JEE Mathematics 2023 Question with Solution

Given:

• $\mathbf{a} = \lambda \hat{i} + \mu \hat{j} + 4\hat{k}$
• $\mathbf{b} = -2\hat{i} + 4\hat{j} - 2\hat{k}$
• $\mathbf{c} = 2\hat{i} + 3\hat{j} + \hat{k}$
• Projection of $\mathbf{a}$ on $\mathbf{b}$ is $\sqrt{54}$
• The vectors are coplanar

Find: The sum of all possible values of $\lambda + \mu$.

From the solution, coplanarity gives

and the projection formula gives

The dot product shown is

so

Also,

Substituting into the projection relation,

the solution then simplifies this using

and states

Using the coplanarity condition together with this equation, the solution concludes that

Therefore, the source solution concludes that the required value corresponds to option A.

However, the solution's is internally inconsistent: the question block marks $(3)\;24$, while the solution says A, and the worked conclusion states $\lambda + \mu = 24$. Since the worked conclusion gives the numerical result $24$, the defensible option from the listed choices is C.

Given: The source solution explicitly concludes $\lambda + \mu = 24$.

Find: Which option matches that value.

Comparing with the options:

• A = $0$
• B = $6$
• C = $24$
• D = $18$

So the numerical conclusion in the working matches option C.

This shows the header statement "The Correct Option is A" is inconsistent with the actual worked result. Therefore, using the worked conclusion as the authority, the correct option should be C.