The area of a triangle whose vertices are
 is

The edges of a parallelepiped are of unit length and are parallel to non-coplanar unit vectors  such that . Then the volume of the parallelepiped is

a)

b)

c)

d)

Consider the lines

 ;

 
The unit vector perpendicular to both L₁ and L₂ is

a)

b)

c)

d)

Consider the lines

 ;

 
The shortest distance between L₁ and L₂ is

a) 0

b)

c)

d)

Consider the lines

 ;

 
The distance of the point (1, 1, 1) from the plane through the point (−1, −2, −1) and whose normal is perpendicular to both lines L₁ and L₂ is

a)

b)

c)

d)

Consider three planes
P₁ : x – y + z = 1

P₂ : x + y – z = −1

P₃  : x – 3y + 3z = 2

Let L₁, L₂, L₃ be lines of intersection of planes P₂ and P₃, P₃ and P_1, and P₁ and P₂ respectively.

Statement 1 – At least two of the lines L₁, L₂, L₃ are non parallel

Statement 2 – The three planes do not have a common point.

a) Statement 1 is true. Statement 2 is true. Statement 2 is a correct explanation of statement 1.

b) Statement 1 is true. Statement 2 is true. Statement 2 is not a correct explanation of statement 1.

c) Statement 1 is true. Statement 2 is false.

d) Statement 1 is false. Statement 2 is true.

Let ABCD is the base of parallelopiped T and Aʹ.BʹCʹDʹ be the upper face. The parallelopiped is compressed so that the vertex Aʹ shifts to Aʹʹ on a parallelepiped S. If the volume of the new parallelopiped is 90% of the parallelopiped T, prove that the locus of Aʹʹ is a plane.

If the lines x = ay + b, z = cy + d and x = aʹy + bʹ, z = cʹy + dʹ are perpendicular to each other, then

a) aaʹ + ccʹ = 1

b)  +  = 1

c) aaʹ + ccʹ = 0

d) aaʹ + ccʹ = –1

The image of the point (−1, 3, 4) in the plane x – 2y = 0 is

a) (15, 11, 4)

b) ( , , 1)

c) (8, 4, 4)

d) ( , , 4)

If the plane 2ax – 3ay + 4az + 6 = 0 passes through the mid-points of the line joining the centre of spheres x² + y² + z² – 10x + 4y – 2z = 8 and x² + y² + z² + 6x – 8y – 2z =13 then a equals

a) 2

b) – 2

c) 1

d) –1

If the angle θ between the line  =  =   and the plane 2x – y +   z + 4 = 0 is such that sin θ =, then the value of  is

a)

b)   

c)

d)

The angle between the lines 2x = 3y = – z and 6x = – y = – 4z is

a) 30˚

b) 45˚

c) 90˚

d) 0˚

The plane x + 2y – z = 4 cuts the sphere x² + y² + z² – x + z – 2 = 0 in a circle of radius

a)  

b) 2

c) 1

d) 3

A line makes same angle θ with each of the x and z axis. If the angle β, which it makes with the y axis, is such that sin²β = 3 sin²θ then cos² θ equals

a)

b)

c)

d)

Distances between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is

a)

b)

c)

d)

A line with direction cosines proportional to 2, 1, 2 meets each of the lines x = y + a = z and x + a = 2y = 2z then the co-ordinates of each of the points of intersection are given by

a) (3a, 3a, 3a), (a, a, a)

b) (3a, 2a, 3a), (a, a, a)

c) (3a, 2a, 3a), (a, a, 2a)

d) (2a, 3a, 3a), (2a, a, a)

The intersection of the spheres
x² + y² + z² + 7x – 2y – z = 13 and
x² + y² + z² – 3x + 3y + 4z = 8
is the same as the intersections of one of the spheres and the planes

a) x – y – z = 1

b) x – 2y – z = 1

c) x – y – 2z = 1

d) 2x – y – z = 1

The radius of the circle into which the sphere x² + y² + z² + 2x – 2y – 4z – 19 = 0 is cut by the plane x + 2y + 2z + 7 = 0 is

a) 1

b) 2

c) 3

d) 4

The shortest distance of the plane 12x + 4y + 3z = 327 to the sphere x² + y² + z² + 4x – 2y – 6z = 155 is

a) 26

b) 11

c) 13

d) 39

Two systems of rectangular axes have the same origin. If a plane cuts them at distances a, b, c and aʹ, bʹ, cʹ, from the origin, then

a)  +  + +  +  +  = 0

b)  +  - +  +  -  = 0

c)  -  - +  -  -  = 0

d)  +  + -  -  -  = 0

A parallelopiped is formed by planes drawn through the points (2, 3, 5) and (5, 9, 7) parallel to the coordinate planes. Then the length of the diagonal of the parallelopiped is

a) 7 units

b)  units

c)  units

d) None of these

The equation of the plane containing the line  =  =   is a (x – x₁) + b (y – y₁) + c (z – z₁) = 0 where

a) ax₁ + by₁ + cz₁ =0

b) al + bm + cn = 0

c)   =  =   

d) lx₁ + my₁ + nz₁ = 0

The centre of the circle given by r. (î +2ĵ + 2k̂) = 15 and |r – (j + 2k̂)| = 4 is

a) (0, 1, 2)

b) (1, 3, 4)

c) (–1, 3, 4)

d) None of these