A line makes the angle with each of the x and z axes. If the angle which it makes with y-axis is such that ^2 = 3^2, then the value of ^2 is Option 1: 2/5 Option 2: 1/5 Option 3: 3/5 Option 4: 2/5

Answer: 3/5. Solution: Direction-cosine identity: ^2 + ^2 + ^2 = 1 with = = . So 2^2 + ^2 = 1, i.e., ^2 = 2^2. Given ^2 = 3^2 = 3(1-^2). Equate: 2^2 = 3 - 3^2, so ^2 = 3/5. Note: Options (1) and (4) appear identical in the paper; we have selected option (3) which matches the computed value.

Geometry

3D Geometry

Medium

A line makes the angle $\theta$ with each of the $x$ and $z$ axes. If the angle $\beta$ which it makes with $y$ -axis is such that $\sin^2\beta = 3\sin^2\theta$ , then the value of $\cos^2\theta$ is

$\frac{2}{5}$

$\frac{1}{5}$

$\frac{3}{5}$

$\frac{2}{5}$

✅ Correct Option: 3

Direction-cosine identity: $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$ with $\alpha = \gamma = \theta$ .

So $2\cos^2\theta + \cos^2\beta = 1$ , i.e., $\sin^2\beta = 2\cos^2\theta$ .

Given $\sin^2\beta = 3\sin^2\theta = 3(1-\cos^2\theta)$ .

Equate: $2\cos^2\theta = 3 - 3\cos^2\theta$ , so $\cos^2\theta = \frac{3}{5}$ .

Note: Options (1) and (4) appear identical in the paper; we have selected option (3) which matches the computed value.

2026: 14 May Shift 2

The shortest distance between the lines whose vector equations are $\vec r=(\hat i+2\hat j+3\hat k)+\lambda(2\hat i+3\hat j+4\hat k)$ and $\vec r=(2\hat i+4\hat j+5\hat k)+\mu(4\hat i+6\hat j+8\hat k)$ is:

2025: 3 June Shift 1

The Cartesian equation of the line passing through the point (1, 2, -1) and parallel to the line $5x - 25 = 14 - 7y = 35z$ is

2026: 22 May Shift 2

Match List-I with List-II

List-I	List-II
(A) The distance of a point $P(a,b,c)$ from x-axis	(I) $\sqrt{a^2+b^2+c^2}$
(B) The distance of a point $P(a,b,c)$ from y-axis	(II) $\sqrt{b^2+c^2}$
(C) The distance of a point $P(a,b,c)$ from Z-axis	(III) $\sqrt{a^2+c^2}$
(D) The distance of a point $P(a,b,c)$ from origin	(IV) $\sqrt{a^2+b^2}$

Choose the correct answer from the options given below:

2026: 18 May Shift 1

Equation of a line passing through (1, 2, -3) which is parallel to the line $\frac{x+2}{3} = \frac{y-3}{-4} = \frac{z+5}{5}$ is

2026: 18 May Shift 1

The cartesian equation of a line is $x = ay+b$ , $z = cy+d$ . It's direction ratios are

2025: 15 May Shift 2

If the lines $\frac{x-5}{7} = \frac{y+2}{-5} = \frac{z}{\lambda}$ and $\frac{x}{1} = \frac{y}{2\lambda} = \frac{z}{3}$ are perpendicular to each other, then $\lambda$ is equal to

A line makes the angle θ\thetaθ with each of the xxx and zzz axes. If the angle β\betaβ which it makes with yyy-axis is such that sin⁡2β=3sin⁡2θ\sin^2\beta = 3\sin^2\thetasin2β=3sin2θ, then the value of cos⁡2θ\cos^2\thetacos2θ is

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

A line makes the angle $\theta$ with each of the $x$ and $z$ axes. If the angle $\beta$ which it makes with $y$ -axis is such that $\sin^2\beta = 3\sin^2\theta$ , then the value of $\cos^2\theta$ is