The foot of perpendicular from the point P (1, 2, -3) to the line (x+1)/2 = (y-3)/(-2) = z/(-1) is Option 1: (1, 1, 1) Option 2: (1, -1, 1) Option 3: (1, 1, -1) Option 4: (-1, 1, 1)

Answer: (1, 1, -1). Solution: General point on line: Q = (2t-1, -2t+3, -t). PQ = (2t-2, -2t+1, -t+3). PQ perpendicular to direction (2, -2, -1): 2(2t-2) - 2(-2t+1) - 1(-t+3) = 9t - 9 = 0, so t = 1. Q = (1, 1, -1).

Geometry

3D Geometry

Medium

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

$(1, 1, 1)$

$(1, -1, 1)$

$(1, 1, -1)$

$(-1, 1, 1)$

✅ Correct Option: 3

General point on line: $Q = (2t-1, -2t+3, -t)$ .

$\vec{PQ} = (2t-2, -2t+1, -t+3)$ .

$\vec{PQ}$ perpendicular to direction $(2, -2, -1)$ :

$2(2t-2) - 2(-2t+1) - 1(-t+3) = 9t - 9 = 0$ , so $t = 1$ .

$Q = (1, 1, -1)$ .

2025: 16 May Shift 1

If a line makes angles α, β and γ with positive x-axis, y-axis and z-axis respectively, then the value of

$sin^2\frac{α}{2}cos^2\frac{α}{2} + sin^2\frac{β}{2}cos^2\frac{β}{2} + sin^2\frac{γ}{2}cos^2\frac{γ}{2}$ is

2026: 30 May Shift 1

Given a line $L:\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ , then

Match the LIST-I with LIST-II

LIST-I	LIST-II
A. The foot of perpendicular from point (1, 6, 3) on line L	I. $2\sqrt{13}$
B. Perpendicular distance from point (1, 6, 3) to line L	II. (1, 0, 7)
C. Image of point (1, 6, 3) with respect to line L	III. (1, 3, 5)
D. Distance between points (1, 6, 3) and (1, 0, 7)	IV. $\sqrt{13}$

Choose the correct answer from the options given below:

2026: 22 May Shift 1

If the lines $\frac{1-x}{2}=\frac{2y}{p}=\frac{z-1}{1}$ and $\frac{3-2x}{p}=\frac{y-1}{2}=\frac{z}{1}$ are parallel then value of 'p' is

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: 7 Aug Shift 2

The distance of the point $(2, 3, -5)$ from the plane $x + 2y - 2z = 9$ is :

2022: 23 Aug Shift 1

The angle between the pairs of lines

$\vec{r} = (3\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(\hat{i} + 2\hat{j} + 2\hat{k})$

$\vec{r} = (5\hat{i} - 2\hat{j}) + \mu(3\hat{i} + 2\hat{j} + 6\hat{k})$ is :

The foot of perpendicular from the point P (1, 2, -3) to the line x+12=y−3−2=z−1\frac{x+1}{2} = \frac{y-3}{-2} = \frac{z}{-1}2x+1​=−2y−3​=−1z​ is

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

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