Test: Dimensional Geometry - 3

Test: Dimensional Geometry - 3

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20 Questions MCQ Test Topic-wise Tests & Solved Examples for IIT JAM Mathematics | Test: Dimensional Geometry - 3

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Test: Dimensional Geometry - 3 - Question 1

The area of a triangle whose vertices are given by A(r1, θ1), B(r2, θ2), C(r3, θ3), respectively equals

Detailed Solution for Test: Dimensional Geometry - 3 - Question 1

Point in polar coordiantes

Points in Cartesian coordinates

Test: Dimensional Geometry - 3 - Question 2

The area of a quadrilateral, the polar coordinates of whose vertices are given by A(r1, θ1,), B(r2, θ2). C(r3, θ3) and D(r4, θ4) is given by

Test: Dimensional Geometry - 3 - Question 3

The equation of a straight line which makes an angle of 90° with the positive direction of the x-axis and passes through the origin is given by

Detailed Solution for Test: Dimensional Geometry - 3 - Question 3

Clearly the straighi line which makes an angle of 90°' with the positive direction of the x-axis and passes through the origin is ij-nxis and hence its equations given by x = 0

Test: Dimensional Geometry - 3 - Question 4

The equation of the straight line which makes an angle of 135° with the positive direction of the a--axis and cuts off an intercept equal to 5 on y-axis is given by

Detailed Solution for Test: Dimensional Geometry - 3 - Question 4

Note that the equation of a straight line which makes an angle 0 with the positive direction of the x-axis and cuts an intercept equal to c on y-axis, is given by

y= mx + c,
where m = tan θ.
In this problem,
c = 5
and
θ= 135° ⇒ w = tan 135° = -1 Required equation is given by
y = -x+5
or y + x = 5

Test: Dimensional Geometry - 3 - Question 5

Which of the following polar coordinates are asspcoated to the same point?

Detailed Solution for Test: Dimensional Geometry - 3 - Question 5

and so on.

Test: Dimensional Geometry - 3 - Question 6

Let P be the perpendicular from the origin on a straight line and a be the angle which this perpendicular makes with the positive direction of the axis of x, then the straight line cuts off an intercom equal to ... on the y-axis.

Detailed Solution for Test: Dimensional Geometry - 3 - Question 6

Figure shows the straight line such that
(i) P is the perpendicular from the origin on the straight line.

(ii) Perpendicular makes an angle α with positive direction of the axis of x.
Its equation is given by x cos α + y sin α = p ...(i)
The required intercept on y-axis is obtained by substituting x = 0 in equation (i).
⇒ y sin a = p
=> y = p cosecor.

Test: Dimensional Geometry - 3 - Question 7

Which of the following statements is correct?

Detailed Solution for Test: Dimensional Geometry - 3 - Question 7

The general equation of a straight line is given by
Ax + By = C,
where A. B and C are constants.
Remark: Note that the equation

does represents a straight line. But it is not correct to say that only the equations of this type will represent a straight line. This is the intercept form of a straight line.

Test: Dimensional Geometry - 3 - Question 8

The length of the perpendicular from the origin or. the line Ax + By + C = 0 is given b

Test: Dimensional Geometry - 3 - Question 9

The equation of the straight line which makes equal intercepts on the coordinate axes and passes through the poijnt (2, 3), is given b

Detailed Solution for Test: Dimensional Geometry - 3 - Question 9

Proof: The equation of a st raight line having equal intercepts on the coordinate axis is given by

where a is the intercept on x or y-axis.
Since the. straight line (i) passes through the point (2, 3), therefore the coordinates of this point will satisfy equation (i).

Test: Dimensional Geometry - 3 - Question 10

The length of ihe. perpendicular from the origin on the straight line 5x+12y=0 is given by

Detailed Solution for Test: Dimensional Geometry - 3 - Question 10

If p is the length of the perpendicular from the origin on the line
5x + 12y - 26 = 0
then

Test: Dimensional Geometry - 3 - Question 11

The projection of line segment joining (1, 2, 3) and (3,4, 5) along the line which makes an angle of 60° with the given line segment will be

Test: Dimensional Geometry - 3 - Question 12

If the projection of a line segment on the X- axis , Y-axis and Z axis be 12, 4 and 3 respectiveIv, then it’s length will be

Detailed Solution for Test: Dimensional Geometry - 3 - Question 12

let the given line joins the points P(x1 ,y1, z1 ) and Q(x2 ,y2, z2)
Then since the d.c.'s of x-axis are [1, 0,0]. therefore the projection of PQ on x-axis = 1 (x2 - x1 )
=> x2 - x1 =12
Similarly y2 - y1 = 4
and z2 - z1= 3
Required length r of the line PQ is given by

Test: Dimensional Geometry - 3 - Question 13

Direction ratios of line joining the point s (4, 3, -5) and (-2. 1, -8) are

Detailed Solution for Test: Dimensional Geometry - 3 - Question 13

D.C.'s of the line joining the points (4, 3, -5) and (-2, 1,-8) are

d.r.’s are 6,2,3

Test: Dimensional Geometry - 3 - Question 14

Projection on line segment joining P(6, 3, 2) and Q(5, 1, 4) on the line AB, where A and B are (3, -4, 7) and (0, 2, 5) respectively is

Detailed Solution for Test: Dimensional Geometry - 3 - Question 14

Direction cosines of the line joining A{3, -4, 7) and B(0, 2, 5) are given by

Now the required projection of line joining P(6, 3, 2) and Q(5, 1. 4) on the line AB is given by

Test: Dimensional Geometry - 3 - Question 15

Cosine of the angle between lines

Detailed Solution for Test: Dimensional Geometry - 3 - Question 15

Let the lines OP and OQ be parallel to the given lines, so that the direction cosines of OP are [l1m1n1] and the direction cosines of OQ are [l2m2n2]
Clearly

Remark-3: Condition of two lines being perpendicular.
If the two lines are perpendicular, then cosθ = l1l2+m1m2+n1n2= o
Remark-4: Condition of two lines being parallel. If the two lines are parallel, then

Test: Dimensional Geometry - 3 - Question 16

The three concurrent lines, whose direction cosines are given by l1, m1, n1 : l2, m2, n2, and l3, m3, n3 are coplanar if

Detailed Solution for Test: Dimensional Geometry - 3 - Question 16

Proof- Let (l,m,n) be the direction cosines of a line which is normal to the plane containing the lines with d.c.'s (l1,m1,n1) and (l2,m2,n2) since a plane can always be drawn containing any two concurrent lines]
=> l1l + m1m + n1n = 0 ...(i)
and l2l + m2m + n2n = 0 ...(ii)

If the line with d.c.'s (l3,m3,n3) also lies on this plane, then
l3l + m3m + n3n = 0 ...(iii)
For a non-trivial solution of equations (i), (ii) and (iii) for l,m,n we get

Test: Dimensional Geometry - 3 - Question 17

The direction cosines of the line which is perpendicular to the lines whose direction ratios are (1, -2. -2) and (0. 2. 1) are given by

Detailed Solution for Test: Dimensional Geometry - 3 - Question 17

Let the d.c.’s of the line which is perpendicular to the two lines with d.r.’s (1,-2, -2) and (0, 2, 1) be [l,m,n]
Now the d.c.'s of the line with d.r.'s (1,-2,-2)

Test: Dimensional Geometry - 3 - Question 18

The angle between the two diagonals of a cube are

Test: Dimensional Geometry - 3 - Question 19

If (1/√2,1/√2,0) and (-2/3, 2/3,-1/3) are the direction cosines of two mutually perpendicular lines, then the direction ratios of the line which is perpendicular to both of them are given by

Detailed Solution for Test: Dimensional Geometry - 3 - Question 19

Let[l,m,n] be the d.c.'s of the line which is perpendicular to the two line having d,c.'s as

Test: Dimensional Geometry - 3 - Question 20

Let the coordinates of three points A, B, C be given by (1 ,8 ,4) , (0, 11,4), (2, -3 ,1) respectively. Then the coordinates of a point D which is the foot of the perpendicular from A on BC. are given by

Detailed Solution for Test: Dimensional Geometry - 3 - Question 20

Let D(a,b,c) be foot of perpendicular from A(1, 8, 4) on the line joining B(0, -11, 4) and C(2, -3. 1). Then the direction ratio of DC are proportional to the d.r.'s of BC.

∴ The coordinates of D are obtained by substituting k=1 in (1)
∴ Coordinates of D are (4,5,-2)

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