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JEE Mathematics Solved Question Paper 2009

Multiple Choice Questions

1.
If $α, β, γ$ are the cube roots of unity, then the value of the determinant $|\begin{matrix} e^{α} & e^{2 α} & (e^{3 α} - 1) \\ e^{β} & e^{2 β} & (e^{3 β} - 1) \\ e^{γ} & e^{2 γ} & (e^{3 γ} - 1) \end{matrix}|$ is equal to

- 2

- 1

0

1

Given $α, β, γ$ are the cube roots of unity, then assume $α = 1, β = w and γ = w^{2}$ .

$∴ |\begin{matrix} e^{α} & e^{2 α} & (e^{3 α} - 1) \\ e^{β} & e^{2 β} & (e^{3 β} - 1) \\ e^{γ} & e^{2 γ} & (e^{3 γ} - 1) \end{matrix}| = |\begin{matrix} e^{α} & e^{2 α} & e^{3 α} \\ e^{β} & e^{2 β} & e^{3 β} \\ e^{γ} & e^{2 γ} & e^{3 γ} \end{matrix}| + |\begin{matrix} e^{α} & e^{2 α} & - 1 \\ e^{β} & e^{2 β} & - 1 \\ e^{γ} & e^{2 γ} & - 1 \end{matrix}| = e^{α} e^{β} e^{γ} |\begin{matrix} 1 & e^{α} & e^{2 α} \\ 1 & e^{β} & e^{2 β} \\ 1 & e^{γ} & e^{2 γ} \end{matrix}| - |\begin{matrix} 1 & e^{α} & e^{2 α} \\ 1 & e^{β} & e^{2 β} \\ 1 & e^{γ} & e^{2 γ} \end{matrix}| = |\begin{matrix} 1 & e^{α} & e^{2 α} \\ 1 & e^{β} & e^{2 β} \\ 1 & e^{γ} & e^{2 γ} \end{matrix}| [e^{α} e^{β} e^{γ} - 1] = 0 [∵ e^{α} e^{β} e^{γ} = e^{1 + w + w^{2}} = e^{0} = 1]$

If B is a non-singular matrix and A is a square matrix such that B^-1AB exists, then det (B^-1AB) is equal to

det(A^-1)
det (B^-1)
det(B)
det(A)

If 1, w, w² are the cube roots of unity and if

, a² + b² is equal to

1 + w²
w² - 1
1 + w
(1 + w)²

If the three linear equations

x + 4ay + az = 0

x + 3by + bz = 0

x + 2cy + cz = 0

have a non-trivial solution, where a $a \neq 0, b \neq 0, c \neq 0$ , then ab + bc is equal to

2ac
- ac
ac
- 2ac

If A = $|\begin{matrix} 1 & 0 & 0 \\ x & 1 & 0 \\ x & x & 1 \end{matrix}| and I = |\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}|$ , then A³ - 4A² + 3A + I is equal to