Q.1 Find the distance between the points A(1,2,3) and B(4,6,8).
7
√50
√35
9
Explanation - Distance formula in 3D: √((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2) = √((4-1)^2 + (6-2)^2 + (8-3)^2) = √(9+16+25) = √50.
Correct answer is: √50
Q.2 The coordinates of the midpoint of the line segment joining P(2,3,4) and Q(6,7,8) are:
(4,5,6)
(3,4,5)
(5,6,7)
(2,3,4)
Explanation - Midpoint formula: ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2) = ((2+6)/2, (3+7)/2, (4+8)/2) = (4,5,6).
Correct answer is: (4,5,6)
Q.3 The vector equation of a line passing through point P(1,2,3) and parallel to vector v = 2i + 3j + 4k is:
r = (1,2,3) + t(2,3,4)
r = (1,2,3) + t(1,1,1)
r = (2,3,4) + t(1,2,3)
r = (0,0,0) + t(2,3,4)
Explanation - Vector form: r = r0 + tv, where r0 is position vector of a point on the line and v is the direction vector.
Correct answer is: r = (1,2,3) + t(2,3,4)
Q.4 The angle between vectors a = i + j + k and b = 2i + 3j + 4k is:
cos⁻¹(0.9)
cos⁻¹(0.8)
cos⁻¹(0.7)
cos⁻¹(0.6)
Explanation - cos θ = (a·b)/(|a||b|) = (1*2 + 1*3 + 1*4)/ (√3 * √(4+9+16)) = 9 / (√3*√29) ≈ 0.8.
Correct answer is: cos⁻¹(0.8)
Q.5 Equation of the plane passing through point (1,2,3) with normal vector n = i + 2j + 3k is:
x + 2y + 3z = 14
x + 2y + 3z = 12
x + 2y + 3z = 10
x + 2y + 3z = 6
Explanation - Equation of plane: n·(r - r0) = 0 → 1(x-1) + 2(y-2) + 3(z-3) = 0 → x + 2y + 3z = 12.
Correct answer is: x + 2y + 3z = 12
Q.6 Distance from point P(1,2,3) to the plane x + y + z = 6 is:
√3
3/√3
2/√3
1/√3
Explanation - Distance formula: |ax1 + by1 + cz1 + d| / √(a²+b²+c²) = |1+2+3-6| / √(1+1+1) = 0/√3 = 0.
Correct answer is: 3/√3
Q.7 If a line intersects xy-plane at point (2,3,0) and is parallel to z-axis, its parametric equations are:
x=2, y=3, z=t
x=t, y=t, z=0
x=2t, y=3t, z=t
x=2, y=t, z=t
Explanation - For line parallel to z-axis: x and y are constant; z varies with parameter t.
Correct answer is: x=2, y=3, z=t
Q.8 The scalar triple product of vectors a = i + j, b = j + k, c = k + i is:
0
1
2
-1
Explanation - Scalar triple product: a·(b×c) = 0. Compute b×c = determinant and dot with a → 0.
Correct answer is: 0
Q.9 The line joining points (1,0,0) and (0,1,1) has direction ratios:
1,-1,-1
-1,1,1
1,1,1
1,0,1
Explanation - Direction ratios: difference of coordinates = (0-1,1-0,1-0) = (-1,1,1).
Correct answer is: -1,1,1
Q.10 The projection of vector a = 3i + 4j + k on vector b = i + j + k is:
4
8/3
8/√3
3
Explanation - Projection = (a·b)/|b| = (3+4+1)/√(1+1+1) = 8/√3.
Correct answer is: 8/√3
Q.11 The plane passing through origin and perpendicular to vectors a = i + j and b = j + k has equation:
x - z = 0
x - y + z = 0
x - y - z = 0
x + y + z = 0
Explanation - Plane perpendicular to a and b: normal vector = a×b = |i j k;1 1 0;0 1 1| = i - j + k → equation: x - y + z = 0.
Correct answer is: x - y + z = 0
Q.12 The shortest distance between two skew lines is given by:
|(a2 - a1)·(b1×b2)| / |b1×b2|
|(a1 - a2)·(b1+b2)| / |b1+b2|
|(a1 - a2)×(b1×b2)| / |b1×b2|
|(a2 - a1)×(b1×b2)| / |b1×b2|
Explanation - Shortest distance formula for skew lines: d = |(r2-r1)·(b1×b2)| / |b1×b2|.
Correct answer is: |(a2 - a1)·(b1×b2)| / |b1×b2|
Q.13 If a = i + 2j + 3k, b = 4i + 5j + 6k, the vector perpendicular to both a and b is:
i - 2j + k
i - 2j - k
-3i + 6j - 3k
i + j + k
Explanation - Vector perpendicular to both = a×b = |i j k;1 2 3;4 5 6| = -3i + 6j - 3k.
Correct answer is: -3i + 6j - 3k
Q.14 Equation of sphere with center (1,2,3) and radius 5 is:
(x-1)^2 + (y-2)^2 + (z-3)^2 = 25
(x+1)^2 + (y+2)^2 + (z+3)^2 = 5
(x-1)^2 + (y-2)^2 + (z-3)^2 = 5
(x+1)^2 + (y-2)^2 + (z+3)^2 = 25
Explanation - Equation of sphere: (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 → (x-1)^2 + (y-2)^2 + (z-3)^2 = 25.
Correct answer is: (x-1)^2 + (y-2)^2 + (z-3)^2 = 25
Q.15 A line passes through (1,0,0) and (0,1,1). Its symmetric form is:
(x-1)/-1 = y/1 = z/1
x/1 = y/1 = z/1
x = y = z
(x+1)/1 = (y-1)/1 = z/1
Explanation - Direction ratios: (-1,1,1), symmetric form: (x-x1)/l = (y-y1)/m = (z-z1)/n → (x-1)/-1 = y/1 = z/1.
Correct answer is: (x-1)/-1 = y/1 = z/1
Q.16 The volume of the parallelepiped formed by vectors a = i + j, b = j + k, c = k + i is:
0
1
2
√2
Explanation - Volume = |a·(b×c)| = 0, because vectors are coplanar (scalar triple product = 0).
Correct answer is: 0
Q.17 The plane 2x - y + 2z = 5 intersects x-axis at:
(5/2,0,0)
(2,0,0)
(5,0,0)
(0,5/2,0)
Explanation - Intersection with x-axis: y=0, z=0 → 2x =5 → x=5/2 → point (5/2,0,0).
Correct answer is: (5/2,0,0)
Q.18 If two lines are perpendicular, then the dot product of their direction vectors is:
1
0
-1
Undefined
Explanation - Two vectors are perpendicular if a·b = 0.
Correct answer is: 0
Q.19 The distance between parallel planes x + y + z = 6 and x + y + z = 12 is:
6/√3
12/√3
√3
1
Explanation - Distance = |d2 - d1|/√(a²+b²+c²) = |12-6|/√(1+1+1) = 6/√3.
Correct answer is: 6/√3
Q.20 If a line has parametric equations x=1+2t, y=3-t, z=4+3t, a point on the line is:
(3,2,7)
(1,3,4)
(0,4,1)
(2,3,5)
Explanation - Take t=1 → x=1+2=3, y=3-1=2, z=4+3=7 → point (3,2,7).
Correct answer is: (3,2,7)
Q.21 Vector perpendicular to plane 2x - y + z = 3 is:
2i - j + k
i + 2j + k
-2i + j - k
i - j + k
Explanation - Normal vector to plane ax+by+cz+d=0 is n = ai + bj + ck → 2i - j + k.
Correct answer is: 2i - j + k
Q.22 The centroid of tetrahedron with vertices A(0,0,0), B(1,0,0), C(0,1,0), D(0,0,1) is:
(1/4,1/4,1/4)
(1/3,1/3,1/3)
(1/2,1/2,1/2)
(0,0,0)
Explanation - Centroid = average of coordinates = ((0+1+0+0)/4,(0+0+1+0)/4,(0+0+0+1)/4) = (1/4,1/4,1/4).
Correct answer is: (1/4,1/4,1/4)
Q.23 Equation of plane passing through three points (1,0,0),(0,1,0),(0,0,1) is:
x + y + z = 1
x + y + z = 0
x - y + z = 1
x + y - z = 1
Explanation - Plane through three points: x + y + z =1.
Correct answer is: x + y + z = 1
Q.24 If r1 = i + 2j + k and r2 = 2i + j + k, angle between them is:
cos⁻¹(4/√14)
cos⁻¹(5/√14)
cos⁻¹(6/√14)
cos⁻¹(3/√14)
Explanation - cos θ = (r1·r2)/(|r1||r2|) = (1*2 + 2*1 + 1*1)/(√6 * √6) = 5/6 ≈ cos⁻¹(0.833).
Correct answer is: cos⁻¹(4/√14)