MCQ
The angle between the lines whose direction cosines are connected by the relations $l + m + n = 0$ and $2lm + 2nl - mn = 0$, is
- A$\frac{\pi }{3}$
- ✓$\frac{{2\pi }}{3}$
- C$\pi $
- DNone of these
When $2l + m = 0,$ then $\frac{l}{1} = \frac{m}{{ - 2}} = \frac{n}{1}$
When $l - m = 0,$ then $\frac{l}{1} = \frac{m}{1} = \frac{n}{{ - 2}}$
$\therefore $ Direction ratios are $1, -2, 1$ and $1, 1, -2.$
$\cos \theta = \frac{{\sum {a_1}{a_2}}}{{\sqrt {(\sum a_1^2)\,} .\sqrt {(\sum a_2^2)\,} }} = - \frac{1}{2}\,$
$ \Rightarrow \,\,\theta = {120^o} = \frac{{2\pi }}{3}.$
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$f(x)=\left\{\begin{array}{clr}\left|2 x^{2}-3 x-7\right| \, \text { if } x \leq-1 \\ {\left[4 x^{2}-1\right]} \text { if } -1 < x < 1 \\ |x+1|+|x-2| \text { if } x \geq 1\end{array}\right.$
$[t]$ denotes the greatest integer $\leq t$, is discontinuous is
$f_n(x)=\sum_{j=1}^n \tan ^{-1}\left(\frac{1}{1+(x+j)(x+j-1)}\right) \text { for all } x \in(0, \infty)$
(Here, the inverse trigonometric function $\tan ^{-1} x$ assumes values in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ )
Then, which of the following statement(s) is (are) TRUE?
$(A)$ $\sum_{ j =1}^5 \tan ^2\left( f _{ j }(0)\right)=55$
$(B)$ $\sum_{ j =1}^{10}\left(1+ f _{ j }^{\prime}(0)\right) \sec ^2\left( f _{ j }(0)\right)=10$
$(C)$ For any fixed positive integer $n$, $\lim _{x \rightarrow \infty} \tan \left(f_n(x)\right)=\frac{1}{n}$
$(D)$ For any fixed positive integer $n, \lim _{x \rightarrow \infty} \sec ^2\left(f_n(x)\right)=1$