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Let $\{Δ_{1}, Δ_{2}, Δ_{3}, ...... , Δ_{k}\}$ be the set of all third-order determinants that can be made with the distinct non-zero real numbers $a_{1}, a_{2}, a_{3}, . . . . . . . ., a_{9}$ . Then

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Important Questions on Determinants

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Mathematics>Algebra>Determinants>Properties of Determinants

Let

a, b, c

be such that

(b + c) \neq 0

and

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

+ |\begin{matrix} a + 1 & b + 1 & c - 1 \\ a - 1 & b - 1 & c + 1 \\ {(- 1)}^{n + 2} a & {(- 1)}^{n - 1} b & {(- 1)}^{n} c \end{matrix}| = 0

Then the value of

n

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Mathematics>Algebra>Determinants>Properties of Determinants

α, β \neq 0

f (n) = α^{n} + β^{n}

and

|\begin{matrix} 3 & 1 + f (1) & 1 + f (2) \\ 1 + f (1) & 1 + f (2) & 1 + f (3) \\ 1 + f (2) & 1 + f (3) & 1 + f (4) \end{matrix}| = K {(1 - α)}^{2} {(1 - β)}^{2} {(α - β)}^{2}

, then

K

is equal to

HARD

Mathematics>Algebra>Determinants>Properties of Determinants

If area of triangle is

35

sq. units with vertices

(2, - 6), (5, 4)

and

(k, 4),

then

k

HARD

Mathematics>Algebra>Determinants>Properties of Determinants

Let

α

and

β

be the roots of the equation

x^{2} + x + 1 = 0 .

Then for

y \neq 0

R, |\begin{matrix} y + 1 & α & β \\ α & y + β & 1 \\ β & 1 & y + α \end{matrix}|

is equal to

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Mathematics>Algebra>Determinants>Properties of Determinants

∆ (x) = |\begin{matrix} 1 & \cos x & 1 - \cos x \\ 1 + \sin x & \cos x & 1 + \sin x - \cos x \\ \sin x & \sin x & 1 \end{matrix}|

,

then

\int_{0}^{\frac{π}{2}} ∆ (x) d x =

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Mathematics>Algebra>Determinants>Properties of Determinants

A value of

θ \in (0, \frac{π}{3}),

for which

|\begin{matrix} 1 + {c o s}^{2} θ & {s i n}^{2} θ & 4 c o s 6 θ \\ {c o s}^{2} θ & 1 + {s i n}^{2} θ & 4 c o s 6 θ \\ {c o s}^{2} θ & {s i n}^{2} θ & 1 + 4 c o s 6 θ \end{matrix}| = 0,

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Mathematics>Algebra>Determinants>Properties of Determinants

Δ_{r} = |\begin{matrix} r & 2 r - 1 & 3 r - 2 \\ \frac{n}{2} & n - 1 & a \\ \frac{1}{2} n (n - 1) & {(n - 1)}^{2} & \frac{1}{2} (n - 1) (3 n + 4) \end{matrix}|

, then the value of

\sum_{r = 1}^{n - 1} Δ_{r}

EASY

Mathematics>Algebra>Determinants>Properties of Determinants

Three vertices of

∆ A B C

are

A (1, 4), B (- 2, 2)

and

C (3, 2) .

Then the area of

∆ A B C

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Mathematics>Algebra>Determinants>Properties of Determinants

The value of

x

obtained from the equation

|\begin{matrix} x + α & β & γ \\ γ & x + β & α \\ α & β & x + γ \end{matrix}| = 0

will be

EASY

Mathematics>Algebra>Determinants>Properties of Determinants

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

HARD

Mathematics>Algebra>Determinants>Properties of Determinants

Let two points be

A (1, - 1)

and

B (0, 2) .

If a point

P (x^{'}, y^{'})

be such that the area of

∆ P A B = 5

sq. units and it lies on the line

3 x + y - 4 λ = 0,

then a value of

λ

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Mathematics>Algebra>Determinants>Properties of Determinants

a_{1}, a_{2}, \dots . ., a_{n}, \dots .

are in G.P. then

|\begin{matrix} l o g a_{n} & l o g a_{n + 1} & l o g a_{n + 2} \\ l o g a_{n + 3} & l o g a_{n + 4} & l o g a_{n + 5} \\ l o g a_{n + 6} & l o g a_{n + 7} & l o g a_{n + 8} \end{matrix}|

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Mathematics>Algebra>Determinants>Properties of Determinants

Let

a - 2 b + c = 1 .

f (x) = |\begin{matrix} x + a & x + 2 & x + 1 \\ x + b & x + 3 & x + 2 \\ x + c & x + 4 & x + 3 \end{matrix}|,

then:

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Mathematics>Algebra>Determinants>Properties of Determinants

A = [\begin{matrix} - 4 & - 1 \\ 3 & 1 \end{matrix}]

, then the determinant of the matrix

(A^{2016} - 2 A^{2015} - A^{2014})

is :

EASY

Mathematics>Algebra>Determinants>Properties of Determinants

Let

A = [a_{i j}]

and

B = [b_{i j}]

be two

3 \times 3

real matrices such that

b_{i j} = {(3)}^{(i + j - 2)} a_{i j}

, where

i, j = 1,2, 3

. If the determinant of

B

81

, then determinant of

A

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The value of the determinant

|\begin{matrix} bc & ca & ab \\ a^{3} & b^{3} & c^{3} \\ \frac{1}{a} & \frac{1}{b} & \frac{1}{c} \end{matrix}|

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Mathematics>Algebra>Determinants>Properties of Determinants

A = [\begin{array}{l} 1 & 3 \\ 4 & 2 \end{array}], B = [\begin{array}{l} 2 & - 1 \\ 1 & 2 \end{array}],

then

|A B B^{'}| =

EASY

Mathematics>Algebra>Determinants>Properties of Determinants

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

,

then

f (50) =

EASY

Mathematics>Algebra>Determinants>Properties of Determinants

Let

f (x) = |\begin{matrix} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{matrix}|

.

x = - 9

is a root of

f (x) = 0

,

then the other roots are

HARD

Mathematics>Algebra>Determinants>Properties of Determinants

Let

a_{1}, a_{2}, a_{3} \dots, a_{10}

be in

G . P .

with

a_{i} > 0

for

i = 1, 2, \dots, 10

and

S

be the set of pairs

(r, k), r, k \in N

(the set of natural numbers) for which

|\begin{matrix} \log_{e} a_{1}^{r} a_{2}^{k} & \log_{e} a_{2}^{r} a_{3}^{k} & \log_{e} a_{3}^{r} a_{4}^{k} \\ \log_{e} a_{4}^{r} a_{5}^{k} & \log_{e} a_{5}^{r} a_{6}^{k} & \log_{e} a_{6}^{r} a_{7}^{k} \\ \log_{e} a_{7}^{r} a_{8}^{k} & \log_{e} a_{8}^{r} a_{9}^{k} & \log_{e} a_{9}^{r} a_{10}^{k} \end{matrix}| = 0

Then the number of elements in

S,

is:

Let Δ1,Δ2,Δ3,...... ,Δk be the set of all third-order determinants that can be made with the distinct non-zero real numbers a1, a2, a3,........,a9. Then

Important Questions on Determinants

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Let $\{Δ_{1}, Δ_{2}, Δ_{3}, ...... , Δ_{k}\}$ be the set of all third-order determinants that can be made with the distinct non-zero real numbers $a_{1}, a_{2}, a_{3}, . . . . . . . ., a_{9}$ . Then