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1.

In Fig 4 the coordinates of point D areA. `(b,a,0)`B. `(a,b,0)`C. `(b,c,0)`D. `(0,b,c)`

Answer» Correct Answer - B
Clearly, point D lies on XY-plane such that `OA=a` and `AD=OB=b` So the coordinates of D are `(a,b,0)`.
Discussion
2.

The point equidistant from the `O(0,0,0),A(a,0,0),B(0,b,0)` and `C(0,0,c)` has the coordinatesA. `(a,b,c)`B. `(a//2,b//2,c//2)`C. `(a//3,b//3,c//3)`D. `(a//4,b//4,c//4)`

Answer» Correct Answer - B
Let `P(x,y,z)` be the required point. Then
`OP=AP=BP=CP`
Now `OP=AP`
`impliesOP^(2)=AP^(2)`
`impliesx^(2)+y^(2)+z^(2)=(x-a)^(2)+y^(2)+z^(2)`
`implies-2ax+a^(2)=0impliesx=a/2`
Similarly, `OP=BP` and `OP=CP`implies `y=b/2` and `z=c/2`
Hence required point has the coordinates `(a//2,b//2,c//2)`.
Discussion
3.

In fig. 28.7 if the coordinates of point `P a r e (a , b , c)`thenWrite the coordinates of points A, B, C, D, E and F.Write the coordinates of the feet of the perpendiculars from the pointP to the coordinate axes.Write the coordinates of the feet of the perpendicular from the point `P`on the coordinate planes `X Y , Y Z a n d Z Xdot`Find the perpendicular distances of point `P`from `X Y , Y Z a n d Z X-`planes.Find the perpendicular distances of the point `P`fro the coordinate axes.Find the coordinates of the reflection of `P`are `(a , b , c)`. Therefore `O A=a , O B=b n d O C=cdot`A. `a,b,c`B. `b,c,a`C. `c,a,b`D. none of these

Answer» Correct Answer - C
`PD,PE` and `PF` are the perpendicular distances of `P` from `XY,YZ` and `ZX`- planes respectively.
We have
`PD=OC=c,PE=OA=a` and `PF=OB=b`
Hence, the perpendicular distances of `P(a,b,c)` from `XY,YZ` and `ZX` lanes are `c,a` and `b` respectively.
Discussion
4.

Find the ratio in which the line joining thepoints `(1,2,3)a n d(-3,4,-5)`is divided by the `x y-p l a n e`. Also, find the coordinates of the point ofdivision.A. 3:5 internallyB. 5:3 externallyC. 3:5 externallyD. 5:3 internally

Answer» Correct Answer - A
Suppose the line joining the points `P(1,2,3)` and `Q(-3,4,-5)` is divided by the xy-plane at a point `R` in the ratio `lamda:1`. Then the coordinates of `R` are
`((-3lamda+1)/(lamda+1),(4lamda+2)/(lamda+1),(-5lamda+3)/(lamda+1))`…………..i
Since `R` lies on xy plane i.e. `z=0`
`:.(-5 lamda+3)/(lamda+1)=0implieslamda=3/5`
So, the required ratio is `3/5:1` or `3:5` internally.
Discussion
5.

Let `P(2,-1,4)` and `Q(4,3,2)` are two points and as point `R` on `PQ` is such that `3PQ=5QR`, then the coordinates of `R` areA. `(14/5,3/5,16/5)`B. `(16/5,7/5,14/5)`C. `(11/4,1/2,13/4)`D. none of these

Answer» Correct Answer - A
Clearly, `R` diveides `PQ` internally in the ratio `2:3`. So the coordinates of `R` are
`((2xx4+3xx2)/(2+3),(2xx3+3xx(-1))/(2+3),(2xx2+3xx4)/(2+3))` or `(14/5,3/5,16/5)`
Discussion
6.

If a line makes angle `alpha, beta` and `gamma` with the coordinate axes respectively, then `cos2alpha+cos 2 beta+cos 2gamma=`A. 2B. -1C. 1D. 2

Answer» Correct Answer - B
We have
`l=cos alpha, cosb` and `n=cos gamma`
`:.l^(2)+m^(2)+n^(2)=1`
`impliescos^(2) alpha+cos^(2)beta+cos^(2)gamma=1`
`implies(1+cos 2alpha)/2+(1+cos 2 beta)/2+(1+cos 2gamma)/2=1`
`impliescos 2 alpha+cos 2 beta+cos 2gamma=-1`
Discussion
7.

The direction cosines of the line passing through `P(2,3,-1)` and the origin areA. `2/(sqrt(14)),3/(sqrt(14)),1/(sqrt(14))`B. `2/(sqrt(14)),-3/(sqrt(14)),1/(sqrt(14))`C. `-2/(sqrt(14)),-3/(sqrt(14)),1/(sqrt(14))`D. `-2/(sqrt(14)),-3/(sqrt(14)),-1/(sqrt(14))`

Answer» Correct Answer - D
The direction cosines of `OP` are proportional to 2,3,-1. So directions cosines of OP are:
`2/(sqrt(14)),3/(sqrt(14)),-1/(sqrt(14))`
Discussion
8.

If the direction ratios of a lines are proportional to `1,-3,2` then its direction cosines areA. `1/(sqrt(14)),(-3)/(sqrt(14)),2/(sqrt(14))`,B. `1/(sqrt(14)),2/(sqrt(14)),3/(sqrt(14))`C. `-1/(sqrt(14)),3/(sqrt(14)),2/(sqrt(14))`D. `-1/(sqrt(14)),(-2)/(sqrt(14)),(-3)/(sqrt(14))`

Answer» Correct Answer - A
The direction ratios are proportional to 1,-3,2.
So direction cosines are
`1/(sqrt(1^(2)+(-3)^(2)+2^(2))),(-3)/(sqrt(1^(2)+(-3)^(2)+2^(2))),2/(sqrt(1^(2)+(-3)^(2)+2^(2)))`
or `1/(sqrt(14)),(-3)/(sqrt(14)),2/(sqrt(14))`
Discussion
9.

If `P(x,y,z)` is a point on the line segment joining `Q(2,2,4)` and `R(3,5,6)` such that projections of `vec(OP)` on the axes are `13/5,19/5,26/5` respectively, then `P` divides `QR` in the ratioA. `1:2`B. `3:2`C. `2:3`D. `3:1`

Answer» Correct Answer - B
The coordinates of `P` are `(13/5,19/5,26/5)`
Suppose, `P` divides `QR` in the ratio `lamda:1`. Then the coordinates of `P` are `((3lamda+2)/(lamda+1),(5lamda+2)/(lamda+1),(6lamda+4)/(lamda+1))`
`implies (3lamda+2)/(lamda+1)=13/5,(5lamda+2)/(lamda+1)=19/5,(6lamda+4)/(lamda+1)=26/5implieslamda=3/2`
Hence required ratios is `3:2`.
Discussion
10.

The projections of a line segment on the coordinate axes are 12,4,3 respectively. The length and direction cosines of the line segment areA. `13,12/13,4/13,3/13`B. `19,12/19,4/19,3/19`C. `11,12/11,14/11,3/11`D. none of these

Answer» Correct Answer - A
Length of the line segment `=sqrt(12^(2)+4^(2)+3^(2))=13`
`:.` Direction cosines are `12/13,4/13,3/13`
Discussion
11.

If OA is equally inclined to OX,OY ,OZ and if A is `sqrt(3)` units from the origin then the cordinates of A areA. `(3,3,3)`B. `(-1,1,-1)`C. `(-1,1,1)`D. `(1,1,1)`

Answer» Correct Answer - D
We have
`l=m=n=1/(sqrt(3))`
`:.vec(OA)=|vec(OA)|(lhati+mhatj+nhatk)`
`impliesvec(OA)=sqrt(3)(1/(sqrt(3))hati+1/(sqrt(3))hatj+1/(sqrt(3))hatk)=hati+hatj+hatk`
So, coordinates of A are `(1,1,1)`.
Discussion