The vertices of a ΔABC are A (1, 0, 2), B (−2, 1, 3) and C (2, 1, −1). If D be the feet of perpendicular drawn from A on BC the vector equation of the bisector of angle A is given by

a) r = (î + 2ĵ) + λ()

b) r = (î + 2) + λ()

c) r = (î + 2ĵ) + λ()

d) None of these

If the axes are rectangular, the distance from the point (3, 4, 5) to the point where the line  =   meets the plane x + y + z = 17 is

a) 1

b) 2

c) 3

d) None of these

There is a point P (a, a, a) on the line passing through the origin and equally inclined with axes. The equation of the plane perpendicular to OP and passing through P cuts the intercepts on the axes, the sum of whose reciprocal is

a)  

b)

c)

d)

Find the equation of the bisector of the obtuse angle between the lines 3x – 4y + 7 = 0 and – 12x – 5y + 2 = 0.

a) 21x + 77y – 101 = 0

b) 99x – 27y + 81 = 0

c) 21x – 77y + 101 = 0

d) None of these

Let OA, OB are drawn from origin with direction cosines proportional to <1, – 2, –1> and <3, – 2, 3> respectively. The direction ratio of the normal to the plane AOB are

a) <4, 3, 2>

b) <4, –3, –2>

c) <–4, 3, –2>

d) <4, 3, –2>

The shortest distance between the lines r =  + t and r =  + s where  and  are non-collinear is

a)   0

b)    .

c)

d) None of these

The distance between the line r = 2î – 2ĵ + 3k̂ + λ (î – ĵ + 4k̂) and the plane r. (î + 5ĵ + k̂) = 5 is

a)  

b)

c)

d)

The equation of the line of intersection of planes 4x + 4y – 5z = 12 and 8x + 12y – 13z = 32 can be written as

a)   =  =

b)   =  =

c)   =  =

d)   =  =

If a plane passing through the point (1, 1, 1) and is perpendicular to the line  =  , then its perpendicular distance from the origin is

a)   

b)  

c)

d)  1

The reflection of the point (2, –1, 3) in the plane 3x – 2y – z = 9 is

a) (, , )

b) (, , )

c) (, , )

d) (, , )

The equation of the plane through the point (–1, 2, 0) and parallel to the lines  =   and  =   is

a) x – 2y + 3z + 5 = 0

b) x + y + 3z –1 = 0

c) 2x + 3y +6x –4 =0

d) x + y – 3z + 1 =0

A variable plane at a constant distance p from the origin cuts the axes at A, B, C then the locus of the centroid of tetrahedron is

a) x⁻² + y⁻² + z⁻² = 16p²

b) x⁻² + y⁻² + z⁻² = 16p⁻¹

c) x⁻² + y⁻² + z⁻² = 16 p⁻²

d) x⁻² + y⁻² + z⁻² = 16 p

The cartesian equation of plane r = (1 + λ  µ) î + (2 – λ) ĵ + (3  2λ + 2µ) k̂ is

a) 2x + y =5

b) 2x – y = 5

c) 2x + z = 5

d) 2x – 2 = 5

The line joining the points (4, -1, 2) and (-3, 2, 3) meet the plane at (-10, 5, 4) at 90˚, then the equation of the plane is

a) 7x - 3y + z + 89 = 0

b) 7x - 3y - z + 89 = 0

c) 7x + 3y + z +89 =0

d) None of these

The shortest distance between the skew lines  =  =  and   =  = is 9

a) True

b) False

Let two planes P₁ : 2x – y + z = 2 and P₂ : x + 2y – z = 3 are given. The equation of the bisector of angles of the planes P₁ and P₂ not containing the origin is

a) x – 3y + 2z + 1 = 0

b) x + 3y = 5

c) x + 3y + 2z + 2 = 0

d) 3x + y = 5

A line AB in three dimensional space makes angles 45˚ and 120˚ with the positive X- axis and the positive Y- axis respectively. If AB makes an acute angle θ with the positive Z– axis, then θ equals

a) 30˚

b) 45˚

c) 60˚

d) 75˚

The line   =  =   lies in the plane x + 3y – αz + β = 0.  Then, (α, β) equals

a) (6, –17)

b) (– 6, 7)

c) (5, –15)

d) (– 5, 15)

Statement 1 – The points (2, 1) and (−3, 2) lie on opposite sides of the line
Statement 2 – The algebraic perpendicular distance from the given points to the line have opposite signs

a) Statement 1 is false, Statement 2 is true

b) Statement 1 is true, Statement 2 is true. Statement 2 is correct explanation of Statement1

c) Statement 1 is true, Statement 2 is true. Statement 2 is not a correct explanation of Statement1

d) Statement 1 is true, Statement 2 is false

Statement 1 – The equation  represents an ellipse
Statement 2 – The only condition for a general equation of second degree to be an ellipse is that the square of the coefficient of xy is less than the product of coefficients of  and

a) Statement 1 is false, Statement 2 is true

b) Statement 1 is true, Statement 2 is true. Statement 2 is correct explanation of Statement1

c) Statement 1 is true, Statement 2 is true. Statement 2 is not a correct explanation of Statement1

d) Statement 1 is true, Statement 2 is false

If a > 0, b > 0 the maximum area of the triangle formed by the points O(0, 0), A(acosθ, bsinθ), B(acosθ, −bsinθ) is (in square units)

a)  when

b)  when

c)  when

d)

If the lines x = 1 + s, y = – 3 – λ s, and x = , y = 1 + t, z = 2 – t with parameters s, t respectively are coplanar, then λ equals

a) – 2

b) – 1

e) –

c) 0

e) –

c) 0

e) –

c) 0

e) –

c) 0

e) –

c) 0

e) –

c) 0

e) –

c) 0

e) –

c) 0