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Determinants

Multiple Choice Questions

51.

If x, y, z are different from zero and $∆ = |\begin{matrix} a & b - y & c - z \\ a - x & b & c - z \\ a - x & b - y & c \end{matrix}|$ = 0, then the value of the expression $\frac{a}{x} + \frac{b}{y} + \frac{c}{z}$ is

52.

If x = - 5 is a root of $|\begin{matrix} 2 x + 1 & 4 & 8 \\ 2 & 2 x & 2 \\ 7 & 6 & 2 x \end{matrix}|$ = 0, then the other roots are :

3, 3.5
1, 3.5
1, 7
2, 7

53.

The simultaneous equations Kx + 2y - z = 1, (K - I)y - 2z = 2 and (K + 2)z = 3 have only one solution when :

K = - 2
K = - 1
K = 0
K = 1

54.

If the matrix M_r is given by Mr = $[\begin{matrix} r & r - 1 \\ r - 1 & r \end{matrix}]$ , r = 1, 2, 3, ..., then the value of det(M₁) + det(M₂) + ... + det(M₂₀₀₈) is

2007
2008
(2008)²
(2007)²

55.
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 $α = 1, β = w and γ = w^{2}$ .

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

56.

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

57.

If A = $[\begin{matrix} 1 & 2 \\ 3 & 5 \end{matrix}]$ , then the value of the determinant $|A^{2009} - 5 A^{2008}|$ is

58.

The value of the determinant $|\begin{matrix} 15! & 16! & 17! \\ 16! & 17! & 18! \\ 17! & 18! & 19! \end{matrix}|$ is equal to

15! + 16!
2(15!)(16!)(17!)
15! + 16! + 17!
16! + 17!

59.

If A = $[\begin{matrix} \log (x) & - 1 \\ - \log (x) & 2 \end{matrix}]$ and if det(A) = 2, then the value of x is equal to

60.

If $|\begin{matrix} \sin^{2} (α) & \cos^{2} (α) \\ \cos^{2} (α) & \sin^{2} (α) \end{matrix}|$ = 0, $α \in (0, π)$ , then the value of $α$ are

$\frac{π}{2} and \frac{π}{12}$
$\frac{π}{2} and \frac{π}{6}$
$\frac{π}{4} and \frac{3 π}{4}$
$\frac{π}{6} and \frac{π}{3}$

Previous Year Papers

Test Series

Test Yourself

Multiple Choice Questions

55. If α, β, γ are the cube roots of unity, then the value of the determinant eαe2αe3α - 1eβe2βe3β - 1eγe2γe3γ - 1 is equal to - 2 - 1 0 1

55.
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