Test: Dimensional Geometry - 4

# Test: Dimensional Geometry - 4

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## 20 Questions MCQ Test Topic-wise Tests & Solved Examples for IIT JAM Mathematics | Test: Dimensional Geometry - 4

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Test: Dimensional Geometry - 4 - Question 1

### The equation of a straight line which makes an angle of 60° with x-axis and passes through the point (√3, 2) is given by

Detailed Solution for Test: Dimensional Geometry - 4 - Question 1

The required equation
(i) makes an angle of 60° with x-axis
(ii) passes through the point (√3 , 2)
Now
(i) ⇒ equation of the straight line will be
y = tan 60° x + c
or y = √3 x + c ...(i)
(ii) ⇒ the coordinates of the point will satisfy equation (i) Test: Dimensional Geometry - 4 - Question 2

### The line y - x + 2 = 0 cuts the line joining (3, -1) and (8, 9) in the ratio

Detailed Solution for Test: Dimensional Geometry - 4 - Question 2

First remember the following points:
l.The equation of a line passing through two p o in ts (x1,y1) and (x2, y2) is given by 2. The. coordinates of the point R which divides the line joining P(x1,y1) and Q(x2, y2) internally in the ratio m1 : m2 are given by 3. If R divides PQ externally in the ratio m1 : m2, then Now the given straight line is
y - x + 2 - 0 ...(iv)
The equation of the straight line joining the points (3,-1) and (8, 9) is given by (see (i)) or y - 2x + 7 = 0 ...(v)
The point of intersection of (iv) and (v) is obtained on solving these equations and is given by (5,3)
Let the points of intersection (5, 3) divide the line joining (3. -4) and (8. 9) in the ratio m : n.
Then Test: Dimensional Geometry - 4 - Question 3

### The equation of the straight line passing through (4, 5) and parallel to the line 2x - 3y = 5 is given by

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

The given straight line is The equation of a straight line parallel to (i) will be Since (ii) passes through P(4, 5), therefore the coordinates of P A\*ill satisfy equation (ii) ∴ Required equation of the straight line is Test: Dimensional Geometry - 4 - Question 4

The condition that the three straight lines
ax + by+ c = 0
a1x + b1y + c = 0
a2x + b2y + c2 = 0
may meet in a point, is that, the value of the determinant must be

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

The condition for three lines lines to be concurrent is that Test: Dimensional Geometry - 4 - Question 5

The point (x1,y1) lies on the positive side of the straight line Ax + By + C = 0

Detailed Solution for Test: Dimensional Geometry - 4 - Question 5

First of all learn the following:
Division of a plane by the line
L : Ax + By + C = 0
As shown in the adjoining Figure, the line L divides the plane into three parts, namely I, L and II, where
Ax1 + By1 + C > 0 for all poins (x1, y1) in l
Ax2 + By2 + C = 0 for all points (x2, y2) on L
Ax3 + By3 + C < 0 for all points (x3, y3) in II

Test: Dimensional Geometry - 4 - Question 6

The point (3, 2) lies on the

Detailed Solution for Test: Dimensional Geometry - 4 - Question 6

Given point is (3, 2)
Given straight lines are   Test: Dimensional Geometry - 4 - Question 7

When two straight lines 2x- 3y + 1 = 0 and 3x -6y + 2 = 0 are traced, we get four different compartments. Which of the following four points lie in the same compartment? A(0, 0), B(-1, 1), C(-7, -4), D(9, 6)

Detailed Solution for Test: Dimensional Geometry - 4 - Question 7

The given straight lines are   Test: Dimensional Geometry - 4 - Question 8

The angle between the lines ax + by + c = 0 and a'x  + b'x + c' = 0 is given by

Detailed Solution for Test: Dimensional Geometry - 4 - Question 8

Comments. The angle φ between two straight lines
y = mx + c and y = m'x + c‘
is given by Since the straight lines are ax + by + c = 0 and a'x +b'y + c' = 0
therefore Both φ and π - φ shall be the angles between the given lines.

Test: Dimensional Geometry - 4 - Question 9

The straight lines ax + by + c = 0 and a'x + b'y + c' = 0 arc perpendicular if

Detailed Solution for Test: Dimensional Geometry - 4 - Question 9

The angle φ between the lines is given by  Test: Dimensional Geometry - 4 - Question 10

The line passing through the origin is given by

Detailed Solution for Test: Dimensional Geometry - 4 - Question 10

Remark: A straight line passing through the origin shall be of the form
y = mx
and there will not bo any constant term in the eouation.

Test: Dimensional Geometry - 4 - Question 11

The four points (0, 4, 1), (2, 3, -1), (4, 5, 0). (2, 6, 2) are the vertices of a

Test: Dimensional Geometry - 4 - Question 12

Let A(3, 2, 0), B(5, 3, 2), C( 9, 6, -3) be the three points forming a triangle. Let the bisector of the angle BAC meet BC in D. Then D divides BC in the ratio of

Test: Dimensional Geometry - 4 - Question 13

Let A(-1,2, -3), B(5, 0, -6), C(0, 4, -1) be the three points. Then the direction cosines of the internal bisector of the angle BAC are proportional to

Detailed Solution for Test: Dimensional Geometry - 4 - Question 13

Proof in short : verify that
AB = 7 and AC = 3  where θ is the angle between AB (with d.c.'s [l1, m1, n1]) and AC (with d.c.’s [l2, m2, n2)
∴ the d.c.’s of the internal bisector are proprotional to l1+l2, m1+m2, n1+n2
or proportional to or proportional to or proportional to 25, 8, 5

Test: Dimensional Geometry - 4 - Question 14

Which of the following is incorrect?
If l1, m1, n1, : l2, m2, n2, : l3, m3, n3 be the direction cosines of three mutually perpendicular lines then

Detailed Solution for Test: Dimensional Geometry - 4 - Question 14

In fact, we shall have Test: Dimensional Geometry - 4 - Question 15

The direction ratios of the line, which is equally inclined to the three mutually perpendicular lines with direction cosines l1, m1, n1: l2, m2, n2 : l3, m3, n3; are given by

Detailed Solution for Test: Dimensional Geometry - 4 - Question 15

Proof: Let the three mutually perpendicular lines be   Test: Dimensional Geometry - 4 - Question 16

“If O, A, B, C be the four points not lying in the same plane such that OA ⊥ BC, OB ⊥ CA, then OC ⊥ AB". if the points O, A, B, C are coplanar, then the above reduces to

Test: Dimensional Geometry - 4 - Question 17

If a1, b1, c1 and a2, b2, c2 are the direction ratios of two lines, then the angle between them is not given by

Test: Dimensional Geometry - 4 - Question 18

If the edges of a rectangular parallelepiped are OA , OB and OC along coordinate axes such that OA = a, OB = b, OC = c and PL, PM and PN are the perpendiculars drawn from P(a, b, c) on XY, ZX and YZ planes respectively, then angle between OP and AN is given by

Test: Dimensional Geometry - 4 - Question 19

If a variable line in two adjacent positions has direction cosines as then the small angle dθ between those two positions.in given by

Detailed Solution for Test: Dimensional Geometry - 4 - Question 19   Test: Dimensional Geometry - 4 - Question 20

The area of the triangle with vertices (0, 0, 0), (0, b, 0) and (0, 0, c) is given by

Detailed Solution for Test: Dimensional Geometry - 4 - Question 20

Proof: The area of the triangle OAB, where O is the origin and the coordinates of A and B are (x1, y1, z1) a n d [x2, y2, z2) respectively, Here since the coordinates of A and B are (0,b,0) and (0.0,c), therefore the required area A is given by

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