Question
Find the acute angle between the lines whose direction ratios are 5, 12, -13 and 3, -4, 5
$\cos \theta=\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right|$
$=\left|\frac{15-48-65}{\sqrt{25+144+169} \sqrt{9+16+25}}\right|$
$=\left|-\frac{98}{13 \sqrt{2} \times 5 \sqrt{2}}\right|$
$=\left|-\frac{98}{13 \times 5 \times 2}\right|$
$=\frac{49}{65}$
$\theta=\cos ^{-1}\left(\frac{49}{65}\right)$
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Following is the probability distribution of a r.v.X.
| X | – 3 | – 2 | –1 | 0 | 1 | 2 | 3 |
| P(X = x) | 0.05 | 0.1 | 0.15 | 0.20 | 0.25 | 0.15 | 0.1 |
Find the probability that X is positive.
$\log \left[\cos \left(x^3-5\right)\right]$