JEE Mathematics 2025 Question with Solution

Step-by-step solution

Standard Method

Given: A line passes through $P(4,1,0)$ and intersects $L_1$ at $A(\alpha,\beta,\gamma)$ and $L_2$ at $B(a,b,c)$.

Find: The value of

$$\begin{vmatrix} 1 & 0 & 1 \\ \alpha & \beta & \gamma \\ a & b & c \end{vmatrix}$$

Let the point $A$ on $L_1$ be represented by parameter $p$:

$$\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} = p$$

So,

$$A(2p + 1, 3p + 2, 4p + 3)$$

Let the point $B$ on $L_2$ be represented by parameter $q$:

$$x - 6 = y = -z + 4 = q$$

So,

$$B(q + 6, q, 4 - q)$$

Now the direction ratios of $\overrightarrow{PA}$ and $\overrightarrow{PB}$ are:

$$\overrightarrow{PA} = (2p - 3, 3p + 1, 4p + 3)$$ $$\overrightarrow{PB} = (q + 2, q - 1, 4 - q)$$

Since $P, A, B$ are collinear, these vectors are proportional:

$$\frac{2p - 3}{q + 2} = \frac{3p + 1}{q - 1} = \frac{4p + 3}{4 - q}$$

Equating pairwise and solving, the solution gives:

$$pq + 8p + 4q - 1 = 0$$ $$7pq - 16p + 4q - 7 = 0$$

Hence,

$$pq = -3, \quad p = -1, \quad q = 3$$

Therefore,

$$A(-1,-1,-1), \quad B(9,3,1)$$

Now compute the determinant:

$$\begin{vmatrix} 1 & 0 & 1 \\ -1 & -1 & -1 \\ 9 & 3 & 1 \end{vmatrix}$$

Expanding along the first row,

$$= 1 \cdot \begin{vmatrix} -1 & -1 \\ 3 & 1 \end{vmatrix} - 0 + 1 \cdot \begin{vmatrix} -1 & -1 \\ 9 & 3 \end{vmatrix}$$ $$= 1((-1)(1) - (-1)(3)) + 1((-1)(3) - (-1)(9))$$ $$= ( -1 + 3 ) + ( -3 + 9 )$$ $$= 2 + 6 = 8$$

Therefore, the determinant is $8$ and the correct option is A.

Common mistakes

• Students often parametrize $L_2$ incorrectly. From $x - 6 = y = -z + 4 = q$, the correct coordinates are $B(q+6, q, 4-q)$, not $B(q+6, q, q-4)$. Write each coordinate separately from the common parameter.

• A common mistake is to use the direction ratios of the given lines $L_1$ and $L_2$ directly. The required collinearity is for $\overrightarrow{PA}$ and $\overrightarrow{PB}$, because $P, A, B$ lie on the same line.

• While expanding the determinant, sign errors are frequent. In cofactor expansion along the first row, keep track of minors carefully and evaluate each $2 \times 2$ determinant before adding the terms.