CUET Mathematics 2024 - Which of the following cannot be the direction ratios of the straight line x-3/2=2-y/3=z+4/-1 ? | PYQs + Solutions

CUET Mathematics 2024

Algebra

Vector Algebra

Easy

Which of the following cannot be the direction ratios of the straight line $\frac{x-3}{2}=\frac{2-y}{3}=\frac{z+4}{-1}$ ?

2,-3,-1

-2,3,1

2,3,-1

6,-9,-3

✅ Correct Option: 3

Direction ratios are the components of a vector that gives the direction of a line in 3D space. For a line in symmetric form like ours, the denominators directly give us the direction ratios.

From our equation

\dfrac{x-3}{2} = \dfrac{2-y}{3} = \dfrac{z+4}{-1}

, let's rewrite the middle term to match the standard form:

\dfrac{2-y}{3} = \dfrac{-1(y-2)}{3} = \dfrac{y-2}{-3}

So our line can be written as:

\dfrac{x-3}{2} = \dfrac{y-2}{-3} = \dfrac{z+4}{-1}

This means the direction ratios are proportional to

(2, -3, -1)

A key concept to understand: Any scalar multiple of these ratios represents the same direction. This is because direction ratios define the orientation of the line, not its magnitude.

Think of it like this: If you have a vector pointing northeast, doubling or tripling its length doesn't change which way it points - it's still northeast.

Now let's check each option:

Option 1:

(2, -3, -1)

\newline

These are exactly our identified direction ratios, so they're valid.

Option 2:

(-2, 3, 1)

\newline

These are

(-1) \times (2, -3, -1)

, so they're valid direction ratios.

Option 3:

(2, 3, -1)

\newline

These numbers aren't proportional to our direction ratios.

Option 4:

(6, -9, -3)

\newline

These are

3 \times (2, -3, -1)

, so they're valid direction ratios.

Therefore, the option that cannot be the direction ratios of the given line is

(2, 3, -1)

Additional Learning: the direction ratios tell us how much we move in each coordinate direction as we travel along the line. If we move 2 units in the x-direction, we must move -3 units (not 3 units) in the y-direction to stay on our line. The middle value having the wrong sign means it points in the opposite direction from what our line requires.

CUET Mathematics 2024 PYQs

Which of the following cannot be the direction ratios of the straight line x−32=2−y3=z+4−1\frac{x-3}{2}=\frac{2-y}{3}=\frac{z+4}{-1}2x−3​=32−y​=−1z+4​ ?

Keyboard Shortcuts

Which of the following cannot be the direction ratios of the straight line $\frac{x-3}{2}=\frac{2-y}{3}=\frac{z+4}{-1}$ ?