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QUESTION: 1

A line makes angles α,β,γ with the coordinates axes. If α+β = 90°, then (gamma) equal to

Solution:

If a line makes the angles α,β,γ,

Then, cos^{2 }α+cos^{2 }β+cos2 γ=1

It is given that, α+β=90°

⇒ α=β−90°

⇒ cosα=cos(90o−β)

⇒ cosα=sinβ

⇒ cos^{2 }α = sin^{2 }β=1−cos^{2} β

⇒ cos^{2}α+cos^{2}β=1

As, cos^{2 }α+cos^{2} β+cos^2γ=1

⇒ 1+cos^{2} γ=1

⇒ cos^{2} γ=0

hence, γ=π/2=90°

QUESTION: 2

The coordinates of the point A, B, C, D are (4, α, 2), (5, –3, 2), (β, 1, 1) & (3, 3, – 1). Line AB would be perpendicular to line CD when

Solution:

QUESTION: 3

The locus represented by xy + yz = 0 is

Solution:

QUESTION: 4

The equation of plane which passes through (2, –3, 1) & is normal to the line joining the points (3, 4, –1) & (2, – 1, 5) is given by

Solution:

A(3,4,−1) and B(2,−1,5)

Vector AB is normal to the required plane

⇒ Directions of normal (−1,−5,6)

∴ Equation ,−x−5y+6z=k

Point (2,−3,1) passes through the plane,

∴ −2+15+6=k⇒k=19

∴ −x−5y+6z = 19

x+5y−6z+19 = 0

QUESTION: 5

If the sum of the squares of the distances of a point from the three coordinate axes be 36, then its distance from the origin is

Solution:

Let (x,y,z) be the point.

Given sum of the squares of distance from point to the axes is 36.

⇒(x^{2}+y^{2})+(y^{2}+z^{2})+(z^{2}+x^{2})=36

⇒2(x^{2}+y^{2}+z^{2})=36⇒x^{2}+y^{2}+z^{2}=18

So the distance of the point from the origin is =3(2)^{1/2}

QUESTION: 6

The locus of a point P which moves such that PA^{2} – PB^{2} = 2k^{2} where A and B are (3, 4, 5) and (–1, 3, –7) respectively is

Solution:

QUESTION: 7

The equation of the plane passing through the point (1, – 3, –2) and perpendicular to planes x + 2y + 2z = 5 and 3x + 3y + 2z = 8, is

Solution:

The normals to the planes x+2y+2z=5 and 3x+3y+2z=8 are their respective unit vectors ie

Since the required plane is perpendicular to the planes

x+2y+2z=5 and 3x+3y+2z=8,

its normal would be perpendicular to the normals to the planes

The cross product of is normal to the required plane

The equation of the required plane would be

-2x+4y-3z=2+12-6 ie 2x-4y+3z-8 = 0

QUESTION: 8

A variable plane passes through a fixed point (1, 2, 3). The locus of the foot of the perpendicular drawn from origin to this plane is

Solution:

α(x − α) + β(y − β) + γ(z − γ) = 0

α(1 − α) + β (2 − β) + γ ( 3 − γ) = 0

α + 2β + 3γ = α^{2 }+ β^{2 }+ γ^{2}

α^{2 }+ β^{2 }+ γ^{2 }− α − 2β − 3γ = 0

x^{2 }+ y^{2 }+ z^{2 }− x − 2y− 3z = 0

QUESTION: 9

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

Solution:

line AB = (x-2)/3 = (y+1)/-2 = (z-3)/-1 = λ

(x,y,z)=(3λ+2,−2λ−1,−λ+3)

3x−2y−z=9

3(3λ+2)−2(−2λ−1)−3+λ=9

9λ+6+4λ+2−3+λ=9

14λ=4

λ=2/7

C(x,y,z)=(207, -117, 19/7)

A(2,-1,3)

C is MP of A and B

C= (A+B)/2, B= (2C-A)

B=(26/7, −22/7+1, 38/7−3)

B=(26/7,−15/7,17/7)

QUESTION: 10

The distance of the point (–1, –5, –10) from the point of intersection of the line, and the plane, x – y + z = 5, is

Solution:

QUESTION: 11

The distance of the point (1, –2, 3) from the plane x – y + z = 5 measured parallel to the line,

Solution:

QUESTION: 12

The straight l ines and

Solution:

QUESTION: 13

If plane cuts off intercepts OA = a, OB = b, OC = c from the coordinate axes, then the area of the triangle ABC equal to

Solution:

=AC = −ai^ + ck^

AB = −ai^ + bj^

Area of △ABC= ½|AB × AC∣

|AB × AC∣ =

−(bc)i^− (ac)j^ − (ab)k^

∣AB × AC∣ = (b^{2}c^{2} + a^{2}c^{2} + a^{2}b^{2})^{1/2}

Area = 1/2(a^{2}b^{2} + b^{2}c^{2} + c^{2}a^{2})^{1/2}

QUESTION: 14

A point moves so that the sum of the squares of its distances from the six faces of a cube given by x = ± 1, y = ± 1, z = ± 1 is 10 units. The locus of the point is

Solution:

Let P(x,y,z) be any point on the locus, then the distances from the six faces are ∣x+1∣, ∣x−1∣, ∣y−1∣, ∣z−1∣

According to the given condition, we have

∣x+1∣^{2} + ∣x−1∣^{2} + ∣y+1∣^{2} + ∣y−1∣^{2} + ∣z+1∣^{2}+ ∣z−1∣^{2}=10

⇒ 2(x^{2}+y^{2}+z^{2})=10−6=4

⇒ x^{2 }+ y^{2 }+ z^{2 }= 2

QUESTION: 15

A variable plane passes through a fixed point (a, b, c) and meets the coordinate axes in A, B, C. Locus of the point common to the planes through A, B, C and parallel to coordinate plane, is

Solution:

Let equation of plane is

x/α + y/β + z/γ = 1.............(1)

Plane passes through point (a,b,c)

a/α + b/β + c/γ = 1................(2)

Again plane meets the coordinate points.

Coordinates of points are (α,0,0)

Coordinates of point B are (0, β, 0) and

coordinates of point C are (0, 0, γ)

∴ Equation of planes, parallel to coordinate axis and passing through

Point A is x = α …..(3)

Point B is y = β …..(4)

Point C is z = γ …..(5)

∴ Locus of the point of intersection is

a/x + b/y + c/z = 1

QUESTION: 16

Two systems of rectangular axes have same origin. If a plane cuts them at distances a, b, c and a_{1}, b_{1}, c_{1 }from the origin, then

Solution:

QUESTION: 17

The angle between the plane 2x – y + z = 6 and a plane perpendicular to the planes x + y + 2z = 7 and x – y = 3 is

Solution:

QUESTION: 18

The non zero value of ‘a’ for which the lines 2x – y + 3z + 4 = 0 = ax + y – z + 2 and x – 3y + z = 0 = x + 2y + z + 1 are co-planar is

Solution:

2x - y +3z + 4 = 0

x - 3y + z = 0

x + 2y + z + 1 = 0

x = 12/5

y = -1/5

z= -3

this point also satisfied by

ax - y + z + 2 = 0

a(12 / 5) - (-1/5) + (-3) = 0

⇒ 12a/5 + 1/5 -3 = 0

a= -2

QUESTION: 19

If the lines and are concurrent then

Solution:

Let the point of intersection is (λ, 2λ, 3λ).

Clearly, λ = 3μ + 1, 2λ = 2 – μ

Solving, we get λ = 1, μ = 0

Hence, the point of intersection is (1, 2, 3).

Therefore, (1+k)/3 = (2-1)/2 = (3-2)/h

= (1+k)/3 = 1/2 = 1/h

⇒ h = 2 k = 1/2

QUESTION: 20

The coplanar points A, B, C, D are (2 – x, 2, 2), (2, 2 – y, 2), (2, 2, 2 – z) and (1, 1, 1) respectively. Then

Solution:

We have four coplanar points.

The three vectors connecting two of them at a time are thus coplanar.

⇒{(−x, y, 0) (-x, 0, z) (1−x, 1, 1)} = 0

⇒−x (−z) −y (−x−z+xz)+0=0

⇒xz + xy + yz = xyz

⇒1/y + 1/z + 1/x = 1

QUESTION: 21

The direction ratios of a normal to the plane through (1, 0, 0), (0, 1, 0), which makes an angle of π/4 with the plane x + y = 3 are

Solution:

QUESTION: 22

Let the points A(a, b, c) and B(a^{'}, b^{'}, c^{'}) be at distances r and r^{'} from origin. The line AB passes through origin when

Solution:

QUESTION: 23

Let L be the line of intersection of the planes 2x + 3y + z = 1 and x + 3y + 2z = 2. If L makes an angle ? with the positive x-axis, the cos α equals

Solution:

QUESTION: 24

If a line makes an angle of π/4 with the positive directions of each of x-axis and y-axis, then the angle that the line makes with the positive direction of the z-axis is

Solution:

QUESTION: 25

If the angle θ between the line and the plane is such that sinθ = 1/3 The value of λ is

Solution:

QUESTION: 26

A line makes the same angle θ with each of the x and z-axis. If the angle θ, which it makes with y-axis is such that sin^{2} β = 3 sin^{2} θ, then cos^{2}θ equals

Solution:

QUESTION: 27

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

Solution:

Let x_{1}, y_{1}, z_{1} be any point on the plane

QUESTION: 28

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

Solution:

The co-ordinates of any point on L1 in terms of parameter r1 are given by,x=y+a=z=r1

⇒x=r1,y=r1 - a,z=r1....(1)

Similarly the co-ordinates of any point on L2 in terms of parameter 2r2 are given by

⇒x+a=2y=2z=2r2

⇒ x=2r2−a,y=r2, z=r2...(2)

Let 'A' be a point on L1 and 'B' be a point on L2

Using (1) and (2), the direction ratios of AB are

2r2−a−r1,r2−r1+a,r2−r1

If the above line is same as the line whose direction cosines are proportional to (2,1,2) as given in the question, then

(2r2−r1−a)/2= (r2−r1+a)/1 = (r2−r1)/2

Solving the first two of the above equation, we get r1=3a

Again solving the last two, we get r2=a

Using these values in (1) and (2), we get the coordinates of points as

(3a,2a,3a) and (a,a,a).

QUESTION: 29

The lines and are coplanar if

Solution:

(x-x1)a1 = (y-y1)b1 = (z-z1)c1 &

(x-x2)a2 = (y-y2)b2 = (z-z2)c2 are coplanar if

k^{2} + 3k = 0

k = 0 or -3

QUESTION: 30

The equation of plane which meet the co-ordinate axes whose centroid is (a, b, c)

Solution:

A(a,0,0),B(0,b,0),C(0,0,c) are the points on coordinate axis and centriod (α,β,γ).

According to centroid formula,

α= a/3

a=3α

b=3β

c=3γ

Equation of plane is,x/a+y/b+z/c =1

x/α+y/β+z/γ=3

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