JEE Mains · Maths · STD 12 - 11. three dimension geometry
Let a unit vector \(\hat{ OP }\) make angle \(\alpha, \beta, \gamma\) with the positive directions of the co-ordinate axes \(OX , OY\), \(OZ\) respectively, where \(\beta \in\left(0, \frac{\pi}{2}\right) \hat{ OP }\) is perpendicular to the plane through points \((1,2,3)\), \((2,3,4)\) and \((1,5,7)\), then which one of the following is true ?
- A \(\alpha \in\left(\frac{\pi}{2}, \pi\right)\) and \(\gamma \in\left(\frac{\pi}{2}, \pi\right)\)
- B \(\alpha \in\left(0, \frac{\pi}{2}\right)\) and \(\gamma \in\left(0, \frac{\pi}{2}\right)\)
- C \(\alpha \in\left(\frac{\pi}{2}, \pi\right)\) and \(\gamma \in\left(0, \frac{\pi}{2}\right)\)
- D \(\alpha \in\left(0, \frac{\pi}{2}\right)\) and \(\gamma \in\left(\frac{\pi}{2}, \pi\right)\)
Answer & Solution
Correct Answer
(A) \(\alpha \in\left(\frac{\pi}{2}, \pi\right)\) and \(\gamma \in\left(\frac{\pi}{2}, \pi\right)\)
Step-by-step Solution
Detailed explanation
Equation of plane :- \(\left|\begin{array}{ccc}x-1 & y-2 & z-3 \\ 1 & 1 & 1 \\ 0 & 3 & 4\end{array}\right|=0\) \(\Rightarrow[x-1]-4[y-2]+3[z-3]=0\) \(\Rightarrow x-4 y+3 z=2\) \(D.R's\) of normal of plane \( < 1,-4,3 > D.C's\) of…
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
- If \(\frac{1}{2 \cdot 3^{10}}+\frac{1}{2^{2} \cdot 3^{9}}+\ldots \frac{1}{2^{10} \cdot 3}=\frac{K}{2^{10} \cdot 3^{10}}\), then the remainder when \(K\) is divided by \(6\) isJEE Mains 2022 Hard
- If \(\displaystyle\int_{\pi/6}^{\pi/4}\left(\cot\left(x-\dfrac{\pi}{3}\right)\cot\left(x+\dfrac{\pi}{3}\right)+1\right)dx = \alpha\log_e(\sqrt{3}-1)\), then \(9\alpha^2\) is equal to ________.JEE Mains 2026 Hard
- Let \(\alpha, \beta \in \mathrm{N}\) be roots of equation \(\mathrm{x}^2-70 \mathrm{x}+\lambda=0\), where \(\frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathrm{N}\). If \(\lambda\) assumes the minimum possible value, then \(\frac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{|\alpha-\beta|}\) is equal to :JEE Mains 2024 Hard
- If \(\int \frac{1}{ x } \sqrt{\frac{1- x }{1+ x }} dx = g ( x )+ c , g (1)=0\), then \(g \left(\frac{1}{2}\right)\) is equal toJEE Mains 2022 Hard
- The area (in square units) of the region enclosed by the ellipse \(x^2+3 y^2=18\) in the first quadrant below the line \(y=x\) is :JEE Mains 2024 Hard
- For \(I(x)=\int \frac{\sec ^{2} x-2022}{\sin ^{2022} x} d x\), if \(I\left(\frac{\pi}{4}\right)=2^{1011}\), thenJEE Mains 2022 Hard
More PYQs from JEE Mains
- Let \(y=y(x)\) be the solution curve of the differential equation
\(x\left(x^2+e^x\right) d y+\left(e^x(x-2) y-x^3\right) d x=0, x \gt 0\) passing through the point \((1,0)\). Then \(y(2)\) is equal to :JEE Mains 2025 Medium - Let \(\ell\) be a line which is normal to the curve \(y=2 x^{2}+x+2\) at a point \(P\) on the curve. If the point \(Q(6,4)\) lies on the line \(\ell\) and \(O\) is origin, then the area of the triangle \(OPQ\) is equal to.......JEE Mains 2022 Hard
- Let the sets \(A\) and \(B\) denote the domain and range respectively of the function \(f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}}\) where \(\lceil x \rceil\) denotes the smallest integer greater than or equal to \(x\). Then among the statements \(( S 1): A \cap B =(1, \infty)-N\) and \(( S 2): A \cup B=(1, \infty)\)JEE Mains 2023 Hard
- Let the line \(x - y = 4\) intersect the circle \(C: (x-4)^2 + (y+3)^2 = 9\) at the points \(Q\) and \(R\). If \(P(\alpha, \beta)\) is a point on \(C\) such that \(PQ = PR\), then \((6\alpha + 8\beta)^2\) is equal to __________.JEE Mains 2026 Hard
- Let a relation \(R\) on \(\mathbb{N} \times \mathbb{N}\) be defined as : \(\left(\mathrm{x}_1, \mathrm{y}_1\right) \mathrm{R}\left(\mathrm{x}_2, \mathrm{y}_2\right)\) if and only if \(\mathrm{x}_1 \leq \mathrm{x}_2\) or \(\mathrm{y}_1 \leq \mathrm{y}_2\) Consider the two statements : (\(I\)) \(\mathrm{R}\) is reflexive but not symmetric. (\(II\)) \(\mathrm{R}\) is transitive Then which one of the following is true?JEE Mains 2024 Medium
- The number of distinct real roots of the equation \(3 x^{4}+4 x^{3}-12 x^{2}+4=0\) is ..... .JEE Mains 2021 Hard