MHT CET · Maths · Vector Algebra
\(\bar{a}\) and \(\bar{b}\) are non-collinear vectors. If \(\bar{p}=(2 x+1) \bar{a}-\bar{b}\) and \(\bar{q}=(x-2) \bar{a}+\bar{b}\)
are collinear vectors, then \(x=\)
- A \(-3\)
- B \(\frac{1}{3}\)
- C \(-\frac{1}{3}\)
- D 3
Answer & Solution
Correct Answer
(B) \(\frac{1}{3}\)
Step-by-step Solution
Detailed explanation
(A)
The vector \(\bar{q}\) is non-zero since the coefficient in \(\bar{b}\) is different from zero and so the vectors \(\bar{q}\) and \(\bar{p}\) are collinear if for some number \(y\) we have
\(\overline{\mathrm{p}}=\mathrm{c} \overline{\mathrm{q}}\) that is
\((2 x+1) \bar{a}-\bar{b}=y[(x-2) \bar{a}+\bar{b}]\)
\(\therefore(2 x+1)=y(x-2)\) and \(-1=y \Rightarrow y=-1\)
\(2 x+1=-(x-2) \Rightarrow 3 x=1 \Rightarrow x=\frac{1}{3}\)
The vector \(\bar{q}\) is non-zero since the coefficient in \(\bar{b}\) is different from zero and so the vectors \(\bar{q}\) and \(\bar{p}\) are collinear if for some number \(y\) we have
\(\overline{\mathrm{p}}=\mathrm{c} \overline{\mathrm{q}}\) that is
\((2 x+1) \bar{a}-\bar{b}=y[(x-2) \bar{a}+\bar{b}]\)
\(\therefore(2 x+1)=y(x-2)\) and \(-1=y \Rightarrow y=-1\)
\(2 x+1=-(x-2) \Rightarrow 3 x=1 \Rightarrow x=\frac{1}{3}\)
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
- \(\lim _{x \rightarrow \infty} \frac{(2 x+1)^{50}+(2 x+2)^{50}+(2 x+3)^{50}+\cdots \cdots+(2 x+100)^{50}}{(2 x)^{50}+(10)^{50}}=\cdots\)MHT CET 2025 Medium
- Given \(P(A)=0.5, P(B)=0.4, P(A \cap B)=0.3\),
then \(P\left(A^{\prime} / B^{\prime}\right)\) is equal toMHT CET 2009 Medium - A line drawn from the point \(\mathrm{A}(1,3,2)\) parallel to the line \(\frac{x}{2}=\frac{y}{4}=\frac{z}{1}\), intersects the plane \(3 x+y+2 z=5\) in point \(\mathrm{B}\), then co-ordinates of point \(\mathrm{B}\) areMHT CET 2023 Medium
- \(\mathrm{f}(x)=\left\{\begin{array}{ll}\frac{1-\cos k x}{x^2}, & \text { if } x \leq 0 \ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & \text { if } x>0\end{array}\right.\) is continuous at \(x=0\), then the value of \(\mathrm{k}\) isMHT CET 2023 Medium
- Water is being poured at the rate of \(36 \mathrm{~m}^3 / \mathrm{min}\). into a cylindrical vessel, whose circular base is of radius \(3 \mathrm{~m}\). then the water level in the cylinder is rising at the rate ofMHT CET 2021 Easy
- If \(a, b, c\) and \(d\) are positive, then
is equalMHT CET 2011 Easy
More PYQs from MHT CET
- Define \(\mathrm{f}(x)=\left\{\begin{array}{cl}\mathrm{b}-\mathrm{a} x & , \text { if } x < 2 \\ 3 & , \text { if } x=2 \\ \mathrm{a}+2 \mathrm{~b} x & , \text { if } x>2\end{array}\right.\) and if \(\lim _{x \rightarrow 2} \mathrm{f}(x)\) exists, then \(\frac{\mathrm{a}}{\mathrm{b}}=\)MHT CET 2025 Medium
- The length of a sonometer wire ' AB ' is 110 cm . Where should the two bridges be placed from end ' A ' to divide the wire in three segments whose fundamental frequencies are in the ratio \(1: 2: 3\) ?MHT CET 2024 Medium
- What is Henry's law constant of a gas if solubility of gas in water at \(25^{\circ} \mathrm{C}\) is \(0.028 \mathrm{~mol} \mathrm{dm}^{-3}\) ?
[Partial pressure of gas \(=0.346 \mathrm{bar}]\)MHT CET 2023 Medium - A gaseous mixture of \(\mathrm{O}_2\) and \(\mathrm{CH}_4\) are in the ratio \(1: 4\) by mass. Find the ratio of their molecules.MHT CET 2025 Easy
- A thin uniform metal rod of mass ' \(M\) ' and length ' \(L\) ' is swinging about a horizontal axis passing through its end. Its maximum angular velocity is ' \(\omega\) '. Its centre of mass rises to a maximum height of ( \(\mathrm{g}=\) Acceleration due to gravity)MHT CET 2024 Medium
- Identify the product obtained when benzonitrile is reduced by stannous chloride in presence of hydrochloric acid followed by acid hydrolysis.MHT CET 2023 Medium