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

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IMPORTANT

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

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Important Questions on Determinants

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IMPORTANT

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)______.

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 12 - Determinants>Exercises>Q 3

The sum of values of

p

for which the equations

x + y + z = 1, x + 2 y + 4 z = p,

and

x + 4 y + 10 z = p^{2}

have a solution is ____.

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IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 12 - Determinants>Exercises>Q 4

The sum of roots of the equation

|\begin{matrix} x + 2 & 2 x + 3 & 3 x + 4 \\ 2 x + 3 & 3 x + 4 & 4 x + 5 \\ 3 x + 5 & 5 x + 8 & 10 x + 17 \end{matrix}| = 0

is ____.

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IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 12 - Determinants>Exercises>Q 5

If

|\begin{matrix} {(β + γ - α - δ)}^{4} & {(β + γ - α - δ)}^{2} & 1 \\ {(γ + α - β - δ)}^{4} & {(γ + α - β - δ)}^{2} & 1 \\ {(α + β - γ - δ)}^{4} & {(α + β - γ - δ)}^{2} & 1 \end{matrix}|

= - k (α - β) (α - γ) (α - δ) (β - γ) (β - δ) (γ - δ),

then the value of

k

is _____.

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IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 12 - Determinants>Exercises>Q 6

If

f (x) = |\begin{matrix} 1 & x & x + 1 \\ 2 x & x (x - 1) & (x + 1) x \\ 3 x (x - 1) & x (x - 1) (x - 2) & (x + 1) x (x - 1) \end{matrix}|

, then the value of

f (500) =______

.

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 12 - Determinants>Exercises>Q 7

If

|\begin{matrix} x & x + y & x + y + z \\ 2 x & 3 x + 2 y & 4 x + 3 y + 2 z \\ 3 x & 6 x + 3 y & 10 x + 6 y + 3 z \end{matrix}| = 64

, then the real value of

x

is _____.

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IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 12 - Determinants>Exercises>Q 8

Let

D_{1} = |\begin{matrix} a & b & a + b \\ c & d & c + d \\ a & b & a - b \end{matrix}|

and

D_{2} = |\begin{matrix} a & c & a + c \\ b & d & b + d \\ a & c & a + b + c \end{matrix}|

then the value of

|\frac{D_{1}}{D_{2}}|

, where

b \neq 0

and

a d \neq b c

, is_____.