Let a, b be unit vectors. If c be a vector such that the angle be… - Vidyadip

MCQ

Let $\hat{a}, \hat{b}$ be unit vectors. If $\vec{c}$ be a vector such that the angle between $\hat{ a }$ and $\overrightarrow{ c }$ is $\frac{\pi}{12}$, and $\hat{ b }=\overrightarrow{ c }+2(\overrightarrow{ c } \times \hat{ a })$, then $|6 \overrightarrow{ c }|^{2}$ is equal to

A
$6(3-\sqrt{3})$
B
$3+\sqrt{3}$
✓
$6(3+\sqrt{3})$
D
$6(\sqrt{3}+1)$

✓

Answer

Correct option: C.

$6(3+\sqrt{3})$

c
$|\hat{b}|^{2}=|\vec{c}+2(\vec{c} \times \hat{a})|^{2}$

$|\hat{ b }|^{2}=| c |^{2}+4|\overrightarrow{ c } \times \hat{ a }|^{2}+4 \overrightarrow{ c } \cdot(\overrightarrow{ c } \times \hat{ a })$

$1=|c|^{2}+4|c|^{2} \sin ^{2} \frac{\pi}{12}+0$

$1=|c|^{2}+4|c|^{2}\left(\frac{\sqrt{3}-1}{2 \sqrt{2}}\right)^{2}$

$|c|^{2}=\frac{1}{3-\sqrt{3}}=\frac{3+\sqrt{3}}{6}$

So $6^{2}|c|^{2}=6(3+\sqrt{3})$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from STD 12 - 10. vector algebra→STD 12 - 10. vector algebra — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

Let $f(x) = \left\{ {\begin{array}{*{20}{c}}
{{x^2}\ln x,\,x > 0} \\
{0,\,\,\,\,\,\,\,\,\,\,\,\,\,x = 0}
\end{array}} \right\}$, Rolle’s theorem is applicable to $ f $ for $x \in [0,1]$, if $\alpha = $

If $\int\frac{2^{\frac{1}{\text{x}}}}{\text{x}^2}\text{dx}=\text{k }2^{\frac{1}{\text{x}}}+\text{C},$ then k is equal to:

Let $\vec p = 2\hat i + 3\hat j + a\hat k,\,\vec q = b\hat i + 5\hat j - \hat k,\,\vec r = \hat i + \hat j + 3\hat k$ . If $\vec p,\vec q,\vec r$ are coplanar and $\vec p.\vec q = 20$ , then the ordered pair $(a, b)$ is

If $y = y ( x )$ is the solution of the differential equation $\left(1+ e ^{2 x }\right) \frac{ dy }{ dx }+2\left(1+ y ^{2}\right) e ^{ x }=0$ and $y (0)=0$, then $6\left( y ^{\prime}(0)+\left( y \left(\log _{ e } \sqrt{3}\right)\right)^{2}\right)$ is equal to

Subtraction of integers is:

Let $A= \{1, 2, 3, 4\}$ and $R : A \to A$ be the relation defined by $R = \{ (1, 1), (2, 3), (3, 4), ( 4, 2) \}$. The correct statement is

If $\int_0^k {\frac{{dx}}{{2 + 8{x^2}}}} = \frac{\pi }{{16}}\,,$ then $k = $

Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation, $\frac{2+\sin x}{y+1} \cdot \frac{d y}{d x}=-\cos x, y>0, y(0)=1,$ If $y(\pi)=a$ and $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{x}=\pi$ is $b$, then the ordered pair $(a, b)$ is equal to :

The minimum value of the function $y = 2{x^3} - 21{x^2} + 36x - 20$ is

If $A$ and $B$ are two matrices such that $AB = B$ and $BA = A, A^2 + B^2$ is equal to: