Geometry

3D Geometry

Hard

core

The angle between the two lines whose direction cosines are given by the equations $4l+m+n=0$ and $2mn+24nl+4lm=0$ , is

$\pi/2$

$\cos^{-1}\left(\dfrac{19}{3\sqrt{41}}\right)$

$\cos^{-1}\left(\dfrac{12}{7\sqrt{10}}\right)$

✅ Correct Option: 3

From the first equation $n = -4l - m$ . Substituting into the second and simplifying gives $m^2 + 14lm + 48l^2 = 0$ , i.e. $(m+6l)(m+8l)=0$ .

If $m=-6l$ then $n=2l$ , giving direction ratios $(1,-6,2)$ . If $m=-8l$ then $n=4l$ , giving $(1,-8,4)$ .

$\cos\theta = \dfrac{1+48+8}{\sqrt{41}\sqrt{81}} = \dfrac{19}{3\sqrt{41}}$ .

2022: 17 Aug Shift 2

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

2022: 7 Aug Shift 2

Equation of y-axis in space, in vector form is :

2026: 29 May Shift 1

A line in space is uniquely determined if

(A) It passes through a given point and has a given direction.

(B) It passes through two given points.

(D) It has a given direction.

Choose the correct answer from the options given below:

2025: 14 May Shift 2

Consider a line $\vec{r} = (\hat{i} + 4\hat{j}) + \lambda(2\hat{i} - 2\hat{j} + 3\hat{k})$ , then which of the following statements are correct?

(A) it passes through point (9, -4, 12)

(B) it passes through point (1, 4, -1)

(D) its Cartesian equation is $\frac{x - 1}{2} = \frac{y - 4}{-2} = \frac{z}{3}$

Choose the correct answer from the options given below:

2023: 23 May Shift 3

Match List I with List II

LIST I	LIST II
A. $lx + my + nz = d$ is	I. Equation of plane passing through a given point and normal to given vector
B. $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ is	II. Equation of plane in normal form
C. $(\vec{r} - \vec{a}) \cdot \vec{n} = 0$	III. Plane passing through the intersection of two planes
D. $(a_1x + b_1y + c_1z + d_1) + \lambda(a_2x + b_2y + c_2z + d_2) = 0$	IV. Intercept from of plane

Choose the correct answer from the options given below:

2025: 14 May Shift 2

The angle between the lines

$l_1: \frac{x + 1}{1} = \frac{2 - y}{2} = \frac{z - 1}{1}$ and $l_2: \frac{x - 1}{4} = \frac{2y - 4}{6} = \frac{z - 1}{2}$ is

The angle between the two lines whose direction cosines are given by the equations 4l+m+n=04l+m+n=04l+m+n=0 and 2mn+24nl+4lm=02mn+24nl+4lm=02mn+24nl+4lm=0, is

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The angle between the two lines whose direction cosines are given by the equations $4l+m+n=0$ and $2mn+24nl+4lm=0$ , is