JEE 2023 — Mathematics PYQ

Q: How do I solve JEE 2023 Mathematics questions?

Practice JEE 2023 Mathematics previous year questions with step-by-step solutions on Mathem Solvex. Each question includes a detailed explanation and video solution to help you master the concept quickly.

Q: What is the difficulty level of JEE Mathematics questions?

JEE Mathematics questions range from moderate to advanced difficulty. Consistent practice with official previous year papers and topic-wise drills on Mathem Solvex is the most effective preparation strategy.

Q: Are JEE previous year papers available for free?

Yes. Mathem Solvex provides free access to JEE previous year question papers and topic-wise PDFs. You can download them or attempt them as timed live mock tests directly on the platform.

Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

If the vectors $\vec{a} = \lambda\hat{i} + \mu\hat{j} + 4\hat{k}$ , $\vec{b} = -2\hat{i} + 4\hat{j} - 2\hat{k}$ and $\vec{c} = 2\hat{i} + 3\hat{j} + \hat{k}$ are coplanar and the projection of $\vec{a}$ on $\vec{b}$ is $\sqrt{54}$ units, then the sum of all possible values of $\lambda + \mu$ is:

Choose the correct answer:

Explanation

Solution

Coplanar condition: $[\vec{a} \ \vec{b} \ \vec{c}] = 0$ .

$\begin{vmatrix} \lambda & \mu & 4 \\ -2 & 4 & -2 \\ 2 & 3 & 1 \end{vmatrix} = 0$

$\lambda(4 + 6) - \mu(-2 + 4) + 4(-6 - 8) = 0$

$10\lambda - 2\mu - 56 = 0 \implies 5\lambda - \mu = 28 \quad \dots(1)$
Projection condition: $\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} = \sqrt{54}$ .

$|\vec{b}| = \sqrt{(-2)^2 + 4^2 + (-2)^2} = \sqrt{4 + 16 + 4} = \sqrt{24}$ .

$\vec{a} \cdot \vec{b} = -2\lambda + 4\mu - 8$ .

$\frac{-2\lambda + 4\mu - 8}{\sqrt{24}} = \pm\sqrt{54}$ (Projection can be magnitude, so consider $\pm$ ).

$-2\lambda + 4\mu - 8 = \pm \sqrt{54 \times 24} = \pm \sqrt{1296} = \pm 36$ .

$-\lambda + 2\mu - 4 = \pm 18$ .
- Case I: $-\lambda + 2\mu = 22 \quad \dots(2)$
- Case II: $-\lambda + 2\mu = -14 \quad \dots(3)$
Solving for $\lambda + \mu$ :
- From (1) and (2): $\lambda = 6, \mu = 2 \implies \lambda + \mu = 8$ .
- From (1) and (3): $\lambda = 4.66, \mu = -4.66$ (approx) $\implies$ Solving exactly: $10\lambda - 2\mu = 56$ and $-\lambda + 2\mu = -14 \implies 9\lambda = 42 \implies \lambda = 14/3, \mu = -14/3 \implies \lambda + \mu = 10$ .

Sum of all possible values of $\lambda + \mu = 24$ .

Correct Option: (2)

Choose the correct answer:

Explore More