3D Coordinate Geometry Quiz-16

3D Coordinate Geometry Quiz-16

Dear Readers,

As per analysis for previous years, it has been observed that students preparing for JEE MAINS find Mathematics out of all the sections to be complex to handle and the majority of them are not able to comprehend the reason behind it. This problem arises especially because these aspirants appearing for the examination are more inclined to have a keen interest in Mathematics due to their ENGINEERING background. Furthermore, sections such as Mathematics are dominantly based on theories, laws, numerical in comparison to a section of Engineering which is more of fact-based, Physics, and includes substantial explanations. By using the table given below, you easily and directly access to the topics and respective links of MCQs. Moreover, to make learning smooth and efficient, all the questions come with their supportive solutions to make utilization of time even more productive. Students will be covered for all their studies as the topics are available from basics to even the most advanced..

Q1. The equation of the plane through the point (0,-4,-6) and (-2,9,3) and perpendicular to the plane x-4y-2z=8 is:

•  3x+3y-2z=0
•  x-2y+z=2
•  2x+y-z=2
•  5x-3y+2z=0

Solution

Q2.What are the DR’s of vector parallel to (2,-1,1) and (3,4,-1)?:

•  (1,5,-2)
•  (-2,-5,2)
•  (-1,5,2)
•  (-1,-5,-2)

Solution
(a)
Required DR’s are (3-2,4+1,-1-1) ie,(1,5,-2).

Q3.  A plane makes intercepts 3 and 4 respectively on z-axis and x-axis. If plane is parallel to y-axis, then its equation is:

•   3x+4z=12
•  3z+4x=12
•  3y+4z=12
•  3z+4y=12

Solution
(a)
Plane intercept on x-axis at a=4
Plane intercept on z-axis at c=3
Required equation is $$\(\frac{x}{4}+\frac{z}{3}=1\) or 3x+4z=12

Q4. A vector $\overrightarrow{r}$ is equally inclined with the coordinate axes. If the tip of $\overrightarrow{r}$ is in the positive octant and |$\overrightarrow{r}$ |=6, then $\overrightarrow{r}$ is

•  2√3($\hat{i}$-$\hat{j}$+k ̂)
•  2√3(-$\hat{i}$+$\hat{j}$+k ̂)
•  2√3($\hat{i}$+$\hat{j}$-k ̂)
•  2√3($\hat{i}$+$\hat{j}$+k ̂)

Solution

Q5. If the direction ratios of two lines are given by 3lm+4ln+mn=0 and l+2m+3n=0, then the angle between the lines is

•  π/2
•  π/3
•  π/4
•  π/6

Solution
 

Q6. If a line makes angles α,β,γ with the coordinate axe, then

•  cos⁡2α+cos⁡2β+cos⁡2γ-1=0
•  cos⁡2α+cos⁡2β+cos⁡2γ-2=0
• cos⁡2α+cos⁡2β+cos⁡2γ+1=0
•  cos⁡2α+cos⁡2β+cos⁡2γ+1=0

Solution
(c)
Since, cos ² α+cos ² β+cos ² γ=2
[∵ l ²+m ²+n ²=1]
⟹1+cos⁡2α+1+cos⁡2β+1+cos⁡2γ=2
⟹ cos⁡2α+cos⁡2β+cos⁡2γ+1=0

Q7. The distance of the point (1, -2,3) from the planes x-y+z=5 measured along the line $$\( \frac{x}{2}=\frac{y}{3}=\frac{z}{-6}\) is:

•  1
•  $$\(\frac{6}{7}\)
•  $$\(\frac{7}{6}\)
•  None of these

Solution
(c)

Q8.A parallelopiped is formed by planes drown through the points (2, 3, 5) and (5, 9, 7) parallel to the coordinate planes. The length of a diagonal of the parallelopiped is :

•  7
•  √38
•  √155
•  None of these

Solution
(a)
The length of the edges are given by
a=5-2=3
b=9-3=6
c=7-5=2
So, length of the diagonal =√(9+36+4)=7

Q9.The equation of the plane, which makes with coordinate axes, a triangle with its centroid (α, β, γ)is :

•  αx+ βy+γz=3
•  αx+ βy+ γz=1
•  $$\(\frac{x}{α}+\frac{y}{β}+\frac{z}{γ}=3\)
•  $$\(\frac{x}{α}+\frac{y}{β}+\frac{z}{γ}=1\)

Solution

Q10. The shortest distance between the skew lines l₁:$\overrightarrow{r}$=$\overrightarrow{a}$₁+λ$\overrightarrow{b}$₁ l₂:$\overrightarrow{r}$=$\overrightarrow{a}$₂+μ$\overrightarrow{b}$₂ is:

•  |($\overrightarrow{a}$₂-$\overrightarrow{a}$₁ )∙$\overrightarrow{b}$₁×$\overrightarrow{b}$₂ |/(|$\overrightarrow{b}$₁×$\overrightarrow{b}$₂ |)
•  ( |$\overrightarrow{a}$₂-$\overrightarrow{a}$₁ )∙$\overrightarrow{a}$₂×$\overrightarrow{b}$₂ |)/(|$\overrightarrow{b}$₁×$\overrightarrow{b}$₂ |)
•  ( |$\overrightarrow{a}$₂-$\overrightarrow{b}$₂ )∙$\overrightarrow{a}$₁×$\overrightarrow{b}$₁ |)/(|$\overrightarrow{b}$₁×$\overrightarrow{b}$₂ |)
• ( |$\overrightarrow{a}$₁-$\overrightarrow{b}$₂ )∙$\overrightarrow{b}$₁×$\overrightarrow{a}$₂ |)/(|$\overrightarrow{b}$₁×$\overrightarrow{a}$₂ |)

Solution