Question

# Statement-1 : If a line makes acute angles α,β,γ,δ with diagonals of a cube, then cos2α+cos2β+cos2γ+cos2δ=43Statement 2 : If a line makes equal angle (acute) with the axes, then its direction cosine are 1√3,1√2 and 1√3

A
Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
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B
Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
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C
Statement-1 is True, Statement-2 is False
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D
Statement-1 is False, Statement-2 is True
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Solution

## The correct option is A Statement-1 is True, Statement-2 is False Let l,m,n be the direction ratio of a line making angles ,,, with four diagonals of a cube then as four diagonals have direction ratio,(a,a,a);(a,a,−a);(a,−a,a,) ;(−a,a,a)(where a is the side of the cube) Now, cosα=∣∣∣al+am+an√a2+a2+a2√l2+m2+n2∣∣∣=∣∣∣l+m+n√3√l2+m2+n2∣∣∣ cosβ=∣∣∣al+am−an√a2+a2+a2√l2+m2+n2∣∣∣=∣∣∣l+m−n√3√l2+m2+n2∣∣∣ cosγ=∣∣∣al−am+an√a2+a2+a2√l2+m2+n2∣∣∣=∣∣∣l−m+n√3√l2+m2+n2∣∣∣ cosδ=∣∣∣al+am+an√a2+a2+a2√l2+m2+n2∣∣∣=∣∣∣−l+m+n√3√l2+m2+n2∣∣∣ and cos2α+cos2β+cos2γ+cos2δ =13×(4l2+4m2+4n2)l2+m2+n2=43 So statement 1 is true and statement-2 is not true . Hence, option 'C' is correct.

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