If the points (-1, -1, 2), (2, m, 5) and (3, 11, 6) are collinear, then m equals Option 1: 4 Option 2: 8 Option 3: 0 Option 4: -2

Answer: 8. Solution: Given points: - A = (-1, -1, 2) - B = (2, m, 5) - C = (3, 11, 6) For three points to be collinear, vectors AB and AC must be parallel. Vector AB = B - A AB = (2 - (-1), m - (-1), 5 - 2) AB = (3, m+1, 3) Vector AC = C - A AC = (3 - (-1), 11 - (-1), 6 - 2) AC = (4, 12, 4) For collinearity, AB = k x AC for some constant k. (3, m+1, 3) = k(4, 12, 4) From the first component: 3 = 4k k = 3/4 From the third component: 3 = 4k k = 3/4 Both components give the same value of k. Using the second component: m + 1 = 12k m + 1 = 12 x 3/4 m + 1 = 9 m = 8 Therefore, m = 8.

Geometry

3D Geometry

Medium

core

If the points (-1, -1, 2), (2, m, 5) and (3, 11, 6) are collinear, then m equals

-2

✅ Correct Option: 2

Given points:

A = (-1, -1, 2)
B = (2, m, 5)
C = (3, 11, 6)

For three points to be collinear, vectors AB and AC must be parallel.

Vector AB = B - A

$AB = (2 - (-1), m - (-1), 5 - 2)$

$AB = (3, m+1, 3)$

Vector AC = C - A

$AC = (3 - (-1), 11 - (-1), 6 - 2)$

$AC = (4, 12, 4)$

For collinearity, $AB = k \times AC$ for some constant $k$ .

$(3, m+1, 3) = k(4, 12, 4)$

From the first component:

$3 = 4k$

$k = \dfrac{3}{4}$

From the third component:

$3 = 4k$

$k = \dfrac{3}{4}$

Both components give the same value of $k$ .

Using the second component:

$m + 1 = 12k$

$m + 1 = 12 \times \dfrac{3}{4}$

$m + 1 = 9$

$m = 8$

Therefore, $m = 8$ .

2026: 26 May Shift 2

The value of $\lambda$ so that the lines $\dfrac{1-x}{3} = \dfrac{7y-14}{2\lambda} = \dfrac{z-3}{2}$ and $\dfrac{7-7x}{3\lambda} = \dfrac{y-5}{1} = \dfrac{6-z}{5}$

are at right angles is:

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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:

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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 :

If the points (-1, -1, 2), (2, m, 5) and (3, 11, 6) are collinear, then m equals

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