Among the statements : I : If \(\left|\begin{array}{ccc}1 & \cos \alpha & \cos \beta \\ \cos \alpha & 1 & \cos \gamma \\ \cos \beta & \cos \gamma & 1\end{array}\right|=\left|\begin{array}{ccc}0 & \cos \alpha & \cos \beta \\ \cos \alpha & 0 & \cos \gamma \\ \cos \beta & \cos \gamma & 0\end{array}\right|\), then \(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=\frac{3}{2}\), and II : If \(\left|\begin{array}{ccc}x^{2}+x & x+1 & x-2 \\ 2 x^{2}+3 x-1 & 3 x & 3 x-3 \\ x^{2}+2 x+3 & 2 x-1 & 2 x-1\end{array}\right|=p x+q\), then \(\mathrm{p}^{2}=196 \mathrm{q}^{2}\),

JEE Main · 2026 · 23 Jan 2026 · Shift 1 · Determinants

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} + x 2 x^{2} + 3 x - 1 x^{2} + 2 x + 3 x + 1 3 x 2 x - 1 x - 2 3 x - 3 2 x - 1 = p x + q

, then

p^{2} = 196 q^{2}

Answer: (A)

For statement I put

x = cos α, y = cos β, z = cos γ

. The first determinant expands to

1 - x^{2} - y^{2} - z^{2} + 2 x y z

and the second to

2 x y z

; equating them forces

x^{2} + y^{2} + z^{2} = 1

, so

cos^{2} α + cos^{2} β + cos^{2} γ = 1

, not

\frac{3}{2}

, and statement I is false. For statement II the determinant is linear in

x

. Its values at

x = 0, 1, 2

are

- 12, 12, 36

, which lie on the line

24 x - 12

, so

p = 24

and

q = - 12

. Then

p^{2} = 576

while

196 q^{2} = 196 \cdot 144 = 28224

, so

p^{2} \neq = 196 q^{2}

and statement II is false as well. Both statements are false.

JEE Main · 2026 · 23 Jan 2026 · Shift 1 · Single correct option