JEE Mains · Maths · STD 12 - 10. vector algebra
Let \(\sqrt 3 \hat i + j,\hat i + \sqrt 3 \hat j\) and \(\beta \hat i + \left( {1 + \beta } \right)\hat j\) respectively be the position vectors of the points \(A,B\) and \(C\) with respect to the origin \(O\). If the distance of \(C\) from the bisector of the acute angle between \(OA\) and \(OB\) is \(\frac{3}{{\sqrt 2 }}\) , then the sum of all possible values of \(\beta \) is
- A \(4\)
- B \(3\)
- C \(2\)
- D \(1\)
Answer & Solution
Correct Answer
(D) \(1\)
Step-by-step Solution
Detailed explanation
Equation of angle bisector of DA and OB is \(y=x\) Given that, \(\left|\frac{\beta-(1-\beta)}{\sqrt{2}}\right|=\frac{3}{\sqrt{2}}\) \(2 \beta-1=\pm 3\) \(\Rightarrow \beta=2,-1\) Sum of values of \(\beta=1\)
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 value of the integral \(\int_{-1}^{1} \log \left(x+\sqrt{x^{2}+1}\right)\, d x\) is:JEE Mains 2021 Easy
- The value of \(\cot \frac{\pi}{24}\) is :JEE Mains 2021 Hard
- Let \([t]\) be the greatest integer less than or equal to \(t\). Let \(A\) be the set of al prime factors of \(2310\) and \(f: A \rightarrow \mathbb{Z}\) be the function \(f(x)=\left[\log _2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]\). The number of one-to-one functions from \(A\) to the range of \(f\) is :JEE Mains 2024 Hard
- If a circle \(C\) passing through the point \((4, 0)\) touches the circle \(x^2 + y^2 + 4x -6y = 12\) externally at the point \((1, -1)\), then the radius of \(C\) isJEE Mains 2019 Hard
- Let \(f: \mathbf{R} \rightarrow \mathbf{R}\) be a function such that \(f(x) + 3f\left(\dfrac{\pi}{2} - x\right) = \sin x\), \(x \in \mathbf{R}\). Let the maximum value of \(f\) on \(\mathbf{R}\) be \(\alpha\). If the area of the region bounded by the curves \(g(x) = x^2\) and \(h(x) = \beta x^3\), \(\beta > 0\), is \(\alpha^2\), then \(30\beta^3\) is equal to _______.JEE Mains 2026 Hard
- The first term of an A.P. of \(30\) non-negative terms is \(\dfrac{10}{3}\). If the sum of this A.P. is the cube of its last term, then its common difference is:JEE Mains 2026 Medium
More PYQs from JEE Mains
- The equations of the sides \(AB\) and \(AC\) of a triangle \(ABC\) are \((\lambda+1) x +\lambda y =4 \text { and } \lambda x +(1-\lambda) y +\lambda=0\) respectively. Its vertex \(A\) is on the \(y\)-axis and its orthocentre is \((1,2)\). The length of the tangent from the point \(C\) to the part of the parabola \(y^2=6 x\) in the first quadrant isJEE Mains 2023 Hard
- Let \(C\) be a curve given by \(y\left( x \right) = 1 + \sqrt {4x - 3} ,x > \frac{3}{4}\).If \(P\) is a point on \(C\), such that the tangent at \(P\) has slope \(\frac{2}{3}\) , then a point through which the normal at \(P\) passes, isJEE Mains 2016 Hard
- Let \(f(x)=\int_0^t t\left(t^2-9 t+20\right) d t, 1 \leq x \leq 5\). If the range of \(f\) is \([\alpha, \beta]\), then \(4(\alpha+\beta)\) equals :JEE Mains 2025 Easy
- If \(f(\theta ) =\left| {\begin{array}{*{20}{c}}
1&{\cos {\mkern 1mu} \theta }&1\\
{ - \sin {\mkern 1mu} \theta }&1&{ - \cos {\mkern 1mu} \theta }\\
{ - 1}&{\sin {\mkern 1mu} \theta }&1
\end{array}} \right|\) and \(A\) and \(B\) are respectively the maximum and the minimum values of \(f(\theta )\), then \((A , B)\) is equal toJEE Mains 2014 Hard - Let \(\alpha\) and \(\beta\) be the roots of the equation \(\mathrm{x}^{2}-\mathrm{x}-1=0 .\) If \(\mathrm{p}_{\mathrm{k}}=(\alpha)^{\mathrm{k}}+(\beta)^{\mathrm{k}}, \mathrm{k} \geq 1,\) then which one of the following statements is not true?JEE Mains 2020 Hard
- The integral \(80 \int_0^{\frac{\pi}{4}}\left(\frac{\sin \theta+\cos \theta}{9+16 \sin 2 \theta}\right) d \theta\) is equal to :JEE Mains 2025 Medium