MHT CET · Maths · Three Dimensional Geometry
If \(\bar{a}=2 \hat{i}-\hat{j}+\hat{k}, \bar{b}=\hat{i}+2 \hat{j}-3 \hat{k}\) and \(\bar{c}=3 \hat{i}+\lambda \hat{j}+5 \hat{k}\) are coplanar, then \(\lambda\) is the root of the equation
- A \(x^2+3 x=6\)
- B \(x^2+2 x=4\)
- C \(x^2+3 x=4\)
- D \(x^2+2 x=6\)
Answer & Solution
Correct Answer
(C) \(x^2+3 x=4\)
Step-by-step Solution
Detailed explanation
Since given vectors are coplanar, we write
\(
\begin{aligned}
& \left|\begin{array}{ccc}
2 & -1 & 1 \\
1 & 2 & -3 \\
3 & \lambda & 5
\end{array}\right|=0 \end{aligned}
\)
\(\therefore 2(10+3 \lambda)+1(5+9)+1(\lambda-6)=0 \)
\( \therefore 20+6 \lambda+14+\lambda-6=0 \Rightarrow 7 \lambda+28=0 \Rightarrow\) \(\lambda=-4 \)
Which is one of the root of equation in option \(C\)
\(
\begin{aligned}
& \left|\begin{array}{ccc}
2 & -1 & 1 \\
1 & 2 & -3 \\
3 & \lambda & 5
\end{array}\right|=0 \end{aligned}
\)
\(\therefore 2(10+3 \lambda)+1(5+9)+1(\lambda-6)=0 \)
\( \therefore 20+6 \lambda+14+\lambda-6=0 \Rightarrow 7 \lambda+28=0 \Rightarrow\) \(\lambda=-4 \)
Which is one of the root of equation in option \(C\)
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 Maths
- The integrating factor of the differential equation \(x \frac{d y}{d x}+y \log x=x^{2}\) isMHT CET 2020 Medium
- If the equation \(\cos ^4 \theta+\sin ^4 \theta+\lambda=0\). has real solutions for \(\theta\), then \(\lambda\) lies in the intervalMHT CET 2024 Hard
- The equation of tangent to the circle \(x^2+y^2=64\) at the point \(\mathrm{P}\left(\frac{2 \pi}{3}\right)\) isMHT CET 2021 Easy
- The perimeter a square whose two sides have equations \(\frac{x-1}{2}=\frac{y+2}{3}=\frac{z-3}{4}\) and \(\frac{x}{2}=\frac{y-1}{3}=\frac{z+1}{4}\) isMHT CET 2025 Hard
- If \((a,-2 a), a>0\) is the midpoint of a line segment intercepted between the co-ordinate axes, then the equation of the line isMHT CET 2020 Easy
- With usual notations in \(\triangle A B C\) if \(a^2+b^2-c^2=a b\), then measure of angle \(C\) isMHT CET 2022 Medium
More PYQs from MHT CET
- Calculate the PV type of work for the following reaction at 1 bar pressure. \(\underset{(150 \mathrm{~mL})}{\mathrm{C}_3 \mathrm{H}_{6(\mathrm{~g})}}+\underset{(150 \mathrm{~mL})}{\mathrm{HCl}_{(\mathrm{g})}} \longrightarrow \underset{(150 \mathrm{~mL})}{\mathrm{C}_3 \mathrm{H}_7 \mathrm{Cl}_{(\mathrm{g})}}\)MHT CET 2023 Medium
- What is standard reduction potential of \(\mathrm{Cu}^{2+} \mid \mathrm{Cu}_{(\mathrm{s})}\) if \(\mathrm{E}^{\circ}\) of following cell is 0.46 ?
\(\mathrm{Cu}_{(\mathrm{s})}\left|\mathrm{cu}_{(\mathrm{aq})}^{2+} \| \mathrm{Ag}_{(\mathrm{aq})}^{+}\right| \mathrm{Ag}_{(\mathrm{s})} ; \mathrm{E}_{\mathrm{Ag}^{+} / \mathrm{Ag}}^{\circ}=0.80 \mathrm{v} \mid\)MHT CET 2022 Medium - A simple pendulum has length \(2 \mathrm{~m}\) and a bob of mass 100 gram. It is whirled in a horizontal plane. If the string breaks under a tension of \(10 \mathrm{~N}\), the angle made by the string with vertical is \(\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)\)MHT CET 2020 Medium
- A metal ball of mass \(2 \mathrm{~kg}\) moving with a speed of \(10 \mathrm{~ms}^{-1}\) had a head-on collision with a stationary ball of mass \(3 \mathrm{~kg}\). If after collision, both the balls move together, then the loss in kinetic energy due to collision isMHT CET 2020 Medium
- P and Q are two non-zero vectors inclined to each other at an angle ‘p’ and ‘q’ are unit vectors along P and Q respectively. The component of Q in the direction of Q will beMHT CET 2019 Medium
- Calculate the concentration of weak monobasic acid if its degree of dissociation and dissociation constant are \(5.0 \times 10^{-4}\) and \(5.0 \times 10^{-9}\) respectively.MHT CET 2024 Medium