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A line making angles $45^{\circ}$ and $60^{\circ}$ with the positive directions of the axis of $X$ and $Y$, makes with the positive direction of $Z$-axis, an angle of
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The correct answer is:
$60^{\circ}$
Let the line makes an angle $x$ with positive direction of $\angle$-axis. Then, we have
$$
\begin{array}{ll}
& \cos ^{2} 45^{\circ}+\cos ^{2} 60^{\circ}+\cos ^{2} x=1 \\
\Rightarrow & \left(\frac{1}{\sqrt{2}}\right)^{2}+\left(\frac{1}{2}\right)^{2}+\cos ^{2} x=1 \\
\Rightarrow & \cos ^{2} x=1-\frac{1}{2}-\frac{1}{4} \\
\Rightarrow & \cos ^{2} x=\frac{1}{4} \Rightarrow \cos x=\frac{1}{2} \\
& \text { Neglecting } \cos x=\frac{-1}{2}, \text { because line is in I-octant) } \\
\Rightarrow \quad & x=60^{\circ}
\end{array}
$$
$$
\begin{array}{ll}
& \cos ^{2} 45^{\circ}+\cos ^{2} 60^{\circ}+\cos ^{2} x=1 \\
\Rightarrow & \left(\frac{1}{\sqrt{2}}\right)^{2}+\left(\frac{1}{2}\right)^{2}+\cos ^{2} x=1 \\
\Rightarrow & \cos ^{2} x=1-\frac{1}{2}-\frac{1}{4} \\
\Rightarrow & \cos ^{2} x=\frac{1}{4} \Rightarrow \cos x=\frac{1}{2} \\
& \text { Neglecting } \cos x=\frac{-1}{2}, \text { because line is in I-octant) } \\
\Rightarrow \quad & x=60^{\circ}
\end{array}
$$
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