b∧2 ⃗a + ⃗b are parallel
B. ⃗ b⃗ are perpendicular
2 ⃗a + b∧ D.a⃗ . ⃗b−⃗b . ⃗b=0
⃗
28. Let PQ be a vector with initial point P= (1 , 5 ) and terminal point Q= ( 4 , 0 ) . If ⃗ V =xi+2 j is
⃗
parallel to PQ , then what is the value of x?
−6 −2
A. B. −3 C. D. 3
5 5

29. Let l be the line given by the vector equation ( x , y ) =(−2 ,1 ) + λ (1 , 1 ) , λ ∈ R . Which one of
the following is the image of l after being translated by the vector u=( 2 ,−1 ) followed by a
rotation through 45° about the origin?
A. y= √2 x B. y= √ 2 C. x=0 D. y=−2 √ 2x
2 2
30. What is the image of the circle x + y −4 x−6 y +12=0 when it is reflected with respect the
line y=− x?
A. ( x +3 )2 + ( y +2 )2=1 C. ( x +2 )2+ ( y +3 )2=1
B. ( x−2 )2 + ( y−3 )2=1 D. ( x−3 )2 + ( y −2 )2=1
31. If the image of the line 2 x−3 y=7 under a translation is 2 x−3 y=0 , which one of the
following is a translation vector of the translation line?
A. u=( 2 ,−1 ) C. u=(−1 ,2 )
B. u=(−2 ,1 ) D. u=( 1,−2 )
32. If in, ∆ ABC , AB=3, BC=4 and m( ¿ B )=60 ° , then what are the length of AC and the cosine
of ¿ A , respectively?
1 6
A. √ 13 and 13 C. √ 13 and
√ 5 √ 13
−1 −6
B. √ 13 and D. √ 13 and
√13 5 √13
33. What is the standard equation of the line passing through the point (2, 3) and parallel to the
line given by {x=1+2 λ
y=−2−λ
, λ∈R ?
x−2 y −3 x−2 y −3
A. = C. =
−1 2 2 −1
x−1 y −3 x−1 y −3
B. = D. =
2 −2 1 −2

34. Consider a rectangle ABCD with base vertices A¿(0, 3) and B¿ ( 4 , 0 ) and the other vertices,
C and D, in the first quadrant of the coordinate plane. If its height BC is half of the length
of its base, then which of the following indicates the coordinate of the vertex C?
A. ( )
4,
5
2 ( )
B. 6 ,
3
2
5
C. ,−2
2 ( ) D.
11
2
,2 ( )
35. What is the image of the circle x 2+ y 2−4 x−6 x+ 11=0 when the origin is shifted to the
point (1, 1) after translation of axes?
2 2
A. x + y −6 x−8 y +23=0 C. x 2+ y 2+ 6 x+ 8 y −23=0
2 2
B. x + y −4 x−6 y +3=0 D. x 2+ y 2−4 x−2 y+3=0
36. When the plane is rotated 45° about the point (1,−2), then what would be the image of the
point (2, 4)?

(
A. 1−
5 √2
2 )
,−2+ √
7 2
2
C.
5 √2 7 √ 2
2
,
2 ( )
B. (1− ,−2+ ) ( √22 , √22 )
√2 √2 D.
2 2
37. Let l be a line given by the equation ( x , y ) =( 1 ,1 ) +t ( √ 3 , 1 ) , t ∈ R , what is the equation of the
image of l after being rotated 15° about (1, 1) and then translated by the vector u¿ (−1 , 1 )?
A. −x + y=2 C. √ 3 x− y=2
B. x− y =2 D. −x + √ 3 y=1
38. If a sphere with center C(0, 1, 1) intersects the z-axis at p(0, 0, 3), then the radius of the
sphere is equal to:
A. 5 B. 3 C. √ 3 D. √ 5
39. Let 1 and 1 be two lines in space intersecting at the origin, (0, 0, 0). If 1 and l 1 passes
l l l
through A(1, 1, 0) and B(0, 1, 1), respectively, then the angle between l 1 and l 1 is equal to:
A. 30° B. 45° C. 60° D. 90°

40. Let ⃗
V =3i−4 k , and ⃗AB is a vector from point A(0, 1, 2) to a point B in space. If ⃗
AB is
V and |⃗
parallel to ⃗ AB=10|, then point B is at:
 A. (−6 ,−1 ,10 ) C. ( 6 , 1 ,−6 )
B. ( 6 ,−1 ,−10 ) D. (−6 ,−1 ,6 )
⃗
41. Let a⃗ =−i+3 k and b=−i+ j be vectors in space. Which one of the following is the cosine of
the angle between a⃗ and a⃗ −⃗b?
9 3 3 −9
A. B. C. D.
10 5 √10 10
42. Suppose A and B are the end points of a diameter of the sphere whose equation is
2 2 2
x + y + ( z +2 ) =1. If A¿(1, 0, -2), then B is equal to:
A. (0, 1, −2) C. (−1 , 0 ,−2)
B. (0, 0, −1 ) D. (0 ,−1,−2)
43. Suppose P(1, 2, 1) and Q(1, 0, 2) are points in space and ⃗ A=⃗ B is parallel to ⃗
PQ .if ⃗ PQ and
⃗ ⃗
A . B=−10 , then which one of the following is true?
1
A. ⃗A and ⃗ B have the same direction. C. |⃗B|= |⃗ A|
10
B. |⃗B|=10|⃗ A| D. |⃗ B|=2|⃗ A|

44. Which one of the following points is closer to the sphere x 2+ y 2+ z 2−2 x +6 z +9=0 ?
A. (1, 0, 0) B. (0, 0, 0) C. (0 ,−1, 0 ) D. (0, 0, −1)
45. Suppose ⃗ A=2 j−k and ⃗ B=5 i+15 k where i , j and k are the standard unit vectors in the
direction of positive x, y and z-axis, respectively. Which one of the following is the unit
1
vector in the direction of ⃗A+ ⃗ B?
5
3 4 4 3
A. i+ k C. j− k
5 5 5 5
1 2 2 2 1 2
B. i+ j+ k D. i− j+ k
3 3 3 3 3 3
46. Let a⃗ =2 i+ ( x−1 ) j+ k and c⃗ =i− j+ yk be vectors. If a⃗ .c⃗ =0 and |⃗a|=3 , which one of the
following is possible value of y?
A. −4 B. −1 C. 3 D. 4

47. Suppose ⃗ A=2i− j+2 k and ⃗ B is a vector in space such that |⃗B|=⃗A .⃗B . if u⃗ is a unit vector in
the direction of |⃗B|, then |⃗
2
A + u⃗ | is equal to:
A. 16 B. 12 C. 10 D. 14
⃗
48. Suppose P and Q are points in space such that the midpoint of PQ is on the negative z-axis
and the distance between P and Q is 6.if P¿(2,−1 ,0), then what is the coordinate of Q?
A. (−2 , 1, 4 ) C. ( 2 ,−1 ,−6 )
B. ( 2 ,−1 , 6 ) D. (−2 , 1,−4 )

49. If p= (3 ,α −1 , α + 2 ) and Q=(2 α+1 , 3 , 3 α ) are points in space, what should be the value(s)
of α so that the distance between the two points is 6?
A. α =−2 or α =¿ 5 C. α =−1 or α =3
B. α =0 or α =5 D. α =¿ -3 or α =2
50. If (−1 , 2 ,2) and ( 1 , 0 ,−2 ) are end points of a diameter of a sphere, then which one of the
following is true about the sphere?
 A. (0, 1, 0) is a point on the sphere.
B. The equation of the sphere is x 2+ ( y−1 )2 + z 2=6.
C. The equation of the sphere is x 2+ ( y−2 )2 + z 2=6 .
D. The radius of the sphere is 6.
51. Suppose that the equation x 2+ y 2+ z 2 +2 x+ 8 z=6 ( y +1 ) represents a sphere. Where is the
point ( 1 ,−1 , 4 ) located relative to the sphere?
A. Inside the sphere. C. At the center of the sphere.
B. On the sphere. D. outside the sphere.
52. Suppose l is the line through the center of the sphere x 2+ y 2+ ( z −2 )2=9 and intersects the
sphere at (1, 2, 4). What is the cosine of the angle between l and positive z-axis?
2 1 3 4
A. B. C. D.
3 3 5 5

53. Let the angle between ⃗ V =−2i− j+2 k and ⃗ PQ be 60° , where P and Q are points in space. If
⃗ ⃗
V . PQ=2 , the what is the distance between P and Q?
3 4 4 5
A. B. C. D.
4 5 3 4

( ) ( )
1 2k
54. What is the value of k, for which the two vectors u⃗ = k and ⃗ V = −5 are perpendicular?
−3 4
A. 4 B. −4 C. 3 D. −3
55. If one of the end point of the line segment is (3 , 2 ,−4 ) and the mid-point is (4 , 1,−2), then
the coordinate of the other end point is:
A. (5, 0, 0) B. (2, 0, 5) C. (5, 1, 2) D. (3, 1, 0)

56. Let ⃗ A and ⃗ B be vectors in space such that ⃗ A and ⃗B be vectors in space such that ⃗ A.⃗ B=−2
and B=6 i−7 j+ √ 15 k . If θ is an angle between A and B, then what is the value of |⃗
⃗ ⃗ ⃗ A|?
1 1 −1 −1
A. cos θ B. C. cos θ D.
5 5 cos θ 5 5 cos θ
57. If P( 2 , √ 5 , 1 ) and Q( 3 , 0 , 9 ) are points on a sphere whose center is on z-axis, then which one
of the following points is outside of the sphere?
A. (−4 , 3 ,5 ) B. ( 2 ,−2 , 1 ) C. ( 3 , 1 ,1 ) D. ( 0 , 0 , 0 )
58. If A ( x ,0 , 2 ) , B ( 3 ,0 , 2 ) and C ( 2 , √ 3 ,2 ) are vertices of equilateral triangle in space, then what
is the value of x?
A. 5 B. 3 C. 2 D. 1

59. Let A( a , 2 , 5 ) , for a> 0, be a point on the sphere x 2+ y 2+ z 2−6 z=0 and C be center of the
sphere. If P ( k , 2 , 4 )
Is a point in space such that ⃗ PA is perpendicular to ⃗ CA , what is the cosine of the angle
between ⃗ PA and PC ? ⃗
 5 7 7
A. B. C. D.
√35 √35 √70
5
√70
60. If the dot product of a vector ⃗A with the vectors i− j+ k ,2 i+ j−3 k and i+ j+k are 4, 0 and
⃗
2, respectively, what is A ?
A. ⃗A=( 2 ,1 , 1 ) C. ⃗A=(−2 ,−1 , 1 )
⃗
B. A=(−2 ,1 ,−1 ) ⃗
D. A=( 2 ,−1 ,1 )
61. Let P= (1 , α ,α ) and Q= ( α −1, 1 , 1 ) be two points in space and the distance between P and Q
is 3. Then what is the value(s) of α ?
−1
A. α =−1 , α =9 C. α =3 , α =
3
1
B. α =1 , α =−9 D. α =−3 , α =
3
2011
π
62. If the angle between the vectors ⃗
A=( 2 ,−1 ,1 ) and ⃗
B=( 1 , 1 ,α ) is , then what is the value
3
of α ?
A. 1 B. −1 C. 2 D. −2
63. If the point ( α ,0 ,3 ) is on the sphere whose center is ( 1 , 2, 3 ) and radius 2, what is the value
of α ?
A. 0 B. 1 C. 2 D. −3
64. If u⃗ =2 j−k and ⃗v =i−8 j+3 k , then what is the unit vector in the direction of 5 u+ v ?
1 2 2
A. i+2 j−2 k C. i+ j− k
3 3 3
2 1 2 1 2 2
B. i+ j− k D. i+ j+ k
3 3 3 3 3 3