JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $\vec{a} = 4\hat{i} + 3\hat{j}$ and $\vec{b} = 3\hat{i} - 4\hat{j} + 5\hat{k}$ . If $\vec{c}$ is a vector such that $\vec{c} \cdot (\vec{a} \times \vec{b}) + 25 = 0$ , $\vec{c} \cdot (\hat{i} + \hat{j} + \hat{k}) = 4$ , and projection of $\vec{c}$ on $\vec{a}$ is $1$ , then the projection of $\vec{c}$ on $\vec{b}$ equals:

Choose the correct answer:

Explanation

Solution

Maana ki vector $\vec{c} = x\hat{i} + y\hat{j} + z\hat{k}$ hai.

1. $\vec{a} \times \vec{b}$ nikalne par:

\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 4 & 3 & 0 \\ 3 & -4 & 5 \end{vmatrix} = \hat{i}(15) - \hat{j}(20) + \hat{k}(-16 - 9) = 15\hat{i} - 20\hat{j} - 25\hat{k}

Pehli condition: $\vec{c} \cdot (\vec{a} \times \vec{b}) = -25$

(x\hat{i} + y\hat{j} + z\hat{k}) \cdot (15\hat{i} - 20\hat{j} - 25\hat{k}) = -25

15x - 20y - 25z = -25 \implies 3x - 4y - 5z = -5 \quad \dots \text{(Eq. 1)}

2. Dusri condition se:

$\vec{c} \cdot (\hat{i} + \hat{j} + \hat{k}) = 4$

x + y + z = 4 \quad \dots \text{(Eq. 2)}

3. Teesri condition (Projection on $\vec{a}$ ) se:

Projection of $\vec{c}$ on $\vec{a} = \frac{\vec{c} \cdot \vec{a}}{|\vec{a}|} = 1$

\frac{4x + 3y}{\sqrt{4^2 + 3^2}} = 1 \implies \frac{4x + 3y}{5} = 1 \implies 4x + 3y = 5 \quad \dots \text{(Eq. 3)}

4. Equations solve karne par:

(Eq. 2) se $z = 4 - x - y$ , ise (Eq. 1) mein rakhein:

3x - 4y - 5(4 - x - y) = -5 \implies 3x - 4y - 20 + 5x + 5y = -5

8x + y = 15 \quad \dots \text{(Eq. 4)}

Ab (Eq. 3) aur (Eq. 4) ko solve karte hain:

(Eq. 4) se $y = 15 - 8x$ , ise (Eq. 3) mein rakhein:

4x + 3(15 - 8x) = 5 \implies 4x + 45 - 24x = 5 \implies -20x = -40 \implies x = 2

y = 15 - 8(2) = 15 - 16 = -1

z = 4 - 2 - (-1) = 3

Toh, $\vec{c} = 2\hat{i} - \hat{j} + 3\hat{k}$ .

5. Final Step: Projection of $\vec{c}$ on $\vec{b}$ :

\text{Projection} = \frac{\vec{c} \cdot \vec{b}}{|\vec{b}|} = \frac{(2\hat{i} - \hat{j} + 3\hat{k}) \cdot (3\hat{i} - 4\hat{j} + 5\hat{k})}{\sqrt{3^2 + (-4)^2 + 5^2}}

= \frac{6 + 4 + 15}{\sqrt{9 + 16 + 25}} = \frac{25}{\sqrt{50}} = \frac{25}{5\sqrt{2}} = \frac{5}{\sqrt{2}}

Correct Option: (2) $\frac{5}{\sqrt{2}}$

Choose the correct answer:

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