Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 12 - Determinants>Exercises>Q 19

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If determinant $|\begin{array}{ccc} \cos (θ + ϕ) & - \sin (θ + ϕ) & \cos 2 ϕ \\ \sin θ & \cos θ & \sin ϕ \\ - \cos θ & \sin θ & \cos ϕ \end{array}|$ is

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Important Questions on Determinants

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 12 - Determinants>Exercises>Q 20

If

g (x) = |\begin{matrix} a^{- x} & e^{x \log_{e} a} & x^{2} \\ a^{- 3 x} & e^{3 x \log_{e} a} & x^{4} \\ a^{- 5 x} & e^{5 x \log_{e} a} & 1 \end{matrix}|

, then

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 12 - Determinants>Exercises>Q 21

If

∆ (x) = |\begin{matrix} x^{2} + 4 x - 3 & 2 x + 4 & 13 \\ 2 x^{2} + 5 x - 9 & 4 x + 5 & 26 \\ 8 x^{2} - 6 x + 1 & 16 x - 6 & 104 \end{matrix}| = a x^{3} + b x^{2} + c x + d,

then

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 12 - Determinants>Exercises>Q 22

If

|\begin{matrix} y z - x^{2} & z x - y^{2} & x y - z^{2} \\ x z - y^{2} & x y - z^{2} & y z - x^{2} \\ x y - z^{2} & y z - x^{2} & z x - y^{2} \end{matrix}| = |\begin{matrix} r^{2} & u^{2} & u^{2} \\ u^{2} & r^{2} & u^{2} \\ u^{2} & u^{2} & r^{2} \end{matrix}|,

then

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 12 - Determinants>Exercises>Q 23

Which of the following is/ are true for $∆ = |\begin{matrix} a^{2} & 1 & a + c \\ 0 & b^{2} + 1 & b + c \\ 0 & b + c & c^{2} + 1 \end{matrix}| ?$

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 12 - Determinants>Exercises>Q 24

The values of

k \in R

for which the system of equations

x + k y + 3 z = 0, k x + 2 y + 2 z = 0, 2 x + 3 y + 4 z = 0

has non-trivial solution are

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 12 - Determinants>Exercises>Q 25

The system of equations $- 2 x + y + z = a x - 2 y + z = b x + y - 2 z = c$ has

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 12 - Determinants>Exercises>Q 26

Let

α, β

and

γ

be the roots of the equation,

x^{3} + a x^{2} + b x + c = 0, (a \neq 0) .

If the system of equations

α x + β γ + γ z = 0, β x + γ y + α z = 0

and

γ x + α y + β z = 0

has non-trivial solution, then

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 12 - Determinants>Exercises>Q 2

If

a_{1}, a_{2}, a_{3}, 5, 4, a_{6}, a_{7}, a_{8}, a_{9}

are in

H . P .,

and

D = |\begin{matrix} a_{1} & a_{2} & a_{3} \\ 5 & 4 & a_{6} \\ a_{7} & a_{8} & a_{9} \end{matrix}|

, then the value of

[D]

is (where

[.]

represents the greatest integer function)______.