Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 2 - Determinant>Target Exercises>Q 22

EASY

JEE Advanced

IMPORTANT

Earn 100

If $[.]$ denotes the greatest integer less than or equal to the real number under consideration and $- 1 \leq x < 0; 0 \leq y < 1; 1 \leq z < 2,$ then the value of the determinant $|\begin{matrix} [x] + 1 & [y] & [z] \\ [x] & [y] + 1 & [z] \\ [x] & [y] & [z] + 1 \end{matrix}|$ is

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Important Questions on Determinant

EASY

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 2 - Determinant>Target Exercises>Q 23

The determinant

|\begin{matrix} a & b & a α + b \\ b & c & b α + c \\ a α + b & b α + c & 0 \end{matrix}|

is equal to zero, if

EASY

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 2 - Determinant>Target Exercises>Q 24

Value of

θ

lying between

θ = 0

and

θ = \frac{π}{2}

and

|\begin{matrix} 1 + \sin^{2} θ & \cos^{2} θ & 4 \sin 4 θ \\ \sin^{2} θ & 1 + \cos^{2} θ & 4 \sin 4 θ \\ \sin^{2} θ & \cos^{2} θ & 1 + 4 \sin 4 θ \end{matrix}| = 0

is

HARD

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 2 - Determinant>Target Exercises>Q 26

If

D_{r} = |\begin{matrix} 2^{r - 1} & 2 \cdot 3^{r - 1} & 4 \cdot 5^{r - 1} \\ α & β & γ \\ 2^{n} - 1 & 3^{n} - 1 & 5^{n} - 1 \end{matrix}|,

then the value of

\sum_{r = 1}^{n} D_{r}

is

EASY

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 2 - Determinant>Target Exercises>Q 27

The determinant

|\begin{matrix} \cos (α + β) & - \sin (α + β) & \cos 2 β \\ \sin α & \cos α & \sin β \\ - \cos α & \sin α & \cos β \end{matrix}|

is independent of

EASY

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 2 - Determinant>Target Exercises>Q 28

|\begin{array}{l} a \sin^{2} x & a^{2} + \cos^{2} x & 1 \\ a \sin^{2} y & a^{2} + \cos^{2} y & 1 \\ a \sin^{2} z & a^{2} + \cos^{2} z & 1 \end{array}|

is equal to

MEDIUM

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 2 - Determinant>Target Exercises>Q 29

If

α, β, γ \in R,

then the determinant

Δ = |\begin{array}{l} {(e^{i α} + e^{- i α})}^{2} & {(e^{i α} - e^{- i α})}^{2} & 4 \\ {(e^{i β} + e^{- i β})}^{2} & {(e^{i β} - e^{- i β})}^{2} & 4 \\ {(e^{i γ} + e^{- i γ})}^{2} & {(e^{i γ} - e^{- i γ})}^{2} & 4 \end{array}|

is

EASY

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 2 - Determinant>Target Exercises>Q 30

The value of the determinant

|\begin{matrix} \sqrt{6} & \sqrt{3} - i \sqrt{2} & 2 + i \sqrt{6} \\ \sqrt{3} + i \sqrt{2} & 0 & 3 - i \sqrt{8} \\ 2 - i \sqrt{6} & 3 + i \sqrt{8} & \sqrt{11} \end{matrix}|

is

MEDIUM

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 2 - Determinant>Target Exercises>Q 32

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

f (100)

is equal to