JEE Mains · Maths · STD 12 - 10. vector algebra
Let the arc \(A C\) of a circle subtend a right angle at the centre \(O\). If the point \(B\) on the arc \(A C\), divides the arc \(A C\) such that \(\frac{\text { length of } \operatorname{arc} A B}{\text { length of } \operatorname{arc} B C}=\frac{1}{5}\), and \(\overrightarrow{O C}=\alpha \overrightarrow{O A}+\beta \overrightarrow{O B}\), then \(\alpha+\sqrt{2}(\sqrt{3}-1) \beta\) is equal to
- A \(2 \sqrt{3}\)
- B \(2-\sqrt{3}\)
- C \(5 \sqrt{3}\)
- D \(2+\sqrt{3}\)
Answer & Solution
Correct Answer
(B) \(2-\sqrt{3}\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \overrightarrow{\mathrm{c}}=\alpha \overrightarrow{\mathrm{a}}+\beta \overrightarrow{\mathrm{b}} \ldots . .(1) \\ & \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=\alpha \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{a}}+\beta…
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
- A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the work in 24 days less than mason B alone. Then mason A alone will complete the work in:JEE Mains 2026 Hard
- Let \(X\) be a set containing \(10\) elements and \(P(X)\) be its power set. If \(A\) and \(B\) are picked up at random from \(P(X),\) with replacement, then the probability that \(A\) and \(B\) have equal number elements, isJEE Mains 2015 Hard
- Let \(0 < \alpha < 1\), \(\beta = \dfrac{1}{3\alpha}\) and \(\tan^{-1}(1-\alpha) + \tan^{-1}(1-\beta) = \dfrac{\pi}{4}\). Then \(6(\alpha + \beta)\) is equal to:JEE Mains 2026 Medium
- If \(1+\left(2+{ }^{49} C _{1}+{ }^{49} C _{2}+\ldots .+{ }^{49} C _{49}\right)\left({ }^{50} C _{2}+{ }^{50} C _{4}+\right.\) \(\ldots . .+{ }^{50} C _{ so }\) ) is equal to \(2^{ n } . m\), where \(m\) is odd, then \(n\) \(+m\) is equal to.JEE Mains 2022 Hard
- Let \(f : R \rightarrow R\) be continuous function satisfying \(f ( x )+ f ( x + k )= n\), for all \(x \in R\) where \(k >0\) and \(n\)is a positive integer. If \(I _{1}=\int\limits_{0}^{4 n k} f ( x ) dx\) and \(I _{2}=\int\limits_{- k }^{3 k } f ( x ) dx\), thenJEE Mains 2022 Hard
- The temperature \(\mathrm{T}(\mathrm{t})\) of a body at time \(\mathrm{t}=0\) is \(160^{\circ}\) \(\mathrm{F}\) and it decreases continuously as per the differential equation \(\frac{\mathrm{dT}}{\mathrm{dt}}=-\mathrm{K}(\mathrm{T}-80)\), where \(\mathrm{K}\) is positive constant. If \(\mathrm{T}(15)=120^{\circ} \mathrm{F}\), then \(\mathrm{T}(45)\) is equal to ...........JEE Mains 2024 Medium
More PYQs from JEE Mains
- Each of the persons \(\mathrm{A}\) and \(\mathrm{B}\) independently tosses three fair coins. The probability that both of them get the same number of heads is :JEE Mains 2021 Medium
- Let \(A=\left[a_{i j}\right]\) be \(3 \times 3\) matrix such that \(A\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], A\left[\begin{array}{l}4 \\ 1 \\ 3\end{array}\right]=\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]\) and \(A\left[\begin{array}{l}2 \\ 1 \\ 2\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\), then \(a_{23}\) equals :JEE Mains 2025 Easy
- Suppose that two chords, drawn from the point \((1, 2)\) on the circle \(x^2 + y^2 + x - 3y = 0\) are bisected by the \(y\)-axis. If the other ends of these chords are \(R\) and \(S\), and the mid point of the line segment \(RS\) is \((\alpha, \beta)\), then \(6(\alpha + \beta)\) is equal to:JEE Mains 2026 Medium
- If the system of linear equations \(2x + 2y + 3z = a\) ; \(3x - y + 5z = b\) ; \(x - 3y + 2z = c\) Where \(a, b, c\) are non zero real numbers, has more than one solution, thenJEE Mains 2019 Hard
- The mean and standard deviation of \(15\) observations were found to be \(12\) and \(3\) respectively. On rechecking it was found that an observation was read as \(10\) in place of \(12\) . If \(\mu\) and \(\sigma^2\) denote the mean and variance of the correct observations respectively, then \(15\left(\mu+\mu^2+\sigma^2\right)\) is equal to ...........JEE Mains 2024 Hard
- The relation \(R =\{( a , b ): \operatorname{gcd}( a , b )=1,2 a \neq b , a , b \in Z \}\) is:JEE Mains 2023 Hard