JEE Main 2026 Jan 23 Shift 1, Mathematics Q11: Algebra Of Determinants

JEE Main2026Jan 23, Shift 1Mathematics

Among the statements :

I : If $1 cos α cos β cos α 1 cos γ cos β cos γ 1 = 0 cos α cos β cos α 0 cos γ cos β cos γ 0$ , then $cos^{2} α + cos^{2} β + cos^{2} γ = \frac{3}{2}$ , and

II : If $x^{2} + xx + 1 x - 2 2 x^{2} + 3 x - 13 x 3 x - 3 x^{2} + 2 x + 32 x - 12 x - 1 = p x + q$ , then $p^{2} = 196 q^{2}$ ,

A
both are false
B
only II is true
C
both are true
D
only I is true

Solution

Statement I. Let $x = cos α$ , $y = cos β$ , $z = cos γ$ . Expanding both determinants and equating:

LHS $= 1 - x^{2} - y^{2} - z^{2} + 2 x y z$ ; RHS $= 2 x y z$ .

The equality forces $x^{2} + y^{2} + z^{2} = 1$ , not $3/2$ . So I is false.

Statement II. Evaluate the determinant at $x = 0$ and $x = 1$ .

At $x = 0$ : determinant $= 01 - 2 - 10 - 3 3 - 1 - 1 = - 12$ , so $q = - 12$ .

At $x = 1$ : determinant $= 22 - 1 430611 = 42$ , so $p + q = 42 \Rightarrow p = 54$ .

Then $\frac{p ^{2}}{q ^{2}} = (\frac{54}{12})^{2} = \frac{2916}{144} \neq = 196$ . So II is false.

Both statements are false.

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JEE Main 2026 Jan 23 Shift 1, Mathematics Q11: Algebra Of Determinants

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