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Let $D_{k} = |\begin{matrix} \begin{matrix} 1 & 2 k & 2 k - 1 \\ n & n^{2} + n + 2 & n^{2} \\ n & n^{2} + n & n^{2} + n + 2 \end{matrix} \end{matrix}|$ . If $\sum_{k = 1}^{n} D_{k} = 96$ , then $n$ is equal to _________.

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Important Questions on Matrices and Determinants

EASY

Mathematics>Algebra>Matrices and Determinants>Evaluation of Determinants

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

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Evaluation of Determinants

a, b, c

are in

A . P .

then the value of the determinant

|\begin{matrix} x + 2 & x + 3 & x + 2 a \\ x + 3 & x + 4 & x + 2 b \\ x + 4 & x + 5 & x + 2 c \end{matrix}|

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Evaluation of Determinants

s = p + q + r,

then value of

|\begin{matrix} s + r & p & q \\ r & s + p & q \\ r & p & s + q \end{matrix}|

is ...........

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Evaluation 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 =

HARD

Mathematics>Algebra>Matrices and Determinants>Evaluation 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:

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Evaluation of Determinants

What is the value of the determinant

|\begin{matrix} 1 & 1 & 1 \\ 1 & 1 + x y z & 1 \\ 1 & 1 & 1 + x y z \end{matrix}| ?

HARD

Mathematics>Algebra>Matrices and Determinants>Evaluation of Determinants

|\begin{matrix} 10! & 11! & 12! \\ 11! & 12! & 13! \\ 12! & 13! & 14! \end{matrix}| =______

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Evaluation of Determinants

Δ (n) = |\begin{matrix} x^{n} & \sin x & \cos x \\ n! & \sin \frac{n π}{2} & \cos \frac{n π}{2} \\ α & α^{2} & α^{3} \end{matrix}|

, then the value of

x = 0

is,

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Evaluation of Determinants

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}|

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Evaluation of Determinants

If for some

c < 0,

the quadratic equation,

2 c x^{2} - 2 (2 c - 1) x + 3 c^{2} = 0

has two distinct real roots,

\frac{1}{a}

and

\frac{1}{b},

then the value of the determinant,

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

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Evaluation of Determinants

Let

f (x) = |\begin{matrix} \sin^{2} x & - 2 + \cos^{2} x & \cos 2 x \\ 2 + \sin^{2} x & \cos^{2} x & \cos 2 x \\ \sin^{2} x & \cos^{2} x & 1 + \cos 2 x \end{matrix}|, x \in [0, π] .

Then the maximum value of

f (x)

is equal to

EASY

Mathematics>Algebra>Matrices and Determinants>Evaluation 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>Matrices and Determinants>Evaluation of Determinants

Δ = |\begin{array}{l} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{array}| = K (a - b) (b - c) (c - a),

then

K =

EASY

Mathematics>Algebra>Matrices and Determinants>Evaluation 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

EASY

Mathematics>Algebra>Matrices and Determinants>Evaluation of Determinants

Without expanding at any stage find the value of

|\begin{matrix} a & b & c \\ a + 2 x & b + 2 y & c + 2 z \\ x & y & z \end{matrix}|

EASY

Mathematics>Algebra>Matrices and Determinants>Evaluation of Determinants

The equation

|\begin{matrix} 1 & x & x^{2} \\ x^{2} & 1 & x \\ x & x^{2} & 1 \end{matrix}| = 0

has

HARD

Mathematics>Algebra>Matrices and Determinants>Evaluation of Determinants

Let

ω

be the complex number

\cos \frac{2 π}{3} + i \sin \frac{2 π}{3}

. Then the number of distinct complex numbers

z

satisfying

|\begin{matrix} z + 1 & ω & ω^{2} \\ ω & z + ω^{2} & 1 \\ ω^{2} & 1 & z + ω \end{matrix}| = 0

is equal to

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Evaluation of Determinants

a, b, c

are different and

|\begin{array}{l} a & a^{2} & a^{3} - 1 \\ b & b^{2} & b^{3} - 1 \\ c & c^{2} & c^{3} - 1 \end{array}| = 0

, then

a b c

is equal to

HARD

Mathematics>Algebra>Matrices and Determinants>Evaluation 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}

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Evaluation 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}|

Let Dk=12k2k-1nn2+n+2n2nn2+nn2+n+2. If ∑k=1nDk=96, then n is equal to _________.

Important Questions on Matrices and Determinants

Learn and Prepare for any exam you want !

Let $D_{k} = |\begin{matrix} \begin{matrix} 1 & 2 k & 2 k - 1 \\ n & n^{2} + n + 2 & n^{2} \\ n & n^{2} + n & n^{2} + n + 2 \end{matrix} \end{matrix}|$ . If $\sum_{k = 1}^{n} D_{k} = 96$ , then $n$ is equal to _________.