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 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)

Without changing the direction of coordinate axes, origin is transferred to (h, k) so that the linear terms in the equation  are eliminated, then the point (h, h) is

a) (3, 2)

b) (3, 2)

c) (2, 3)

d) None of these

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