MHT CET · Maths · Vector Algebra
If \(a, b, c\) are distinct positive numbers and vectors \(a \hat{\imath}+a \hat{\jmath}+c \hat{k}, \hat{\imath}+\hat{k}\)
and \(c \hat{\imath}+c \hat{\jmath}+b \hat{k}\) lie in a plane, then
- A \(c\) is A.M. of a and b
- B \(\mathrm{c}^{2}=0\)
- C \(\mathrm{c}\) is \(\mathrm{H} . \mathrm{M}\). of \(\mathrm{a}\) and \(\mathrm{b}\)
- D \(c\) is G.M. of a and b
Answer & Solution
Correct Answer
(D) \(c\) is G.M. of a and b
Step-by-step Solution
Detailed explanation
(B)
since, three vectors are coplanar
\(\left|\begin{array}{lll}
\mathrm{a} & \mathrm{a} & \mathrm{c} \\
1 & 0 & 1 \\
\mathrm{c} & \mathrm{c} & \mathrm{b}
\end{array}\right|=0\)
Applying \(\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}-\mathrm{C}_{2}\)
\(\left|\begin{array}{lll}
0 & a & c \\
1 & 0 & 1 \\
0 & c & b
\end{array}\right|=0\)
Expanding along \(C_{1}\), we get
\(-1\left(a b-c^{2}\right)=0 \Rightarrow a b=c^{2} \Rightarrow c\) is G.M. of a and b
since, three vectors are coplanar
\(\left|\begin{array}{lll}
\mathrm{a} & \mathrm{a} & \mathrm{c} \\
1 & 0 & 1 \\
\mathrm{c} & \mathrm{c} & \mathrm{b}
\end{array}\right|=0\)
Applying \(\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}-\mathrm{C}_{2}\)
\(\left|\begin{array}{lll}
0 & a & c \\
1 & 0 & 1 \\
0 & c & b
\end{array}\right|=0\)
Expanding along \(C_{1}\), we get
\(-1\left(a b-c^{2}\right)=0 \Rightarrow a b=c^{2} \Rightarrow c\) is G.M. of a and b
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 equivalent form of the statement is ________MHT CET 2019 Easy
- The equation of the circle passing through the point \((1,1)\) and having two diameters along the pair of lines \(x^2-y^2-2 x+4 y-3=0\) isMHT CET 2025 Medium
- \(\int_{\log \frac{1}{2}}^{\log 2} \sin \left(\frac{e^x-1}{e^x+1}\right) d x=\)MHT CET 2025 Easy
- The line \(\mathrm{y}=\mathrm{m} x+3\) is tangent to the parabola \(\mathrm{y}^2=4 x\), if the value of m isMHT CET 2025 Easy
- The particular solution of the differential equation, \(x y \frac{\mathrm{~d} y}{\mathrm{~d} x}=x^2+2 y^2\) when \(y(1)=0\) isMHT CET 2024 Hard
- If \(\mathrm{A}\) and \(\mathrm{B}\) are two events such that \(\mathrm{P}(\mathrm{A})=\frac{1}{3}\), \(\mathrm{P}(\mathrm{B})=\frac{1}{5}, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{1}{3}\), then the value of \(\mathrm{P}\left(\mathrm{A}^{\prime} / \mathrm{B}^{\prime}\right)+\mathrm{P}\left(\mathrm{B}^{\prime} / \mathrm{A}^{\prime}\right)\) isMHT CET 2023 Easy
More PYQs from MHT CET
- Which carbon atom of glucose, numbered from \(\mathrm{C}-1\) to \(\mathrm{C}-6\) converts its function group to \(-\mathrm{COOH}\) group when heated with \(\mathrm{Br}_2\) water?MHT CET 2022 Easy
- A plate of refractive index 1.6 is introduced in the path of light from one of the slits in Young's double slit experiment thenMHT CET 2024 Easy
- In assigning R-S configuration which among the following groups has highest priority?MHT CET 2017 Easy
- The radius of hydrogen a tomin its ground state is \(5.3 \times 10^{-11} \mathrm{~m}\). After collision with an electron
it is found to have a radius of \(21.2 \times 10^{-11} \mathrm{~m}\).
What is the prindpal quantum number \(n\) of the final state of atom?MHT CET 2009 Medium - \(\int e^{x} \frac{(x-1)}{x^{2}} d x\) is equal toMHT CET 2009 Easy
- Which from following coloured light has the highest energy?MHT CET 2023 Easy