NEET Test Series from KOTA - 10 Papers In MS WORD
WhatsApp Here
Three Dimensional Geometry
121137
A line makes equal angles with the diagonals of a cube. What is the sine of the angle?
1 \(\sqrt{\frac{2}{3}}\)
2 \(\sqrt{\frac{1}{3}}\)
3 \(\sqrt{\frac{1}{2}}\)
4 None of the above
Explanation:
A Let \(l, \mathrm{~m}, \mathrm{n}\) be the \(\mathrm{DC}\) 's of the line. \(\therefore\) DR's of diagonal of the cube are-
\((1,1,1),(-1,11),(1,1,-1)\) and \((1,-1,1)\)
\(\therefore \cos \theta=\frac{ \vertl+\mathrm{m}+\mathrm{n} \vert}{\sqrt{l^2+\mathrm{m}^2+\mathrm{n}^2}}=\left \vert \pm \frac{1}{\sqrt{3}}\right \vert\)
\(\therefore \sin \theta=\sqrt{1-\cos ^2 \theta}=\sqrt{1-\frac{1}{3}}=\sqrt{\frac{2}{3}}\)
SCRA-2012
Three Dimensional Geometry
121139
If a line makes angles \(\alpha, \beta, \gamma\) with \(\mathbf{x}\)-axis, \(\mathbf{y}\)-axis and z-axis respectively, \(\sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma\)
121137
A line makes equal angles with the diagonals of a cube. What is the sine of the angle?
1 \(\sqrt{\frac{2}{3}}\)
2 \(\sqrt{\frac{1}{3}}\)
3 \(\sqrt{\frac{1}{2}}\)
4 None of the above
Explanation:
A Let \(l, \mathrm{~m}, \mathrm{n}\) be the \(\mathrm{DC}\) 's of the line. \(\therefore\) DR's of diagonal of the cube are-
\((1,1,1),(-1,11),(1,1,-1)\) and \((1,-1,1)\)
\(\therefore \cos \theta=\frac{ \vertl+\mathrm{m}+\mathrm{n} \vert}{\sqrt{l^2+\mathrm{m}^2+\mathrm{n}^2}}=\left \vert \pm \frac{1}{\sqrt{3}}\right \vert\)
\(\therefore \sin \theta=\sqrt{1-\cos ^2 \theta}=\sqrt{1-\frac{1}{3}}=\sqrt{\frac{2}{3}}\)
SCRA-2012
Three Dimensional Geometry
121139
If a line makes angles \(\alpha, \beta, \gamma\) with \(\mathbf{x}\)-axis, \(\mathbf{y}\)-axis and z-axis respectively, \(\sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma\)
121137
A line makes equal angles with the diagonals of a cube. What is the sine of the angle?
1 \(\sqrt{\frac{2}{3}}\)
2 \(\sqrt{\frac{1}{3}}\)
3 \(\sqrt{\frac{1}{2}}\)
4 None of the above
Explanation:
A Let \(l, \mathrm{~m}, \mathrm{n}\) be the \(\mathrm{DC}\) 's of the line. \(\therefore\) DR's of diagonal of the cube are-
\((1,1,1),(-1,11),(1,1,-1)\) and \((1,-1,1)\)
\(\therefore \cos \theta=\frac{ \vertl+\mathrm{m}+\mathrm{n} \vert}{\sqrt{l^2+\mathrm{m}^2+\mathrm{n}^2}}=\left \vert \pm \frac{1}{\sqrt{3}}\right \vert\)
\(\therefore \sin \theta=\sqrt{1-\cos ^2 \theta}=\sqrt{1-\frac{1}{3}}=\sqrt{\frac{2}{3}}\)
SCRA-2012
Three Dimensional Geometry
121139
If a line makes angles \(\alpha, \beta, \gamma\) with \(\mathbf{x}\)-axis, \(\mathbf{y}\)-axis and z-axis respectively, \(\sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma\)
121137
A line makes equal angles with the diagonals of a cube. What is the sine of the angle?
1 \(\sqrt{\frac{2}{3}}\)
2 \(\sqrt{\frac{1}{3}}\)
3 \(\sqrt{\frac{1}{2}}\)
4 None of the above
Explanation:
A Let \(l, \mathrm{~m}, \mathrm{n}\) be the \(\mathrm{DC}\) 's of the line. \(\therefore\) DR's of diagonal of the cube are-
\((1,1,1),(-1,11),(1,1,-1)\) and \((1,-1,1)\)
\(\therefore \cos \theta=\frac{ \vertl+\mathrm{m}+\mathrm{n} \vert}{\sqrt{l^2+\mathrm{m}^2+\mathrm{n}^2}}=\left \vert \pm \frac{1}{\sqrt{3}}\right \vert\)
\(\therefore \sin \theta=\sqrt{1-\cos ^2 \theta}=\sqrt{1-\frac{1}{3}}=\sqrt{\frac{2}{3}}\)
SCRA-2012
Three Dimensional Geometry
121139
If a line makes angles \(\alpha, \beta, \gamma\) with \(\mathbf{x}\)-axis, \(\mathbf{y}\)-axis and z-axis respectively, \(\sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma\)
