JEE Mains · Maths · STD 12 - 11. three dimension geometry
If the points with vectors \(\alpha \hat{i}+10 \hat{j}+13 \hat{k}\), \(6 \hat{i}+11 \hat{j}+11 \hat{k}, \frac{9}{2} \hat{i}+\beta \hat{j}-8 \hat{k}\) are collinear, then \((19 \alpha-6 \beta)^2\) is equal to \(...........\).
- A \(36\)
- B \(16\)
- C \(25\)
- D \(49\)
Answer & Solution
Correct Answer
(A) \(36\)
Step-by-step Solution
Detailed explanation
\((\alpha, 10,13) ;(6,11,11),\left(\frac{9}{2}, \beta,-8\right)\) \(\frac{\alpha-6}{3 / 2}=\frac{-1}{11-\beta}=\frac{2}{19} -19=22-2 \beta\) \(\alpha-6=\frac{3}{19} 2 \beta=41\) \(\alpha=6+\frac{3}{19}=\frac{117}{19}\) \(\therefore(19 \alpha-6 \beta)^2=(117-123)^2=36\)
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
- Let the tangent to the parabola \(y^2=12 x\) at the point \((3, \alpha)\) be perpendicular to the line \(2 x+2 y=3\).Then the square of distance of the point \((6,-4)\)from the normal to the hyperbola \(\alpha^2 x^2-9 y^2=9 \alpha^2\)at its point \((\alpha-1, \alpha+2)\) is equal to \(........\).JEE Mains 2023 Hard
- A function \(f\) is defined on \([-3,3]\) as \(f(x)=\left\{\begin{array}{cc}\min \left\{|x|, 2-x^{2}\right\} & , \quad-2 \leq x \leq 2 \\ {[|x|]} & , \quad 2<|x| \leq 3\end{array}\right.\) where \([x]\) denotes the greatest integer \(\leq x .\) The number of points, where \(f\) is not differentiable in \((-3,3)\) isJEE Mains 2021 Hard
- If \((a, b)\) be the orthocentre of the triangle whose vertices are \((1,2),(2,3)\) and \((3,1)\), and \(I_1=\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{x} \sin \left(4 \mathrm{x}-\mathrm{x}^2\right) \mathrm{dx}, \mathrm{I}_2=\int_{\mathrm{a}}^{\mathrm{b}} \sin \left(4 \mathrm{x}-\mathrm{x}^2\right) \mathrm{dx}\) , then \(36 \frac{\mathrm{I}_1}{\mathrm{I}_2}\) is equal to :JEE Mains 2024 Hard
- If all the words, with or without meaning, are written using the letters of the word \(QUEEN\) and are arranged as in English dictionary, then the position of the word \(QUEEN\) isJEE Mains 2017 Hard
- Let a circle \(C\) in complex plane pass tltrough the points \(z _{1}=3+4 i , z _{2}=4+3 i\) and \(z _{3}=5 i\). If \(z \left(\neq z _{1}\right)\) is a point on \(C\) such that the line through \(z\) and \(z _{1}\) is perpendicular to the line through \(z _{2}\) and \(z _{3}\), then \(\arg ( z )\) is equal toJEE Mains 2022 Hard
- If \(m\) and \(n\) respectively are the number of local maximum and local minimum points of the function \(f ( x )=\int_{0}^{ x ^{2}} \frac{ t ^{2}-5 t +4}{2+ e ^{ t }} dt\), then the ordered pair \(( m , n )\) is equal toJEE Mains 2022 Hard
More PYQs from JEE Mains
- For \(\alpha, \beta \in \mathrm{R}\) and a natural number \(\mathrm{n}\), let \(A_r=\left|\begin{array}{ccc}r & 1 & \frac{n^2}{2}+\alpha \\ 2 r & 2 & n^2-\beta \\ 3 r-2 & 3 & \frac{n(3 n-1)}{2}\end{array}\right|\). Then \(2 A_{10}-A_8\)JEE Mains 2024 Hard
- If the system of linear equations \(2 x+3 y-z=-2\) ; \(x+y+z=4\) ; \(x-y+|\lambda| z=4 \lambda-4\) (where \(\lambda \in R\)), has no solution, thenJEE Mains 2022 Medium
- A circle passes through the points \((2, 3)\) and \((4, 5)\). If its centre lies on the line, \(y- 4x + 3 = 0\) , then its radius is equal toJEE Mains 2018 Hard
- If the coefficients of \(x^4, x^5\) and \(x^6\) in the expansion of \((1+x)^n\) are in the arithmetic progression, then the maximum value of \(n\) is :JEE Mains 2024 Hard
- Let \( f(x)=\int\frac{(2-x^{2})e^{x}}{(\sqrt{1+x})(1-x)^{\frac{3}{2}}}dx \). If \( f(0)=0 \), then \( f(\frac{1}{2}) \) is equal to:JEE Mains 2026 Easy
- Let \(X=\{11,12,13, \ldots ., 40,41\}\) and \(Y=\{61,62\), \(63, \ldots ., 90,91\}\) be the two sets of observations. If \(\bar{x}\) and \(\bar{y}\) are their respective means and \(\sigma^2\) is the variance of all the observations in \(X \cup Y\), then \(\left|\overline{ x }+\overline{ y }-\sigma^2\right|\) is equal to \(.................\).JEE Mains 2023 Hard