JEE Mains · Maths · STD 12 - 10. vector algebra
Two adjacent sides of a parallelogram PQRS are given by \(\vec{PQ} = \hat{j} + \hat{k}\) and \(\vec{PS} = \hat{i} - \hat{j}\). If the side PS is rotated about the point P by an acute angle \(\alpha\) in the plane of the parallelogram so that it becomes perpendicular to the side PQ, then \(\sin^2\left(\dfrac{5\alpha}{2}\right) - \sin^2\left(\dfrac{\alpha}{2}\right)\) is equal to:
- A \(\dfrac{1}{2}\)
- B \(\dfrac{\sqrt{3}}{2}\)
- C \(\dfrac{\sqrt{3}}{4}\)
- D \(\dfrac{2\sqrt{3}}{5}\)
Answer & Solution
Correct Answer
(B) \(\dfrac{\sqrt{3}}{2}\)
Step-by-step Solution
Detailed explanation
Let \(\theta\) be the angle between the vectors \(\vec{PQ}\) and \(\vec{PS}\). \(\cos \theta = \dfrac{\vec{PQ} \cdot \vec{PS}}{|\vec{PQ}| |\vec{PS}|}\) \(\cos \theta = \dfrac{(\hat{j} + \hat{k}) \cdot (\hat{i} - \hat{j})}{\sqrt{1^2 + 1^2} \sqrt{1^2 + (-1)^2}}\)…
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 \(f\) be a differentiable function \(R\) to \(R\) such that \(\left| {f\,(x)\, - \,f(y)} \right|\, \le \,2\,{\left| {x - y} \right|^{\frac{3}{2}}},\) for all \(x,y\,\in R .\) If \(f\,(0)=1\) then \(\int\limits_0^1 {{f^2}\,(x)\,dx} \) is equal toJEE Mains 2019 Hard
- If \(\frac{d y}{d x}=\frac{2^{x} y+2^{y} \cdot 2^{x}}{2^{x}+2^{x+y} \log _{e} 2}, y(0)=0\), then for \(y=1\) the value of \(x\) lies in the interval:JEE Mains 2021 Hard
- The value of \(12 \int \limits_0^3\left|x^2-3 x+2\right| d x\) is \(.............\)JEE Mains 2023 Medium
- The distance of the point \((1,1,9)\) from the point of intersection of the line \(\frac{x-3}{1}=\frac{y-4}{2}=\frac{z-5}{2}\) and the plane \(x+y+z=17\) isJEE Mains 2021 Medium
- Let \(Q(a,b,c)\) be the image of the point \(P(3,2,1)\) in the line \(\frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1}.\) Then the distance of Q from the line \(\frac{x-9}{3}=\frac{y-9}{2}=\frac{z-5}{-2}\) isJEE Mains 2026 Hard
- Let each of the two ellipses \(E _1: \frac{ x ^2}{ a ^2}+\frac{ y ^2}{b^2}=1,( a > b )\) and \(E _2: \frac{ x ^2}{A^2}+\frac{ y ^2}{B^2}=1,(A< B )\) have eccentricity \(\frac{4}{5}\). Let the lengths of the latus recta of \(E_1\) and \(E_2\) be \(\ell_1\) and \(\ell_2\), respectively, such that \(2 \ell_1^2=9 \ell_2\). If the distance between the foci of \(E_1\) is 8 , then the distance between the foci of \(E _2\) isJEE Mains 2026 Hard
More PYQs from JEE Mains
- The line \(2 x - y +1=0\) is a tangent to the circle at the point \((2,5)\) and the centre of the circle lies on \(x-2 y=4\). Then, the radius of the circle isJEE Mains 2021 Hard
- The sum to \(20\) terms of the series \(2.2^2-3^2+2.4^2-5^2+2.6^2-\ldots \ldots\) is equal to \(........\).JEE Mains 2023 Hard
- Let the length of the focal chord \(P Q\) of the parabola \(y^2=12 x\) be \(15\) units. If the distance of \(P Q\) from the origin is \(\mathrm{p}\), then \(10 \mathrm{p}^2\) is equal to ...........JEE Mains 2024 Hard
- Let \(P=\{z \in \mathbb{C}:|z+2-3 i| \leq 1\}\) and \(Q=\{z \in \mathbb{C}: z(1+i)+\bar{z}(1-i) \leq-8\}\). Let in \(\mathrm{P} \cap \mathrm{Q},|\mathrm{z}-3+2 \mathrm{i}|\) be maximum and minimum at \(z_1\) and \(z_2\) respectively. If \(\left|z_1\right|^2+2|z|^2=\alpha+\beta \sqrt{2}\), where \(\alpha, \beta\) are integers, then \(\alpha+\beta\) equals ...........JEE Mains 2024 Hard
- Coasider a cuboid of sides \(2 x , 4 x\) and \(5 x\) and a closed hemisphere of radius \(r\). If the sum of their surface areas is a constant \(k\), then the ratio \(x: r\), for which the sum of their volumes is maximum, isJEE Mains 2022 Hard
- Let the solution curve \(y=y(x)\) of the differential equation \(\left(1+ e ^{2 x }\right)\left(\frac{ dy }{ dx }+ y \right)=1\) pass through the point \(\left(0, \frac{\pi}{2}\right)\). Then, \(\lim _{x \rightarrow \infty} e ^{x} y(x)\) is equal to.JEE Mains 2022 Hard