CUET PG 2022 — Mathematics PYQ

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Question

CUET PG 2022 — Mathematics PYQ

CUET PG | Mathematics | 2022

Let $a = \hat{i} - \hat{j}$ and $b = \hat{i} + \hat{j} + \hat{k}$ and $c$ be a vector such $(a \times c) + b = 0$ and $a \cdot c = 4$ , then $∣ c ∣^{2}$ equal to:

Choose the correct answer:

Explanation

1. Calculate the magnitudes of $\vec{a}$ and $\vec{b}$ :

$|\vec{a}|^2 = (1)^2 + (-1)^2 + (0)^2 = 2$
$|\vec{b}|^2 = (1)^2 + (1)^2 + (1)^2 = 3$

2. Use the Lagrange Identity:

The relation between the dot product and cross product of two vectors $\vec{a}$ and $\vec{c}$ is:

|\vec{a} \times \vec{c}|^2 + (\vec{a} \cdot \vec{c})^2 = |\vec{a}|^2 |\vec{c}|^2

3. Substitute the known values into the identity:

From the given equations:

$\vec{a} \times \vec{c} = -\vec{b} \implies |\vec{a} \times \vec{c}|^2 = |-\vec{b}|^2 = |\vec{b}|^2 = 3$
$(\vec{a} \cdot \vec{c})^2 = (4)^2 = 16$
$|\vec{a}|^2 = 2$

Now, substitute these into the identity formula:

3 + 16 = 2 \cdot |\vec{c}|^2

4. Solve for $|\vec{c}|^2$ :

19 = 2 |\vec{c}|^2

|\vec{c}|^2 = \frac{19}{2}