The direction ratios of two lines are given byp,q,r  and  1qr,1rp,1pq. Ifis the angle between them such that 0≤α<π, then the value of  cosα+cos2α+cos3α+....+cospα⋅cosqα⋅cosrα=

The direction ratios of two lines are given by$〈p,q,r〉\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}〈\frac{1}{qr},\frac{1}{rp},\frac{1}{pq}〉$. Ifis the angle between them such that $0\le \alpha <\pi$, then the value of  $\mathrm{cos}\alpha +{\mathrm{cos}}^{2}\alpha +{\mathrm{cos}}^{3}\alpha +....+{\mathrm{cos}}^{p}\alpha \cdot {\mathrm{cos}}^{q}\alpha \cdot {\mathrm{cos}}^{r}\alpha =$

1. A

$pqr$

2. B

$p+q+r$

3. C

$p-q-r$

4. D

None

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Solution:

The direction rations of two lines are given as $〈p,q,r〉\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}〈\frac{1}{qr},\frac{1}{rp},\frac{1}{pq}〉$

Multiply the second set of direction rations with $pqr$, then the direction ratios are$〈p,q,r〉$

It implies that both lines are parallel.

Hence$\alpha =0°$

Therefore, the required value is  $1+1+1+....+\left(p+q+r\text{\hspace{0.17em}times}\right)=p+q+r$

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