Question
If $|\vec{\text{a}}|=\big|\vec{\text{b}}\big|,$ then $\big(\vec{\text{a}}+\vec{\text{b}}\big).\big(\vec{\text{a}}-\vec{\text{b}}\big)=$
- Positive
- Negetive
- 0
- None of these
Solution:
Given that
$|\vec{\text{a}}|=|\vec{\text{a}}|$
$\Rightarrow\big(\vec{\text{a}}+\vec{\text{b}}\big).\big(\vec{\text{a}}-\vec{\text{b}}\big)=|\vec{\text{a}}|^2-\big|\vec{\text{b}}\big|^2$
$|\vec{\text{a}}^2-|\vec{\text{a}}|^2$
$=0$
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$\left\{(\mathrm{x}, \mathrm{y}): \frac{\mathrm{a}}{\mathrm{x}^2} \leq \mathrm{y} \leq \frac{1}{\mathrm{x}}, 1 \leq \mathrm{x} \leq 2,0<\mathrm{a}<1\right\}$ is
$\left(\log _e 2\right)-\frac{1}{7}$ then the value of $7 a-3$ is equal to:
$\alpha=\sum_{ k =1}^{\infty} \sin ^{2 k}\left(\frac{\pi}{6}\right)$
Let $g:[0,1] \rightarrow R$ be the function defined by
$g( x )=2^{\alpha x }+2^{\alpha(1- x )}$
Then, which of the following statements is/are $TRUE$?
$(A)$ The minimum value of $g( x )$ is $2^{\frac{7}{6}}$
$(B)$ The maximum value of $g( x )$ is $1+2^{\frac{1}{3}}$
$(C)$ The function $g( x )$ attains its maximum at more than one point
$(D)$ The function $g( x )$ attains its minimum at more than one point