If, and are angle of inclinations of a line with x, y and z axes respectively, then the value of 2( 2 + 2 + 2) is Option 1: 1 Option 2: -1 Option 3: 2 Option 4: -2

Answer: -2. Solution: When a line is drawn in 3D space, it makes angles with all three axes: - = angle with x-axis - = angle with y-axis - = angle with z-axis The direction cosines satisfy: ^2 + ^2 + ^2 = 1 Using the double angle formula 2 = 2^2 - 1: 2 = 2^2 - 1 2 = 2^2 - 1 2 = 2^2 - 1 Adding these expressions: 2 + 2 + 2 = (2^2 - 1) + (2^2 - 1) + (2^2 - 1) = 2^2 + 2^2 + 2^2 - 3 = 2(^2 + ^2 + ^2) - 3 Substituting ^2 + ^2 + ^2 = 1: = 2(1) - 3 = -1 Therefore: 2( 2 + 2 + 2) = 2 x (-1) = -2 The answer is -2.

Geometry

3D Geometry

Medium

core

If $\alpha, \beta$ and $\gamma$ are angle of inclinations of a line with x, y and z axes respectively, then the value of $2(\cos 2\alpha + \cos 2\beta + \cos 2\gamma)$ is

-1

-2

✅ Correct Option: 4

When a line is drawn in 3D space, it makes angles with all three axes:

$\alpha$ = angle with x-axis
$\beta$ = angle with y-axis
$\gamma$ = angle with z-axis

The direction cosines satisfy:

$\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$

Using the double angle formula $\cos 2\theta = 2\cos^2\theta - 1$ :

$\cos 2\alpha = 2\cos^2\alpha - 1$

$\cos 2\beta = 2\cos^2\beta - 1$

$\cos 2\gamma = 2\cos^2\gamma - 1$

Adding these expressions:

$\cos 2\alpha + \cos 2\beta + \cos 2\gamma$

$= (2\cos^2\alpha - 1) + (2\cos^2\beta - 1) + (2\cos^2\gamma - 1)$

$= 2\cos^2\alpha + 2\cos^2\beta + 2\cos^2\gamma - 3$

$= 2(\cos^2\alpha + \cos^2\beta + \cos^2\gamma) - 3$

Substituting $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$ :

$= 2(1) - 3$

$= -1$

Therefore:

$2(\cos 2\alpha + \cos 2\beta + \cos 2\gamma)$

$= 2 \times (-1)$

$= -2$

The answer is $-2$ .

2022: 6 Aug Shift 2

The foot of perpendicular from the point P (1, 2, -3) to the line $\frac{x+1}{2} = \frac{y-3}{-2} = \frac{z}{-1}$ is

2024: 16 May Shift 1

The direction cosines of the line which is perpendicular to the lines with direction ratios $1,-2,-2$ and $0,2,1$ are :

2026: 22 May Shift 1

Match List-I with List-II

List-I (Lines)	List-II (Direction Ratios)
(A) $2x+3=y+1=z-1$	(I) -1, 2, 0
(B) $\frac{1-2x}{2}=\frac{y}{2}, z=2$	(II) 1, 1, 1
(C) $x=\frac{3y-1}{3}=z+1$	(III) 2, 1, 1
(D) $x=2y+3, z=y+1$	(IV) 1, 2, 2

Choose the correct answer from the options given below:

2022: 16 July Shift 2

The two lines given by $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \mu(\hat{i} - \hat{j} + \hat{k})$ and $\vec{r} = (2\hat{i} - \hat{j} - \hat{k}) + \mu(-\hat{i} + \hat{j} - \hat{k})$

A. are perpendicular

B. are parallel.

C. have shortest distance 0.

D. have shortest distance $\sqrt{26}$ .

E. have shortest distance $\sqrt{78}$ .

Which of the above statements are true?

Choose the correct answer from the options given below:

2026: 22 May Shift 2

The shortest distance (in units) between the lines $L_1:\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+4}{6}$ and $L_2:\frac{x-3}{2}=\frac{3-y}{-3}=\frac{z+5}{6}$ is:

2025: 21 May Shift 2

The angle between the line $2x = 3y = z$ and $x$ - axis is:

If α,β\alpha, \betaα,β and γ\gammaγ are angle of inclinations of a line with x, y and z axes respectively, then the value of 2(cos⁡2α+cos⁡2β+cos⁡2γ)2(\cos 2\alpha + \cos 2\beta + \cos 2\gamma)2(cos2α+cos2β+cos2γ) is

Solution

Related questions:

Keyboard Shortcuts

CUET Mathematics Past Year Questions (Topic-Wise)

If $\alpha, \beta$ and $\gamma$ are angle of inclinations of a line with x, y and z axes respectively, then the value of $2(\cos 2\alpha + \cos 2\beta + \cos 2\gamma)$ is