JEE Advanced · Mathematics · 31. 3D Geometry
Consider the planes \(3 x-6 y-2 z=15\) and \(2 x+y-2 z=5\).
Statement I The parametric equations of the line of intersection of the given planes are \(x=3+14 t, y=1+2 t\) and \(z=15 t\).
Statement II The vectors \(14 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+15 \hat{\mathbf{k}}\) is parallel to the line of intersection of the given planes.
- A Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I
- B Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I
- C Statement I is true, Statement II is false
- D Statement I is false, Statement II is true
Answer & Solution
Correct Answer
(D) Statement I is false, Statement II is true
Step-by-step Solution
Detailed explanation
Given planes are \(3 x-6 y-2 z=15\) and \(2 x+y-2 z=5\)
For \(z=0\), we get \(x=3, y=-1\)
Direction ratios of planes are
\[
< 3-6-2>\text { and } < 2 \quad 1-2>
\]
then the dr's of line of intersection of planes is \(\langle 14215\rangle\) and line is
\[
\begin{aligned}
\frac{x-3}{14} & =\frac{y+1}{2}=\frac{z-0}{15}=\lambda \text { (say) } \\
\Rightarrow \quad x & =14 \lambda+3, y=2 \lambda-1, z=15 \lambda
\end{aligned}
\]
Hence, Statement I is false.
But Statement II is true.
For \(z=0\), we get \(x=3, y=-1\)
Direction ratios of planes are
\[
< 3-6-2>\text { and } < 2 \quad 1-2>
\]
then the dr's of line of intersection of planes is \(\langle 14215\rangle\) and line is
\[
\begin{aligned}
\frac{x-3}{14} & =\frac{y+1}{2}=\frac{z-0}{15}=\lambda \text { (say) } \\
\Rightarrow \quad x & =14 \lambda+3, y=2 \lambda-1, z=15 \lambda
\end{aligned}
\]
Hence, Statement I is false.
But Statement II is true.
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Mathematics
- If is a differentiable function such that for all and thenJEE Advanced 2017 Hard
- A rectangular sheet of fixed perimeter with sides having their lengths in the ratio is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is , the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are:JEE Advanced 2013 Hard
- Let \(S_n=\sum_{k=0}^n \frac{n}{n^2+k n+k^2}\) and \(T_n=\sum_{k=0}^{n-1} \frac{n}{n^2+k n+k^2}\), for \(n=1,2,3, \ldots\), thenJEE Advanced 2008 Medium
- Paragraph:
A box \(B_{1}\) contains \(1\) white ball, \(3\) red balls and \(2\) black balls. Another box \(B_{2}\) contains \(2\) white balls, \(3\) red balls and \(4\) black balls. A third box \(B_{3}\) contains \(3\) white balls, \(4\) red balls and \(5\) black balls.
Question:
If \(2\) balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these \(2\) balls are drawn from box \(B_{2}\) isJEE Advanced 2013 Medium - The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball isJEE Advanced 2012 Hard
- Consider the hyperbola and a circle S with center Suppose that H and S touch each other at point with The common tangent to H and S at P intersects the x - axis at point M. If is the centroid of the triangle then the correct expression(s) is(are)JEE Advanced 2015 Medium
More PYQs from JEE Advanced
- Three randomly chosen non negative integers are found to satisfy the equation . Then the probability that is even, is
JEE Advanced 2017 Medium - The differential equation \(\frac{d y}{d x}=\frac{\sqrt{1-y^2}}{y}\) determines a family of circles withJEE Advanced 2007 Easy
- The equation of the plane passing through the point and perpendicular to the planes and isJEE Advanced 2017 Medium
- In the figure, the inner (shaded) region represents a sphere of radius , within which the electrostatic charge density varies with the radial distance from the center as , where is positive. In the spherical shell of outer radius , the electrostatic charge density varies as . Assume that dimensions are taken care of. All physical quantities are in their SI units.
Which of the following statement(s) is/(are) correct?JEE Advanced 2022 Medium - A monochromatic light is travelling in a medium of refractive index . It enters a stack of glass layers from the bottom side at an angle The interfaces of the glass layers are parallel to each other. The refractive indices of different glass layers are monotonically decreasing as where is the refractive index of the slab and (see the figure). The ray is refracted out parallel to the interface between the and slabs from the right side of the stack. What is the value of m?
JEE Advanced 2017 Hard - Paragraph:
Schemes \(1\) and \(2\) describe sequential transformation of alkynes \(\mathbf{M}\) and \(\mathbf{N}\). Consider only the major products formed in each step for both the schemes.
Question:
The correct statement with respect to product \(\mathbf{Y}\) isJEE Advanced 2014 Medium