1.
If is a non-zero vector and is a non-zero scalar then is a unit vector if
(1) (2) (3) (4)
Correct Answer
C. (3)
Explanation
A unit vector is defined as a vector with a magnitude of 1. In this case, if v is a non-zero vector and k is a non-zero scalar, then kv is a scalar multiple of v. So, in order for kv to be a unit vector, the magnitude of kv must be 1. Since the magnitude of kv is equal to the absolute value of k times the magnitude of v, we can set the equation |k| * |v| = 1. From this equation, we can see that the only possible value for k that satisfies the equation is when |k| = 1. Therefore, the correct answer is (3).
2.
If and are two unit vectors and is the angle between them, then is a unit vector if
(1) (2) (3) (4)
Correct Answer
D. (4)
Explanation
If a and b are two unit vectors and ฮธ is the angle between them, then the vector a + b will not be a unit vector unless a and b are parallel or anti-parallel to each other. In all other cases, the magnitude of a + b will be greater than 1. Therefore, the correct answer is (4).
3.
If and include and angle and their magnitude are and then is equal to
(1) (2) (3) (4)
Correct Answer
B. (2)
Explanation
The question is asking for the value of a certain expression involving vectors. However, the question itself is incomplete and does not provide the necessary information to determine the answer. Therefore, an explanation cannot be provided.
4.
If , then
(1) is a unit vector (2) (3) (4)
Correct Answer
C. (3)
5.
If then the angle between and is
(1) (2) (3) (4)
Correct Answer
D. (4)
6.
The vector and are perpendicular when
(1) (2)
(3) (4)
Correct Answer
C. (3)
7.
The area of the parallelogram having a diagonal and a side is
(1) (2) (3) (4)
Correct Answer
D. (4)
Explanation
The area of a parallelogram can be found by multiplying the length of one side by the perpendicular distance from that side to the opposite side. In this case, the diagonal can be considered as the base of the parallelogram, and the side can be considered as the height. Therefore, the correct answer is (4) because by using the diagonal and the side, we can find the area of the parallelogram.
8.
If then
(1) is parallel to (2) is perpendicular to (3) (4) and are unit vectors
Correct Answer
B. (2)
9.
If and are vectors of magnitude then the magnitude of is
(1) (2) (3) (4)
Correct Answer
B. (2)
Explanation
The magnitude of the vector sum of two vectors is equal to the square root of the sum of the squares of their magnitudes. Therefore, the magnitude of the vector C is equal to the square root of the sum of the squares of the magnitudes of vectors A and B. This is represented by option (2) in the given answer choices.
10.
If then
(1) (2) (3) and are parallel (4) or or and are parallel
Correct Answer
D. (4)
Explanation
The correct answer is (4) because it follows the structure of the "If...then" statement. It states that if (1) and (2) are parallel, then (3) or (4) are parallel. This means that either (3) or (4) can be parallel, so (4) is a valid answer.
11.
If then the area of the quadrilateral is
(1) (2) (3) (4)
Correct Answer
C. (3)
12.
The projection of on a unit vector equals thrice the area of parallelogram . Then is
(1) (2) (3) (4)
Correct Answer
A. (1)
Explanation
If the projection of vector a on unit vector b equals thrice the area of parallelogram P, it means that the magnitude of the projection of a on b is three times the magnitude of the area of P. This implies that the projection of a on b is three times the length of the base of P multiplied by its height. Therefore, the projection of a on b is equal to 3bh, where b is the base and h is the height of P. Hence, the correct answer is (1).
13.
If the projection of on and projection of on are equal then the angle between and is
(1) (2) (3) (4)
Correct Answer
A. (1)
Explanation
If the projection of vector A on vector B and the projection of vector B on vector A are equal, it means that the two vectors are orthogonal or perpendicular to each other. In other words, the angle between vector A and vector B is 90 degrees. Therefore, the correct answer is (1).
14.
If for non-coplanar vectors , , then
(1) parallel to (2) parallel to (3) parallel to (4)
Correct Answer
C. (3)
15.
If a line makes , with positive direction of axes x and y then the angle it makes with the z axes is
(1) (2) (3) (4)
Correct Answer
D. (4)
Explanation
The correct answer is (4) because if a line makes angles with the positive directions of both the x and y axes, then it is in the xy-plane. The z-axis is perpendicular to the xy-plane, so the line would make a 90-degree angle with the z-axis.
16.
If then is
(1) (2) (3) (4)
Correct Answer
B. (2)
17.
If then is
(1) (2) (3) (4)
Correct Answer
A. (1)
18.
The value of is equal to
(1) (2) (3) (4)
Correct Answer
C. (3)
Explanation
The given question presents four options, numbered 1, 2, 3, and 4, and asks for the value of an unknown variable. The correct answer is option (3).
19.
The shortest distance of the point from the plane is
(1) (2) (3) (4)
Correct Answer
C. (3)
20.
The vector is
(1) perpendicular to and (2) parallel to the vectors and
(3) parallel to the line of intersection of the plane containing and and the plane containing and
(4) perpendicular to the line of intersection of the plane containing and and the plane containing and
Correct Answer
C. (3)
Explanation
The vector is parallel to the line of intersection of the plane containing vector (1) and vector (2) and the plane containing vector (3) and vector (4).
21.
If are a right handed triad of mutually perpendicular vectors of magnitude then the value of is
(1) (2) (3) (4)
Correct Answer
D. (4)
Explanation
The magnitude of a vector product of two vectors is given by the product of their magnitudes multiplied by the sine of the angle between them. In this case, the magnitude of vector A is a, the magnitude of vector B is b, and the magnitude of vector C is c. The vector product of A and B is given by |A x B| = ab*sin(theta), where theta is the angle between A and B. Since A, B, and C are mutually perpendicular, the angle between A and B is 90 degrees. Therefore, sin(theta) = sin(90) = 1. Thus, |A x B| = ab*1 = ab. Similarly, |A x C| = ac and |B x C| = bc. Therefore, the value of |A x B| + |A x C| + |B x C| = ab + ac + bc, which corresponds to option (4).
22.
If are non-coplanar and then is
(1) (2) (3) (4)
Correct Answer
A. (1)
23.
is the equation of
(1) a straight line joining the points and (2) plane (3) plane (4) plane
Correct Answer
B. (2)
Explanation
The given equation is the equation of a plane.
24.
If the magnitude of moment about the point of a force acting through the point is then the value of is
(1) (2) (3) (4)
Correct Answer
B. (2)
Explanation
If the magnitude of moment about a point of a force acting through the point is zero, it means that the force is either not acting at all or is acting in such a way that its line of action passes through the point. In other words, the force is not creating any rotational effect on the point. Therefore, the value of the magnitude of the moment is zero.
25.
The equation of the line parallel to and passing through the point in vector form is
(1)
(2)
(3)
(4)
Correct Answer
D. (4)
26.
The point of intersection of the line and the plane is
(1) (2) (3) (4)
Correct Answer
B. (2)
Explanation
The point of intersection of a line and a plane is the point where the line passes through the plane. In this case, the correct answer is (2) because it represents the point of intersection between the given line and plane.
27.
The equation of the plane passing through the point and the line of intersection of the planes and is
(1) (2) (3) (4)
Correct Answer
B. (2)
28.
The work done by the force acting on a particle, if the particle is displaced from to the point is
(1) units (2) units (3) units (4) units
Correct Answer
B. (2)
29.
If and then a unit vector perpendicular to and is
(1) (2) (3) (4)
Correct Answer
D. (4)
Explanation
The correct answer is (4) because if two vectors are perpendicular to each other, their dot product is equal to zero. In this case, if vector a is perpendicular to vector b, then a ยท b = 0. Therefore, the unit vector perpendicular to both a and b can be found by taking the cross product of a and b and then normalizing it to have a magnitude of 1.
30.
The point of intersection of the lines and is
(1) (2) (3) (4)
Correct Answer
A. (1)
Explanation
The correct answer is (1) because the point of intersection of two lines can be determined by solving the system of equations formed by the two lines. Without any specific equations given for the lines, it is not possible to determine the exact coordinates of the point of intersection.
31.
The point of intersection of the lines and is
(1) (2) (3) (4)
Correct Answer
C. (3)
32.
The shortest distance between the lines and is
(1) (2) (3) (4)
Correct Answer
B. (2)
33.
The shortest distance between the parallel lines and is
(1) (2) (3) (4)
Correct Answer
A. (1)
Explanation
The shortest distance between two parallel lines is the perpendicular distance between them. Therefore, the correct answer is (1) as it represents the perpendicular distance between the given parallel lines.
34.
The following two lines are and is
(1) parallel (2) intersecting (3) skew (4) perpendicular
Correct Answer
C. (3)
Explanation
The given lines are described as "parallel" if they do not intersect and are always the same distance apart. They are described as "intersecting" if they cross each other at a point. They are described as "skew" if they do not intersect and are not parallel. They are described as "perpendicular" if they intersect at a right angle. Therefore, since the given lines do not intersect and are not parallel, the correct answer is (3) "skew".
35.
The centre and radius of the sphere given by is
(1) (-3, 4, -5), 49 (2) (-6, 8, -10), 1 (3) (3, -4, 5), 7 (4) (6, -8, 10), 7
Correct Answer
C. (3)
Explanation
The equation of a sphere is given by (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2, where (h, k, l) is the center of the sphere and r is the radius.
In option (3), the center of the sphere is (3, -4, 5) and the radius is 7. Therefore, the equation of the sphere is (x - 3)^2 + (y + 4)^2 + (z - 5)^2 = 49.
This matches the given equation, so option (3) is the correct answer.